Notes on Modulation techniques - Davide Micheli

1

Notes on Modulation techniques

by

Davide Micheli

2

Table of Contents

NOTES ON MODULATION TECHNIQUES.......................................................................................................... 1 TABLE OF CONTENTS................................................................................................................................................2

1 CANNEL CAPACITY AND IDEAL COMMUNICATION SYSTEMS ...................................................... 6 1.1 CHANNEL CAPACITY DEFINITION.................................................................................................................6

2 CODING........................................................................................................................................................... 14 2.1 CODE PERFORMANCE ................................................................................................................................14 2.2 SPECTRAL EFFICIENCY..............................................................................................................................12

3 INTERSYMBOL INTERFERENCE ............................................................................................................. 19 3.1 SPECTRAL PROPERTY REMINDER (SQUARE WAVE SPECTRUM) ...................................................................22 3.2 EQUALIZING FILTER ..................................................................................................................................23 3.3 NYQUIST’S FIRST METHOD (ZERO ISI) .....................................................................................................25 3.4 RAISED COSINE-ROLLOFF NYQUIST FILTERING ........................................................................................28

4 BANDPASS SIGNALING............................................................................................................................... 32 4.1 COMPLEX ENVELOPE RAPPRESTNATION OF BANDPASS WAVEFORMS ................................32

4.1.1 Definitions: Baseband, Bandpass, and modulation ............................................................................. 32 4.1.2 Complex Envelope Representation ...................................................................................................... 33 4.1.3 Theorem............................................................................................................................................... 33

4.2 REPRESENTATION OF MODULATED SIGNALS..............................................................................35 4.3 SPECTRUM OF BANDPASS SIGNALS................................................................................................36

Theorem ............................................................................................................................................................. 36 5 AM, FM, PM MODULATED SYSETMS.................................................................................................... 37

5.1 DEFINITIONS .............................................................................................................................................37 5.2 AMPLITUDE MODULATION ...............................................................................................................37

5.2.1 Normalized AM average power ........................................................................................................... 39 5.2.2 Definition: The modulation efficiency ................................................................................................. 40

6 PHASE MODULATION AND FREQUENCY MODULATION................................................................ 41 6.1 REPRESENTATION OF PM AND FM SIGNALS .............................................................................................41

6.1.1 Definition for peak phase deviation and peak frequency deviaton. ..................................................... 45 6.2 SPECTRA OF ANGLE-MODULATED SIGNALS ..............................................................................................46

6.2.1 Spectrum of a PM or FM signal with Sinusoidal Modulation ............................................................. 47 6.3 NOISE AND FREQUENCY MODULATION ......................................................................................................51

6.3.1 Noise triangle ...................................................................................................................................... 51 6.4 PREEMPHASIS AND DEEMPHASIS IN ANGLE MODULATED SYSTEMS...........................................................52

6.4.1 De-Emphasis response table................................................................................................................ 54 6.4.2 Why use “Roofed” Pre-Enhasis .......................................................................................................... 55

6.5 FREQUENCY DIVISION MULTIPLEXING.......................................................................................................55 7 OUTPUT SIGNAL-TO NOISE RATIOS FOR ANALOG SYSTEMS....................................................... 57

7.1 COMPARISON WITH BASEBAND SYSTEMS .................................................................................................58 7.2 AM SYSTEMS WITH PRODUCT DETECTION................................................................................................59 7.3 SSB SYSTEMS............................................................................................................................................60 7.4 PM SYSTEMS .............................................................................................................................................60 7.5 FM SYSTEMS ............................................................................................................................................62

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7.6 FM SYSTEMS WITH THRESHOLD EXTENSION ............................................................................................63 7.7 FM SYSTEM WITH DE-EMPHASIS...............................................................................................................66

8 COMPARISON OF ANALOG SIGNALING SYSTEMS............................................................................ 67 8.1 IDEAL SYSTEMS PERFORMANCE.................................................................................................................68

9 BINARY MODULATED BANDPASS SIGNALING................................................................................... 70 9.1 BINARY PHASE-SHIFT KEYING (BPSK) ....................................................................................................71

9.1.1 BPSK Generation ................................................................................................................................ 71 9.1.2 BPSK Detection by a Correlation Receiver......................................................................................... 78 9.1.3 With Noise ........................................................................................................................................... 80

9.2 MAXIMUM LIKELIHOOD DETECTION.........................................................................................................83 9.3 BIT ERRORS ..............................................................................................................................................83

9.3.1 Q-Function reminder ........................................................................................................................... 85 9.3.2 Bit Error Probability in terms of Eb and N0......................................................................................... 87

10 DIFFERENTIAL PHASE-SHIFT KEYING (DPSK)................................................................................... 89 10.1 DIFFERENTIAL CODING..............................................................................................................................90

11 FREQUENCY SHIFT KEYING (FSK)......................................................................................................... 91 11.1 DISCONTINUOUS FSK ...............................................................................................................................91 11.2 CONTINUOUS FSK.....................................................................................................................................92 11.3 FSK DETECTION........................................................................................................................................93

12 MULTILEVEL MODULATED BANDPASS SIGNALING ....................................................................... 95 12.1 QUADRATURE PHASE-SHIFT KEYNG (QPSK) AND M-ARY PHASE-SHIFT KEYNG (MPSK) .......................95 12.2 OQPSK AND π/4 QPSK ..........................................................................................................................102 12.3 QUADRATURE AMPLITUDE MODULATION (QAM)..................................................................................106 12.4 PSD FOR MPSK, QAM, OQPSK, AND π/4 QPSK WITHOUT PRE-MODULATION FILTERING ....................107 12.5 SPECTRAL EFFICIENCY FOR MPSK, QAM,OQPSK, AND π/4 QPSK WITH RAISED COSINE FILTERING...109 12.6 RECEIVER QPSK, MSK AND PERFORMANCE ..........................................................................................117

13 FEHER-PATENTED QUADRATURE PHASE-SHIFT KEING.............................................................. 121 13.1 INTRODUCTION........................................................................................................................................121 13.2 SIGNAL MODEL FOR FQPSK ...................................................................................................................121 13.3 SIGNAL MODEL FOR FQPSK-B................................................................................................................131 13.4 SPECTRAL EFFICIENCY COMPARISON.......................................................................................................133

14 EFFICIENT MODULATION METHODS STUDY AT NASA/JPL ........................................................ 135 14.1 FQPSK-B MODULATION BIT-ERROR-RATE (BER) ................................................................................135 14.2 FQPSK-B MODULATION SPECTRA .........................................................................................................136

14.2.1 Hardware Spectrum Measurements.............................................................................................. 136 14.3 FQPSK-B MODULATION POWER CONTAINMENT ...................................................................................138 14.4 FQPSK-B MODULATION STUDY CONCLUSIONS .....................................................................................138

15 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS............................................................... 140 15.1.1 SUMMARY.................................................................................................................................... 140

15.2 CONCLUSIONS....................................................................................................................................144 15.2.1 Filtering Conclusions.................................................................................................................... 144 15.2.2 Loss Conclusions .......................................................................................................................... 145 15.2.3 Modulation Methods Conclusions ................................................................................................ 146 15.2.4 Spectrum Improvement Conclusions............................................................................................. 147

15.3 RECOMMENDATIONS .......................................................................................................................147 15.3.1 Mission Classification................................................................................................................... 147

16 8PSK MODULATION (EXAMPLE IMPLEMENTED IN MOBILE TELEPHONE NETWORK)..... 151 16.1 INTRODUCTION........................................................................................................................................151 16.2 EDGE SIGNAL DESCRIPTION: MODULATING SYMBOL RATE AND SYMBOL MAPPING .............................153

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16.3 SYMBOL ROTATION.................................................................................................................................158 16.4 (8PSK EDGE) MODULATION AM DISTORTION .......................................................................................165

16.4.1 First problem (ISI):....................................................................................................................... 165 16.4.2 Second problem (AM):.................................................................................................................. 166

16.5 USED GAUSSIAN EDGE FILTER ..............................................................................................................167 16.6 EFFECT DUE TO GAUSSIAN EDGE FILTERING IN 3Π/8 SHIFTED 8PSK .....................................................170 16.7 MODULATION..........................................................................................................................................173 16.8 CONCLUSION...........................................................................................................................................173

17 EFFICIENT MODULATION METHODS STUDY AT NASA/JPL ........................................................ 174 17.1 PHASE SHIFT KEYED (8-PSK) MODULATION ...............................................................................174 17.2 PSK MODULATION BIT-ERROR-RATE (BER) .........................................................................................174 17.3 8-PSK MODULATION SPECTRA ...............................................................................................................175 17.4 PSK MODULATION POWER CONTAINMENT.............................................................................................177 17.5 PSK MODULATION STUDY CONCLUSIONS ..............................................................................................177

18 MINIMUM-SHIFT KEYNG (MSK) AND GMSK ..................................................................................... 178 18.1 GMSK ....................................................................................................................................................185

18.1.1 How to implement GMSK modulator............................................................................................ 190 18.1.2 How to implement GMSK demodulator ........................................................................................ 192

19 EFFICIENT MODULATION METHODS STUDY AT NASA/JPL ........................................................ 194 19.1 MSK AND GMSK MODULATION BIT-ERROR-RATE (BER) ....................................................................194 19.2 MSK AND GMSK MODULATION SPECTRA .............................................................................................194 19.3 MSK / GMSK MODULATION POWER CONTAINMENT .............................................................................196 19.4 MSK / GMSK MODULATION STUDY CONCLUSIONS...............................................................................197

20 HISTORY OF SPECTRUM EFFICIENT MODULATION IN TELEMETRY APPLICATIONS....... 198 21 CORRELATED DETECTION .................................................................................................................... 201 22 INTRODUCTION TO CDMA ..................................................................................................................... 206

22.1 MULTIPLE ACCESS ..................................................................................................................................206 22.2 SPREAD SPECTRUM MODULATION ...........................................................................................................207 22.3 TOLERANCE TO NARROWBAND INTERFERENCE ......................................................................................210 22.4 DIRECT SEQUENCE SPREAD SPECTRUM SYSTEM.....................................................................................211

22.4.1 Channelization operation.............................................................................................................. 212 22.4.2 Scrambling operation.................................................................................................................... 214

22.5 ORTHOGONAL SEQUENCES REMINDER.....................................................................................................214 22.6 MODULATION AND TOLERANCE TO WIDEBAND INTERFERENCE .............................................................216 22.7 UPLINK MODULATION.............................................................................................................................219

22.7.1 One UL parallel channel .............................................................................................................. 223 22.7.2 Two UL parallel channel .............................................................................................................. 225 22.7.3 Three UL parallel channel............................................................................................................ 230 22.7.4 Filtering ........................................................................................................................................ 236

22.8 DOWNLINK SPREADING AND MODULATION ............................................................................................238 22.8.1 Downlink Spreading Codes........................................................................................................... 238

23 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING (OFDM) .............................................. 241 23.1 INTRODUCTION........................................................................................................................................241 23.2 DIGITAL AUDIO BROADCASTING.............................................................................................................242 23.3 DIGITAL VIDEO BROADCASTING.............................................................................................................243 23.4 BASIC PRINCIPLE OF OFDM....................................................................................................................244 23.5 ORTHOGONALITY....................................................................................................................................246

23.5.1 FREQUENCY DOMAIN ORTHOGONALITY.............................................................................. 249 23.6 OFDM GENERATION AND RECEPTION .....................................................................................................250

23.6.1 Serial To Parallel Conversion ...................................................................................................... 251 23.6.2 Subcarrier modulation and mapping ............................................................................................ 252 23.6.3 Frequency to time domain conversion .......................................................................................... 253

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23.7 OFDM TRANSMITTER.............................................................................................................................254 23.8 FFT(LINE SPECRA FOR PERIODIC WAVEFORMS).....................................................................................256

23.8.1 Theorem ........................................................................................................................................ 256 23.8.2 Theorem ........................................................................................................................................ 258

24 APPENDIX..................................................................................................................................................... 267 24.1 CONVOLUTION PROCESS..........................................................................................................................267 24.2 DIRAC DELTA FUNCTION AND CONVOLUTION PROCESS ...........................................................................270

25 REFERENCE................................................................................................................................................. 271

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1 CANNEL CAPACITY AND IDEAL COMMUNICATION SYSTEMS

1.1 Channel capacity definition Suppose the following transmission systems

figure 1

Let us define the channel capacity:

• For digital systems, the optimum system might be defined as the system that minimizes the probability of bit error at the system output. The variables are bit error, signal bandwidth, transmitted energy and channel bandwidth.

The question is: is it possible to invent a system with non bit error at the output even when we have noise introduced into the channel ? The answer is yes, under certain assumption Shannon showed that (for the case of signal plus white Gaussian noise) a channel capacity C(bits/s) could be calculated such that if the rate information R(bits/s) was less then C, the probability of error would approach to zero.

Transduction Source information

Antenna and Receiver NOISE added

Transmitter

Destination Receiver Transduction

skay NOISE added

Modulated signal

7

⎟⎠⎞

⎜⎝⎛ +=

NSBC 1log2 eq 1

where B is the channel bandwidth in (Hz) and S/N is the signal-to-noise power ratio(watts/watts, no dB) at the input to the digital receiver.

Shannon gives us a theoretical performance bound that we can strive to achieve with practical communication systems. Systems that approach this bound usually incorporate error correction coding.

• In analog systems, in place of error probability, the optimum system might be

defined as the one that achieves the largest signal-to-noise ratio at the receiver output, subject to design constraints such as channel bandwidth and transmitted power.

In this case the question is: is it possible to design a system with infinite signal to noise ratio at the output when noise is introduced to the channel ? The answerer is no.

The law above, in the first equation, it was found considering that: if S is a continuous signal between that varies s1 and s2 values and, if p(s) is the density of probability for S, than p(s)⋅ds gives the probability for S of felling inside ds interval. Therefore the information bits quantity related to S signal is

dssp

spqs

s∫=2

1)(

1log)( 2 eq 2

where )(

1log2 sp is the number of bits needed to represent all values of signal S, each of

which is multiplied by the density probability p(s) of the signal S In a discrete case we easily have:

∑=

=N

n nn P

Pq1

21log eq 3

where in case of equal probability Pn=1/N of the N symbols we had

∑ ∑ ∑= = =

====N

n

N

n

N

nnn NN

NN

NPPq

1 1 12222 loglog1log11log eq 4

8

As an example the representation of 8 states discrete signal, where all states are supposed with the same probability, require 3 bits (each bit can assume 2 values high, low so we using log2x) because

8log3 2= eq 5

In the continuous scenario, if S is the average signal and σ2 is the variance, then it can be shown that q (the number of bits needed to represent S) became maximum when p(s) has a Gaussian distribution probability:

2

2

2)(

2

2

πσ

σS

esp−

= eq 6

In this case the bound of S are: s1=-∞ , s2=∞. The corresponding quantity of information q (number of bits needed to represent S ) can be obtained substituting this equation in the integral, above:

ebits/sampl )2(log21)2(log

2

1log)( 22

21

22

2

2

22

2 eSeSds

e

spqS

ππ

πσ

σ

=== ∫∞

∞− − eq 7

here S2 is the average signal power. Considering now the Shannon theorem which says that a signal of bandwidth B requires at least 2B sample rate (samples/second). Therefore in a time interval t are necessary 2B⋅t samples of the signal S, here each sample are univocally determined by q bits. So the information quantity bit related to the sampled signal S with q bits quantizing, on the time interval t is:

bits )2(log)2(log22 22

21

22 eSBteSBtqtBQ ππ ==⋅⋅= eq 8

The presence of noise corrupts the receiving information at the receiver input. The relative information associated is:

bits )2(log2)2(log)2(log22 22

221

22 NN ePBteSBteSBtqtBQ πππ ===⋅⋅= eq 9

where PN= 2S is the average noise power . Then at the input of the receiver we have the noise power PN along with the information power signal PS:

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( )[ ]{ } bits 2log2 2 NSNSNS PPeBtqtBQ +=⋅⋅= ++ π eq 10 where PS+PN =S2 is the total power of signal plus noise with Gaussian probability density function The uncorrupted quantity of information will be

( )[ ] ( )[ ]{ }( )( ) bits 1log

22log

2log2log2

22

22

PPt B

PePPet B

PePPetBqtBQQQ

N

S

N

NS

NNSNSNNS

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅=

+⋅=

=−+⋅=⋅⋅=−= ++

ππ

ππ

eq 11

The bitrate Rb will be

bits/s 1log2 RPPB

tQ

bN

S =⎟⎟⎠

⎞⎜⎜⎝

⎛+= eq 12

we can invert the equation to find PS/PN

⎟⎟⎠

⎞⎜⎜⎝

⎛−= ⋅ 12 tB

Q

N

S

PP

eq 13

where

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⇒⎟⎟

⎠

⎞⎜⎜⎝

⎛−=⇒= ⋅⋅ 12 12 tB

Q

StB

QS

N KTBPKTBPKTBP eq 14

This is the Hartely-Shannon law which contains the essential components of transmission systems:

1. bandwidth B 2. power of signal PS 3. power pf noise PN 4. duration of message t 5. temperature of reference T

PS and PN power are reported in figure below as a function of the bandwidth, we have fixed T=300 (K), Q=270833 (bit), K=1.38*10-23 (J/K) , t= 1 (s)

10

PS/PN=0 dB and Q=B

K T tempo t (s) Q (bit) B (Hz) Pn (w) Ps (w) Pn (dBm) Ps (dBm) Ps/Pn (dB)bit rate Q/t (Kbit/sec)

1.38E-23 300 1 270833 10000.0 4.1E-17 5.9E-09 -133.83 -52.30 82 270.83312500.0 5.2E-17 1.7E-10 -132.86 -67.64 65 270.83315625.0 6.5E-17 1.1E-11 -131.89 -79.71 52 270.83319531.3 8.1E-17 1.2E-12 -130.92 -89.18 42 270.83324414.1 1.0E-16 2.2E-13 -129.95 -96.56 33 270.83330517.6 1.3E-16 5.9E-14 -128.98 -102.28 27 270.83338147.0 1.6E-16 2.2E-14 -128.02 -106.67 21 270.83347683.7 2.0E-16 9.9E-15 -127.05 -110.03 17 270.83359604.6 2.5E-16 5.5E-15 -126.08 -112.59 13 270.83374505.8 3.1E-16 3.5E-15 -125.11 -114.53 11 270.83393132.3 3.9E-16 2.5E-15 -124.14 -116.01 8 270.833

116415.3 4.8E-16 1.9E-15 -123.17 -117.13 6 270.833145519.2 6.0E-16 1.6E-15 -122.20 -118.00 4 270.833181898.9 7.5E-16 1.4E-15 -121.23 -118.66 3 270.833227373.7 9.4E-16 1.2E-15 -120.26 -119.18 1 270.833284217.1 1.2E-15 1.1E-15 -119.29 -119.58 0 270.833355271.4 1.5E-15 1.0E-15 -118.32 -119.90 -2 270.833444089.2 1.8E-15 9.7E-16 -117.36 -120.14 -3 270.833555111.5 2.3E-15 9.2E-16 -116.39 -120.34 -4 270.833693889.4 2.9E-15 8.9E-16 -115.42 -120.49 -5 270.833867361.7 3.6E-15 8.7E-16 -114.45 -120.62 -6 270.833

1084202.2 4.5E-15 8.5E-16 -113.48 -120.71 -7 270.8331355252.7 5.6E-15 8.3E-16 -112.51 -120.79 -8 270.8331694065.9 7.0E-15 8.2E-16 -111.54 -120.85 -9 270.8332117582.4 8.8E-15 8.1E-16 -110.57 -120.90 -10 270.8332646978.0 1.1E-14 8.1E-16 -109.60 -120.94 -11 270.8333308722.5 1.4E-14 8.0E-16 -108.63 -120.97 -12 270.8334135903.1 1.7E-14 8.0E-16 -107.66 -121.00 -13 270.8335169878.8 2.1E-14 7.9E-16 -106.70 -121.02 -14 270.8336462348.5 2.7E-14 7.9E-16 -105.73 -121.03 -15 270.833

Singal power Ps (dBm), Noise Power Pn (dBm), Ps/Pn ratio (dB)

-135.0-130.0-125.0-120.0-115.0-110.0-105.0-100.0-95.0-90.0-85.0-80.0-75.0-70.0-65.0-60.0-55.0-50.0-45.0-40.0

1000

0.0

1250

0.0

1562

5.0

1953

1.3

2441

4.1

3051

7.6

3814

7.0

4768

3.7

5960

4.6

7450

5.8

9313

2.3

1164

15.3

1455

19.2

1818

98.9

2273

73.7

2842

17.1

3552

71.4

4440

89.2

5551

11.5

6938

89.4

8673

61.7

1084

202.

2

1355

252.

7

1694

065.

9

2117

582.

4

2646

978.

0

3308

722.

5

4135

903.

1

5169

878.

8

6462

348.

5

bandwidth (Hz)

Ps (d

Bm

), Pn

(dB

m)

-20-15-10-505101520253035404550556065707580

Ps/P

n (d

B)

Pn (dBm)Ps (dBm)Ps/Pn (dB)

figure 2 E:\documenti per

corsi\ELETTRONICA T

11

we can get some note:

• By equation 11 we can observe that the information Q(n°bit) remains constant if the product B⋅t is constant, the exchange between B and t is used in satellite application. In fact the data can be collected, along the orbit, with low B and high period of time t, the data are stored by using a memory device. After, the transmission of data stored before, is possible only in a short period when the satellite is flying above the Earth Station, by using a great bandwidth B.

• The bit-rate Q/t (bit/s) remain constant increasing the bandwidth and simultaneously reducing the PS/PN ratio, this represent a possible application in spread spectrum communication system, the greater the bandwidth the lower PS/PN ratio. Anyway if we are increasing the bandwidth, then the power of the signal PS will reduce to tend asymptotically to a constant while PN will increases more. When

112 =⎟⎟⎠

⎞⎜⎜⎝

⎛−= ⋅tB

Q

N

S

PP

eq 15

i.e. where

( ) bits/s 11log1log 22 RBBPPB

tQ

bN

S ==+=⎟⎟⎠

⎞⎜⎜⎝

⎛+= eq 16

Therefore when PS=PN, then the maximum transmission bandwidth (Hz) needed for a correct received signal, is the same order of the channel capacity (bit/s).

Onboard of a satellite, one of the main problems concerns the heavy of the transmitter systems, and the power required to transmission signaling. Therefore we are looking for a system which can beneficial of exchanging between (PS/PN) and bandwidth between input and output of the receiver. In frequency modulation systems the bandwidth of the carrier channels are greater than RX_output bandwidth channel, so at the input of the receiver can be used a lower PS/PN. Anyway FM must work above FM C/N threshold and since the greater the bandwidth, the greater the noise, then C/N threshold will increase with bandwidth. As a consequence the Carrier power C will be affected by the same amount of growth too, resulting in a bigger amplifier rather then a little amplifier as required. As a conclusion, once we have been fixed PS/PN and the bit_rate Q/t, then we can observe that the lower the PN, the lower PS. So the very important component of a transmission design is trying to reducing the noise power PN.

12

1.2 Spectral Efficiency The spectral efficiency η of a digital signal is given by the number of bits per second of data that can be supported by each hertz of bandwidth, in other words is the bit rate supported by the unit of bandwidth.

z(bits/s)/H BR

=η eq 17

In application in which the bandwidth is limited by physical constraints, the goal is to choose a signaling technique that gives the highest spectral efficiency while achieving a low probability of bit error at the system output. Moreover, the maximum possible spectral efficiency is limited by the channel noise if the error is to be small, this maximum spectral efficiency is given by Shannon’s capacity formula

z(bits/s)/H 1log2 PP

BRη

N

Sb⎥⎦

⎤⎢⎣

⎡+== eq 18

All the binary codes have η ≤ 1. Multilevel signaling can be used to achieve much greeter spectral efficiency.

13

The greater Ps/PN, the more is the spectral efficiency. Plotting this function we have the following graph:

Spectral Efficiency as a function of SNR

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

0.00

8.45

11.1

412

.79

13.9

814

.91

15.6

816

.33

16.9

017

.40

17.8

518

.26

18.6

318

.98

19.2

919

.59

19.8

720

.13

20.3

720

.61

20.8

321

.04

21.2

421

.43

21.6

121

.79

21.9

622

.12

22.2

822

.43

22.5

822

.72

22.8

622

.99

23.1

223

.24

23.3

623

.48

23.6

023

.71

23.8

223

.93

24.0

324

.13

24.2

324

.33

SNR (dB)

Spec

tral

effi

cien

cy

0.00

50.00

100.00

150.00

200.00

250.00

300.00

SNR

(Lin

eare

)

Spectral efficiencySNR (lineare)

figure 3

E:\documenti per corsi\ELETTRONICA T

14

2 CODING If the data at the output of a digital communication system have errors that are too frequent for the desired use, the errors can be often reduced by use either of two main techniques:

• Automatic repeat request (ARQ) • Forward error correction (FEC)

In an ARQ system, when a receiver circuit detects parity errors in a block of data, it requests that the data block be retransmitted. In FEC system, the transmitted data are encoded so that the receiver can correct, as well as detect errors. These procedures are also classified as channel coding because they are used to correct errors caused by channel noise. This is different from source coding, where the purpose of coding is to extract the essential information from the source and encode it into digital form so that it can be efficiently stored or transmitted using a digital techniques (example PCM) The choice between using the ARQ or the FEC technique depends on the particular application.

• ARQ is often used in computer communication systems because is relatively inexpensive to implement and there is usually a duplex (two-way) channel so that the receiving and can transmit back an acknowledgement (ACK) for correctly received data or a requests for retransmission (NAC) when the data are received in error.

• FEC is preferred on systems with large retransmission delays because if the ARQ technique were used, the effective data rate would be small; the transmitter would have long periods while waiting for the ACK/NAC indicator, which is retarded by the long transmission delay.

2.1 Code performance The improvement in the performance of a digital communication system can be achieved by the use of coding as illustrated below:

15

figure 4

It is assumed that a digital signal plus channel noise is present at the receiver input. The performance of a system that uses binary-phase-shift-keyed (BPSK) signaling is shown both for the case when coding is used and for the case when there is no coding. For the no coding case, the optimum (matched filter detector ) circuits is used at the receiver. For the coded case a Golay code is used.

• Pe is the probability of error-also called the Bit Error Rate (BER)- that is measured at the receiver output.

• Eb/N0 is the energy per-bit / noise-density ratio at the receiver input

• The coding gain is defined as the reduction in Eb/N0 (in dB) that is achieved

when coding is used, when compared with required for the uncoded case at some specific level of Pe. This improvement is significant in space communication applications, where every decibel of improvement is valuable since its possible to get the same Pe by using a lower Bit energy ( lower Power) in transmission .

The figure also show noted that there is a coding threshold in the sense that the coded system actually provides poorer performance than the uncoded system when Eb/N0 is less than the threshold value.

16

For optimum coding, Shannon’s channel capacity theorem already seen, gives the Eb/N0 required.

bits/s 1log2 RPPB

tQ

bN

S =⎟⎟⎠

⎞⎜⎜⎝

⎛+= eq 19

That is, if the source rate is below the channel capacity Rb, the optimum code will allow the source information to be decoded at the receiver with Pe→0. (i.e.10-∞) , even though there is some noise in the channel. We will now find the required Eb/N0 so that Pe→0 with the optimum (unknown) code. Assume that the optimum encoded signal is not restricted in bandwidth, i.e. assuming an optimum encoder formed by a sequence of infinite redundant bit such that in order to transmit signal plus redundant coding bit, the required bandwidth became infinite. Then from equation above where it has been posed C( as channel capacity), PS=S ( as power signal) and PN=N (as Noise power):

bit/s 1log2 ⎟⎠⎞

⎜⎝⎛ +=

NSBC eq 20

[ ][ ]

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧⎟⎟⎠

⎞⎜⎜⎝

⎛+

=⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

⇒=

=⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

→→

→∞→∞

x

xTN

E

xN

TEx

C

xB

BNTEB

WattNWattSBC

b

b

xbb

x

bbBB

02

00

20

022

1loglim/1log1lim

:1 if

/1loglim1loglim

eq 21

where : Tb is the time needed to send one bit and N is the noise power that occurs within the bandwidth of the signal:

BNdfNNB

B 00

2== ∫− eq 22

where B is the signal bandwidth and N0/2 is the noise power spectral density (W/Hz).

Channel capacity

17

L’ Hospital’s rule is used to evaluate this limit:

( )2ln

12ln

lnlog1

log1

1

lim

1loglim

002

0

20

0

0

02

0

b

b

b

b

b

b

b

b

b

b

x

b

b

x

TNEe

TNEe

TNE

eTN

E

xTN

E

xx

xTN

Ex

C

=⎟⎠⎞

⎜⎝⎛==

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

∂∂

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

=

→

→

eq 23

where we have used the logarithm property of base changes

( ) ( )( ) ( ) ( )

( )( )( ) ( ) 2ln

1 2ln

ln2log

loglog logloglog 2 ===⇒=

eeeaxx

b

e

b

ba eq 24

and the derivative property of logarithm

( ) ( )

2ln1

2ln1

11log lim

ln1log

00

0

020

b

b

b

b

b

bb

bx

a

TNE

TNE

xTN

Ex

TNE

dxd

dxdu

auu

dxd

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⇒

⇒⋅

=

→

eq 25

If we signal at a rate, approaching the channel capacity, then Pe→0, and we have the maximum information rate allowed for the Pe→0 (i.e. the optimum system). Thus

2ln

11

source ratebit re whe1

0

⇒=

==

b

b

b

bb

TNE

T

TT

C

eq 26

18

dB,NEb 5912ln

0

−== eq 27

This minimum value for Eb/N0 is -1.59 dB and is called Shannon limit. That is, if the optimum coding/decoding is used at the transmitter and receiver, error free data will be recovered at the receiver output, provided that the Eb/N0 at the receiver input is larger than -1.59 dB, assuming that the ideal (unknown )code is used. Any practical systems will perform worse than this ideal system described by Shannon’s limit, thus the goal of digital system designer is to find practical codes that approach the performance of Shannon’s limit. The better code reported in figure above, achieves their coding gains at the expense of bandwidth expansion. That is when redundant bits are added to provide coding gain, the overall data rate and, consequently, the bandwidth of the signal are increased by a multiplicative factor that is the reciprocal of the code rate. Thus, if the uncoded signal takes up all the available bandwidth, coding cannot be added to reduce receivers errors, because the coded signal would take up too much bandwidth.

19

3 INTERSYMBOL INTERFERENCE The absolute bandwidth of rectangular multilevel pulses tend to infinity; on the contrary the bandwidth available in communication system is always limited. Therefore in order to limit the bandwidth and reducing the PSD (Power Spectral Density) of signaling we needed some kind of filtering system for the binary pulses. When these pulses are filtered improperly as they pass through a communication system, they will spread in time, and the pulse for each symbol may be smeared into adjacent time slot and cause intersymbol interference (ISI).

figure 5

Now, how do we can restrict the bandwidth and still not introduce ISI ? This problem was first studied by Nyquist who discovered three different methods for pulse shaping that could be used to eliminate/reducing ISI. Let us consider a digital signaling system as shown below, in which the flat-topped multilevel signal at the input is:

1 0 0

t

0 0

t0

Input weaveform, win(t)

1 1 1

t

0 0

t0

Sampling points (trasmitter clock)

tt0

Sampling points (receiver clock)

tt0

Sampling points (receiver clock)

Individual pulse response

tt0

Ts

tt0

Received waveform, wout(t) (sum of pulse responses)

Intersymbol interference

20

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-20 -15 -10 -5 0 5 10 15 20

freq -Ts/2 Ts/2

1

t

2π/Ts

figure 6

Where at the input we have a series of flat-top impulse of a symbol rate D=1/Ts (pulses/s):

( )∑ −=n

snin nTthatw )( eq 28

where an is the amplitude of multilevel signal (for a binary signal there are only two levels permitted) and, where the shaping of a single flat top-impulse is

shaping) inpulsear (rectangul inpulse flat top a is

2t se 0,

2t se 1,

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

>

≤≡⎟⎟

⎠

⎞⎜⎜⎝

⎛=∏

s

s

ST

T

Tth(t) eq 29

figure 7 : time domain figure 8: frequency domain

Trasmitting filter HT(f)

Channel (filter)charateristics

HC(f)

Receiver filter HR(f)

Win(t) Wout(t)Wc(t)

Flat top pulses

Recovered rounded Pulse (to sampling

and decoding cicuits)

21

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-20 -15 -10 -5 0 5 10 15 20

The frequency domain of a single flat-top impulse is obtained by Fourier transform

2

22

2

2

2

11)(

2/2/2/2/

2/2/2/

2/

2/

2/

2/

2/

s

s

s

TjTj

ss

sTjTj

s

s

TjTjT

T

tjT

T

tjT

T

tj

T

TsenT

jee

TT

jee

T

T

jee

jedtej

jdtefH

ssss

sss

s

s

s

s

s

ω

ω

ωω

ωωω

ω

ωωωω

ωωωωω

⎟⎠⎞

⎜⎝⎛

=−

=−−

=

=−−

=⎥⎦

⎤⎢⎣

⎡−

=⋅−−

=⋅=

−++−

+−+

−

−

−

−

−

− ∫∫

eq 30

The PSD of single flat top impulse is then obtained by :

( )( )

2

2

22/2/22/

2/

22/

2/

22/

2/

2

2

2

2/2/

221

11)(

s

s

s

s

sTjTjT

T

tj

T

T

tjT

T

tj

T

TsenT

TT

jeee

j

dtejj

dtefH

sss

s

s

s

s

s

ω

ω

ωω

ωω

ωωω

ωω

⎟⎠⎞

⎜⎝⎛

=

=−−

=⎥⎦

⎤⎢⎣

⎡−

=

=⋅−−

=⋅=

−

−

−

−

−

−

− ∫∫

eq 31

which is plotted in figure below

figure 9

22

3.1 Spectral property reminder (square wave spectrum) Let us consider a square wave signal ( )ts as shown below

figure 10

The relative ( )ts0 is shown in figure below where it is represented by a unitary impulse ( )tsI

figure 11

Then ( )ω0S has the shape reported below

figure 12

s(t)

t -T/2 -∆t/2 T/2 ∆t/2

t

s0(t) = sI(t)

-T/2 -∆t/2 T/2 ∆t/2

S0(ω) = SI(ω)

ω

2π/∆t 4π/∆t-2π/∆t-4π/∆t

23

As a consequence ( )ωS has the following shape

Figura 1

3.2 Equalizing filter Using the Dirac delta function and the convolved product we can also rewrite the flat-top impulse series as:

( ) ( ) ( ) ( ) ( )thnTtanTtthanTthatw sn

nsn

nn

snin ∗⎥⎦

⎤⎢⎣

⎡−=−∗=−= ∑∑∑ δδ )( eq 32

The output of the linear system would be just the input impulse train convolved with the equivalent impulse response of the overall system; that is

( ) ( ) ( ) ( ) ( )ththththnTtatw RCTsn

nout ∗∗∗⎭⎬⎫

⎩⎨⎧

∗⎥⎦

⎤⎢⎣

⎡−= ∑ δ)( eq 33

Calling the equivalent impulse response of the overall system as:

( ) ( ) ( ) ( ) ( )ththththth RCTe ∗∗∗= eq 34

S(ω)

ω

2π/∆t 4π/∆t-2π/∆t-4π/∆t

ω0=2π/T0

24

then

( ) ( ) ( )sen

nesn

nout nTthathnTtatw −=∗⎥⎦

⎤⎢⎣

⎡−= ∑∑ δ)( eq 35

Note that he(t) is also the pulse shape that will appear at the output of the receiver filter when a single Dirac pulse is fed into the transmitting filter. The equivalent system transfer function is in frequency domain and it is represented by the product of the corresponding Fourier transform of each single element:

)()()()()( fHfHfHfHfH RCTe = eq 36

H(f) is the Fourier transform already seen that define the pulse shape

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛

=

2

2)(s

s

ss

s T

TsenTfH

ω

ω eq 37

Then the receiving filter is given by

)()()()()(

fHfHfHfHfH

CT

eR = eq 38

When He(f) is chosen to minimize the ISI, HR(f) is called an equalizing filter. The equalizing filter characteristic depends on Hc(f), the channel frequency response, as well as on the required He(f). When the channel consists of dial-up telephone lines, the channel transfer function changes from call to call and the equalizing filter may need to be an adaptive filter. In this case, the equalizing filter adjusts itself to minimize the ISI. In some adaptive schemes, each communication session is preceded by a test pattern that is used to adapt the filter electronically for the maximum eye opening (minimum ISI). Such sequence are called learning or training sequence and preambles. As an example of training sequence we report those used in GSM mobile network Um-Interface for a burst of 148 bits:

25

figure 13

The pulse train at the output of the receiver filter is

( ) )( ∑ −=n

senout nTthatw eq 39

The output pulse shape is affected by the input pulse shape (flat-topped in this case), the transmitter filter, the channel filter, and the receiving filter. Because, in practice, the channel filter is already specified, the problem is to determine the transmitting filter and the receiving filter that will minimize the ISI on the rounded pulse at the output of the receiving filter.

3.3 Nyquist’s First Method (Zero ISI) Nyquist’s first method for eliminating ISI is to use an equivalent transfer function He(f), such that the impulse response satisfies the condition

⎭⎬⎫

⎩⎨⎧

≠=

=+0kfor 00kfor

)(C

kTh se τ eq 40

where k is an integer, Ts is the symbol (sample) clocking period, τ is the offset in the receiver sampling clock times due to propagation delay that should be compared with the times of the input symbols, and C is a nonzero constant.

Encrpted bits 57

TB 3

Training sequence 26

1 Encrpted bits 57

1 TB 3

GP 8.25

0.577 ms 156.25 bit

Normal burst (NB): TCH end other control channel, except. RACH, SCH e FACH

Fixed bits 142

TB 3

TB 3

GP 8.25

0.577 ms 156.25 bit

Freq. correction burst (FB): Is equivalent to unmodulated carrier . I used to synchronize the mobile in frequency

Synchronization burst (SB): Is used to temporally synchronize the mobile, it contains TDMA frame number and and BSIC

0.577 ms 156.25 bit

TB 3

Synchronization sequence 64

Encrpted bits 39

TB 3

GP 8.25

Encrpted bits 39

26

That is, for a single flat-top pulse of level a at the input to the transmitting filter at t=0, the received pulse would be a⋅he(t) and it would have a value of a⋅C at t=τ (i.e. for K=0) but it would not cause interference at any other sampling time because he(kTs+τ)=0 when k≠0.

figure 14

figure 15

Improper filtering with ISI

1 1 0 0

Input weaveform, win(t)

Individual Nyquist filter pulse response is 0 in KTs

NO ISI

t

Tss

Ts 2Ts 3Ts

1 0 0

t

0 0

t0

Input weaveform, win(t)

1 1 1

t

0 0

t0

Sampling points (trasmitter clock)

tt0

Sampling points (receiver clock)

tt0

Sampling points (receiver clock)

Individual Nyquist filter pulse response is 0 in KTs

tt0

Ts

s

tt0

Received waveform, wout(t) (sum of filter pulse responses)

No Intersymbol interference Ts

Improper filtering with ISI

27

2

2DBDB =+−=−

freq

A rectangular filter function can be used for this purpose, therefore suppose that we chose a (sinx)/x function for he(t), in particular, let τ =0, and chose

ss

s

se

Tf

tftfth

/1

sin)(

=

=ππ

KTs=0 KTs≠0 eq 41

Then this impulse response satisfies zero ISI Nyquist’s first criterion because in t=KTs=0 we have max amplitude of the output function he(t) whereas in t=KTs≠0 we have zero amplitude of function.

11coslim sinlim 00sinlim)(lim

0000=−=

−=⎥

⎦

⎤⎢⎣

⎡⇒==

→→→→s

s

ts

s

ts

s

tet ftff

tftf

dtd

tftfth

πππ

ππ

ππ

eq 42

Consequently, if the transmit and receive filters are designed so that the overall transfer function is

⎟⎟⎠

⎞⎜⎜⎝

⎛∏=

sse f

ff

fH 1)( eq 43

i.e. like a rectangular frequency response, there will be no ISI. Furthermore, the absolute bandwidth of this transfer function is 2B=fs=D . This is the optimum filtering method to produce a minimum bandwidth system. It will allow signaling at a baud rate of D=1/Ts=2B (pulses/s), where B=D/2 is the absolute bandwidth of the system. As an example, if D=271 ks/s then the bandwidth of filter is B=D/2=135.5 KHz. However, the (sinx/x) type of overall pulse shape has two practical difficulties:

• The overall amplitude transfer characteristic He(f) has to be flat over a bandwidth –B<f<B and zero elsewhere. This is physically unrealizable (i.e. the impulse response would be non casual and of infinite duration)

• The synchronization of the clock in the decoding sampling circuit has to be almost perfect, since the (sinx/x) pulse decays only as 1/x and is zero in adjacent time slot only when is at the exactly correct sampling time. Thus, inaccurate sync will cause ISI.

Because of these difficulties, we are forced to consider other pulse shapes that have a slightly wider bandwidth then a rectangular –B<f<B.

28

The idea is to find pulse shapes that go through zero at adjacent sampling points and yet have an envelope that decays much faster than 1/x so that clock jitter in the sampling times KTs≠0 does not cause appreciable ISI. One solution for the equivalent transfer function, which has many desirable future, is the raised cosine-rolloff Nyquist filter.

3.4 Raised Cosine-Rolloff Nyquist Filtering DEFINITION: the raised cosine-rolloff Nyquist filter has the transfer function:

( )

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

>

<<⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡ −+

<

=∆

Bf

Bfff

ffff

fHe

,0

,2

cos121

,1

)( 11

1

π eq 44

Where B is the absolute bandwidth and the parameters f∆ and f1 are

∆

∆

−=

−=

fff

fBf

01

0

eq 45

f0 is the 6-dB Raised Cosine filter bandwidth

00

rf fffr ∆ =⇒= ∆ eq 46

Where r is the rolloff factor of the filter. Consequently

)1(

)1(

00001

0000

rfrfffff

rfrffffB

−=−=−=

+=+=+=

∆

∆

eq 47

The filter characteristics are illustrated in figure below:

29

figure 16

For several value of the rolloff factor r the corresponding required signaling transmission bandwidth B are:

figure 17

For rolloff factors r=0 we have f∆=0 and we obtain the minimum-bandwidth required case, where f0=B. The corresponding impulse shape response is

r0=0, min bandwidth

r1=0.5

0.5

1.0

Freq.

He(f)

r2=1

f0=fB

f∆ f∆

f0 B -B f1 -f1 -f0

0.5

1.0

Freq.

He(f)

6-db bandwidth

30

( ) ( )[ ] ( ) ( )( ) ⎥⎦

⎤⎢⎣

⎡

−⎟⎟⎠

⎞⎜⎜⎝

⎛==

∆

∆−2

0

00

1

412cos

22sin2

tftf

tftfffHFth ee

πππ

eq 48

when r = 0, the corresponding impulse pulse shape response became like (sinx/x):

( ) ( )[ ] ( ) ( )( )

( )⎟⎟⎠

⎞⎜⎜⎝

⎛=⎥

⎦

⎤⎢⎣

⎡

−⎟⎟⎠

⎞⎜⎜⎝

⎛== −

tftff

tftfffHFth ee

0

002

0

00

1

22sin2

010cos

22sin2

ππ

ππ

eq 49

Time Response for different rolloff factor: r=0 r= 0.5 r=1

-5

0

5

10

15

20

-0.4

00

-0.3

50

-0.3

00

-0.2

50

-0.2

00

-0.1

50

-0.1

00

-0.0

50

0.00

0

0.05

0

0.10

0

0.15

0

0.20

0

0.25

0

0.30

0

0.35

0

0.40

0

Ts (symbol time)

impu

lse

resp

once

he_(r=0)he_(r=0.5)he_(r=1)

figure 18

E:\documenti per corsi\ELETTRONICA T As the absolute bandwidth is increased (e.g. r=0, r=0.5 or r = 1.0) the filtering requirements are relaxed, the clock timing requirements are relaxed too, since the envelope of the impulse response decays faster than 1/t (on the order of 1/t3 for large value of t).

31

Let us now develop a formula which gives the baud rate that the raised cosine-rolloff system can support without ISI. From figure above, the zeros in the system impulse response occur at t =n/2f0 where n≠0. Therefore, data pulses may be inserted at each of these zero points without causing ISI. That is, referring to

⎭⎬⎫

⎩⎨⎧

≠=

=+0kfor 00kfor

)(C

kTh se τ eq 50

with τ =0, we see that the raised cosine-rolloff filter, satisfies Nyquist’s first criterion (for the absence of ISI) if the symbol clock period is equal to Ts=1/(2f0). The corresponding baud rate is

( )2

/ 2100

DfsSymbolfT

Ds

=⇒==

That is, the 6-dB bandwidth of the raised cosine-rolloff filter, f0, is designed to be half the symbol (baud) rate.

12 2

D

2

2

22

2

2 0

0

0

rB DDBr D

DBD

DB

D

DB

ffB

ffr

+=⇒−=⋅⇒

⇒−

=

−

=−

=−

== ∆

eq 51

where B is the absolute bandwidth of the system and r is the system rolloff factor.

( )rDB += 12

eq 52

The greater r the grater B, and as a function of B and r we can aspect a maximum value of symbol rate D. Comparing to first criteria where B=D/2 now the bandwidth is increased when r≠0

32

4 BANDPASS SIGNALING

4.1 COMPLEX ENVELOPE RAPPRESTNATION OF BANDPASS WAVEFORMS

4.1.1 Definitions: Baseband, Bandpass, and modulation Definition. A baseband waveform has a spectral magnitude that is nonzero for frequencies in the vicinity of the origin (i.e. f=0) and negligible elsewhere.

figure 19

Definition. A bandpass waveform has a spectral magnitude that is non zero for frequencies in some band concentrated about a frequency f =±fc, where fc>>0. the spectral magnitude is negligible elsewhere. fc is called the carrier frequency.

figure 20

Definition. Modulation is the process of imparting the source information on to a bandpass signal with a carrier frequency fc by the introduction of amplitude or phase perturbations or both. This bandpass signal is called the modulated signal s(t), and the baseband source signal is called the modulating signal m(t). As the modulated signal passes through the channel, noise corrupts it. The result is a bandpass signal-plus noise waveform that is available at the receiver input, r(t). (see figure below) the receiver has the job of trying to recover the information that was sent from the source; m* denote the corrupted version of m.

+fC -fC

f

f0 f

33

4.1.2 Complex Envelope Representation All bandpass waveform, whether they arise from a modulated signal, interfering signals, or noise, may be represented in a convenient form by the theorem that follows. v(t) will be used to denote the bandpass waveform canonically; it can represent the signal when s(t)=v(t), the noise when n(t)=v(t), the signal plus noise at the channel end when r(t)=v(t).

Figure 21: communication system

4.1.3 Theorem

Any physical bandpass waveform can be represented by

{ } [ ]{ }( ) Re ( ) Re ( ) cos sincj tc cv t g t e g t t j tω ω ω= = + eq 53

here, Re(⋅) denotes the real part of (⋅), g(t) is called the complex envelope of v(t), and fc is the associated carrier frequency (in Hertz) where ωc=2πfc is the radian frequency. Furthermore, two other equivalent representations are

[ ]( ) ( ) cos ( )cv t R t t tω θ= + eq 54

and

( ) ( ) cos ( )sinc cv t x t t y t tω ω= − eq 55

where

( ) ( )( ) ( ) ( ) ( ) ( )j g t j tg t x t jy t g t e R t e θ∠= + = = eq 56

Signal processing

Carrier circuits

Trasmission medium

(channel) Carrier circuits

Signal processing

m g(t) s(t) r(t) g*(t) m*

Trasmitter Receiver

34

{ }( ) Re ( ) ( ) cos ( )x t g t R t tθ= = eq 57

{ }( ) Im ( ) ( )sin ( )y t g t R t tθ= = eq 58

2 2( ) ( ) ( ) ( )R t g t x t y t= + eq 59

and

1 ( )( ) ( ) tan( )

y tt g tx t

θ − ⎛ ⎞∠ = ⎜ ⎟

⎝ ⎠ eq 60

Consequently we obtain:

{ } [ ]{ } [ ][ ]{ }{ }

( ) Re ( ) Re ( ) cos sin Re ( ) ( ) cos sin

Re ( )cos ( )sin ( )cos ( )sin ( ) cos ( )sin

cj tc c c c

c c c c

c c

v t g t e g t t j t x t jy t t j t

x t t jx t t jy t t y t tx t t y t t

ω ω ω ω ω

ω ω ω ω

ω ω

= = + = + + =

= + + − =

= −

eq 61

In communication systems, the frequencies in the baseband signal g(t) are said to be heterodyned up to fc. The complex envelope, g(t), is usually a complex function of time, it is the baseband equivalent of the bandpass signal v(t), and it is a generalization of the phasor concept. In this case x(t) is said to be the in-phase modulation component also colled I(t), and y(t) is said to be quadrature modulation component Q(t) associated with v(t). Alternatively, the polar form of g(t) is represented by R(t) and θ(t); here R(t) is always non negative and is said to be amplitude modulation (AM) on v(t), while θ(t) is said to be the phase modulation (PM) on v(t).

35

4.2 REPRESENTATION OF MODULATED SIGNALS Modulation is the process of encoding the source information m(t) (modulating signal) into a bandpass signal s(t) (modulated signal). Consequently, the modulated signal is just a special application of the bandpass representation. The modulated signal is given by

{ }tj cetgts ω)(Re)( = eq 62

where ωc=2πfc is the carrier frequency. The complex envelope g(t) is a function of modulating signal m(t). That is,

[ ])()( tmgtg = eq 63

36

4.3 SPECTRUM OF BANDPASS SIGNALS The spectrum of a bandpass signal is directly related to the spectrum of its complex envelope. Theorem. If a bandpass waveform is represented by

{ }tj cetgtv ω)(Re)( = eq 64

than the Spectrum of the bandpass waveform is

( ) ( )[ ]cc ffGffGfV −−+−= *

21)( eq 65

where

[ ])()( tgFfG = eq 66

is the Fourier transform of g(t) the Power Spectral Density (PSD) of the waveform is

[ ])()(41)( cgcgv ffPffPfP −−+−= eq 67

where

)( fPg is the PSD of g(t)

figure 22

G(-f-fc) G(f-fc)

G(f) G(f)

37

5 AM, FM, PM MODULATED SYSETMS

5.1 Definitions

Amplitude Modulation (AM) is a system where the frequency of a carrier wave is held constant while the amplitude is varied in sympathy with the voltage of the modulating signal.

Frequency Modulation (FM) is a system where the amplitude of a carrier wave is held constant while the frequency is varied in sympathy with the voltage of the modulating signal.

Phase Modulation (PM) is a similar system where the phase of the carrier wave is varied in sympathy with the voltage of the modulating signal, and as in frequency modulation, the amplitude of a carrier is held constant.

Phase is the position of a rotating vector or phasor.

Angular Velocity is the rate of change of phase (usually expressed in radians per second).

The Radian is a unit of angular displacement (as is degrees), there are 2*π radians in a full circle (or 360°), so a radian is approximately 57°.

Frequency is a measure of the number of repetitions of a periodic waveform in unit time (1 second). Frequency of a carrier wave is related to Angular Velocity, there are 2*π radians in each cycle of a carrier wave, so the Angular Velocity is 2*π * frequency.

Pi is a numeric constant, it's value is approximately 3.1411592654. (You can approximate it by using 22/7 - the error is less than 0.05%).

5.2 AMPLITUDE MODULATION The complex envelope of an AM signal is given by

[ ]( ) 1 ( )cg t A m t= + eq 68

Where the constant Ac, has been included to specify the power level and m(t) is the modulating signal (which may be analog or digital). These equations reduce to the following representation for AM signal:

38

[ ] [ ][ ]ttmjAttmA

etmAeTgts

cccc

tjc

tj cc

ωω

ωω

sin))(1(cos))(1(Re ))(1(Re)(Re)(+++=

=+== eq 69

[ ]( ) 1 ( ) cosc cs t A m t tω= + eq 70

For convenience, it is assumed that the modulating signal m(t) is a sinusoid. The modulating signal corresponds to the in-phase component x(t) of the complex envelope; it also correspond to the real envelope ( )g t when m(t)≥-1 (the usual case).

If ( ) cos( )mm t m tω= eq 71

Then recalling that ( ) ( )[ ]βαβαβα −++= coscos21coscos we have

[ ] [ ]

( ) ( )

( ) 1 ( ) cos 1 cos cos cos cos cos1 1 cos cos cos2 2

c c c m c c c c m c

c c c c m c c m

s t A m t t A m t t A t A m t t

A t A m t A m t

ω ω ω ω ω ω

ω ω ω ω ω

= + = + = + =

= + + + − eq 72

in figure below is shown a sinusoidal modulating wave and the resulting modulated AM signal.

figure 23

1/Tc=fc Tc

[ ]( ) 1 ( )cg t A m t= +

( )s t Amin

envelope = information associated with modulating signal

Ac Amax

t

m(t)

39

The overall modulation percentage is:

[ ] [ ]max min max ( ) -min ( )A -A%modulation= 100 1002Ac 2Ac

m t m t= eq 73

5.2.1 Normalized AM average power The normalized average power of an AM signal is:

[ ] )(21)(

21)(2)(1

21)(1

21

)(21)(

22222222

22

tmAtmAAtmtmAtmA

tgts

ccccc ++=++=+=

== eq 74

If modulation m(t) contains no dc level, then 0)( =tm and the normalized power of an AM signal is

The voltage magnitude spectrum of the AM signal is given by:

( ) ( ) ( ) ( )[ ]ccccc ffMffffMffAfS ++++−+−= δδ

2)( eq 76

)(21

21)( 2222 tmAAts cc += eq

discrete carrier power

sideband power

40

figure 24

5.2.2 Definition: The modulation efficiency The modulation efficiency is the percentage of the total power of the modulated signal that conveys information. In AM signal, only the sideband components conveys information, so the modulation efficiency is

100)(1

)(100

)(21

21

)(21

2

2

222

22

tm

tm

tmAA

tmAE

cc

c

+=

+= eq 77

The highest efficiency that can be attained for a 100% AM signal would be 50%, (for the case when square-wave modulation is used m=1).

)( fS A

2cA

fc-B fc+B fc -fc-B -fc+B -fc

-B

)( fM

Upper sideband

Lower sideband

Discrete carrier term with weight=1/2Ac

a) Magnitude spectrum of modulation signal

f

f

b) Magnitude spectrum of AM signal

B

δ(f-fc) δ(f+fc)

41

6 PHASE MODULATION AND FREQUENCY MODULATION

6.1 Representation of PM and FM Signals Phase Modulation (PM) and Frequency modulation (FM) are special cases of angle-modulated signaling. In this kind of signaling the complex envelope is

)()( tjceAtg θ= eq 78

here θ(t) is a linear function of the modulating signal m(t), while g(t) is a non linear function of the modulation. The resulting angle-modulated signal is:

{ } { } [ ]{ } [ ])(cosReRe)(Re)( )()( ttAeAeeAetgts ccttj

ctjtj

ctj ccc θωωθωθω +==== + eq 79

The relation between phase θ(t) and the instantaneous frequency fi is:

2)( )(21

∫∞−

=⇒=t

ii dtftdt

tdf πθθπ

eq 80

1. for PM, the phase is directly proportional to the modulating signal m(t);

)()( tmDt p=θ eq 81

Where the proportionally constant Dp is the phase sensitivity(phase deviation constant) of the phase modulator, having units of radians per volt [assuming that m(t) is a voltage waveform].

2. For FM, the phase is proportional to the integral of m(t), so that

σσθ dmDtt

f ∫∞−

= )()( eq 82

where the frequency deviation constant Df, has units of radians/volt-second.

42

We can develop an example for a PM first and FM later. suppose

tAtm mm ωcos)( = eq 83

• in the PM case we have

ttADtmDt mpmmpp ωβωθ coscos)()( === eq 84

Where

mpp AD=β is the phase modulation index • in the FM case we have

ttDAdADdmDt mfmfm

t

mmf

t

f ωβωω

σσωσσθ sinsin1cos)()( ==== ∫∫∞−∞−

eq 85

Where

mfmf DAω

β 1= is the frequency modulation index

The complex envelope is:

tjc

tjc

mfeAeAtg ωβθ sin)()( == eq 86

Therefore the modulated bandpass signal is:

{ }{ } [ ]{ } [ ]

⎥⎦

⎤⎢⎣

⎡+=⎥

⎦

⎤⎢⎣

⎡+=

=+===

==

∫∫∞−∞−

+

t

mm

mfcc

t

mfcc

cmfcttj

ctjtj

c

tj

tAD

tAttA

ttAeAeeA

etgtscmfcmf

c

ωω

ωωβω

ωωβωωβωωβ

ω

coscoscoscos

sincosReRe

)(Re)(sinsin

eq 87

43

the last term: [ ]tt mfc ωβω sincos + can be studied by Bessels function

figure 25

Definition. If a bandpass signal is represented by

[ ])(cos)()(cos)()( tttRttRts c θωψ +== eq 88

then the instantaneous frequency (Hertz) of s(t) is [Boashash, 1992]

[ ]

dttdf

dttd

dtttd

dttdttf

cc

cii

)(21)(

21

21

)(21)(

21)(

21)(

θπ

θπ

ωπ

θωπ

ψπ

ωπ

+=+=

=⎥⎦⎤

⎢⎣⎡ +

=⎥⎦⎤

⎢⎣⎡==

eq 89

Therefore using σσθ dmDtt

f ∫∞−

= )()(

)(21

)(

21)(

21)( tmDf

dt

dmDdf

dttdftf fc

t

f

cci π

σσ

πθ

π+=

⎥⎦

⎤⎢⎣

⎡

+=+=∫∞− eq 90

44

Of course, this is the reason for calling this type of signaling frequency modulation—the instantaneous frequency varies about the assigned carrier frequency fc, in a manner that is directly proportional to the modulating signal m(t).

figure 26

figure above show how the instantaneous frequency varies when a sinusoidal modulation (for illustrative purposes ) is used. The frequency deviation from the carrier frequency is

dttdftftf cid)(

21)()( θπ

=−= eq 91

and the peak frequency deviation is

[ ] ⎥⎦⎤

⎢⎣⎡==∆

dttdtff d)(

21max)(max θπ

eq 92

fc+∆f

fc-∆f

fc

fi(t)---istantaneous frequency of the corresponding FM signal

m(t)---sinusoidal modulating signal

s(t)---correspondign FM signal

vp

Ac

s(t)---has constant amplitude

45

note that ∆f is a non negative number. In some applications, such as unipolar digital modulation, the peak to peak deviation is used:

⎥⎦⎤

⎢⎣⎡−⎥⎦

⎤⎢⎣⎡=∆

dttd

dttdf pp

)(21min)(

21max θ

πθ

π eq 93

For FM signaling, the peak frequency deviation is related to the peak modulating voltage by:

[ ] [ ] pfffd VDtmDtmDdt

tdtffπππ

θπ 2

1)(max21)(

21max)(

21max)(max ==⎥⎦

⎤⎢⎣⎡=⎥⎦

⎤⎢⎣⎡==∆ eq 94

An increase in the amplitude of the modulation signal Vp will increase ∆f. This in turn will increase the bandwidth of the FM signal, but will not affect the average power level of the FM signal, which is AC

2/2. As Vp is increased, spectral components will appear farther and farther away from the carrier frequency, and the spectral components near the carrier frequency will decrease in magnitude, since the total power in the signal remains constant. This situation is distinctly different from AM signaling, where the level of the modulation affects the power in the AM signal, but does not affect its bandwidth. In a similar way, the peak phase deviation may be defined by:

[ ])(max tθθ =∆ eq 95

which for PM, is related to the peak modulation voltage by

[ ])(max tmDVD ppp ==∆θ eq 96

6.1.1 Definition for peak phase deviation and peak frequency deviaton.

• The phase modulation index is given by

θβ ∆=p eq 97

where ∆θ is the peak phase deviation.

• The frequency modulation index is given by:

46

BF

f∆

=β eq 98

where ∆F is the peak frequency deviation and B is the bandwith of the modulating signal, which, for the case of sinusoidal modulation, is fm, i.e the sinusoid sinusoid. Using the deviation frequency expression found above we can rewrite:

m

f

m

f

m

f

mf

tmDf

tmDf

tmD

fF

ωππβ

)(2

)()(21

===∆

= eq 99

Therefore the more greater m(t) the more greater βf, and the more greater ωm, the lower βf. These facts have fundamental implication when we will speak about noise effect in FM / PM modulation. For digital signals, an alternative definition of modulation index is sometimes used and is denoted by h in the literature. This digital index is:

πθ∆

=2h eq 100

where 2∆θ is the maximum peak to peak phase deviation change during the time that it takes to send one symbol, Ts. Strictly speaking, the FM index is different only for the case of single-tone (i.e. sinusoidal) modulation. However, it is often used for other waveshapes, where B is chosen to be the highest frequency or the dominant frequency in the modulating waveform.

6.2 Spectra of Angle-Modulated signals We found that the spectrum of an angle modulated signal is given by

( ) ( )[ ]cc ffGffGfS −−+−= *21)( eq 101

where

[ ] [ ] [ ][ ])(()()()( tmfjc

tjc eAFeAFtgFfG === θ eq 102

47

Since g(t) is a non linear function of m(t) a general formula relating G(f) cannot be obtained, that is G(f) must be evaluated case by case basis for the particular modulating waveshape of interest, furthermore superposition does not hold, and the FM spectrum for the sum of two modulating waveshapes is not the same as summing the FM spectra that were obtained when the individual waveshapes were used. The evaluation into a closed form is not easy, one often has to use a numerical techniques to approximate the Fourier transform integral. An example for the case of sinusoidal waveshape will now be worked out.

6.2.1 Spectrum of a PM or FM signal with Sinusoidal Modulation Let us assume as an example

ttDAdADdmDt mfmfm

t

mmf

t

f ωβωω

σσωσσθ sinsin1cos)()( ==== ∫∫∞−∞−

eq 103

then the complex envelope is

tjc

tjc

meAeAtg ωβθ sin)()( == eq 104

which is periodic with period

mmm f

Tωπ21

== eq 105

consequently g(t) could be represented by a Fourier series that is valid over all time (-∞<t<∞);

∑∞

−∞=

=n

tjnn

mectg ω)( eq 106

where:

[ ] ( )[ ]dteTAdteeA

Tdtetg

Tc

m

m

mmm

m

m

mm

m

m

T

T

tntj

m

ctjn

T

T

tjc

m

tjn

T

Tmn ∫∫∫

+

−

−−

+

−

−

+

−

===2

2

sin2

2

sin2

2

1)(1 ωωβωωβω eq 107

Calling:

48

tmωϑ = eq 108

ππωϑ

ππωϑ

+=⎟⎠⎞

⎜⎝⎛−=⎟

⎠⎞

⎜⎝⎛+=

−=⎟⎠⎞

⎜⎝⎛−=⎟

⎠⎞

⎜⎝⎛−=

22

2

22

2

2

1

m

m

mm

m

m

mm

TT

T

TT

T

eq 109

ϑπ

πωϑ dTdtdtT

dtd m

mm 2

2=⇒== eq 110

Substituting we obtain

( )[ ] ( )[ ] ( )[ ] ϑπ

ϑπ

πϑ

πϑ

ϑϑβ

πϑ

πϑ

ϑϑβωωβ deTTAdTe

TAdte

TAc njm

m

cm

TT

TT

nj

m

c

T

T

tntj

m

cn

m

m

m

m

m

m

mm ∫∫∫=

−=

−

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛ −=

−

+

−

− === sin2

2

22

sin2

2

sin

22 eq 111

( )[ ] ( )βϑπ

πϑ

πϑ

ϑϑβnc

njcn JAdeAc =

⎭⎬⎫

⎩⎨⎧

= ∫=

−=

−sin

21

eq 112

This integral [ known as the Bessel function of the first kind of the nth order, Jn(β) ] cannot be evaluated in closed form, but it has been evaluated numerically. Taking the Fourier transform of g(t) we obtain:

[ ] ( ) ( ) ( )∑∑+∞=

−∞=

+∞=

−∞=

−=−==n

nmnc

n

nmn nffJAnffctgFfG δβδ)()( eq 113

The spectrum is a series of Dirac-impulse spaced by nfm and multiplied by the amplitude of Bessel function. Therefore the discrete carrier term (at f=fc) is proportional to )(0 βJ ; consequently, the level (magnitude) of the discrete carrier depends on the modulation index, it will be zero if 0)(0 =βJ . The bandwidth of the bandpass angle-modulated signal depend on β and fm. In fact it can be shown that 98% of the power is approximately contained in the bandwidth

49

BBT )1(2 += β eq 114

where β is either the phase modulation index or the frequency modulation index and B is the bandwidth of the modulating signal (which is fm for sinusoidal modulation). This formula is called Carson’s rule. Using this result, we get the spectrum of FM or PM with sinusoidal modulation for various modulation indexes as reported in figure below.

Fig below shows the spectral distribution of an FM wave, The column labelled fc is the carrier and the other columns are the nth sidebands. Note the increasing in large number of sidebands and the simultaneously carrier amplitude decreasing (in the centre of the diagram) close to zero, when β change up to β =2, we can see how most of the FM wave energy is in the sidebands.

The values of Bessel functions are commonly published in tables as in Table below. In the table, mf is the modulation index and J0..Jn are the amplitude coefficients for the carrier and sidebands

mf J0 J1 J2 J3

0.00 1.000 0.000 0.000 0.0000.05 0.999 0.025 0.000 0.0000.50 0.938 0.242 0.031 0.0031.00 0.765 0.440 0.115 0.0201.50 0.512 0.558 0.232 0.0612.00 0.224 0.577 0.353 0.1292.50 -0.048 0.497 0.446 0.2173.00 -0.260 0.339 0.486 0.309

4.00 -0.397 -0.066 0.364 0.430

Table 1

The greater mf the lower j0.

50

figure 27

1.0

J0(0,2)

J1(0,2) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

cA

fS

21

)(

f β=0,2

1.0

J0(1) J1(1)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

cA

fS

21

)(

f β=1

J2(1)

fc BT

fc

fc+fm

BT

fc+2fm

1.0

J0(2) J1(2)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

cA

fS

21

)(

f β=2

J2(2)

fc

fc+fm

BT

fc+3fm

J2(2)

1.0

J0(5) J1(5)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

cA

fS

21

)(

f β=5

J2(5)

fc

fc+fm

BT

fc+6fm

J3(5) J4(5) J5(5) J6(5)

51

6.3 Noise and frequency modulation

6.3.1 Noise triangle

To understand the effect of noise on an FM signal, it helps to consider a single noise frequency vector added to the FM signal vector Ec (see figure below). Since it is at a different frequency, the noise vector will rotate about it with an angular velocity equal to the difference between the noise frequency and the carrier frequency. This will produce a variation in amplitude and phase of the resultant vector. The amplitude variation can be largely eliminated in a limiter stage, but the phase variation (shown as Ø) remains

figure 28 : Effect of a noise phasor on an FM carrier phasor

The modulation index due to the noise voltage is constant, whereas the modulation index for the desired modulation signal decreases with the increasing of the modulating frequency fm or with the band B of the modulating signal (for FM).

mm

mff ff

BfAD ∆≅

∆==

ωβ 1

eq 115

Or the similar expression already found

m

f

m

f

m

f

mf

tmDf

tmDf

tmD

ff

ωππβ

)(2

)()(21

===∆

= eq 116

Consequently the lower the βf, the lower the phase variation θ(t) and the lower the deviation frequency ∆f:

ttDAdADdmDt mfmfm

t

mmf

t

f ωβωω

σσωσσθ sinsin1cos)()( ==== ∫∫∞−∞−

eq 117

This means that the noise degrades the Signal-to-Noise Ratio for an FM / PM modulation more at higher modulating frequencies than at lower modulating frequency.

Resulting Vector

52

A plot of the noise vs frequency has a triangular shape, hence the term noise triangle

figure 29: FM noise sideband distribution (mf=1).

Figure above shows the distribution of noise sidebands compared with the AM case.

6.4 Preemphasis and Deemphasis in angle modulated Systems

As described, with FM reception, noise contributes more to the high frequency portions of the spectrum than to the lower frequency portions. The higher frequency portions therefore tend to have a lower Signal-to-Noise Ratio than the lower frequency portions.

The noise contribution of the high frequency region can be reduced by transmitting the highs at increased relative levels and then reducing the level by the same amount at the receiver. This boosting of the highs at the transmitter is known as Pre-emphasis and the reduction of the highs at the receiver is called De-emphasis.

For realistic reproduction, the amount of de-emphasis at the receiver must equal the pre-emphasis at the transmitter. Simple networks are utilized to achieve this. The networks are typically a single RC filter stage, and are characterised by the time constant of the filter section

In a modulator the audio modulating signal is boosted with Pre-emphasis prior to modulation. In the receiver, De-emphasis is used after demodulation to recover a flat audio frequency response. This results in a much improved Signal-to-Noise Ratio for any given FM transmission system. This gives an overall baseband frequency response that is flat, while improving the signal to noise ratio at the receiver output.

FM noise triangle AM noise rectangular

f fc

noise

53

figure 30

The standard in the USA for FM Radio is τ =75 microseconds. Be aware that some countries have standardized on τ =25 or τ =50 microseconds and International Satellites use what is called the J-17 standard as well as others.

figure 31

R1

R1

C

C

R2

f1 f2

Log(f)

Log(f)

( )fHLog p

( )fHLog p

Pre-enphasis filter

De-enphasis filter

)/(1)/(1)(

2

1

ffjffjKfH p +

+=

CRRRRf

CRf

21

21

22

111 22

1 2

12

1τπττπτ+

====

)/(11)(

1ffjfH p +=

21

21

111 CR

fτπτ

==

Bode plot Pre-enphasis

Bode plot De-enphasis

Pre-emphasis filter

FM trasmitter

Trasmission medium

(channel) FM

receiver De-emphasis

filter mf s(t) r(t) g*(t) m*

Trasmitter Receiver

Modulating Input signal vin

54

6.4.1 De-Emphasis response table -- 75 MICROSECOND RESPONSE TABLE for broadcasting FM

UNLIMITED 26 dB 26 dB FREQUENCY 75uS 75uS (roofed) 25uS (roofed) 50 Hz -0.00 dB -0.00 dB -0.00 dB 100 Hz -0.01 dB -0.01 dB -0.00 dB 500 Hz -0.23 dB -0.23 dB -0.03 dB 1 KHz -0.87 dB -0.87 dB -0.11 dB 2 KHz -2.76 dB -2.75 dB -0.41 dB 3 KHz -4.77 dB -4.75 dB -0.87 dB 4 KHz -6.58 dB -6.55 dB -1.44 dB 5 KHz -8.16 dB -8.11 dB -2.08 dB 6 KHz -9.54 dB -9.46 dB -2.75 dB 7 KHz -10.75 dB -10.64 dB -3.43 dB 8 KHz -11.82 dB -11.68 dB -4.10 dB 9 KHz -12.78 dB -12.60 dB -4.75 dB 10 KHz -13.66 dB -13.43 dB -5.37 dB 11 KHz -14.45 dB -14.17 dB -5.97 dB 12 KHz -15.18 dB -14.85 dB -6.54 dB 13 KHz -15.86 dB -15.47 dB -7.09 dB 14 KHz -16.49 dB -16.04 dB -7.61 dB 15 KHz -17.07 dB -16.56 dB -8.10 dB 16 KHz -17.62 dB -17.05 dB -8.58 dB 17 KHz -18.14 dB -17.50 dB -8.95 dB 18 KHz -18.63 dB -17.91 dB -9.45 dB 19 KHz -19.09 dB -18.30 dB -9.86 dB 20 KHz -19.53 dB -18.66 dB -10.26 dB 100 KHz -33.47 dB -25.27 dB -21.85 dB 1 MHz -53.47 dB -25.97 dB -25.95 dB

table 2

-- J-17 RESPONSE TABLE for Satellite

FREQUENCY J-17 1.00 Hz +9.38 50.00 Hz +9.32 200.00 Hz +8.68 400.00 Hz +7.10 800.00 Hz +3.72 1.42 KHz +0.00 2.00 KHz -2.40 4.00 KHz -6.28 6.40 KHz -7.89 8.00 KHz -8.37 10.00 KHz -8.70 100.00 KHz -9.38

table 3

Note that all the losses shown on the De-emphasis tables would become gains for the Pre-emphasis network within any FM modulator

55

6.4.2 Why use “Roofed” Pre-Enhasis

The higher audio frequencies are boosted in the pre-emphasis network. Out-of-band RF signals entering with the audio program would also be boosted if they were not limited. Observe that an unlimited 75 microsecond pre-emphasis network would boost a 1 MHz spurious signal by 53.47 dB while the 26 dB "Roofed" 75 microsecond pre-emphasis network would only boost the spurious signal by 25.97 dB. Higher spurious signals would be boosted without limit in an unlimited pre-emphasis network while the "roofed" pre-emphasis network would limit at 26 dB.

6.5 Frequency division multiplexing Frequency-Division multiplexing (FDM) is a technique for transmitting multiple messages simultaneously over a wideband channel by first modulating the message signals on to several subcarriers and forming a composite baseband signal that consists of the sum of these modulated subcarriers. This composite signal may then be modulated onto the main carrier as shown in figure below

figure 32

Subcarrier modulator fSC1

Subcarrier modulator fSC2

Subcarrier modulator fSCN

Σ Transmitter fC

m1(t)

m2(t)

m3(t)

Ssc1(t)

Ssc1(t)

SscN(t)

mb(t) s(t)=FDM

Composite baseband modulating signal

Transmitter

56

Any type of modulation, such as AM, DSB, SSB, PM, FM, and so on, can be used. The types of modulation used on the subcarriers, as well as the type of modulation used on the main carrier, may be different. However, as shown in figure below the composite signal spectrum must consists of modulated signal that do not have overlapping spectra; otherwise, crosstalk will occur between the message signals at the receiver output. The composite baseband signal then modulates a main transmitter to produce the FDM signal that is transmitted over the wideband channel.

figure 33

The received FDM signal, is first demodulated to reproduce the composite baseband signal that is passed through filters to separate the individual modulated subcarriers. Then the subcarriers are demodulated to reproduce the message signals m1(t), m2(t), and so on.

f (Hz)

)( fMb

0

fSC2 fSC3 fSCN

BSC1

fSC1

BSC2 BSC3 BSC

B

57

figure 34

7 OUTPUT SIGNAL-TO NOISE RATIOS FOR ANALOG SYSTEMS

For systems with additive noise channels the input to the receiver is

)()()( tntstr += eq 118

for bandpass communication systems having a transmission bandwidth of BT

( ){ } ( ){ } [ ] ( ){ }cccccc tjns

tjn

tjs etgtgetgetgtr θωθωθω +++ +=+= )()(Re)(Re)(Re)( eq 119

or

( ){ }cctjT etgtr θω += )(Re)( eq 120

where

Badnpass filter fSC1

Badnpass filter fSC2

Badnpass filter fSCN

Main receiver

m1(t)

m2(t)

mN(t)

Ssc1(t)

Receiver

Demodulator fSC1

Demodulator fSC2

Demodulator fSCN

s(t)=FDM

Ssc2(t)

Ssc3(t)

Composite baseband signal

58

[ ] [ ]

TjT

TT

nsns

nsT

etR

tjytxtytyjtxtx

tgtgtg

θ)(

)()( )()()()(

)()()(

=

=+==+++=

=+=

eq 121

)()()( tgtgtR nsT += eq 122

gT(t) denotes the total (i.e. composite) complex envelope at the receiver input; it consists of the complex envelope of the signal plus complex envelope of the noise.

7.1 Comparison with Baseband Systems The noise performance of various types of bandpass systems is examined by evaluating the signal-to-noise power ratio at the receiver output, (S/N)out, when a modulated signal plus noise is present at the receiver input. We would like to see if (S/N)out is larger for an AM system or an FM system. To compare these SNRs, the power of the modulated signals at the inputs of the receivers is set to the same value and the PSD of the input noise is N0/2. (that is , the input noise is white with a spectral level set to N0/2).

To compare the output signal to noise ratio (S/N)out for various bandpass systems we need a common measurement criterion for the receiver input. For analog systems, the criterion is the received signal power Ps, divided by the amount of the power in the white noise that is contained in a bandwidth equal to the message (modulating) bandwidth B of the baseband signal. This is equivalent to the (S/N)out of a baseband transmission system, as illustrated in figure below.

figure 35 : Baseband system

where:

Σ Lowpass filter Bandwidth =B

n(t)

s(t)

PN=N0/2

r(t)=s(t)+n(t)

2B= bandwidth of modulated s(t) signal

( ) ( )BN

PNSNS sbasebandin

0

// == (S/N)in

PN=2BxN0/2= N0B

Receiver ( ){ }cctj

T etgtr θω += )(Re)(

59

BNP

NS s

baseband 0

=⎟⎠⎞

⎜⎝⎛

eq 123

We can compare the performance of different modulated systems by evaluating (S/N)out

for each system as a function of ( )BN

PNS sbaseband

0

/ = , where Ps, is the power of AM or FM

signal at the receiver input, B is chosen to be the bandwidth of the baseband (modulating) signal where the same baseband modulating signal is used for all cases so that the same basis of comparison will be realized.

7.2 AM Systems with Product Detection Figure Below illustrates the receiver for an AM system with coherent detection (correlator receiver).

figure 36 : Coherent receiver

it can be shown that

( )( ) 2

2

*1*

//

mm

NSNS

baseband

out

+= eq 124

for 100% sine-wave modulation, 2/1*2 =m therefore

Lowpass filter Bandwidth =B

r(t) = Modulated signal plus noise in

Product detector

( )outNS / IF filter

{ })(Re)(* tgtm T=

)cos(2 cct θω +

( ){ }cctjT etgtntstr θω +=+= )(Re)()()(

( ) ( )basebandin NSNS // =

60

( )( ) 3

1

211

21

//

=+

=baseband

out

NSNS

eq 125

This illustrates that AM system is worse than a baseband system that uses the same amount of signal power, because of the additional power in the discrete AM carrier that does not contribute to the information ability of the signal but permits AM receivers to use economical envelope detectors

7.3 SSB systems

it can be shown that

( )( ) 1

//

=baseband

out

NSNS

eq 126

7.4 PM systems The modulation on a PM signal is recovered by a receiver that uses a (coherent) phase detector.

figure 37 : Receiver for angle-modulated-signal The PM signal has a complex envelope of

Lowpass filter Bandwidth =B

Modulated signal plus noise in

( )outNS / IF filter

{ })(Re)(* tgtm T=

[ ] ⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

∠=

∠=

FMfor )()(

PMfor )()(

0

0

tgdtdtr

tgtr

T

T

( ){ }cctjT etgtntstr θω +=+= )(Re)()()(

Detector (PM or FM)

61

)()( tj

csseAtg θ= eq 127

where

)()( tmDt ps =θ eq 128

the complex envelope of the complex signal plus noise at the detector inputs is

[ ] )()()( )()()()()( tjn

tjcns

tjTT

ns etReAtgtgetgtg θθθ +=+== eq 129

figure 38 : vector diagram for angle modulation (S/N)in>1

it can be shown that

( )( )

2

2

//

⎟⎟⎠

⎞⎜⎜⎝

⎛=

pp

baseband

out

Vm

NSNS β eq 130

where: βp is the PM modulation index and Vp is the peak value of m(t).

Real

Imaginary

θs θT AC

θn

Rn( )tj

csseAtg θ=)(

( )tjnn

netRtg θ)()( =

)sin( snnR θθ −

62

This equation shows that the improvement of a PM system over a baseband signalling system depends on amount of phase deviation that is used; the larger the phase deviation, the better the signal-to-noise-ratio. It seems to indicate that we can make the improvement as large as we wish simply by increasing βp. This depends on the types of circuits used. If the peak phase deviation exceed π radians, special “phase unwrapping” techniques have to be used in some circuits to obtain the true value (as compared to the relative value) of the phase output.

Thus the maximum value of )()(

tmDV

tmp

p

p =β

might be taken to be π. For sinusoidal

modulation, this would provide an improvement of [ ] dBmDV

mp

p

p 9.62

22

2

≅==⎥⎥⎦

⎤

⎢⎢⎣

⎡ πβ over

baseband signaling. It is emphasized that the results obtained previously for (S/N)out are valid only when the input signal is above the threshold [i.e. when (S/N)in>1]

7.5 FM Systems The procedure that we will use to evaluate the output SNR for FM systems is essentially the same as that used for PM systems, except that the output for the FM detector is proportional to dθ(t)/dt, whereas the output of the PM detector is proportional to θ(t). assuming that (S/N)in>1 then we can find

( )( )

2

23/

/⎟⎟⎠

⎞⎜⎜⎝

⎛=

pf

baseband

out

Vm

NSNS β eq 131

for the case of sinusoidal modulation 21

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛

pVm we have

( )( )

2

23

//

fbaseband

out

NSNS β=

At first glance, these results seem to indicate that the performance of FM systems can be increased without limit simply by increasing the FM index βf. However, as βf is increased, the transmission bandwidth increases, and consequently, (S/N)in decreases. These equations for (S/N)out are valid only when (S/N)in >>1 (i.e. when the input signal power is above the threshold), so (S/N)out does not increase to an excessively large value simply by increasing the FM index βf. Figure below, by dashed line, show a plot of the preceding equation.

63

figure 39 : noise performance of an FM discriminator for a sinusoidal modulated FM signal plus Gaussian noise (no deemphasis)

A generalized expression to describe S/N near the threshold, for the case of sinusoidal modulation is plotted by the solid line. Figure illustrates that the FM noise performance can be substantially better than baseband performance, particularly when βf increases.

7.6 FM Systems with Threshold Extension A PLL FM detector could be used to extend the threshold below that provided by an FM discriminator. However, when the input SNR is Large, all the FM receiving techniques provide the same performance. An FM receiver with feedback (FMFB) is shown below.

figure 40

FM Discriminator

FM Signal in

outNS )/(

VCO

{ })(Re)(* tgtm T=

)(* te )(te

IF Filter )(tvin

)(0 tv

Demodulated output

inNS )/(

64

the FMFB receiver provides threshold extension by lowering the modulation index for the FM signal that is applied to the discriminator input. That is, the modulation index of e*(t) is smaller than that for vin(t) (that in a normal case would have been applied to the discriminator input). the FM signal at the receiver input is

[ ])(cos)( ttAtv iccin θω += eq 132

where

( ) σσθ dmDtt

fi ∫∞−

=)( eq 133

the output of the VCO is

[ ])(cos)( 0000 ttAtv θω += eq 134

where

( ) σσθ dmDtt

v∫∞−

= *)(0 eq 135

With these representations for vin(t) e v0(t), the output of multiplier (mixer) is:

[ ] [ ]

( ) ( )[ ] ( ) ( )[ ])()(cos21)()(cos

21

)(cos)(cos)(

000000

000

tttAAtttAA

ttttAAte

iccicc

icc

θθωωθθωω

θωθω

++++−+−=

=++= eq 136

If the IF filter is tuned to pass the band of frequencies centered about 0fff cIF −≡ , the IF filter output is

( ) ( )[ ] ( )[ ])()(cos21)()(cos

21)(* 00000 tttAAtttAAte iIFcicc θθωθθωω −+=−+−= eq 137

or

65

[ ] ⎥⎦

⎤⎢⎣

⎡−+= ∫

∞−

t

vfifc dmDmDtAAte σσσω )(*)(cos21)(* 0 eq 138

the FM discriminator output is proportional to the derivative of the phase deviation

[ ]dt

dmDmDdKtm

t

vf⎭⎬⎫

⎩⎨⎧

−

=∫∞−

σσσ

π

)(*)(

2)(* eq 139

Evaluating the derivative and solving the resulting equation for m*(t), we obtain

[ ]

[ ]v

f

v

f

fv

fv

vfvf

KDtmKD

DK

tmDK

tmtmDKDKtm

tmDKtmDKtm

tmDKtmDKtmDtmDKtm

+=

⎥⎦⎤

⎢⎣⎡ +

=⇒=⎥⎦⎤

⎢⎣⎡ +

=+

−=−=

ππ

πππ

ππ

πππ

2 )(

21

)(2)(* )(

221)(*

)(2

)(*2

)(*

)(*2

)(2

)(*)(2

)(*

eq 140

)(2

)(* tmKD

KDtm

v

f⎟⎟⎠

⎞⎜⎜⎝

⎛+

=π

eq 141

Substituting this expression for m*(t) we get

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛+

+=

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+

−++=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+=

∫

∫

∫

∞−

∞−

∞−

t

f

v

IFc

t

v

fvvfIFc

t

v

fvfIFc

dmDDk

tAA

dKD

mKDDmKDmDtAA

dmKD

KDDmDtAAte

σσ

π

ω

σπ

σσσπω

σσπ

σω

)(

21

1cos21

2)()()(2

cos21

)(2

)(cos21)(*

0

0

0

eq 142

66

This demonstrates that the modulation index of e*(t) (i.e of the signal in input to the

discriminator), is exactly ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+ vDk π2/1

1 of the modulation index of vin(t).

Since m

mff ADω

β 1= , the lower the modulation index Df the lower the β, then once,

the lower will be the needed ( )( )

2

23

//

fbaseband

out

NSNS β= .

The threshold extension provided by the FMFB receiver is on the order of 5 dB, whereas that of PLL receiver is on the order of 3 dB (when both are compared with the threshold of an FM discriminator). Although this is not a fantastic improvement, it can be quite significant for systems that operate near the threshold, such as satellite communication systems. A System that uses a threshold extension receiver instead of a conventional receiver may be much less expensive than the system that requires a double-sized antenna to provide the 3-dB signal gain.

7.7 FM System with De-emphasis The noise performance of the FM system can be improved by preemphasizing the higher frequencies of the modulation signal at the transmitter input and deemphasizing the output of the receiver. This improvement occurs because the PSD of the noise at the output of the FM detector has a parabolic shape as a function of the frequency. The improvement when a sinusoidal test tone is transmitted over this FM system is

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

1

2

21

//

fB

NSNS

fbaseband

out β eq 143

of course, each of these results is valid only when the FM signal at the receiver input is above the threshold.

67

figure 41

8 COMPARISON OF ANALOG SIGNALING SYSTEMS Table below compares the analog systems that were analyzed in the previous sections. It has been seen that the non linear modulation systems (FM and PM) provide significant improvement in the noise performance, provided that the input signal is above the threshold. Of course, the improvement in the noise performance is obtained at the expense of having to use a wider transmission bandwidth. If the input SNR is very low, the linear systems outperform (AM)are better than non linear systems. SSB is best in terms of a small bandwidth, and it has one of the best noise characteristics at low input SNR. The selection of a particular system depends on the transmission bandwidth that is allowed and the available receiver input SNR. For the non linear systems a bandwidth spreading ratio of BT/B=12 is chosen for systems comparisons. This corresponds to a βf=5 for FM systems cited in the figure below and corresponds to commercial FM broadcasting.

68

Of course, when operating above the threshold, all the non linear modulation systems have better SNR performance than the linear modulation systems, because the non linear systems have larger transmission bandwidths.

8.1 Ideal systems performance What is the best noise performance that is theoretically possible? How can wide transmission bandwidth be used to gain improved noise performance? The answer is given by Shannon’s channel capacity theorem. The ideal system is defined as one that does not lose channel capacity in the detection process.

outin CC = eq 144

where Cin is the bandpass channel capacity and Cout is the channel capacity after detection. The equation for channel capacity is

⎟⎠⎞

⎜⎝⎛ +=

NSBC 1log2 eq 145

using this equation in the preceding equation we get:

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

outinT N

SBNSB 1log1log 22 eq 146

where BT is the transmission bandwidth of the bandpass signal at the receiver input and B is the bandwidth of the baseband signal at the receiver output. Solving for (S/N)out , we get

out

BB

in

B

out

B

in NS

NS

NS

NS

TT

⎟⎠⎞

⎜⎝⎛+=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⇒⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+ 1 1 11 eq 147

11 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=⎟

⎠⎞

⎜⎝⎛ B

B

inout

T

NS

NS

eq 148

but

69

basebandTT

s

T

s

in NS

BB

BB

BNP

BNP

NS

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛==⎟

⎠⎞

⎜⎝⎛

00

eq 149

thus equation (S/N)out becomes

11 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟

⎠⎞

⎜⎝⎛ B

B

basebandTout

T

NS

BB

NS

eq 150

equation above, which describes the ideal system performance, is plotted in figure below for the case of BT/B=12. As expected, none of the practical signalling systems equals the performance of the ideal system. However, some of the non linear systems (near the threshold) approach the performance of the ideal systems.

figure 42

70

9 BINARY MODULATED BANDPASS SIGNALING For digital modulated signals, the modulating signal m(t) is a digital signal given by the binary or multilevel line codes. The most common binary bandpass signalling techniques are illustrated as follows:

• On-off keing (OOK) also called Amplitude Shift Keying (ASK), which consists of keying (switching) a carrier sinusoid on and off with a unipolar binary signal. Morse code radio transmission is an example of this technique.

figure 43

• Binary phase-shift keying (BPSK), which consists of shifting the phase of sinusoidal carrier 0° or 180° with a unipolar binary signal. BPSK is equivalent to PM signalling with a digital waveform.

figure 44

71

• Frequency shift keying (FSK) which consists of shifting the frequency of sinusoidal carrier from a mark frequency (corresponding, for example, to sending a binary 1) to a space frequency (corresponding to sending a binary 0) according to the baseband digital signal. FSK is identical to modulating an FM carrier with a binary digital signal

figure 45

9.1 Binary Phase-Shift Keying (BPSK)

9.1.1 BPSK Generation A PSK bandpass modulated signal is generally represented by

{ } { } [ ]{ } [ ][ ])(cos

)(cosReRe)(Re)( )()(

tmDtAttAeAeeAetgts

pcc

ccttj

ctjtj

ctj ccc

+==+==== +

ωθωωθωθω

eq 151

Where

)()( tmDt p=θ eq 152

In BPSK m(t) is a polar baseband data signal. For convenience, let m(t) have a peak values of ±1 and a rectangular pulse shape. We now show that BPSK is also a form of AM-type signalling, in fact expanding the preceding equation we get

bababa sinsincoscos)cos( −=+ eq 153

72

( ) ( ) ( ) ( ))(sinsin)(coscos)( tmDtAtmDtAts pccpcc ωω −= eq 154

[ ]{ } ( ) [ ]{ } ( )ttmDAttmDAts cpccpc ωω sin)(sincos)(cos)( −= eq 155

From equation above we can see that it corresponds to a quadrature modulation scheme.

figure 46

Now recalling that m(t)= ±1 and that cos(x) and sin(x) are even and odd functions of x we get:

( )[ ] ( )[ ] [ ]( )[ ] ( )[ ]1sin1sin

cos1cos1cos−⋅−=+⋅

=−⋅=+⋅DpDp

DpDpDp eq 156

we see that the representation of BPSK signal reduces to:

( ) ( ) ( ) ( )( )[ ] ( ) ( )[ ] ( )ttmDAtDA

tmDtADtAts

cpccpc

pccpcc

ωω

ωω

sin)(sincoscos

)(sinsincoscos)(

−=

=⋅−= eq 157

s(t) +

-

Accos[Dpm(t)]

Acsin[Dpm(t)]

cosωct

sinωct

Pilot Carrier Term, m(t) is not present

Data Term: m(t) is present

73

The level of the pilot carrier term is set by the value of the peak phase deviation constant, ∆θ=Dp. For digital angle-modulated signals, the digital modulation index h is defined by

ππϑ pD

h22

=∆

= eq 158

where 2∆θ =2Dp is the maximum peak-to-peak phase deviation (radians) during the time required to send one symbol, Ts.

h 2∆θ=2Dp

2 2π 1 π

For binary signaling, the symbol time is equal to the bit time (Ts=Tb). The level of the pilot carrier term is set by the value of the peak deviation, which is ∆θ=Dp for m(t)=±1. The value of m is determined by the input data bit stream converted in NRZ for example. If Dp is small, the pilot carrier term has a relatively large amplitude compared to the data term; consequently, there is a very little power in the data term (which contains the source information). To maximize the signaling efficiency (so that there is a low probability of error), the power in the data term needs to be maximized. This is accomplished by letting ∆θ=Dp=π/2 radians, which corresponds to a digital modulation index of h=1. For this optimum case of h=1, the BPSK signal becomes

ttmAts cc ωsin)()( −= eq 159

74

The baseband complex envelope for this BPSK signalling is:

)()( tmjAtg c= eq 160

Therefore the modulated signal is

[ ] [ ] [ ]ttmA

ttmAttmjAetmjAetgts

cc

cccctj

ctj cc

ωωωωω

sin)( sin)(cos)(Re)(Re)(Re)(

−=−===

eq 161

1)( sin)(1)( sin)(

2

1

−=+=+=−=

tmtAtstmtAts

cc

cc

ωω

To simplify the explanation, suppose now to 90° rotate all constellation then we can obtain the simulation like as obtained by using WINIQ software as shown below. This means that

)()( tmAtg c= eq 162

Consequently

[ ] [ ] [ ]ttmA

ttmjAttmAetmAetgts

cc

cccctj

ctj cc

ωωωωω

cos)( sin)(cos)(Re)(Re)(Re)(

+=+===

eq 163

In figure below are reported the phase i(t)=±1 and quadrature q(t)=0 components of the baseband modulating signal g(t) plus the constellation diagram.

g2(t)

g1(t)

Real

Imaginary

g1(t) g2(t)

75

figure 47: left i(t)=±1 and q(t)=0, right constellation diagram of baseband i(t) ,q(t) signal

When the modulating wave shape is rectangular and the symbol 0,1 are equally probable, the Power Spectral Density (PSD) for the baseband complex envelope is

22 sin)( ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

b

bbcg Tf

TfTAfPππ

eq 164

The resulting FFT of the bandpass digital signals s(t) can be obtained by Wiener-Khintchine theorem:

[ ] ( )( )( )

( )( )( )

( )( )[ ] ( )( )[ ]{ }222

222

sinsin4

sinsin4

)()(41)]([

bcbcb

bc

bc

bc

bcbcgcg

TffcTffcTA

TffTff

TffTffTAffPffPtsPSD

++−=

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡++

+⎥⎦

⎤⎢⎣

⎡−−

=++−=

ππ

ππ

ππ

figure 48

g2(t) g1(t)

)(4

2

cg

b

ffPTA− )(

4

2

cg

b

ffPTA+

76

Considering the positive frequency only, we can have the FFT reported on figure below:

figure 49

To have an easier BPSK model, we can eliminate the minus sign by a 90° rotation of all signals s(t):

[ ] ttmAttmAttmAts cccccc ωωπω cos)(cos)()2

sin()()( =−−=−−= eq 165

The symbols, s1(t) and s2(t) representing the ones and zeros respectively, are then,

1 when cos)(1 when cos)(

2

1

−=−=+=+=

mtAtsmtAts

cc

cc

ωω

Using the duplication formula

22cos1sin

22cos1cos

2

2

xx

xx

−=

+=

eq 166

the energy transmitted during one bit period b

bTωπ2

= is :

77

( ) ( ) ( ) ( )

( )

( ) ( ) ....1,2,3,4...n with ,2 , where

20

2

22

21

222

21

2

2cos21

222cos1

ˆ

222

2

0

22

2

0

2

0

2

0

22

0

2

====

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−⎟⎟⎠

⎞⎜⎜⎝

⎛

−=⎥⎦

⎤⎢⎣

⎡−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=−

=== ∫∫∫∫

bcb

bcc

bc

c

bc

cb

cc

cc

bc

cb

c

Tc

c

T

cc

T

b

nTtsenAts

TAsen

ATAtsenATA

dttTAdttAdttsenAdttsE

b

bbbb

ωωωπω

ωωπω

ωω

ωωω

ωπ

ωπ

eq 167

Therefore the carrier power, C, during a bit period Tb is:

[ ]WATEC c

b

b 2

2

== eq 168

The value AC of the carrier is given by:

b

bc T

EA 2= [V] eq 169

Substituting this value of Ac into equation of s(t) we get:

1 when cos2cos)()(

1 when cos2cos)()(

2

1

−=−==

+=+==

mtT

EttmAts

mtT

EttmAts

cb

bcc

cb

bcc

ωω

ωω

eq 170

This form of BPSK is referred to as phase-reversal keying since the two carrier signals representing the logic ones and zeros are exactly 180° out of the phase i.e. the phase modulation index h=1. A more general form of BPSK occurs when the phase difference between the two signals is other than 180°. This creates a residual carrier term that allows carrier tracking by a phase-lock loop (PLL). Unless stated otherwise, BPSK will refer to 180° mode.

78

A method of generating BPSK is shown in figure below. A bit sequence represented by ±Ac is applied to a balanced modulator, resulting in an output of ±Accos(ωct) which is a BPSK.

figure 50: BPSK modulator

9.1.2 BPSK Detection by a Correlation Receiver When binary PSK modulation is used at the transmission end, then a receiver employing coherent demodulation must be employed since the information is contained in the carrier phase. A correlation receiver performs coherent demodulation. Correlation, C(t), of two signals, r(t) and s(t), over a period, T, is defined mathematically as:

∫ <<=t

TtdttstrtC0

0 )()()( eq 171

Correlation is implemented in hardware by a multiplier and an integrator, as shown in figure below:

figure 51:hardware implementation of correlation

r(t)

s(t)

Low-pass Filter

∫t

dttrts0

)()(

±Ac

tcωcos

BPSK signal s(t) ±Accos (ωct)

79

A BPSK correlation receiver is shown in figure below, with each block of hardware labeled with its functional purpose.

figure 52

The correlation receiver is so called because it correlates the received signal, composed of signal plus noise, with a replica of the signal. For the correlation to be achieved, it is necessary for the receiver to be phase locked to the carrier as discussed earlier. The purpose of the correlation receiver is to reduce the received symbol to a single point or statistic that will be used by the decision device to determine which symbol has been transmitted. In practice, the single point is a fixed voltage obtained by a S/H device. The decision device is a voltage comparator that is set such that if the voltage point is above a certain level, the comparator indicates a one is received; if the voltage is below this level, a received zero is indicated, the case for no noise will be treated first. Functionally and conceptually, the correlation receiver is composed of the receiver and bit synchronizer. The correlation receiver and the matched filter are equivalent. Specifically, the integrator and the S/H, at t=Tb, are equivalent to a matched filter, sampled at the output. Since there is no discrete carrier term in the ideal BPSK signal, a PLL may be used to extract the carrier reference only if a low level pilot carrier is transmitted together with the BPSK signal otherwise is needed a coherent detection. However, the 180° phase ambiguity must be resolved, this can be accomplished by using differential coding at the transmitter input and differential decoding at the receiver output.

si(t)=BPSK

tT c

b

ωcos2

Low-pass Filter Or Matched Filter

∫t

dttstr0

)()(

Sample & Hold S/H

Threshold device

em(t) )(taZ

Receiver Bit synchronizer

Output of the receiver and input to the bit synchronizer

Output of the bit synchronizer

80

9.1.2.1 No Noise An exact replica of the carrier multiplies the received symbol, and the output of the multiplier is applied to an integrator. The output of the multiplier is given by:

⎥⎦⎤

⎢⎣⎡ +±=⎥⎦

⎤⎢⎣⎡ +

±=±=

=×±=

tT

EtT

EtT

E

tT

tT

Ete

cb

bc

bbc

bb

cb

cb

bm

ωω

ω

ωω

2cos21

212

22cos12cos2

cos2cos2)(

2

eq 172

The output of the multiplier, em(t), is applied to integrator. The integrator will integrate the double frequency term over an integer number of cycles therefore deleting this term. In practice, a low pass filter follows the integrator to ensure that this term is deleted from the output. The output a(t), is:

tT

EdttT

Etab

b

t

cb

b12cos

21

212)(

0

±=⎥⎦⎤

⎢⎣⎡ +±= ∫ ω eq 173

This is the output of the correlator is either a positive- or negative-going ramp function (triangular function). The S/H device is usually set to sample the ramp whenever it reaches a maximum value, which ideally occurs whenever t=Tb. the upper limit of the integrator is also set to Tb. For the no noise case, the output of the integrator a t=Tb is:

bbb

b

T

cb

bb ETT

EdttT

ETtab

±=±=⎥⎦⎤

⎢⎣⎡ +±== ∫

12cos21

212)(

0

ω eq 174

The S/H device is clocked to sample the output of the integrator whenever the maximum voltage is expected. For this case, the S/H samples at t=Tb and the output voltage

bE± is applied to the threshold device, which normally triggers out a crisp waveform representing a one if the voltage is greater than zero or a zero if the voltage is less than zero.

9.1.3 With Noise The case when the received signal is contaminated with additive white Gaussian Noise (AWGN) is now considered. Let the white noise have a power spectral density (PSD) given by N0/2. The received modulated baseband signal in input to the correlator receiver is now given by

t=Tb

bb ETta == )(

81

)(cos2)( tntT

Ets cb

b +±= ω eq 175

Where n(t) is white Gaussian noise. The output of the integrator at t=Tb due to the signal part of s(t) will be the same as before, ±a. For the low-noise case, the output of the integrator might look similar to the ramp shown in figure below on the left; and for the high-noise case, the ramp on the right figure is indicative about what the output might look like.

figure 53: Integrator output of a correlation receiver: (left) low noise, and (right) high noise.

Let the baseband sampled voltage outgoing by the integrator at t=T, can be represented by z, then

NEz b +±= eq 176

It can be shown that N is a Gaussian random variable with zero mean µ and variance σ2 given by:

2/02 N=σ eq 177

Therefore, the output random variable voltage, z, outgoing from the integrator is also a Gaussian random variable. Then z will have a variance of N0/2 and a mean of

bEa ±= , depending upon which symbol has been received. Letting

bEa = eq 178

T T

t t

82

The conditional probability density function for z, gives an information about the correspondence of symbol with ±a. Precisely it gives an indication if a one or a zero has been transmitted:

2

21

221)|(

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

=+ σ

πσ

bEz

eazp eq 179

2

21

221)|(

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

=− σ

πσ

bEz

eazp eq 180

Maximum value for p(z|a) function, is reached when z=a. The plot for these two functions is shown in figure below, this figure shows that the sampled output voltage z will fall somewhere along the x axis. Points ±a represents the more density probability places for z

Figure 54: Conditional probability density functions, gives if a “one” or a “zero” has been transmitted

The baseband signal constellation for a BPSK is shown below, we can observe the jitter around ±a points due to the noise presence.

figure 55: BPSK signal constellation

max p(z | -a) max p(z | +a)

+a -a

-a +a φ(t)

83

9.2 Maximum Likelihood Detection At the end of each symbol period when the integrator output voltage is sampled, the receiver must decide which symbol was sent based on the sampled voltage z. For maximum likelihood detection, conceptually, the statistic z is substituted into conditional probability density function already seen, and the function with the largest value indicates which symbol have the maximum possibility to have been transmitted. The test is implemented by forming a ratio between two densities, such as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟

⎟⎠

⎞⎜⎜⎝

⎛ +

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

==−

+=

−+

=

22

2

2

21

21

2

21

2

21

2

21

21

21

)|()|(

)|()|( σσ

σ

σ

πσ

πσ

πσλbb

b

b

EzEz

Ez

Ez

b

b e

e

e

EzpEzp

azpazp

eq 181

Assuming each symbol is equally likely and the cost of all errors is the same, the received point z is substituted as Gaussian random variable. Therefore a value for λ>1, choose the symbol corresponding to +a; λ<1, choose the symbol corresponding to -a. Fundamentally, this test computes the value for each conditional probability density function at instant and selects the density with the largest value in that time instant.

9.3 Bit Errors A bit error is made if a zero is transmitted and the sampled voltage, z, falls above zero. The probability which this error can happen is the area (i.e. the integral) beneath the Gaussian density function curve, with mean –a. The integral of the probability density function is made starting the integration, from zero toward infinity and is given by

figure 56

p(z | -a) p(z | +a)

+a-a 0

84

( ) ( )

∫∫∫∞

⎥⎦⎤

⎢⎣⎡ −−

−∞⎥⎦⎤

⎢⎣⎡ −−

−∞

==−=−0

21

20

21

20

22

21

21)|()|( σσ

πσπσ

azaz

edzedzazPaaP eq 182

calling

dxσdzazx =+

= n theσ

eq 183

Therefore when z=0 then x=a/σ and when z=∞ then x=∞

( ) ( ) ( ))/(

21

21

21)|(

/

21

/

21

20

21

2

222

σπ

σπσπσ σσ

aQdxedxedzeaaPa

x

a

xx====− ∫∫∫

∞−

∞−

∞−

eq 184

By the symmetry, a bit error is made if a one is transmitted and the sampled voltage, Z, falls under zero. The probability of this occurrences is the area beneath the Gaussian curve, with mean +a, from zero to -infinity, the probability of mistaking –a for +a is

( )

dzeaaPaz

∫∞−

⎥⎦⎤

⎢⎣⎡ +−

−=−

0

21

2

2

21)|( σ

πσ eq 185

dxσdzazx =−

= n theσ

eq 186

( ) ( ) ( ))/(

21

21

21)|(

/21/

21

2

021

2

222

σπ

σπσπσ

σσ

aQdxedxedzeaaPa xa xx

====− ∫∫∫−

∞−

−−

∞−

−

∞−

− eq 187

Considering that P(a)=P(-a)=1/2 then the total error probability is, for the Bayes theorem:

21)|(

21)|()()|()()|( aaPaaPaPaaPaPaaPPe −+−=−+−−= eq 188

)/()/(21)/(

21 σσσ aQaQaQPe =+= eq 189

Since bEa = then

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

00

2

2NEQ

NE

QE

QP bbbe σ

eq 190

85

9.3.1 Q-Function reminder We say that X is a normal or gaussian variable with mean µ =X and variance 2σ if its conditional probability density function is:

( )( )

2

2

222

1| σµ

πσµ

−−

=x

x exf eq 191

One example for Gaussian probability Density function is plotted below:

Probability Density

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

-250 -200 -150 -100 -50 0 50 100 150 200 250

x

f(x)

f(X)

figure 57: Probability density for a Gaussian random variable

The curve is symmetric around the parameterµ and the relative probability distribution function referred to an interval within -∞ to x is given by

( ) ( )( )

∫∫∫∞−

⎥⎦⎤

⎢⎣⎡ −

−

∞−

−−

∞−

===x yx yx

xx dyedyedyyfxF2

2

2

21

22

2 21

21 σ

µσµ

πσπσ eq 192

Using the substitution: σµ−

=yz then dzdyzy σµσ =⇒+= consequently

( )

⎟⎠⎞

⎜⎝⎛ −

===== ∫∫∫∞−

−−

=

∞−

−

∞−

−−

σµ

πσ

πσπσ

σµ

σµ xQzQdzedzedyexF

z zxz zx y

)(21

21

21)( 22

22

2

22

2

2

eq 193

86

The resulting function is that more often tabulated

( ) ∫∞−

−

=z z

dzezQ 2

2

21π

eq 194

One example of cumulative distribution function F(X) relative to a Gaussian density function f(x), is plotted below

Distribution F(X)

0

0.2

0.4

0.6

0.8

1

1.2

-250 -200 -150 -100 -50 0 50 100 150 200 250

X

F(X)

F(X)

figure 58: cumulative distribution of a Gaussian density

The constant 22πσ is a normalization factor which maintain the area under the F(x) unitary within the interval -∞, +∞ When µ=0 then the Q function is defined as:

function density of variance where 21)/()( 2

/

2

2

=== ∫∞

−σ

πσ

σ

dyeaQZQa

y

eq 195

i.e. the probability value F(x) computed within an interval starting from a/σ to +∞ That is, for a Gaussian distribution with mean equal to zero and variance σ2=1, Q(a/σ) i.e. Q(a), is the area beneath the tail of the curve from a to +∞.

87

9.3.2 Bit Error Probability in terms of Eb and N0

Because white noise (AWGN) is presented, it can be shown that the variance of the noise is given by

202 N

=σ eq 196

• N0 = one-sided power spectral density (PSD) of the white noise going into the

IF bandwidth measured in [W/Hz]. Calling:

• Eb= bit energy [J] Eb is a variable that occurs in the theoretical analysis of digital systems, but it is equivalent to the carrier power, C, at the receiver that is measured or determined by link analysis ad is a function of antenna gain, path loss distance. transmitted power, and losses.

bbb R

CTCE == eq 197

Where:

• Tb = bit period = 1/Rb where Rb = bit rate [bit/s] • C = carrier power at the receiver [W] • BIF= bandwidth of the IF [Hz]

The noise power within the IF bandwidth is

IFp BNN 0= eq 198

These parameters can be related as follows:

IF

bb

IF

bb

p BR

NE

BNRE

NC

00

== eq 199

The greater the Rb, the greater the C/Np needed at the receiver side, whereas the greater the bandwidth B, the lower will be the S/N ratio required.

88

C/Np is the carrier-to-noise ratio in the IF, which is determined by link analysis, the BER may be predicted based on this engineering parameter. Consequently :

b

IF

p

b

RB

NC

NE

=0

eq 200

Under the ideal condition and NRZ data format, the assumption is that BIF ≅ Rb ( this is usually not the case unless raised cosine filtering is used); In fact using a RRC filter the since for a BPSK the Symbol rate D is equal to the bit rate Rb, then we can write:

1 when )1(2

)1(2

→≈+=+= rRbrRbrDB eq 201

then the remaining relationship is

p

b

NC

NE

≅0

eq 202

Then using the above relationship, the error probability Pe (BER) may be written as

⎟⎟⎠

⎞⎜⎜⎝

⎛≅⎟

⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

p

bbbe N

CQNEQ

NEQ

EQP 22

200σ

eq 203

This is an important equation that relates the BER to the carrier-to-noise ratio, (C/Np) in the IF bandwidth, which is determined by link analysis.

89

10 Differential Phase-Shift Keying (DPSK) Phase-shift-keyed signals cannot be detected incoherently. However, a partially coherent detection technique can be used to extract the phase reference for the present signalling interval. This is provided by a delayed version of the signal occurred during the previous signalling interval. This is illustrated below where differential coding is provided by (one-bit) delay and the multiplier. If a BPSK signal (no noise) were applied to the receiver input, the output of the Sample-and-Hold circuit, says, r0(t0), would be positive (binary 1) if the present data bit and the previous data bit were of the same sense; while r0(t0) would be negative (binary 0) if the two data bits were different. Consequently, if the data on the BPSK signal are differentially encoded, the decoded data sequence will be recovered at the output of the receiver. This technique consisting of transmitting a differentially encoded BPSK signal is known as DPSK. In practise, DPSK is often used instead of BPSK, because the DPSK receiver does not require a carrier synchronizer circuit. An example is the Bell212 A modem (1200 bits/s)

figure 59 : (partially Coherent detection of DPSK)

The BER of an optimum DPSK receiver is:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

= 0

21 N

Eb

ePe eq 204

Threshold device

DPSK in

Low-pass Filter integrator

∫−

0

0

(..)t

Tt

dt

One-bit Delay, Tb

S&H

Bandpass filter

-fc +fc

H(f) BT

f

r0(t0),Bit sync From bit sync i it

r0(t0)

Binary output

90

10.1 Differential coding When serial data are passed through many circuits along a communication channel, the waveform is often unintentionally inverted (i.e. data complemented). This result can occur in a twisted-pair transmission line channel just by switching the two leads at a connection point where a polar line code is used. (note that such switching would non affect the data on a bipolar signal). To ameliorate the problem, Differential Coding is often employed. The encoded differential data are generated by

1−⊕= nnn ede eq 205

figure 60 : differential coding system

The received encoded data are decoded by:

1−⊕= nnn eed eq 206

where ⊕ is a modulo 2 adder or an exclusive-OR gate (XOR) operation. Each digit in the encoded sequence is obtained by comparing the present input bit with the past encoded bit. A binary 1 is encoded if the present input bit and the past encoded bit are of opposite state, and a binary 0 is encoded if the state are the same (XOR operation) Differential encoding present a great advantage when the waveform is passed through thousand of circuits in a communication system and the positive sense of the output is lost or changes occasionally as the network changes.

One-bit Delay, Tb

Line Encoder circuit

Differential Encoder

Data out

One-bit Delay, Tb

Data in

en-1

en dn

Modulo 2 adder (EX-OR)

Channel Line Decoder circuit

Modulo 2 adder (EX-OR)

Differential Decoder

dn en

en-1

91

11 FREQUENCY SHIFT KEYING (FSK) The FSK signal can be characterized as one of two different types, depending on the used to generate it.

11.1 Discontinuous FSK One type is generated by switching the transmitter output line between two deferent oscillators. This type generates an output waveform that is discontinuous at the switching times, it is called discontinuous phase FSK, because θ(t) is discontinuous at the switching times. It can be represented by:

[ ] [ ][ ] ⎭

⎬⎫

⎩⎨⎧

++

=+=sent is 0binary when cossent is 1binary when cos

)(cos)(22

11

θωθω

θωtAtA

ttAtsc

ccc eq 207

And where θ1 and θ1 are the start-up phases of the two oscillators.

figure 61

Oscillator Freq=f1

Oscillator Freq=f2

Bynary data input m(t)

Control line

Electronic switch

Discontinuous FSK output

92

11.2 Continuous FSK The continuous FSK signal is generated by feeding the data signal into a frequency modulator.

figure 62

The FSK signal is represented by:

[ ] [ ]tjt

fcccccetgdmDtAttAts ωλλωθω )(Re)(cos)(cos)( =⎥

⎦

⎤⎢⎣

⎡+=+= ∫

∞−

eq 208

Where m(t) is a baseband digital modulating signal. Although m(t) is discontinuous at the switching time, the phase function θ(t) is proportional to the integral of m(t). Using the digital modulation index h=2*∆f/ fb then we can rewrite the equation above:

[ ] [ ]tjt

cccccetgdmhtfAttAts ωλλππθω )(Re)(22cos)(cos)( =⎥

⎦

⎤⎢⎣

⎡+=+= ∫

∞−

If the serial data input waveform is binary, such as polar baseband signal, m(t)=±1, the resulting FSK signal is called a binary FSK (BFSK) signal.

In this case the overall phase ∫∞−

+t

c dmhtf λλππ )(22 will be like as

thfhttfdhtfdmhtfy cc

t

c

t

c )(222122)(22 +=+=+=+= ∫∫∞−∞−

πππλππλλππ eq 209

Which is the equation of rect line where the angular coefficient is )(2 hfc +π Of course, a multilevel input signal would produce a multilevel FSK signal. In general, the spectra of FSK signals are difficult to evaluate since the complex envelope g(t), is a non linear function of m(t). The approximate transmission bandwidth BT for FSK signal is given by Carson’s rule:

Bynary data input m(t)

Frequency modulator

(carrier freq.=fc) Continuous FSK output

93

BfBBfBBT 2212)1(2 +∆=⎥⎦

⎤⎢⎣⎡ +∆

=+= β eq 210

Where B is the bandwidth of the signal (e.g. square wave) modulation waveform. If the Bandwidth B is equal to the Bit Rate R i.e. B=R, then BT became:

( )RfRfBfBBfBBT +∆=+∆=+∆=⎥⎦

⎤⎢⎣⎡ +∆

=+= 2222212)1(2 β eq 211

Using a raised-cosine-rolloff premodulation filter and since in a binary signaling D=R then the transmission bandwidth of the FSK signal became:

)1(22

)1(222

)1(2222 rRfrRfrDfBfBT ++∆=+

+∆=+

+∆=+∆= eq 212

11.3 FSK detection FSk can be detected by using either a frequency (noncoherent) detector or two product detectors (coherent detection). In order to obtain the lowest BER when the FSK signal is corrupted by AWGN, coherent detection with matched filter processing and threshold device (comparator) is required.

figure 63: coherent (synchronous) detection

Low-pass filter

cos(ω1t)

FSK in ∑

Low-pass filter

cos(ω2t)

Binary out

94

figure 64: Noncoherent detection

FSK in Frequency detector Binary out

95

12 MULTILEVEL MODULATED BANDPASS SIGNALING With multilevel signalling, digital inputs with more than two levels are allowed on the transmitter input. This technique is illustrated in figure below, which show how multilevel signals can be generated from a serial binary input stream by using a digital-to-analog converter(DAC). For example, suppose that an L=2-bit/symbol DAC is used. Then the number of levels in the multilevel signal is M=2L. The symbol rate (baud) of the multilevel signal is D=R/L where R=1/Tb is the bit rate bits/s.

figure 65: multilevel digital transmission system

12.1 Quadrature Phase-shift Keyng (QPSK) and M-ary Phase-Shift Keyng (MPSK)

If the transmitter is a PM transmitter with a M=4-level digital modulation signal, M-ary phase-shift keing (MPSK) is generated at the transmitter output. Assuming rectangular-shaped data pulses, a plot of the permitted values of the complex envelope, g(t)=Acejθ(t), would contain four points, one value of g(t) (a complex number in general) for each of the four multilevel values, which correspond to the four phase θ permitted. A plot of two possible sets of g(t) is shown in figure below (constellation point).

figure 66

Binary input R bits / sec

Transmitter Modulated output Digital-to-Analog converter L bits

M=2L -level digital signal D(symbol/sec)=R/L

g(t)

Real (in Phase)

Imaginary (Quadrature)

θi

g(t)

Real (in Phase)

Imaginary (Quadrature)

θi

QPSK QPSK

96

The signal constellation are essentially the same, except for a shift in the carrier-shift-keyed (QPSK) signalling. A constellation is an N-dimensional plot for the possible signal vectors corresponding to the possible diagram signals as reported in figure below.

figure 67

For instance, suppose that the permitted multilevel values at the DAC are -3,-1,+1,+3 V; then these multilevel values might correspond to PSK phase of 0°,90°,180°,270°, respectively.

This example of M-ary PSK where M=4 is called quadrature phase-shift-keyed (QPSK) signalling.

MPSK can also be generated by using two quadrature carriers modulated by the x and y components of the complex envelope (instead of using a phase modulator); in that case,

)()()( )( tjytxeAtg tjc +== θ eq 213

Where the permitted values x and y are

ic

ic

AtyAtx

θθ

sin)(cos)(

==

eq 214

and where the permitted phase angles are θI , i=1,2,….,M of the MPSK signal.

97

The output modulated signal v(t)=s(t) is:

{ } [ ]{ } [ ][ ]{ }{ }

( ) Re ( ) Re ( ) cos sin Re ( ) ( ) cos sin

Re ( )cos ( )sin ( ) cos ( )sin ( ) cos ( )sin

cj tc c c c

c c c c

c c

v t g t e g t t j t x t jy t t j t

x t t jx t t jy t t y t tx t t y t t

ω ω ω ω ω

ω ω ω ω

ω ω

= = + = + + =

= + + − =

= −

eq215

This situation is illustrated in figure below

figure 68: modulator for generalized signal constellation

figure 69: Modulator for Rectangular Signal Constellation

x(t) Binary input

2-bit serial-to-parallel converter

L/2 bit DAC

R bits/s

Multilevel digital signal

d1(t) R/2 bits/sec

y(t)

cos(ωct)

Oscillator f=fc

Σ

-90° phase shift

sin(ωct)

s(t)=Re[g(t)ejωt]

QAM signal out

Baseband processing

L/2 bit DAC

d2(t) R/2 bits/sec

g(t)=x(t)+jy(t) M=2L point constellation D=R/L symbol/sec

x(t) Binary input

Digital-to-analog converter L-bits

Signal processing

Oscillator f=fc

Σ

-90° phase shift

R bits/s

Multilevel digital signal

M=2L level D=R/L symbols / sec

y(t)

cos(ωct)

sin(ωct)

s(t)

QAM signal out

Baseband processing

98

For rectangular-shaped data pulses, the envelope of the QPSK signal is constant. That is, there is no AM (Amplitude Modulation) on the signal, even during the transmission times when there is a 180° phase shift, since the data switches value (say, from +1 to -1) instantaneously.

figure 70

As an instance by winiq software simulator, generating a random binary input we can observe the (I,Q) baseband signal diagram and (s(t),ϕ(t)) output modulated signal diagram. From (s(t),ϕ(t)) diagram its possible to see that the module of s(t) is constant. This case is reported as an example in figure below

figure 71

QPSK

180°

Phase shift: 180° 0° -90°

s(t) with constant amplitude

s(t) phase shift

99

The rectangular-shaped data pulse produce a (sinx/x) -type power spectrum for the QPSK signal that has large undesirable spectral sidelobes.

figure 72

These undesirable sidelobes can be eliminated if the data pulses are filtered by a pulse shaping filter corresponding for example to a raised cosine rolloff filter. Unfortunately this produces AM on the resulting QPSK signal, because the filtered data waveform cannot change instantaneously from one peak to another when 180° phase transition occur. On figure below it possible to observe how the spectrum changes, when a raised cosine rolloff filter is used

figure 73

100

Although filtering solves the problem of poor spectral sidelobes, it creates another one: AM on the QPSK signal, see figure below:

figure 74

On figure below are reported the constellation diagram and the vector diagram at the output of the transmitter. We can observe how shape filtering causes a constellation

point position dispersion around the expected position symbol.

figure 75

Due to this AM, low-efficiency linear (Class A or B) amplifiers, instead of high-efficiency nonlinear (Class C) amplifiers, are required for amplifying the QPSK signal without distortion. In portable communication application, these amplifiers increase the battery capacity requirements by as much as 50%. A possible solution to the dilemma is to use Offset QPSK (OQPSK) or π/4 QPSK, each of which has a lower amount of AM. For a QPSK in order to represent the modulated bandpass signal we can use also the following equation where the square of 2 is used as a normalization factor, and π/4 is an initial phase shift on both axes:

s(t) with no constant amplitude

s(t) phase shift

Constellation point position dispersion

101

)4

2sin()(2

1)4

2cos()(2

1)](2cos[)( ππππθπ +++=+= tftdtftdttfts cQcIc

Where

)......(,,)......(,,

531

420

oddbitddddevenbitdddd

Q

I

==

figure 76

In other word, as we have already seen a QPSK signal can be represented as the superposition of two BPSK signal with carrier in quadrature. Each signal is characterized by a 2Tb symbol time, i.e. double time length with respect to original modulating signal.

102

x(t)

y(t)

cos(ωct)

Oscillator f=fc

Σ

-90° phase shift

sin(ωct)

s(t)=Re[g(t)ejωt]

QAM signal out

g(t)=x(t)+jy(t) M=2l point constellation D=R/l symbol/sec

Ts/2 delay

12.2 OQPSK and π/4 QPSK Offset Quadrature Phase-Shift keying (OQPSK) is M=4 PSK in which the allowed data transition times for I and Q components are offset by ½ symbol (i.e. by 1 bit) interval. This offset provides an advantage when nonrectangular (i.e. filtered) data pulses are used, because the offset greatly reduces the AM on the OQPSK signal compared to the AM on the corresponding QPSK signal. The AM is reduced because a maximum phase transition of only 90° occurs for OQPSK signalling (as opposed to 180° for QPSK), since the I and Q data cannot change simultaneously, because the data are offset.

figure 77

In mathematical form we have:

422 2 === lM eq 216

lR

symbolperbitofnumberBitRate

TratesymbolD

s

====____

1_ eq 217

2_ sTdelaySymbol = eq 218

• in a usual QAM the I and Q component are:

103

∑ ⎟⎠⎞

⎜⎝⎛ −=

nln D

nthxtx )( eq 219

∑ ⎟⎠⎞

⎜⎝⎛ −=

nln D

nthyty )( eq 220

where: (xn,yn) denotes one of the permitted (xi,yi) value during the symbol time that is centred on t=nTs=n/D (s) (it takes Ts (s) to send each symbol), hl(t) is the pulse shape that is used for each symbol. If the bandwidth of the QAM signal doesn’t need to be restricted, the pulse shape will be rectangular and of Ts (s) duration.

• In OQPSK the timing between the x(t) and y(t) components is offset by Ts/2=(1/2D) (s). that is

∑ ⎟⎠⎞

⎜⎝⎛ −=

nln D

nthxtx )( eq 221

∑∑ ⎟⎠⎞

⎜⎝⎛ −−=⎟

⎠⎞

⎜⎝⎛ −−=

nln

n

sln DD

nthyTDnthyty

21

2)( eq 222

In Figure Below is reported a comparison between the QPSK and OQPSK (I,Q) signal

figure 78

O-QPSK Shift=1/2

Tsymbol

QPSK Shift=0 Tsymbol

Ts

I

Q

104

In the other figure is shown a comparison between QPSK and OQPSK vector constellation, note that in OQPSK case there is no zero crossing by the modulating vector.

figure 79

At the output of the QPSK and OQPSK transmitter the modulated signal s(t) could be as shown in example below:

figure 80

O-QPSK

90°

QPSK

180°

QPSK

S(t)

OQPSK

Max phase shift 180°

Max phase shift 90°

105

From mathematical point of view the only difference between QPSK and OQPSK is a shift/delay of Ts/2 on Q branches, so for OQPSK we can use the representation reported below:

)24

2sin()(2

1)4

2cos()(2

1)](2cos[)( scQcIc

Ttftdtftdttfts −+++=+=ππππθπ eq 223

Where

)......(,,)......(,,

531

420

oddbitddddevenbitdddd

Q

I

==

This equation is very similar to that used for QPSK which is reported as a reminder:

)4

2sin()(2

1)4

2cos()(2

1)](2cos[)( ππππθπ +++=+= tftdtftdttfts cQcIc eq 224

Using rectangular modulating signal, we must observe that there are no difference between QPSK and OQPSK signal spectrum. This can be explained observing that while the OQPSK amplitude phase shift is only half with respect to QPSK, the transitions can occur more frequently (in each period of 2Tb for aQPSK and in each period of Tb of OQPSK) Anyway thanks to a lower phase transition OQPSK induces lower AM modulation amount when the signal is filtered prior to modulation. Both modulation techniques QPSK and OQPSK, are used in order to reduce BPSK bandwidth to 1/2, and the staggering doesn’t modify the properties.

figure 81

106

12.3 Quadrature Amplitude Modulation (QAM) In general, QAM signal constellation is not restricted to having permitted signalling point only on a unique circle (of radius Ac, as was for the case of MPSK). The general QAM signal is

)()()()()(

sin)(cos)()(tj

cc

etRtjytxtg

ttyttxtsϑ

ωω

=+=

−= eq 225

For example, a popular 16-symbol (M= 16 levels) QAM constellation is shown in figure below, where the relationship between (Ri,θi) and (xi,yi) can readily be evaluated for each of the 16 signal values permitted. This type of signaling is used by 2400-bit/s V.22 modem.

figure 82

The spectrum of the output transmitted signal s(t) and the I,Q modulating signal can be evaluated for a random rectangular binary input signal:

figure 83

107

12.4 PSD for MPSK, QAM, OQPSK, and π/4 QPSK without pre-modulation filtering

The PSD for MPSK and QAM signals for the case of rectangular bit-shape signaling is the same of the BPSK, provided that proper frequency scaling is used.

figure 84

The PSD for the complex envelope of MPSK and QAM signals with data modulation of rectangular bit shape is:

l

b

b

b

bg

M

TRClTK

lTflTfKfP

2

/1

sin)(2

=

==

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ππ

eq 226

C=power (watts) For L=1 we have the BPSK PSD, L=2 (M=4 symbols) describe QPSK, OQPSK, π/4 QPSK PSD.

13,5 dB

f(Hz)=R/l=1/Ts

108

It is also realized that the PSD for the complex envelope of bandpass signals (i.e. envelope of the modulated signal), is essentially the same as the PSD for baseband multilevel signals (i.e. envelope of the modulated signal) when any filtering method is used. From figure above, we can see that the null-to-null transmission bandwidth of MPSK or QAM when rectangular data pulses are used is

( ) ST DTslRB 2/12/2 === eq 227

Therefore ones BT it is fixed we can observe that an increasing of L has as a consequence an increasing of bit rate R. In the same way once R is fixed the an increasing of L produce a decreasing of BT. The spectral efficiency of MPSK or QAM signaling with rectangular pulses is

⎥⎦⎤

⎢⎣⎡===

Hzbit/s

22

l

lR

RBR

T

η eq 228

As a consequence if:

• L=2 then R=B • L=4 then R=2B • L=8 then R=4B

One way to define the transmission bandwidth efficiency for a waveform encoding is the Out-Of-Band-Power (POB), which is defined as:

∫

∫

∫

∫ ∫∞

∞

∞+

∞−

−

∞−

∞

=

+

=

0

2

2

2

22

)(

)(

)(

)()()(

dffS

dffS

dffS

dffSdffSfPOB f

f

f eq 229

Where S(f)=PSD (Power Spectral Density). This method gives the contribution of the PSD, S(f), above a certain frequency, f, when compared with the total PSD for the signal on entire bandwidth. If one wanted a 99% power for the signal, which is a common requirement for regulatory measures, one would find the frequency that gives a POB of 1%.

109

12.5 Spectral efficiency for MPSK, QAM,OQPSK, and π/4 QPSK with Raised Cosine Filtering

The spectrum shown above was obtained for the case of rectangular symbol pulses shaping, and the spectral side lobes was terrible. The first side lobe is attenuated only by ≅ 13.5 dB.

figure 85

The high side lobes can be eliminated if raised cosine filtering is used (since the raised cosine filter has an absolutely band limited frequency response). We should select the 6-dB bandwidth f0 of the raised cosine filter, for the baseband signal, equal to half of the symbol (baud) rate in order for o avoid ISI. That is

sTlRDf 1

21

21

21

0 === eq 230

In practise, a square root raised cosine SRRC frequency response characteristic is often used at the transmitter, along with another SRRC filter at the receiver, in order to simultaneously prevent ISI on the received filtered pulses and minimize the bit errors due to channel noise. However the SRRC filter also introduces AM on the transmitted signal. If the overall pulse shape satisfies the raised cosine-rolloff filter characteristic, then, the absolute bandwidth of the M-level modulating baseband signal signal is

13,5 dB

110

( )[ ]

symbol/s 12

or

Hz 121

lR

rBD

DrB

=⎟⎠⎞

⎜⎝⎛+

=

+=

eq 231

r is the characteristic of the filter called rolloff factor. From AM study modulation we know that the transmission bandwidth BT is related to the modulation bandwidth B by

BBT 2= eq 232

so the overall absolute transmission bandwidth of the QAM signal with raised cosine filtered pulses is:

( )[ ] ( )[ ] ( ) Hz 11 12122 ⎥⎦

⎤⎢⎣⎡ +=+=+==

lRrDrDrBBT eq 233

This bandwidth can be compared to a null bandwidth lRBT 2= of rectangular pulse

shaping i.e. without SRRC filter). We can see that the greeter the r , the greeter the BT, when r=1 then we have again the maximum bandwidth such as we used the rectangular filter shape. On table below are reported some values of the parameter related to the above equations supposing a fixed bit-rate R=812 Kbit/s. Note that when the rolloff factor of the SRRC filter tend to 1 then the bandwidth BT become the same as found for rectangular filter data pulse shape.

111

Transmission Bandwidth Bt as a function of rolloff factor r for SRRC and Rectangular pulse shape filter

0.0

200.0

400.0

600.0

800.0

1000.0

1200.0

1400.0

1600.0

1800.0

0 0.1 0.2 0.4 0.6 0.8 1

Rolloff factor r

Bt K

Hz

BT (KHz) SRRCBT (KHz) rectangBT (KHz) SRRCBT (KHz) rectangBT (KHz) SRRCBT (KHz) rectang

l=1-bit

l=2-bit

l=3-bit

R=812,499 Kbit/s

figure 86

figure 87

E:\documenti per corsi\ELETTRONICA T

112

l=1-bit

l=2-bit

l=3-bit

Transmission Bandwidth Bt as a function of rolloff factor r for SRRC and Rectangular pulse shape filter

0.0

200.0

400.0

600.0

800.0

1000.0

1200.0

1400.0

1600.0

1800.0

BT (KHz) SRRC BT (KHz) rectang BT (KHz) SRRC BT (KHz) rectang BT (KHz) SRRC BT (KHz) rectang

Bandwidth with SRRC and rectangular pulse filter shape

Bt

KH

z

0 0.1 0.2 0.4 0.6 0.8 1

figure 88

Because

lM 2= eq 234

which implies

2lnln

2MMLogl == eq 235

Then the spectral efficiency of QAM-type signaling with raised cosine filtering is

( ) Hzbit/s

2ln1ln

12ln

ln

11 rM

r

M

rl

Rl

rR

BR

T +=

+=

+=

⎟⎠⎞

⎜⎝⎛ +

==η eq 236

113

The above equation can be compared with the rectangular data pulse shape filter efficiency already seen

⎥⎦⎤

⎢⎣⎡===

Hzbit/s

22

l

lR

RBR

T

η eq 237

we can see that ηSRRC is greater than ηrectang if r < 1. This result is important because tell us how fast we can signaling for a prescribed bandwidth. The result also holds for MPSK, since it is a special case of QAM. For example, suppose that we want to signal over a communications satellite that has an available bandwidth of BT=2.4 MHz.

• If we used BPSK (M=2) with a r=50% rolloff factor, we could signal at rate of

sMbitBR T / 60.1677.04.2 =×=×= η eq 238

• If we used QPSK (M=4) with a r=25% rolloff factor, we could signal at a rate of

sMbitBR T / 84.36.14.2 =×=×= η eq 239

Table below illustrates the allowable bit rate per hertz of transmission bandwidth for

a QAM signalling E:\documenti per

corsi\ELETTRONICA T Size of DAC Number ofLevels

l bit per symbol M symbols 0 0.1 0.25 0.5 0.75 11 2 1.00 0.91 0.80 0.67 0.57 0.502 4 2.00 1.82 1.60 1.33 1.14 1.003 8 3.00 2.73 2.40 2.00 1.71 1.504 16 4.00 3.64 3.20 2.67 2.29 2.005 32 5.00 4.55 4.00 3.33 2.86 2.50

Rolloff factor r

figure 89

114

Spectral efficiency for QAM signaling with Raised Cosine-Rolloff Pulse Shaping filtering

0

1

2

3

4

5

6

1 2 3 4 5

l - bit per symbol

effic

ienc

y=R

/Bt (

(bit/

s)/H

z))

00.10.250.50.751

( ) Hzbit/s

2ln1ln

12ln

ln

11 rM

r

M

rl

Rl

rR

BR

T +=

+=

+=

⎟⎠⎞

⎜⎝⎛ +

==η

figure 90

In order to conserve more bandwidth, the number of levels M cannot be increased too much, since for a given peak envelope power (PEP) the spacing between the signal points on the signal constellation will decrease and noise on the received signal will cause errors (Noise moves the received signal vector to a new location that might correspond to a different signal level.) However, we know that R certainly has to be less than C, the channel capacity, if the errors are to be kept small.

⎟⎠⎞

⎜⎝⎛ +=

<

NSLog 12max

max

η

ηη eq 240

115

Spectral Efficiency as a function of SNR

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

0.00

8.45

11.1

412

.79

13.9

814

.91

15.6

816

.33

16.9

017

.40

17.8

518

.26

18.6

318

.98

19.2

919

.59

19.8

720

.13

20.3

720

.61

20.8

321

.04

21.2

421

.43

21.6

121

.79

21.9

622

.12

22.2

822

.43

22.5

822

.72

22.8

622

.99

23.1

223

.24

23.3

623

.48

23.6

023

.71

23.8

223

.93

24.0

324

.13

24.2

324

.33

SNR (dB)

Spec

tral

effi

cien

cy

0.00

50.00

100.00

150.00

200.00

250.00

300.00

SNR

(Lin

eare

)

Spectral efficiencySNR (lineare)

figure 91

116

As an example related to the effect of rolloff factor value on the constellation dispersion and on the PSD, we can consider a QPSK modulation at the transmitter output side. Let us consider two cases: Rectangular filter pulse shaping and SRRC filter pulse shaping. In this last case are being considered the value of rolloff factor: r = 0.5

figure 92

figure 93

The difference on side lobe decaying its clear !!

QPSK constellation With rectangular pulse shaping

QPSK constellation With SRRC pulse shaping and rolloff factor r = 0.5

117

12.6 Receiver QPSK, MSK and performance QPSK is a multilevel signalling technique that uses L=4 levels per symbol. Thus, 2 bits are transmitted during each signalling interval (T seconds). The QPSK signal may be represented by

( ) ( ) TttAtAts cccc ≤<+±−+±= 0 )sin()cos()( θωθω eq 241

Where the (±A) on the cosine carrier is one bit of data and the (±A) factor on the sine carrier is another bit of data. The relevant input noise is represented by

)sin()()cos()()( ncnc ttyttxtn θωθω +−+= eq 242

Total input power it will be given by adding signal and noise: after the demodulation we have ±A+x(t) and ±A+y(t) as input to the integrator. The QPSK signal is equivalent to two BPSK signals-one using a cosine carrier and the other using a sine carrier. The QPSK signal is detected by using a coherent receiver shown in figure below:

figure 94 : Coherent/Matched filter detection of QPSK

Because both the upper and lower channels of the receiver are BPSK receivers, the BER is the same as that for BPSK system. Thus the BER for the QPSK receiver is:

± A+x(t)

Digital output

R/2 bits/sec

2cos(ωct+θc)

Carrier sync f=fc (form carrier sync circuits)

+90° phase shift

QPSK signal + noise Bit sync (from

bit sync circuitry)

R/2 bits/sec

Carrier recovery

Matched Filter: integrator

∫−

0

0

(..)t

Tt

dt

Threshold device

Threshold device

Matched Filter: integrator

∫−

0

0

(..)t

Tt

dt

Parallel To serial converter

-2sin(ωct+θc)

± A+y(t)

S&H

S&H

118

⎟⎟⎠

⎞⎜⎜⎝

⎛=

0

2NEQP b

e eq 243

figure 95 : Comparison of the probability of bit error for several digital signaling schemes

Except for the curves describing the non coherent detection cases, all of these results assume that the optimum filter-the matched filter- is used in the receiver. Comparing the various bandpass signaling techniques, we see that QPSK and MSK give the best overall performance in terms of the minimum bandwidth required for a given signaling rate and one of the smallest Pe for a given Eb/N0. However QPSK is relatively expensive to implement, since it requires coherent detection. Channel coding can be used to reduce the Pe below values given above. The BERs for BPSK and QPSK signaling are identical. But for the same bit rate R, the bandwidth of the QPSK is exactly one-half the bandwidth of the BPSK, i.e. the same information but in half bandwidth.

119

When rectangular data pulses is used, the null-to-null transmission bandwidth is:

RQPSKBRBPSKBlRB TTT 2

1)( )( 2==⇒= eq 244

The spectral efficiency is

2)( 1)( 2

2 QPSK BPSK lB

Bl

BR

T

T

T

==⇒=

⋅

== ηηη eq 245

The bandwidth of π/4 QPSK is identical to that for QPSK. For the same BER, the differentially detected π/4 QPSK requires about 3 dB more Eb/N0 than that for QPSK, but coherently detected π/4 QPSK has the same BER performance as QPSK. The MSK is essentially equivalent to QPSK, except that the data on the x(t) and y(t) quadrature modulation components are normally offset and their equivalent data pulse shape is a positive part of a cosine function instead of rectangular pulse (this gives a PSD for MSK that rolls off faster than that for QPSK). Consequently, because the MSK and QPSK signal representations and the optimum receiver structures are identical except for the pulse shape, the probability of bit error for MSK and QPSK is identical.

120

TYPE OF DIGITAL SIGNALING

MINIMUM TRANSMISSION BANDWIDTH REQUIRED

(R is the bit rate)

ERROR PERFORMANCE

Baseband Signalling

Unipolar R

21

⎥⎥⎦

⎤

⎢⎢⎣

⎡

0NEQ b

Polar R

21

⎥⎥⎦

⎤

⎢⎢⎣

⎡

0

2NEQ b

Bipolar R

21

⎥⎥⎦

⎤

⎢⎢⎣

⎡

0

223

NEQ b

Bandpass Signalling

Coherent detection Non coherent detection

OOK (On Off Keing) R

⎥⎥⎦

⎤

⎢⎢⎣

⎡

0NEQ b

41

21

0

21

0 >⎟⎟⎠

⎞⎜⎜⎝

⎛−

NEe bN

Eb

BPSK R

⎥⎥⎦

⎤

⎢⎢⎣

⎡

0

2NEQ b

Requires coherent detection

PCM/FM R ⎟⎟⎠

⎞⎜⎜⎝

⎛−

021

21 N

Eb

e

FSK Rf +∆2 where

12 fff −=∆ =frequency shift

⎥⎥⎦

⎤

⎢⎢⎣

⎡

0NEQ b ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

021

21 N

Eb

e

DPSK R Not used in practise ⎟⎟⎠

⎞⎜⎜⎝

⎛−

0

21 N

Eb

e

QPSK R

21

⎥⎥⎦

⎤

⎢⎢⎣

⎡

0

2NEQ b

Requires coherent detection

MSK )(5,1 bandwidthnullR

⎥⎥⎦

⎤

⎢⎢⎣

⎡

0

2NEQ b ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

021

21 N

Eb

e

121

13 Feher-Patented Quadrature Phase-Shift Keing

13.1 Introduction The Feher-patented Quadrature-phase-shift Keying, type B (FQPSK-B) modulation scheme is a proprietary bandwidth-efficient modulation technique invented by Dr.Kamilo Feher. FQPSK-B is a variant of the cross-correlated FQPSK scheme (originally referred to as XPSK) which in turn was derived from a previous modulation scheme also invented by Dr. Feher known as Inter-symbol-interference and Jitter-Free (IJF) QPSK, which has a 3-dB envelope fluctuation. With FQPSK as well as FQPSK-B, there is an intentional controlled amount of cross-correlation between the In-phase (Ik) and Quadrature-phase (Qk) channels which allows for a quasi-constant envelope, reducing the envelope fluctuation to 0 dB. This cross-correlation was applied to the IJF-QPSK baseband signals prior to modulation onto In-phase (Ik) and Quadrature-phase (Qk) carriers. This transformation was initially implemented by mapping, in each half symbol, the 16 possible combinations of the (Ik) and (Qk) baseband waveforms present in the IJF-QPSK onto a new set of 16 waveform combinations. Here SI(t) and SQ(t) are the I- and Q-channel baseband signals. These new waveforms were chosen in such a way that the baseband signals are time continuous and the envelope is constant. FQPSK-B improves spectral efficiency over FQPSK because low-pass filtering is applied to the baseband I- and Q-channel waveforms. When properly designed and specified, a system using FQPSK-B is interoperable with other modulation schemes such as OQPSK, GMSK, and MSK.

13.2 Signal model for FQPSK The FQPSK signal (or XPSK signal) can be realized using either a half-symbol cross-correlation mapping or a full-symbol mapping. In this section, the full-symbol cross-correlation mapping is described instead of the half-symbol mapping because the full-symbol mapping facilitates the interpretation of the FQPSK as a TCM (Trellis Code Modulation). The waveform of each baseband signal in a TS symbol interval is chosen from a set of 16 waveforms {si(t) | 0 ≤ i ≤ 15} defined as follows:

Typically 2/12 ≅=b

b

TEA eq 246

And:

122

eq 247

123

124

Example of quasi constant envelope modulation si(t) and sQ(t)

125

Note that for any value of A other then unity, s5(t) and s6(t) as well as their negatives, s13(t) and s14(t), will have a discontinuous slope at their midpoints (i.e., at t = 0), whereas the remaining 12 waveforms all have a continuous slope throughout their defining intervals. Also note that all 16 waveforms have zero slope at their end points and, thus, concatenation of any pair of these will not result in a slope discontinuity. The wavelets are numbered according to a Trellis mapping rule that determines which wavelet is transmitted. Specifically, the mapping rule specifies that during the n-th channel symbol interval, [(n - [1/2])Ts] ≤ t ≤ [(n + [1/2])Ts], the baseband I- and Q-channel waveforms Ik and QK are assigned wavelets sI(t)=si and sQ(t)=sj respectively, where the indices i and j are given by:

00

11

22

33

00

11

22

33

2222

2222

×+×+×+×=

×+×+×+×=

QQQQj

IIIIi eq 248

with

nI

nInI

nQnQ

nQnQ

dIddI

ddI

ddI

,3

1,,2

2,1,1

1,,0

=

⊕=

⊕=

⊕=

−

−−

−

nQ

nQnQ

nInI

nInI

dQ

IddQIddQ

ddQ

,3

01,,2

21,,1

1,1,0

=

=⊕=

=⊕=

⊕=

−

−

+

eq 249

Where { }1,0 and ,, ∈nQnI dd are the n-th I- and Q- channel inputs to the modulator respectively The particular sI(t) and sQ(t) waveforms chosen for any particular Ts signalling interval on each channel depend on the most recent data transition on that channel, as well as the two most recent successive transition on the other channel. Next, define the following mapping function for the baseband I-channel transmitted waveform [yI (t)=sI (t) ] in the n-th signalling interval [(n - [1/2])Ts] ≤ t ≤[(n + [1/2])Ts] in terms of the transition properties of the I and Q data symbol sequences dIn and dQn, respectively.

126

Making use of the signal properties of eq above, the mapping conditions in (1) through (4) for the I-channel baseband output can be summarized in a concise form described by Table 1. A similar construction for the baseband Q-channel transmitted waveform [yQ(t)=sQ(t-Ts/2) ] in the n-th signalling interval nTs ≤ t ≤ (n + 1)Ts in terms of the transition properties of the I and Q data symbol sequences, dIn and dQn, respectively,

127

can be obtained analogously to (1) through (4) above. The results can once again be summarized in the form of a table, as in Table 2. Tables below specify the details where dIK is the data sequence on the I channel, and dQK the data sequence for the Q channel.

128

A block diagram of a FQPSK transmitter based on the reformulation by Simon and Yan is shown in figure below:

figure 96 : The conceptual block diagram of FQPSK(SPSK)

FQPSK-B Transmitter and Receiver schemes are reported below:

figure 97 : FQPSK transmitter

Modulated signal

Ik

cos(ωct+θc)

[ ]sTt 2/cos π= [ ]sTt 2/sin π=

SFQPSK(t) Digital source

sin (ωct+θc) Qin=2 Bin=1/Tb

Encoder

Qout=4 Bout= Bin /2 Ts=2Tb

I and Q Comb logic

Delay=Tb =Ts/2

Qk Q seq. Logic And switch

I seq. Logic And switch

si(t)

sQ(t-T/2)

+

+

QPSK Modulator

129

figure 98: coherent FQPSK receiver

The FQPSK signal obtained by transmitter can be represented by:

)2cos()2

()2sin()()()( ccs

QccIFQPSKXPSK tfTtstftstStS θπθπ +−++== eq 250

Where sI(t) and sQ(t) are the combined cross-correlator output. These are obtained, as we have seen so far, by cross-correlating (i.e. mapping) the sequence dI and dQ baseband-data represented by IK and QK The value of IK and QK depend on the encoder output state k as given by equation below, and is a function of two consecutive encoder inputs bits as specified in table below.

( )

( ) ⎥⎦⎤

⎢⎣⎡ −−=

⎥⎦⎤

⎢⎣⎡ −+=

412sin2

412cos2

π

π

kTEQ

kTEI

s

sk

s

sk

figure 99

EncoderInput

Encoder Input

Encoder Output

I logic Q logic

Ej-1 Ej k IK QK 0 0 0 +1 +1 0 1 1 +1 -1 1 0 2 -1 -1 1 1 3 -1 +1

Ik

Received Modulated signal

cos(ωct+θc) r(t)= SFQPSK(t) Digital

sink

sin (ωct+θc)

ei Decoder mk I and Q Decision logic

Delay=Tb =Ts/2

Qk

∫bT

dt0

(.)

∫bT

dt0

(.)

Decision unit

Decision unit

r2

Decision mapping

0101

1

1

<←−>←+

rr

Decision mapping

0101

2

2

<←−>←+

rr

Correlator

r1

130

As an example for Ik and Qk which in turn are the dI e dQ data flow on I and Q channels, we have:

( )

( )

( )

( ) 1222

45cos2

416cos2

1222

43cos2

414cos2

1222

4cos2

412cos2

1222

4cos2

410cos2

3

2

1

0

−=−=−=⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡ −=

−=−=−=⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡ −=

+===⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡ −=

+===⎥⎦⎤

⎢⎣⎡−=⎥⎦

⎤⎢⎣⎡ −=

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

TE

TE

TE

TEI

TE

TE

TE

TEI

TE

TE

TE

TEI

TE

TE

TE

TEI

ππ

ππ

ππ

ππ

eq 251

( )

( )

( )

( ) 1222

45sin2

416sin2

1222

43sin2

414sin2

1222

4sin2

412sin2

1222

4sin2

410sin2

3

2

1

0

+===⎥⎦⎤

⎢⎣⎡−=⎥⎦

⎤⎢⎣⎡ −−=

−=−=−=⎥⎦⎤

⎢⎣⎡−=⎥⎦

⎤⎢⎣⎡ −−=

−=−==⎥⎦⎤

⎢⎣⎡−=⎥⎦

⎤⎢⎣⎡ −−=

+===⎥⎦⎤

⎢⎣⎡−−=⎥⎦

⎤⎢⎣⎡ −−=

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

TE

TE

TE

TEQ

TE

TE

TE

TEQ

TE

TE

TE

TEQ

TE

TE

TE

TEQ

ππ

ππ

ππ

ππ

eq 252

Consecutive values of Ik and Qk assume A=±1 values, and are then nonlinearity filtered by the I sequential logic and switch, and by the Q sequential logic and switch respectively. The switch outputs are defined in table 1 and table 2. Note that the switch output is determined by three consecutive inputs, and thus has memory. This is in contrast to OQPSK, which is memory less.

131

13.3 Signal model for FQPSK-B The FQPSK-B signal obtained by transmitter can be represented by:

)2cos()2

(~)2sin()(~)()( ccs

QccIBFQPSKXPSK tfTtstftstStS θπθπ +−++== − eq 253

where the filtered signals

∫

∫

∞−

∞−

−=

−=

t

QQ

t

II

dthtsts

dthtsts

ττ

ττ

)()()(~

)()()(~

eq 254

are the low pass filtered version of the sI(t) and sQ(t) seen so far in FQPSK form, and h(t) is the impulse response of the low-pass filter. The functional block diagram below depicts full-symbol cross-correlation mapping followed by the transmission filter and an Offset-QPSK (OQPSK) modulator.

figure 100 : FQPSK-B modulator

Figure below shows the phasor diagrams and the eye diagrams of both FQPSK and FQPSK-B baseband signals.

sI(t)

SQ(t)

132

figure 101: phasor diagrams FQPSK (left) FQPSK-B (right)

figure 102: transmitter eye figure for FQPSK (top) and FQPSK-B (bottom)

133

13.4 Spectral efficiency comparison Required bit rates in range telemetry are increasing dramatically, resulting in research to develop modulation techniques that have greater spectral efficiency than 35-year-old workhorse fo the telemetry industry, NRZ, PCM/FM. In figure below is presented a comparison between several type of modulation techniques.

figure 103

134

figure 104

• The 99.99% bandwidths of filtered FQPSK-B are approximately one-half of the

corresponding bandwidth of optimized PCM/FM, even when the signal is non linearly amplified.

• The EB/N0 required for a BER of 1x10-5 for non optimized FQPSK-S is approximately 12 dB, which is approximately the same as the limiter discriminator detected PCM/FM

135

14 EFFICIENT MODULATION METHODS STUDY AT NASA/JPL CCSDS (RF and Modulation) became aware of a new modulation type at its Spring 1997 meeting. Named FQPSK for its inventor, Dr. Kamilo Feher, it was reported to have a very narrow RF spectrum and only minimal end-to-end system losses. Test data provided by Dr. Feher showed a spectrum narrower than that of GMSK using a BTS = 0.50 filter. Sideband attenuations were tabulated for the several modulation types studied and it was concluded that FQPSK-B could be a very attractive modulation method. FQPSK-B, a specific version of FQPSK, was simulated using SPW. Additionally, Mr. Eugene Law of the Naval Air Warfare Center Weapons Division at Point Mugu obtained an FQPSK-B modulator-demodulator (modem) for hardware tests. NASA witnessed these spectrum tests and obtained copies of the spectra. Note: This is the only modulation type covered in this report for which there are actual hardware verification tests. These tests confirm the simulation results reported here.

• FQPSK-B modulation is a form of OQPSK modulation in which one of 16 wavelets [waveforms] is selected for transmission on the I-channel and another is chosen for transmission on the Q-channel. Wavelet determination depends on the present and previous data bit pair values for the I and Q channels. There is a ½-symbol-time offset between I and Q transmissions.

• FQPSK-B modulates and filters at baseband. There after, the signal is translated to an i.f. frequency and then translated again to the transmitted RF frequency.

14.1 FQPSK-B Modulation Bit-Error-Rate (BER) Simulations of FQPSK-B were conducted at JPL with the assistance of Dr. Feher. Figure below shows the Bit-Error-Rate (BER) performance. Like MSK and GMSK modulation, existing transmitting and receiving equipment simulation models were unsuitable for FQPSK-B. However, BER performance was measured using ESA’s power amplifier operating in full saturation. Comparing FQPSK-B to ideal BPSK/NRZ shows that an additional EB / N0 of 1.7 dB is required to achieve a 1 x 10-3

BER. This is 0.3 dB greater than GMSK with a BTS = 0.5. Dr. Feher commented that additional system optimization might reduce these losses. His suggestions included adding hard limiters to the transmitting system and improving the receiver filter’s phase performance. Supporting his position, Dr. Feher points to BER measurements made at Point Mugu using actual hardware. Dr. Feher’s modem, operating with a 1 Watt SSPA in full saturation, produced a 1 x 10-3 BER at an EB / N0 of 8 dB, about 1.3 dB more than ideal BPSK/NRZ and 0.1 dB less than GMSK with a BTS = 0.5. Further BER tests will be required to verify the better EB / N0 performance using a modulator capable of a 60 dB sideband attenuation.

136

figure 105: FQPSK-B mosulation Bit Error Rate

14.2 FQPSK-B Modulation Spectra Figure below FQPSK-B spectra obtained by simulations. Spectra are obtained using ESA’s 10 Watt SSPA operating in full saturation. However, as with the MSK and GMSK simulations, an ideal modulator and receiver were simulated. FQPSK-B spectra do not have discrete components, giving it a distinct advantage over filtered phase modulation schemes. Sideband attenuation does tend to reach a floor at approximately 75 dB below the peak amplitude where spectral broadening is clearly evident in Figure below. Unlike most of the phase modulation schemes, spectral broadening in the vicinity of fC does not occur. Rather, Figure shows the spectrum width around fC to be significantly narrower than BPSK/NRZ. FQPSK-B has a very compact, bandwidth-efficient spectrum. Simulations show it to be slightly better than GMSK reaching a level 50 dB below the peak sideband amplitude at a bandwidth of 1.7 RB rather than at 1.9 RB for GMSK with a BTS = 0.5. At a sideband attenuation of 60 dB, FQPSK-B and GMSK are within 0.1 RB of one another.

14.2.1 Hardware Spectrum Measurements FQPSK-B is the only modulation type in the Phase 3 Efficient Modulation Methods Study for which there are actual hardware measurements. On 1 July 1997 FQPSK-B hardware tests were conducted at the Naval Air Warfare Center at Point Mugu. Dr. Feher contributed a laboratory model of his FQPSK-B modulator. The test configuration included: a random data generator producing 1 Mb/s, Dr. Feher’s FQPSK-B modulator, a Hewlett Packard (HP) Model 8780A Vector Signal Generator for QPSK modulation, a frequency translator, a 1-Watt SSPA, and an HP spectrum analyzer.

137

Tests were run with the SSPA in full saturation at 2.44 GHz and frequency spectra were plotted by the HP spectrum analyzer. Figure reproduces the HP analyzer’s plot on the same scale as that used for the Fine Detail spectra shown in Figure. Separate figures are provided because the spectrum plotted in Figure is virtually indistinguishable from the FQPSK-B curve in Figure, down to a level 55 dB below the peak sideband amplitude. Below the -55 dB point, the hardware generated spectrum in Figure becomes wider than the SPW computed spectrum in Figure. Readers should understand that no attempt was made to optimize the hardware test configuration at Point Mugu. The test bed was constructed using hardware elements designed for a variety of other uses. These measurements confirm the bandwidth efficiency of FQPSK-B modulation, as predicted by SPW. Neither a 2 GHz receiver nor an FQPSK-B demodulator-symbol synchronizer were available to measure Bit-Error-Rate. Therefore, system losses calculated by SPW could not be confirmed using this test configuration. Additional hardware tests were conducted using an FQPSK-B modem provided by Dr. Feher. The test configuration operated at 70 MHZ. This inexpensive commercially available modem was designed to operate over a more restrictive set of signal levels than the laboratory modulator described above. It did not provide sideband attenuations much below 40 dB.

figure 106: fc=±10 Rb

138

figure 107: Broadband spectra (fc=±250 Rb)

14.3 FQPSK-B Modulation Power Containment FQPSK-B frequency spectrum efficiency is so high that two power containment plots are required. First figure is plotted using a 0 - 20 RB scale for consistency with the other modulation methods. However, virtually all of the transmitted power is contained in such a small bandwidth that a second figure is added. Its scale of 0 - 2 RB clearly shows the occupied bandwidth to be only 0.8 RB. This is significantly better than the 1.0 RB found with GMSK using a filter bandwidth of BTS = 0.5.

14.4 FQPSK-B Modulation Study Conclusions Although FQPSK-B modulation was only recently added to the Efficient Modulation Methods Study, it appears to be one of the most bandwidth-efficient modulation method considered. Because of its proprietary nature, some of its parameters are not apparent from published documents. Whether this proprietary nature would serve as an impediment to universal application by space agencies is also not clear. What is clear is that FQPSK-B modulation must be seriously considered for high and very high data rate missions. With RF spectra valued in the Unites States at several hundred dollars per Hertz, NASA, and probably all space agencies, have a duty to investigate this modulation type further.

139

figure 108: FQPSK-B Power containment (0-20 Rb)

140

figure 109: FQPSK-B Power containment (0-2 Rb)

15 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS The CCSDS - SFCG Efficient Modulation Methods Study measured the RF spectrum’s width and end-to-end system performance using computer simulations. In compliance with the SFCG’s request, the conclusions identify those modulation schemes that are the most bandwidth-efficient and suggest that CCSDS and SFCG Space Agencies adopt recommendations specifying their use.

15.1.1 SUMMARY For each modulation method, it reviews end-to-end system losses, examines RF spectrum bandwidth, and discusses the spectrum improvement factor resulting from baseband filtering.

15.1.1.1 Summary of Losses Table below shows system and filtering losses occurring in the end-to-end system for each modulation type. Column 2 contains losses relative to ideal BPSK/NRZ modulation. Recall that ideal BPSK/NRZ assumes: perfect data (Pm = Ps = 0.5), an ideal system (perfect carrier tracking and symbol synchronization), and no filtering (BT = 4).

141

Filtering losses, inherent in GMSK and FQPSK-B, are included in leftmost column of Table containing losses relative to ideal BPSK/NRZ. NOTE: All modulation types exhibit a loss with respect to ideal BPSK/NRZ. To find the true cost of a modulation method, one should subtract 0.56 dB which is the loss for filtered BPSK. Thus, the true loss for GMSK (BTS = 0.5) is about 0.8 dB and FQPSK-B is about 1.1 dB. Phase 3 studies employed baseband filtering exclusively. A principal objective was the selection of the proper filter bandwidth. Recall that filter selection criteria required using a filter producing the narrowest RF spectrum while introducing only moderate losses. From Table it is clear that filters having a BTS = 1 often exceeded the allowable loss of approximately 1 dB. However, filters having a BTS = 2 generally met the 1 dB loss criterion. 8-PSK was the exception requiring a BTS = 3 filter bandwidth to be acceptable. For the other modulation types, BER curves showed that there was no significant benefit in using a BTS = 3 filter bandwidth. Thus, Butterworth and Bessel baseband filters, with a BTS = 2, were used.

figure 110: System Losses

NOTES: 1. Losses determined at a Bit-Error-Rate of 1 x 10-3

with 0 = 0, Pm = 0.55 (negative numbers indicate a loss). 2. System losses were measured relative to ideal BPSK/NRZ (perfect data, lossless equipment). 3. Filtering Losses include: ISI + Mismatch + Imperfect Carrier Tracking & Symbol Synchronization. 4. BER reached a minimum of 1 x 10-2. 5. Filtering Losses Not Available (N/A) because BER measured with ideal system components. 6. Filter bandwidth BTS = 1 (BTB = 0.5). 7. Filter bandwidth BTS = 0.5 (BTB = 0.25).

142

15.1.1.2 RF Spectrum Efficiency Another Phase 3 study objective was to determine the RF spectral bandwidth of each modulation type. This was necessary to rank the several modulation methods with respect to one-another. Many Spectrum Managers are concerned principally with occupied bandwidth (i.e., 99% power containment). The Efficient Modulation Methods Study was motivated by a desire to pack a substantially greater number of spacecraft into a given frequency allocation, particularly in the 2 and 8 GHz Category A mission bands. Maximum packing density occurs when spectra from two spacecraft, operating on adjacent frequencies, just begin to overlap at n dB below the peak of the data sideband’s spectrum. This follows from a worst-case assumption that the Earth station’s antenna is bore sighted on both spacecraft simultaneously. Where spacecraft are not coincidently within the Earth station antenna’s beamwidth, the interferer’s and victim’s relative signal strengths will determine the special separation necessary to avoid interference. Obviously, even as frequency band usage increases, some spatial separation is expected. This study attempted to determine the value of n. Views differ regarding the optimal value of n. Some believe that spectra from spacecraft on adjacent frequencies could be permitted to intersect at a level of 20 dB below the peak sideband amplitude. Others believe that the number should be greater or less than 20 dB. In any event, Category A missions in highly elliptical orbits can undergo signal level changes of 30 dB or more at the Earth’s surface. Thus, it would seem prudent to prohibit RF spectra, from spacecraft operating on adjacent frequencies, from intersecting at levels higher than 50 dB below the peak of the data sideband generated by the spacecraft having the stronger signal. To provide maximum flexibility, RF spectrum bandwidths have been tabulated at values of n from 20 to 60 (dB) below the data sideband’s peak. RF spectrum width increases as a function of n and each user must select the proper value. A value of n = 50 is recommended for most applications. For a specific value of n, one can calculate the improvement in spectral efficiency. CCSDS concluded that BPSK/NRZ was to be the reference modulation type. A Spectrum Improvement Factor (SIF) can be calculated by comparing the bandwidth of unfiltered BPSK/NRZ to the bandwidth of the modulation method under discussion according to the relationship:

eq 255

Since the bandwidth is a function of n, the SIF will also vary with n. Table below contains the bandwidths and SIFs at several values of n for all modulation types covered in this Phase 3 study.

143

Bandwidths for all phase modulation types were evaluated using a Butterworth, BTS = 2 filter. For symmetrical data, no spikes are present in the spectra of unfiltered BPSK/NRZ. All measurements in Table were made with respect to a continuous unfiltered BPSK/NRZ reference spectrum. Conversely, all phase modulation schemes, which employ baseband filtering, have both continuous and discrete parts to their spectrum. SIF measurements in Table below were made with respect to the discrete part of the baseband filtered modulation spectrum. This represents a worst case bandwidth comparison. Readers should understand that no discrete spectral components exceed the value of n in any of the SIFs shown in Table. Since SPW’s resolution bandwidth was set to 1.33 Hz, one can conclude that the SIFs should be close to those obtained using real hardware viewed on a spectrum analyzer with a 1 Hz resolution. Preferred modulation types become immediately apparent when SIFs are plotted as a function of n as in Figure below. Modulation types fall into two distinct groups FQPSK-B / GMSK and everything else. Even 8-PSK is not a competitor for those two types. The message is clear: If RF bandwidth is important, then the results of this study show that FQPSK-B and GMSK (BTS = 0.5) are the modulation methods of choice.

figure 111: Bandwidth Efficiencies

144

figure 112 : Spectral efficiency relative to unfiltered BPSK/NRZ

Figure above relates the spectral efficiencies of the several modulation methods investigated in the Phase 3 Efficient Modulation Methods Study. SIF, as defined in equation above is plotted as a function of n (number of dB below the peak sideband amplitude). Three classes of bandwidth efficiency are evident: High (FQPSK-B and GMSK); Medium (8-PSK, QPSK/OQPSK, MSK, PCM/PM/NRZ, and BPSK/NRZ); and Low (PCM/PM/Bi-φ, BPSK/Bi-φ). All Phase 3 modulation bandwidth measurements are made using a Butterworth 3RD order BTS= 2 filter. MSK has no filtering and GMSK curves are labeled with the Gaussian filter’s BTs

factor. FQPSK-B measurements are based on a proprietary filter in FQPSK-B modulation.

15.2 CONCLUSIONS Conclusions fall into distinct categories relating to filtering methods, losses, modulation types, and Spectrum Improvement Factors (SIFs). Each conclusion is summarized in the subsections below.

15.2.1 Filtering Conclusions Baseband filtering significantly reduces the transmitted RF spectrum’s width. Study conclusions are:

145

• Filtering of transmitted signals will be required to obtain an acceptably narrow RF spectrum.

• Hardware limitations make post PA filtering impractical at data rates below about 8 Ms/s.

o Realizable Qs limit the filter’s bandwidth to about 1-2% of the transmitted frequency.

o Filtering power losses may be unacceptable, even at a 1-2% bandwidth. o For low data symbol rates, post PA filtering may make turnaround

ranging difficult. • Depending upon its architecture, transponder i.f. filtering may not be practical.

o Q limitations stated above apply if modulation occurs at the transmitting frequency.

o Filtering at i.f. requires transponders be modified for each mission. o Filtering at i.f. makes data rate changes difficult. o For low data symbol rates, i.f. filtering may make turnaround ranging

difficult. o Filtering within the transponder risks introducing spurious emissions

causing lock-up. • Baseband filtering is the only practical alternative to unacceptable post PA and

i.f. filtering. o Baseband filtering can be accomplished with a simple, passive low-pass

filter design. A 3RD order Butterworth filter (BTS = 2) provides the best

performance-simplicity ratio. o Filtering prior to phase modulation produces undesirable spikes in the RF

spectrum. Spikes can only be avoided by using a different modulation method

(GMSK, FQPSK). • Both GMSK and FQPSK-B utilize baseband filtering and do not require i.f. nor

post PA filters.

15.2.2 Loss Conclusions Table above partitions losses into two categories: System (losses relative to ideal BPSK) and Filtering (ISI and Mismatch). One criterion for the Phase 3 study was that end-to-end losses should be reasonable. CCSDS determined that approximately 1 dB was reasonable. The following conclusions regarding losses were reached:

• High system loss (1.5 dB) found for PCM/PM/NRZ, resulted from a 10% data imbalance.

o When a BTS = 2 Butterworth filter is used, data imbalance should not exceed 5%.

• 8-PSK modulation exhibits an excessive system loss (3.4 dB). o Filtering losses decreased for non-constant envelope modulation.

However, spectrum width increased. Losses were not reduced to an acceptable level.

o High losses make 8-PSK modulation unsuitable for power-limited Category A missions.

• GMSK (BTS =0.5) also exhibited high (1.4 dB) system losses.

146

o Increasing filter bandwidth to BTS = 1 reduced system losses to an acceptable level.

o Losses were measured with an ideal (lossless) receiver. • FQPSK-B losses were found to be a high 1.7 dB.

o Losses were also measured with an ideal (lossless) receiver.

15.2.3 Modulation Methods Conclusions Figures above , graphically identify the preferred modulation methods. For the several modulation methods considered, the following conclusions were reached.

• FQPSK-B provides the narrowest RF spectrum of all modulation methods studied.

o FQPSK-B should be considered for all high and very high data rate missions.

Provided that losses are acceptable. • GMSK, with a filter bandwidth BTS = 0.5, produces virtually equivalent results to

FQPSK-B. o Further work is required to validate system losses using real hardware.

• 8-PSK, with its high losses, does not appear useful for most Category A missions.

o Excessive losses and modest performance gains do not provide sufficient advantages.

• QPSK has comparatively poorer bandwidth efficiency than does FQPSK-B and GMSK.

o Its common usage may dictate its consideration in some applications. o Absent spread spectrum, QPSK cannot provide simultaneous telemetry

and ranging. • OQPSK could not be evaluated properly with the UPM.

o OQPSK should be reserved for applications requiring separate, independent data channels.

o Orthogonally phased BPSK/NRZ modulators, with a ½ symbol offset should be used.

• BPSK/NRZ has poor bandwidth efficiency and should not be used if bandwidth is important.

o Bandwidth efficiency is slightly lower than PCM/PM/NRZ modulation. o BPSK/NRZ may be an alternative to PCM/PM/NRZ when:

A residual carrier is not required. The data imbalance is so great that PCM/PM/NRZ would suffer

excessive losses. • PCM/PM/NRZ has poor bandwidth efficiency, but has best efficiency of residual

carrier types. o Applications requiring a residual carrier should consider this modulation

method. o When using PCM/PM/NRZ, care must be taken to ensure proper data

balance. • MSK modulation is not highly spectrum efficient.

147

o No specific advantages were found to MSK, save the lack of spectral spikes.

• Bi-φ modulation has very poor RF spectrum efficiency. o Bi-φ modulation should not be used unless the symbol transition density is

too low. o This conclusion applies to both PCM/PM/Bi-φ and BPSK/Bi-φ modulations.

• Subcarrier modulation tends to waste spectrum and should be avoided whenever possible.

o When used, the subcarrier frequency-to-data symbol rate ratio should be low

o CCSDS virtual channels should be used to separate data types.

15.2.4 Spectrum Improvement Conclusions The following conclusions were reached regarding RF spectrum efficiency improvement:

• Baseband filtering greatly increases the number of spacecraft operating in a frequency band.

o Spectrum utilization efficiency can increase by a factor from 2 to more than 100 times.

o The amount of improvement depends upon modulation method and sideband attenuation.

o This result attains despite non-linear system elements, non-ideal data, and spectral spikes.

• Modulation method should be selected to maximize the Spectrum Improvement Factor.

o Modulation schemes with low Spectrum Improvement Factors should be avoided.

• Modulation method selection should be based on system capabilities, data rates, and SIFs.

15.3 RECOMMENDATIONS Based upon the results of the Phase 3 Efficient Modulation Methods Study, the CCSDS and SFCG are encouraged to create and adopt Recommendations specifying the preferred modulation methods. Because space missions have a broad range of objectives, communication requirements will vary. Some grouping of applications is necessary before assigning a modulation type.

15.3.1 Mission Classification One method for grouping applications is by specific attributes. Missions sharing those attributes are assigned a classification and a modulation method(s) most appropriate to that group are selected. Where RF spectrum and modulation types are of paramount concern, the telemetry data symbol rate appears to be the best discriminator. The following classifications are recommended:

148

15.3.1.1 Low Data Rate (10 s/s - 20 ks/s) This class includes low rate scientific missions as well as the Telemetry, Tracking, and Command (TT&C) services for most missions. Turnaround ranging may be required. If it is, subcarrier modulation may be appropriate (see CCSDS Recommendation 401 (3.3.4) B-1). If ranging is not required, then any appropriate modulation type should be acceptable. All mission types operating in the space services can be found in this class.

15.3.1.2 Modest Data Rate (20 ks/s - 200 ks/s) Most Category A missions fall in this and the following classification. If space agencies are serious about reducing RF spectrum requirements, they must use appropriate filtering and modulation techniques for spacecraft in these classes. Typical missions operate in the SpaceResearch service and include NASA’s ISTP Wind and ESA’s Integral missions. The recommended modulation method depends upon whether or not simultaneous telemetry and turnaround ranging signals are required (see CCSDS Recommendation 401 (3.4.1) B-1). If they are, a residual carrier modulation method is suggested because users can independently control the division of power between the carrier, telemetry, and ranging channels. PCM/PM/NRZ is the most bandwidth-efficient residual carrier modulation method and is recommended provided that the telemetry data imbalance is less than 5% during a time interval equal to one time-constant of the Earth station receiver’s phase-locked-loop. At low data symbol rates, care must be taken with PCM/PM/NRZ modulation to ensure that the Earth station’s receiver can distinguish between the RF carrier and the spectral components of the data sidebands. The spacecraft’s modulation index and the Earth station receiver’s phaselocked-loop bandwidth should be adjusted to ensure proper operation. If simultaneous telemetry and turnaround ranging is required and the data imbalance is greater than 5%, then Unbalanced QPSK (UQPSK) is the recommended modulation type. Within limits, telemetry and ranging powers can be set independently. If simultaneous telemetry and turnaround ranging is not required or where data imbalance exceeds 5%, BPSK/NRZ is recommended.

15.3.1.3 Medium Data Rate (200 ks/s - 2 Ms/s) As noted, most Category A scientific missions fall into this and the prior classification. Generally, such spacecraft operate in the Space Research service allocation. Examples include NASA’s Polar and ESA’s SOHO missions. Because many of these missions are collecting scientific data, simultaneous turnaround ranging is frequently required. In these cases, PCM/PM/NRZ modulation is recommended, providing the telemetry data symbol imbalance does not exceed 5% in one time-constant of the Earth station receiver’s phase-locked-loop. If data imbalance exceeds 5%, then UQPSK can be used. In this classification, data symbol rates can be as high as 2 Ms/s, so bandwidth conservation is important. If simultaneous turnaround ranging is not required, then QPSK modulation is recommended.

149

15.3.1.4 High Data Rate (2 Ms/s - 20 Ms/s) Typically, missions with data symbol rates in this range operate in the Earth Exploration Satellite service. Examples include NASA’s Lewis and the Canadian Space Agency’s (CSA’s) Radarsat projects. In this and the following classification, RF spectrum limiting becomes imperative. Decreasing bandwidth utilization by a factor of 10 saves considerably more RF spectrum when the data symbol rate is 20 Ms/s than in the case when it is 200 ks/s. Both the CCSDS and SFCG should immediately adopt filtering and modulation Recommendations for these last two classes. From Figures above, and Table above, FQPSK-B or GMSK (BTS = 0.5) modulation are the clear choices if RF spectrum conservation is important. Modulator modifications may be required to provide turnaround ranging with either of these modulation types and the ranging signal will have to be sequential, not simultaneous, with the telemetry data.

15.3.1.5 Very High Data Rate (20 Mb/s - and Above) Missions with data symbol rates in this range operate almost exclusively in the Earth Exploration Satellite service. Examples include NASA’s Earth Observation Satellite (EOS) and ESA’s Earth Resources Satellite (ERS-1). Previous comments regarding bandwidth conservation and modulation methods apply emphatically to this class. FQPSK-B or GMSK (BTS = 0.5) are the recommended modulation methods. The CCSDS and SFCG are urged to move with all dispatch to obtain the additional system performance information for both FQPSK-B and GMSK modulation types. The authors recommend that tests, using real hardware, be conducted in a carefully controlled environment to validate these simulations and to measure actual system performance. Recommendations, consistent with Table below, should be adopted at the earliest possible opportunity.

figure 113: Recommended Modulation Methods for Category A Missions

150

GLOSSARY ARX II A Research and Development Earth Station Receiver, (Prototype for DSN Block V) BER Bit-Error-Rate Bi-φ Binary-Phase [Manchester] modulation BL Receiver phase-locked-loop’s bandwidth, expressed in Hz BPSK Bi-Phase Shift Keying [modulation method] BTB Bandwidth • Time Product Based on Bit-Period BTS Bandwidth • Time Product Based on Symbol-Period Category A Space Mission whose distance from Earth is less than 2 • 106

km CCSDS Consultative Committee for Space Data Systems DSN Deep Space Network DTTL Digital Transition Tracking Loop ESA European Space Agency ESOC ESA Operation Center (Darmstadt, Germany) ESTEC ESA Technical Center (Noordwijk, The Netherlands) FQPSK Feher QPSK [modulation method] GMSK Gaussian Minimum Shift Keying HP Hewlett Packard Hz Hertz k Kilo (1,000) kb/s Kilo Bits per Second kHz Kilo Hertz ks/s Kilo Symbols Per Second M Mega (1,000,000) MHZ Mega Hertz MAP Maximum A Posteriori MODEM Modulation-Demodulation MSK Minimum Shift Keying NRZ Non Return to Zero [format] OQPSK Offset QPSK [modulation method] PA Power Amplifier PCS Personal Communications System PM Phase Modulation PSK Phase Shift Keying PT Total Power [transmitted] QPSK Quadrature Phase Shift Keying [modulation method] SAW Surface Acoustic Wave [Filter] SER Symbol-Error-Rate SFCG Space Frequency Coordination Group SPW Cadence Design Systems Inc. Signal Processing Worksystem SRRC Square Root Raised Cosine [Filter] SSPA Solid State Power Amplifier Subpanel 1E CCSDS group concerned with RF and Modulation standards

151

16 8PSK MODULATION (Example implemented in mobile telephone network)

16.1 Introduction After GPRS, the next step in improving the GSM system data rate is to change the signal to a type that has greater bandwidth efficiency, i.e. more bits per second can be supported per unit bandwidth. This is most economically implemented throughout the existing GSM infrastructure when the new signal type has identical bandwidth occupancy characteristics to the original 0.3-GMSK signal. EDGE is a modulation scheme that is more bandwidth efficient than the Gaussian pre-filtered minimum shift keying (GMSK) modulation scheme used in the GSM standard. The technology defines a new physical layer of an 8-Phase-Shift-Keying modulation (8PSK), instead of Gaussian-Minimum-Shift Keying (GMSK). 8PSK enables each pulse to carry 3 bits of information per symbol versus GPRS/GMSK’s 1 bit per symbol per pulse rate. Thus, it has the potential to increase the data rate of existing GSM system by a factor of three. For this reason, it requires a hardware upgrade of the RF part in the base stations and new mobile stations that support EDGE modulation. A traditional 8PSK system uses raised-cosine filtering to remove Inter-symbol Interference (ISI). Although ISI is eliminated by using raised-cosine filtering, (270 Ksymbol/s results in a channl bandwidth greater than 200 KHz) the 8PSK signal with rised-cosine filtering does not fit within 200 KHz of bandwidth. eq 256

In order to achieve the desired symbol rate using 200 KHz of bandwidth, a more severe filtering approach is required. 8PSK-EDGE makes this trade-off, resulting in a conservation of bandwidth versus the increase of ISI. This also necessitates a more complex design for the receiver.

( )

22.0_ 2002

KHz 330 1.22*2700001)/(2

12)/(

≅==

≅=+⋅=⇓

+=

factorrolloffrKhzB

rssymbolsDB

rBssymbolsD

EGPRS e GSM for instead where

filtering cosine raised a for

152

In the EDGE modulation system the serial bit stream is converted into 3 bit words and mapped to the 8PSK modulation using Gray encoding. The symbol are than rotated by 3π/8 radians to ensure that the envelope of the signals does not go to zero. Next, the symbols are up-sampled and filtered using Linearized Gussian Filter (similar to, but different than the method used for GSM). In this manner, the spectrum of 8PSK signal can be restricted to 200 KHz. Linearized Gussian Filtering allows the 8PSK signal spectrum to occupy the same bandwidth as a GPRS/GSM signals. It also introduces a considerable ISI component. EDGE provides nine different coding schemes and is possible to switch a connection between different schemes. The choice of the coding scheme is dynamic and depends on the Carrier to Interference ratio (C/I). As the signal quality deteriorates, switches to a more robust coding scheme with lower throughput are done. User data Rate ( in Kbps)

MCS1 MCS2 MCS3 MCS4 MCS5 MCS6 MCS7 MCS8 MCS9

1 Timeslot 8.4 11.2 14.8 16.8 22.4 29.6 44.8 54.4 59.2 table 4

In this chapter the focus is on generating the EDGE signal, which unlike GMSK, has a time-varying envelope which exposes in turns the signal to AM-PM distortion. For simulations of EDGE I,Q signals we have been using a software program (winiq downloadable by Internet on www.rohde-schwarz.com )

153

16.2 EDGE signal description: Modulating Symbol Rate and Symbol Mapping

Coding, modulation, and filtering of EDGE signal, which is 3π/8-shifted 8-PSK, is shown in figure below. We assume rectangular-shaped input data pulses.

Figure 114

The bit source generates bits dk at a rate of fb=812.5 Kbps. Each group of 3 bits is Gray-coded into an octal-valued symbol cn=[0,1,2,3,4,5,6,7] (see table below). This symbols are produced at rate fs= fb/3=270.833 Ksps, which is identical to the GSM symbol rate.

d1, d2, d3 0,0,0 0,0,1 0,1,0 0,1,1 1,0,0 1,0,1 1,1,0 1,1,1cn 3 4 2 1 6 5 7 0

Table 1. Gray-coding of binary bit triplets into ocatal symbols

table 5

Group into 3-bit

triplets

GRAY Code

encoding

I Q generator

Bit Input source

dk=[0,1] Cn =[0.1,2,3,4,5,6,7]

Complex Filter

quadrature

QAM modulatorCn Sn S(t)

83 nj

e⋅π y(t)

Carrier fc

Rn

82

nC

je

nS

π

=8

3 nje

nS

nR

π=

i(t)

q(t)

+ Σ

-

sin(ωct) carrier

-90°

i(t)

q(t)

S(t) y(t)

modulator

Modulator IQ

154

10.70.74

5sin4

5

11-02

3sin2

3

10.7--0.74

5sin4

5

101sin

10.7-0.74

3sin4

3

1102

sin2

10.70.74

sin4

1010sin

7cos8

727

6cos8

62

6

5cos8

52

5

4cos8

42

4

3cos8

32

3

2cos8

22

2

1cos8

12

1

00cos8

02

0

=⇒−≅+

=⇒≅+

=⇒≅+

=⇒+−≅+

=⇒+≅+

=⇒+≅+

=⇒+≅+

=⇒+=+

=⋅

=

=⋅

=

=⋅

=

=⋅

=

=⋅

=

=⋅

=

=⋅

=

=⋅

=

Sj

eS

Sj

eS

Sj

eS

Sj

eS

Sj

eS

Sj

eS

Sj

eS

Sj

eS

jj

jj

jj

jj

jj

jj

jj

jj

ππ

ππ

ππ

ππ

ππ

ππ

ππ

π

π

π

π

π

π

π

π

Index n in cn, is the n-th step used to send a symbol, for example we can decide to send the same symbol for n-consecutive times. The octal-valued symbols cn are used to phase modulate a carrier, yielding an 8-PSK-EDGE waveform sequence Sn :

jQICjC nnc

Ac

An

Cj

ec

An

S +=+==8

2sin8

2cos8

2ππ

π

eq 257

where Ac is the amplitude, here set to one as an example.

The above formulation of Sn is also called I,Q, form of the base-band binary signal. Values and vector module of Sn for n=0..7 are listed below. Note that each constellation point is on unitary circle (i.e. the Sn vector module is always unitary).

eq 258

155

The baseband unfiltered 8PSK vector constellation implementation on the imaginary unitary plane is shown in figure below:

Figure 115

Figure below shows the i(t),q(t) signal time variation for an unfiltered 8PSK

modulation, when a sequence of 8 symbols, see table 1, is used as a single ordinate, binary input data stream. To each single couple i(t),q(t), corresponds one and only one point on the unitary imaginary plane, as showed above.

S0= (111)

S1= (011)

S2= (010)

S3= (000)

S4= (001)

S5= (101)

S6= (100)

S7= (110)

Real Part (I)

Imaginary Part (Q)

I=0.7

Q=0.7

( ) 00 =Sϕ

( ) 41 πϕ =S

( ) 22 πϕ =S

156

Figure 116

Figure below shows corresponding vector amplitude and phase time variation.

Note that in case of unfiltered constellation, the vector amplitude is constant. This is not true when a complex filter is used, since amplitude modulation AM is introduced .

Figure 117

157

Figure below shows the corresponding vector constellation for unfiltered implementation.

Figure 118

To study either the spectrum or the vector constellation of an unfiltered 8PSK

configuration, we can use a pseudorandom PRBS9 as a binary input sequence data. If the length of the symbols sequence chosen for winiq software program is for example greeter than 1000, than the screen output will be as reported below:

Figure 119

S0(111)

S1(011) S3(000)

S4(001)

S5(101)

S2(010)

S6(100)

158

In so doing we should keep in mind that in ideal case of unfiltered signals there is

no phase jitters, Hence each single constellation point remains easily selectable by the receiver.

Another observation is that vectors constellation does go through origin. This is the main problem since signal filtering introduces an Amplitude Modulation, and hence, also a greeter sensitivity at low C/I with respect to GMSK modulation.

Figure below reports the relative FFT magnitude for unfiltered case we have

showed so far, note the great sidelobes that are a consequence of rectangular data pulses.

Figure 120

What analyzed so far is an 8 PSK standard unfiltered constellation, in which each of 8 constellation points is uniquely identified with a particular symbol value.

16.3 Symbol Rotation In 8PSK-EDGE modulation, in order to ensure that the envelope of the signal

does not go instantaneously close to zero and hence to reduce AM phenomena due to filtering action, the 8PSK symbols are continuously rotated with 3π/8 radians per symbol before pulse shaping (i.e. each phase modulated symbol is additionally phase shifted by 3π/8 radians per symbol).

The cumulatively phase shift (CPS) sample sequence Rn is:

( )nn

nCjn

jC

jn

j

nn jbaeeeeSR nn

+===⋅=+32

883

82

83 ππππ

eq 259

If symbol Sn varies from 0 to 7 than the corresponding Rn shifted symbols are:

≅13dB

159

( )

( )

( )

( )

( )

( )

( )

( )192.00.38

835sin

835

10.710.718

30sin8

30

10.3892.08

25sin8

25

1108

20sin8

20

138.00.928

15sin8

15

171.0-0.718

10sin8

10

10.9238.0-8

5sin8

5

1010sin

7cos

211487

6cos

181286

5cos

151085

4cos

12884

3cos

9683

2cos

6482

1cos

3281

00cos

080

=⇒+≅+

=⇒−≅+

=⇒−−≅+

=⇒+≅+

=⇒−≅+

=⇒−≅+

=⇒+≅+

=⇒+=+

=+

=

=+

=

=+

=

=+

=

=+

=

=+

=

=+

=

==

Rj

eR

Rj

eR

Rj

eR

Rj

eR

Rj

eR

Rj

eR

Rj

eR

Rj

eR

jj

jj

jj

jj

jj

jj

jj

jj

ππ

ππ

ππ

ππ

ππ

ππ

ππ

π

π

π

π

π

π

π

π

eq 260

Table below reports Rn and the corresponding angle α. Note that the difference: Rn- Rn-1 between two consecutive vector constellation point is always 112.5°

Rn angle Rn- Rn-1

R0 0 0R1 112.5 112.5R2 225 112.5R3 337.5 112.5R4 450 112.5R5 562.5 112.5R6 675 112.5R7 787.5 112.5

table 6

The corresponding unfiltered vector implementation on the imaginary unitary plane is shown below

160

Figure 121

Figure 122

Real Part (I)

Rn=0--111

Rn=1--011

Rn=2-010

Rn=3--000

Rn=4--001

Rn=5--101

Rn=6--100

Rn=7--110

Imaginary Part (Q)

Real Part (I)

R0=111

R1=011

R2=010

R3=000

R4=001

R5=101

R6=100

R7=110

Imaginary Part (Q)

112.5° 112.5°

112.5°

112.5° 112.5°

112.5°

112.5°

161

Transmitting at each time step n the same symbol we note that the relative position on the constellation plane is continually rotated by 67.5°

67.5°67.5°

figure 123

Note that with symbol rotation we have no longer 8 constellation point but 16.

figure 124

8PSK 8PSK + shift (CPS)

Fixed position

Shift position

162

Figure below shows i(t), q(t) signals time variation for an 3π/8 shifted 8PSK-

EDGE unfiltered modulating input signals when a single ordinate sequence of 8 symbols (0,1,2,3,4,5,6,7), is used as a binary input data stream.

Figure 125

Figure below shows the corresponding vector amplitude and phase time variation. Note that in case of unfiltered constellation, the vector amplitude is constant. This is not true when a complex filter is used.

Figure 126

163

To study either the spectrum or the vector constellation of an unfiltered π/8 shifted 8PSK-EDGE configuration, we can use a pseudorandom PRBS9 as a binary input sequence data.

If the length of the symbols sequence chosen for winiq software program is for example greeter than 1000, than the screen output will be as reported below:

Figure 127

As shown in figure above, the EDGE signal has a non-zero minimum magnitude when 3π/8 radiant rotation is used, therefore this phase rotation assures that the signal envelope never goes to zero. Without the CPS operation, the vectors constellation goes through origin.

Figure below reports the relative FFT magnitude, note that in case of unfiltered signals there is no spectrum difference with a standard 8PSK. This is because the points of constellation in 8PSK EDGE are only shifted by 3π/8 radiants.

π/8 shifted 8PSK-EDGE 8PSK

Zero

164

Figure 128

For a non shifted standard 8PSK, we can try to explain passages of the vector through the origin on the constellation imaginary plane. For example, if we consider a pseudorandom PRBS9 as a binary input data sequence and if we focusing our attention around a few random points transition, it is possible to observe that some times, vectors i(t),q(t) have a contemporary passage through origin.

Figure 129

In order to switch from one constellation point to another opposite constellation point, the corresponding i(t),q(t) resulting vector, has to pass through the origin. By doing so, the amplitude of i(t),q(t) resulting vector goes to zero. Defining the envelope dynamic range as the ratio between maximum and minimum envelope values, we note that without CPS, the signal envelope goes to zero and the envelope dynamic range becomes infinite.

≅13dB

165

16.4 (8PSK EDGE) modulation AM distortion For unfiltered rectangular-shaped data pulses, as we have seen, the amplitude of envelope of the 8PSK and 8PSK_EDGE signal is constant. That is, there is no AM on the signal even for a 180° phase shift (for a non shifted 3π/8 radiant vector constellation, since the data switches value say, from +1 to -1,instantaneously). We might wonder why baseband filtering signal is then necessary? The rectangular-shaped data produces a (sin(x) / x)^2 type power spectrum for a signal that has large undesirable spectral sidelobes. The absolute bandwidth of rectangular multilevel pulses is therefore infinity ! Because we never have an infinite bandwidth, we must use a filtering system in order to reduce the bandwidth, but when this pulses are filtered improperly they will spread in time, the pulse for each symbol may be smeared into adjacent time slot causing intersymbol interference (ISI). One way to obtain an ISI reduction is by using a raised cosine rolloff filter. At the same time, when a phase transition occurs, the filtering operation causes an amplitude modulation (AM), the larger the phase transition, the greater the amplitude modulation. These AM effects can be reduced with 3π/8 offset 8PSK in which we never have the deep π phase shift transition. Summarizing, we encounter two types of problems:

16.4.1 First problem (ISI):

Although ISI is eliminated using raised-cosine filter, a 200 KHz channel spacing results in a symbol rate less than 200 KSymbol/s. The 8PSK EDGE signal with rised-cosine filtering does not fit within 200 KHz of bandwidth, for example using a raised cosine rolloff filter we could have:

factorrolloffrKhzB

rBssymbolsD

_ 2002

.12)/(

==

+=

EGPRS and GSM for

filter rolloff cosine raised a for eq 261

If r=0.22, the symbols would be only D=200000/1.22= 163.934,4 (Simbols/s) which is less than 270.833 Ksymbol/s required for EDGE

In order to achieve the desired symbol rate, 270.833 Ks/s and constrain the bandwidth of the output signal so that it remains below (or nearly below) the transmit mask defined for EDGE (200 KHz GSM channel bandwidth) a more severe filtering approach is required

166

The transmit filter used is a Linearized Gussian Filter (similar to, but different than the method used for GSM). In this manner, the spectrum of 8PSK signal can be restricted to 200 KHz and the symbols filtering allow the 8PSK signal spectrum to occupy the same bandwidth as a GPRS/GSM signals. In the same time it also introduces a considerable ISI component. Because of the severe ISI introduced by the linearized Gussian Filters, the receiver includes an equalizer, and the Signal-to-Noise Ratio on Co-channel interference of about C/N ≥ 18 dB is quite high if compared to a GMSK modulation where C/N ≥ 9 dB

Since the EDGE transmit mask is nearly identical to the GSM transmit mask, the EDGE signal succeeds in tripling the data throughput within the standard GSM channel.

16.4.2 Second problem (AM):

Filtering produces Amplitude Modulation AM on the resulting EDGE signal because the filtered data waveform cannot change instantaneously from one peak to another, especially if 180° phase transitions occur as in a simple 8PSK modulation. Although filtering solves the problem of poor spectral sidelobes, it creates another: AM on the EDGE signal. AM has several consequences:

• 8PSK EDGE constellations points are no longer on the unitary circle so amplitude jitter experienced by resulting vector occur in a more C/I sensitivity at receiver user side;

• Because (for the reason above), when EDGE modulation is used, the mean output transmitted power is lower compared to GSM modulation (about 2 dB). Than if BCCH (broadcast control channel) carrier is used for EDGE the cell-reselection-algorithm could experiences a fault.

• Due to AM, low-efficiency, linear (class A or class B) amplifiers, instead of high-efficiency non linear (class C) amplifiers, are required for the 8PSK signal without distortion. In portable communication applications, these amplifiers increase the battery capacity requirements.

These AM effects can be reduced with 3π/8 offset 8PSK in which we never have the π phase shift transition.

167

16.5 Used Gaussian EDGE Filter

The complex sequence Rn is next passed through a low-pass filter to produce the filtered complex signal.

∑ −⋅=n

n nTtpRtx )()( eq 262

where T=1/fs is the symbol interval, and p(t) is the filter pulse shape. We can rewrite in the alternative form:

∑∑ −⋅+−⋅=+=n

nn

n nTtpbjnTtpatjQtItx )()()()()( eq 263

with real symbol an, bn and real pulse p(t), for two real filtered signals I(t) and O(t). Each real filtered signals can be visualized as the output of a linear time-invariant (LTI) filter driven by a stream of impulse scaled by an, (or bn), yielding a linear superposition of

scaled time-shifted pulses p(t).

The exact filter pulses shape p(t) defined for EDGE signal is non-zero over the interval -5(T/2) ≤ t ≤ 5(T/2),

⎟⎠⎞

⎜⎝⎛ += Ttctp

25)( 0 eq 264

where c0 is the principal pulse in the Laurent decomposition of the 0.3-GMSK modulation,

( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

≤≤+= ∏=

elsewhere

TtiTtftc i

0

50)(3

00 eq 265

168

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛

≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛

= ∫

∫−

elsewhere

TtTduug

Ttduug

tfTt

t

0

84)(cos

40)(sin

)(4

0

0

π

π

eq 266

The integrand g(t) is defined in terms of the frequency pulse in gGMSK(t) by

( )Ttgtg GMSK 221)( −= eq 267

( )

τπ

ππ

detQ

TTtq

TTtQ

Ttg

t

r

GMSK

∫∞

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −⋅−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −⋅=

2

2

21)(

)2ln(2/33.02

)2ln(2/53.02

21

eq 268

The combination of expressions above is defined as the filter pulse shape.

Following, the base band signal is:

( )∑ +−⋅=n

n TnTtcRty 2')'( 0 eq 269

The time reference t’=0 is the starting of the active part of burst as shown figure this is also the start of the symbol period of symbol 0 (containing the first tail bit)

169

Output phase

The useful part

The active part

111 111 111....... .......111 111 1119 tail bits9 tail bits

1/2 symbol 1/2 symbol

figure above show Relation between active part of burst and tail bits. For the normal burst the useful part lasts for 147 modulating symbols. Before the first bit of the bursts

as defined in GSM 05.02 [3] enters the modulator, the state of the modulator is undefined. Also, after the last bit of the burst, the state of the modulator is undefined. The tail bits (see GSM 05.02) define the start and the stop of the active and the useful

part of the burst as illustrated in figure above. Nothing is specified about the actual phase of the modulator output signal outside the useful part of the burst.

170

16.6 Effect due to Gaussian EDGE filtering in 3π/8 shifted 8PSK

Figure below shows i(t),q(t) baseband time variation and the output modulated r(t) signal, when a single ordinate sequence of 8 symbols, is used as a binary input data

stream.

Figure 130

Figure above shows the corresponding amplitude vector and phase time variation

for Gaussian EDGE filtered signals. Note that in case of an unfiltered constellation, the vector’s amplitude is constant, this is not true when a complex filter is used having Amplitude Modulation AM.

Using a PBR9 input data stream, we can evaluate The spectrum:

Amplitude modulation

(AM)

I

Q

r

Ф

I

Q

r

Ф

Gaussian EDGE filter No Gaussian edge filter

171

figure 131

Focusing out attention at the transmitter output, we can se haw the baseband signal filtering produces ISI. Obviously the same filter applied at the receiver side reduces these phenomena.

Amplitude of modulated

signal

Phase of modulated

signal

Bandwidth of modulated

signal

Gaussian EDGE filter No Gaussian EDGE filter

172

figure 132

Gaussian EDGE filter cause:Amplitude and phase Jitter

Increasing ISI

Gaussian EDGE filter No Gaussian EDGE filter

173

16.7 Modulation Let us indicate the frequency carrier fc. Follows that the QAM modulation is:

( ) ( )( )tjttyetyts cctj c ωωω sincos)(Re)(Re)( +⋅=⋅= eq 270

because y(t)=i(t)+jq(t) than the real part of output modulated signal becomes:

( ) ( ) ( )( )( )

ttqttits

ttqttjqttjittitjttjqtietyts

cc

cccc

cctj c

ωω

ωωωωωωω

sin)(cos)()(

sin)(cos)(sin)(cos)(Re sincos)()(Re)(Re)(

−=

=−++==+⋅+=⋅=

eq 271

using the filtering equation already seen before for a single burst:

( )∑ +−⋅=n

n TnTtcRty 2')'( 0 eq 272

the modulated RF carrier during the useful part of the burst became:

( )[ ]0'2)'(Re2)'( ϕπ +⋅= tfjs cetyTEts eq 273

where Es is the energy per modulating symbol, fc is the center frequency and φ0 is a random phase and is constant during one burst (see TS 05.04).

16.8 Conclusion The 3π/8 shifted 8PSK-EDGE signals succeeds in it’s primary goals of tripling the on-air data rate while maintaining nearly the same spectral occupancy as the original GSM signal. The 3π/8 shift assures that the signal envelope never falls below a certain level. Nonetheless, this signal has a significant envelope variation, which exposes the signal to AM-PM distortion impairments that don’t degrade the signal quality of GMSK.

174

17 EFFICIENT MODULATION METHODS STUDY AT NASA/JPL

17.1 PHASE SHIFT KEYED (8-PSK) MODULATION 8-PSK modulation is not currently used by CCSDS Space Agencies. Inserting a filter in the modulator further degrades system performance because non-orthogonality increases crosstalk between phase states.

17.2 PSK Modulation Bit-Error-Rate (BER) Losses are evident in Figure below showing the Bit-Error-Rate performance for 8-PSK modulation. Relative to ideal BPSK/NRZ modulation, even an ideal (lossless) 8-PSK system imposes heavy performance penalties. Ideal 8-PSK requires an EB / N0 of 9.5 - 10 dB to attain a BER = 1 x 10-3 When a Butterworth BTS = 3 filter is added, the required EB / N0 rises to 11.5 dB. Compared to the EB / N0 of about 8 dB, needed for a filtered non-ideal QPSK system at the same BER, it is clear that 8-PSK is not a useful modulation method in power limited applications. Losses using a Square Root Raised Cosine (r = 1) filter were so great that the plot is not even included in this report. Excessive losses result from the non-orthogonal relationship between phase states. This simulation shows that inherent 8-PSK modulation losses are unlikely to be acceptable in most applications, even without filtering.

175

figure 133: 8-PSK modulation Bit Error Rate

17.3 8-PSK Modulation Spectra Not with standing the system losses, spectrum advantages of simultaneously transmitting three data bits is clearly evident in Figure below 8-PSK modulation with a Butterworth filter having a BTS = 2. A BTS =2 was used for consistency with studies of the other modulation types. Figure below also demonstrates that filtering will be needed. The unfiltered spectrum (top) is very similar to that for unfiltered QPSK. 8-PSK provides a 1.8 dB improvement in data rate over QPSK and the spectral improvement appears to be on the same order.

176

figure 134

177

17.4 PSK Modulation Power Containment Power Containment curves, Figure below show the occupied bandwidth to be about 2.4 RB when using a Butterworth BTS = 2 filter. This bandwidth will increase with a BTS = 3 filter which is required to avoid the additional 1 dB loss.

figure 135

17.5 PSK Modulation Study Conclusions Results of this study show 8-PSK modulation to be of little value for most space telemetry data transmissions. While 8-PSK does provide a marginally narrower spectrum, system losses make the modulation type unsuitable for most Category A missions. 8-PSK modulation may be attractive in strong signal applications where system losses are of little importance.

178

18 MINIMUM-SHIFT KEYNG (MSK) AND GMSK MSK has the advantages of producing a constant-amplitude signal and, consequently, can be amplified with Class C amplifiers without distortion. As we will see, MSK is equivalent to OQPSK with sinusoidal pulse shaping [for hi(t)].

Definition: Minimum-shift keying (MSK) is a continuous-phase FSK with a minimum modulation index (h=0.5) that will produce orthogonal signalling.

First, let us shown that h=0.5 is the minimum index allowed for orthogonal continuous-phase FSK.

For the binary 1 to be transmitted over the bit interval 0<t<Tb, the FSK signal would be :

)cos()( 111 θω += tAts c eq 274

and for 0 to be transmitted the FSK signal would be

)cos()( 222 θω += tAts c eq 275

where θ1=θ2 for the continuous phase condition at the switching time t=0. For orthogonal signaling, we require the integral of the product of the two signals over the bit period 2Tb to be zero. Thus, we require:

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ]

[ ]

[ ] [ ] 0)()(sin21

)(1)()(sin

21

)(1

)()(cos21)(

)(1

)()(cos21)(

)(1

)()(cos)()(

21 )()(cos

)()(

21

)()(cos)()(

21)()(cos

)()(

21

)()(cos21)()(cos

21

i.e. 0)cos()cos()()(

2

02121

21

22

02121

21

2

2

0212121

2

21

2

2

0212121

21

2

2

02121

21

2122

02121

21

212

2

02121

21

212121

21

212

2

021212121

2

2

0

2

02211

221

=⎥⎦⎤

⎢⎣⎡ −+−

−+⎥⎦

⎤⎢⎣⎡ +++

+=

=⎥⎦⎤

⎢⎣⎡ −+−−

−+

+⎥⎦⎤

⎢⎣⎡ ++++

+=

=⎥⎦

⎤⎢⎣

⎡−+−

−−

+⎥⎦

⎤⎢⎣

⎡+++

++

=

=⎥⎦

⎤⎢⎣

⎡−+−

−−

++++++

=

=⎥⎦⎤

⎢⎣⎡ −+−++++=

=++=

∫

∫

∫∫

∫

∫

∫ ∫

Tb

c

Tb

c

Tb

cc

Tb

c

Tb

c

Tb

c

Tb

c

Tb

c

Tb Tb

c

tAtA

dttAA

dttA

dttAdttA

dtttA

dtttA

dtttAdttsts

θθωωωω

θθωωωω

θθωωωωωω

θθωωωωωω

θθωωωωωωθθωω

ωωωω

θθωωωωωω

θθωωωωωω

θθωωθθωω

θωθω

179

[ ] [ ]

[ ] [ ] 0)(sin21)(2)(sin

21

)(1

)(sin21)(2)(sin

21

)(1)()(

21212121

2

21212121

22

021

=⎥⎦⎤

⎢⎣⎡ −−−+−

−+

+⎥⎦⎤

⎢⎣⎡ +−+++

+=∫

θθθθωωωω

θθθθωωωω

bc

bc

Tb

TA

TAtdtsts eq 276

The first term is negligible, because (ω1+ω2) is large, so the requirements reduces to

[ ] [ ] 0)(sin)()(2sin2

)()(21

21212122

021 =⎥

⎦

⎤⎢⎣

⎡−

−−−+−=∫ ωω

θθθθωω bcTb TAtdtsts eq 277

for the continuous phase case the phase must be θ1=θ2; and the equation above is satisfied (i.e. equal to zero) for a minimum value h=0.5 .

in fact [ ] [ ] 0)(sin)()(2sin 212121 =−−−+− θθθθωω bT is verified when

⇒=− )(2 21 πωω bT [ ] [ ] 212121 when 0)(sin)(sin θθθθθθπ ==−−−+

but 21

21)( 2 )( 22 )(2 212121 hTffTffT bbb =⇒=−⇒=−⋅⇒=− πππωω

Therefore calling the modulation index h as:

( ) ( ) ( )πθ

πω

πωω ∆

=⋅∆

=−

=∆=−=2

2222 21

21bb

bbTTTfTffh then when

( ) ( )

22

212

41f

212 2/1 2121

πϑπϑ

=∆⇒=∆

=−=∆⇒=−=b

b TffTffh

eq 278

i.e. the peak to peak phase variation is 90° while the peak to peak frequency shift is ¼ Tb.

Now we will demonstrate that MSK signal is also a form OQPSK with sinusoidal pulse shaping. First, consider the FSK signal over the signalling interval (0,Tb). Then using

equation:

])(Re[)( tj cetgts ω= eq 279

the complex envelope is:

180

∫==

∆t

dmfj

ctj

c eAeAtg 0

)(2)()(

λλπθ eq 280

where m(t)=±1, and 0<t<Tb , therefore using h=0.5 and ∆f=1/4Tb the complex envelope became:

Tbtj

c

tTb

j

ctfj

ctj

c eAeAeAeAtg 24122)()(

πππθ ±±⋅∆± ==== eq 281

where the ± signs denote the possible data during the (0,Tb) interval. Thus,

bbb

cTbtj

ctj

c TttjytxTtj

TtAeAeAtg <<±=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛±⎟⎟

⎠

⎞⎜⎜⎝

⎛===

±0 ),()(

2sin

2cos)( 2)( πππ

θ eq 282

and the MSK signal is therefore

tTtt

Ttttyttxetgts c

bc

bcc

tj c ωπωπωωω sin2

sincos2

cossin)(cos)(])(Re[)( ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=−== m eq 283

This type of FSK modulation with modulation index h=0.5 is a form of orthogonal FSK with minimum bandwidth required, called MSK (Minimum Shift Keying). This type of

modulation is also a particular case of CPM (Continuous Phase Modulation).

In CPM techniques the phase is slowly varied in each symbol interval starting from the phase value assumed in the preceding symbol interval. Consequently are also called

modulation technique with memory.

Since a frequency shift produces an advancing or a retarding in phase, frequency shifts can be detected by sampling phase at each symbol period. Phase shifts of (2N + 1)π/2 radians are easily detected with an I/Q demodulator. At even numbered symbols, the polarity of the I channel conveys the transmitted data, while at odd numbered symbols the polarity of the Q channel conveys the data. This orthogonally between I and Q simplifies detection algorithms and hence reduces power consumption in a mobile receiver. The minimum frequency shift which yields orthogonality of I and Q is that which results in a phase shift of ± π/2 radians per symbol (90 degrees per symbol). FSK The binary data of m(t) alternatively modulate the x(t) and y(t) components, and the pulse shape for the x(t) and y(t) symbols( which are 2Tb wide instead of Tb) is a sinusoid. Thus MSK is equivalent to OQPSK with sinusoidal pulse shaping.

181

FSK and MSK produce constant envelope carrier signals, which have no amplitude variations. This is a desirable characteristic for improving the power efficiency of transmitters. Amplitude variations can exercise nonlinearities in an amplifier’s amplitude-transfer function, generating spectral regrowth, a component of adjacent channel power. Therefore, more efficient amplifiers (which tend to be less linear) can be used with constant-envelope signals, reducing power consumption. MSK has a narrower spectrum than wider deviation forms of FSK. The width of the spectrum is also influenced by the waveforms causing the frequency shift. If those waveforms have fast transitions or a high slew rate, then the spectrum of the transmitter will be broad. In practice, these Waveforms are filtered with a Gaussian filter, resulting in a narrow spectrum. In addition, the Gaussian filter has no time-domain overshoot, which would broaden the spectrum by increasing the peak deviation. MSK with a Gaussian filter is termed GMSK (Gaussian MSK). An MSK modulator scheme is reported in figure below:

figure 136

A block scheme of MSK modulator is reported below

In OQPSK the rectangular pulse modulates the carrier directly:

MSK

Delay

Sinusoid filter

Cosinusoid filter

X

X

+cos(fc)

sen(fc)

I

Q

Baseband processing

dI

dQ

modulator f=fc

OQPSK

Delay

X

X

+cos(fc)

sen(fc)

I

Q

dI

dQ

modulator f=fc

Baseband processing

182

figure 137: parallel generation of MSK

In order to make the difference between FSK and MSK we are going to show an example for CPM using a Trellis phase transitions representation where phase transitions permitted are only of 90°. Particularly a phase shift of +90 degrees represents a data bit equal to “1”, while –90 degrees represents a “-1(or 0 in a binary form)”. Because CPM can be represented also as an OQPSK modulation and, an MSK as a pre-filtered OQPSK, by now on will call the two modulation types using the quadrature meaning title. Supposing an input NRZ data format, then one possible Trellis phase representation could be the following:

Serial-to-parallel

converter (2bit)

-90° phase shift

Oscillator f0=∆f=1/4R

Carrier oscillator

fc

-90° phase shift

X

X X

X

+

Sync input

y data

x data

y(t)

x(t)

s(t)

MSK signal

m(t)

input

cos(πt/2Tb)

sin(πt/2Tb)

Accos(ωct)

Acsin(ωct)

Baseband

183

figure 138: Trellis tree phase representation for a CPM

Although there are only π/2 phase shift, these discontinuity on the phase transitions requiring a large bandwidth. In order to limit this bandwidth a filtering method is needed. Anyway this filtering method introduce ISI.

figure 139

NRZ data

t

t

t

-1

+1

MSK phase

0 2Tb 4Tb 6Tb 8Tb 10Tb 12Tb

2Tb 4Tb 6Tb 8Tb 10Tb 12Tb

-π/2

-π

-3/2π-2π

π/2π

3/2π

2π

+1 Tx

Sharp slope variation

(discontinuity)

-1 Tx

OQPSK no filtered OQPSK filtered

Phase transition rounded

It does not a shift phase

184

The main difference between MSK and OQPSK is the sinusoidal pre-filtering method used into MSK:

)2sin()()2cos()()( tfctdtfctdts QI ππ += eq 284

)2sin(2

sin)()2cos(2

cos)()( tfcTttdtfc

Tttdts

bQ

bI ππππ

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡= eq 285

As an example we report a plot of the MSK signalling and that for the corresponding RF output:

figure 140

I(t)

Q(t)

I MSK Modulated

signal

Q MSK Modulated

signal

Σ

MSK output modulated signal

I baseband

Q baseband

Q modulated signal

I modulated signal

OQPSK

MSK

185

The peculiarity of MSK is not in the main lobe which is 50% more greater than a OQPSK. Instead what is worth of consideration is that MSK has a fast side lobes decaying. Spectrum of MSK decays with the four power of the frequency, while in the case of QPSK and OQPSK the law is modestly of quadratic type. Therefore we can say that MSK is characterized by a better interference control on adjacent channels.

figure 141

18.1 GMSK Another form of MSK is Gaussian-filtered MSK (GMSK). For GMSK, the data (rectangular-shaped pulses) are filtered by a filter having a Gaussian-shaped frequency response characteristic before the data are frequency modulated onto the carrier. The transfer function of the Gaussian low pass filter is

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−

=22ln2

)( Bf

efH eq 286

Where B is the 3-dB bandwidth of the filter. This filter reduces the spectral sidelobes on the transmitted MSK signal. The product of B with bit period Tb:

186

bTB ⋅=α eq 287

is called normalized bandwidth; BTb=0.3 used in GSM services, gives a good compromise between relatively low sidelobes and tolerable acceptable ISI. For example in GSM mobile we have the figure on the right which represent the time filter response, the smaller time response, the greater frequency bandwidth is required. Ideally MSK needed for a great bandwidth, since a time rectangular pulse shaping is equivalent to an infinite frequency bandwidth.

For BTb=0.3 GMSK has lower spectral sidelobes than those for MSK, QPSK, or OQPSK (with rectangular-shaped data pulses). In addition GMSK has a constant envelope, since it is a form of FM. Consequently GMSK can be amplified without distortion by high efficiency Class C amplifiers. In figure below is reported an example of power efficiency

figure 142

187

In figure below are reported an example for amplitude and phase vector representation and the vector and constellation diagrams. We can see how the amplitude remains constant while the phase change continuously. In this way the vector representation is a circle which has been made of all possible phase point transitions, for the modulated signal s(t), from a symbol to another symbol. In the other figures are reported the spectrum and bandwidth and the ISI representation of modulated signal as a function of normalized bandwidth bTB ⋅=α . Its possible to observe that the lower bTB ⋅=α the greater the ISI.

figure 143

No phase discontinuity

Constant amplitude

Non sono salti di fase ma è la modalità di rappresentazione tra -180° e 180°del programma

Non ci sono salti di fase pertanto i punti infiniti della simulazione giacciono sulla circonferenza unitaria (cioè ampiezza costante )

188

figure 144

BTb=0.5

BTb=0.2

BTb=0.5

BTb=0.2

189

figure 145

BTb=0.1

190

18.1.1 How to implement GMSK modulator An algorithm for a GMSK modulator is described below

1. Create NRZ (-1,1) from the binary (0,1) input sequence 2. Create N samples per symbols 3. Integrate the NRZ sequence, in such a way to have an FSK (CPM) 4. Convolute with Gaussian function filter 5. Compute the corresponding I and Q component. At this stage we have the

quadrature components of the baseband GMSK equivalent signal 6. multiplying the I and Q component by the corresponding cos(ωct), -sin(ωct)

carriers 7. add the two resulting flow

figure 146

Baseband process Quadrature modulation

GaussiaLP

∫NRZ code

BTb is Integrator

b(t) )(tθ

)(cos tθ

)(sin tθ

s(t) ∑

L

090

0,1.. -1,+1

I(t)

Q(t)

NRZ code transformer

RF modulated

output

191

Mathematically:

1 , )( ±=⎟⎟⎠

⎞⎜⎜⎝

⎛ −−Π=∑

∞

∞−i

b

bi m

TiTt

mtmτ

2( )( )

t

Gh t eπαπ

α−

=

NRZ Data:

Gaussian Filter:

After Filter:

2( ) 2

2

( ) ( ) ( ) ( ) ( )

b

b

bb

G G

Tt iT

Ti t iT

b t h t m t h m t d

m e dπ τα

τ τ τ

π τα

∞ −− +

− −−∞

= ∗ = −

=

∫

∑ ∫

After integrator:

2

( ) 2

- 2

2

2

-2 2 -1 1 0 0 1 1 2 2

1( ) ( ) 4

1 4

1 ( ) 4

( )

m ( ) m ( ) m ( ) m ( ) m ( )

bb

bb

bb

bb

t

b

Tt z iT

Ti z iTb

Tt z iT

Ti Gz iTb

i i

t b z dzT

m e d dzT

m h d dzT

m t

t t t t t

π τα

θ

π τα

τ τ

θ

θ θ θ θ θ

−∞

∞ −− +

∞ − −−∞

∞ − +

−∞ − −−∞

∞

−∞

− −

=

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

=

=

= + + + + +

∫

∑∫ ∫

∑ ∫ ∫

∑LL +LL

192

18.1.2 How to implement GMSK demodulator A coherent detector can be used to demodulate a GMSK signal:

figure 147 : MODULATOR

figure 148 : DEMODULATOR

However to avoid the receiver to have its own reference (i.e. the necessity of coherent demodulation), a differential encoding can be used to create NRZ signal at the input of demodulator. Table below show as an example how to convert binary signal (0,1) to differential NRZ symbol. It can be easily computed using the following operations:

193

table 7

figure 149

Modulating binary signal at instant x[n-1], x[n]

X

Y NRZ

194

19 EFFICIENT MODULATION METHODS STUDY AT NASA/JPL

19.1 MSK and GMSK Modulation Bit-Error-Rate (BER) Figure below contains Bit-Error-Rate curves for MSK and GMSK. GMSK studies included two separate filters with BTS = 0.5 (equivalent to BTB = 0.25) and BTS = 1 (equivalent to BTB = 0.5). MSK is unfiltered and GMSK includes a Gaussian filter with two bandwidths (BTS = 0.5). For simplicity, these, along with an ideal BPSK/NRZ reference curve, are placed on a single BER graph. Note that the EB / N0 required for a 1 x 10-3 BER is 7.3 - 8.2 dB which compares favourably with BPSK/NRZ, even with a Gaussian filter bandwidth BTS = 0.5. Losses can be expected to increase when a non-ideal modulator and receiver are employed; however, Figure was generated using the ESA power amplifier operating in full saturation.

figure 150

19.2 MSK and GMSK Modulation Spectra Most MSK and GMSK applications have been applied in Personal Communication Systems (PCSs). Spacecraft telemetry transmission systems have avoided GMSK because of demodulation and synchronization difficulties. Often termed frequency modulation, MSK and GMSK were included because of their inherently narrow spectral bandwidths. Unlike the other modulation types, MSK is unfiltered and sidelobes are reduced by avoiding phase change discontinuities. Figure below shows spectra for unfiltered, ideal BPSK/NRZ (reference), MSK, and GMSK using the two filter bandwidths. No discrete components are present in MSK or GMSK spectra despite baseband filtering.

195

figure 151

figure 152

196

Figure above shows MSK modulation to be significantly more bandwidth-efficient than the unfiltered BPSK/NRZ reference, reaching a level 60 dB below the peak sideband amplitude at ± 8 RB. Its lack of discrete spectral components makes it attractive for space telemetry applications. However, from Figure it is apparent that MSK modulation is of little interest when compared to GMSK. GMSK modulation is significantly more bandwidth-efficient than any other method considered previously except of FQPSK. For example, it is 2 to 6 times more bandwidth-efficient than filtered QPSK modulation, depending upon the specific filter bandwidths selected. When coupled with its BER performance, GMSK should be seriously considered for high and very high data rate missions.

19.3 MSK / GMSK Modulation Power Containment Figure below, concern Power Containment, show that GMSK has a high bandwidth efficiency. Occupied bandwidth is difficult to read, because of its small value, but it appears to be less than 1.2 RB for both filter bandwidths. This represents a 16-times improvement over the unfiltered [reference] BPSK/NRZ modulation and a 5-fold efficiency increase over filtered BPSK/NRZ.

figure 153

197

19.4 MSK / GMSK Modulation Study Conclusions Clearly, space agencies interested in RF spectrum efficiency should seriously consider GMSK modulation. This is particularly true for high and very high data rate missions. Unlike the phase modulation types described above, GMSK requires new modulator, demodulator, and symbol synchronizer designs. In that respect, this recommendation departs from one of the Efficient Modulation Methods Study guidelines: that only simple modifications to existing Earth station equipment are permitted for any recommended modulation method. However, GMSK’s bandwidth efficiency is too great to be ignored and a departure from the guideline is warranted.

198

20 HISTORY OF SPECTRUM EFFICIENT MODULATION IN TELEMETRY APPLICATIONS

Since 1992 the amount of available aeronautical telemetry spectrum has been decreasing. Efforts by the national and international communities have reallocated 25 MHz of telemetry spectrum and efforts are under way to reallocate even more. This is completely opposite of the requirements of the telemetry community as can be seen

from figure below. The chart clearly shows the data requirements of the test community are increasing at an almost exponential rate. This increase in requirements is being driven by the increasing complexity of the test articles coupled with compressed test schedules. The increasing requirements for data coupled with the reduction in the

amount of available spectrum is causing the major test centers to have serious concerns if there will be sufficient spectrum available to support all their programs.

These trends of increasing data requirements and decreasing availability of spectrum have focused attention on the efficient use of the remaining spectrum. It is the

responsibility of all programs to insure that their use of spectrum is as efficient as possible and that spectrum is not needlessly being wasted. Some of the major test

ranges have already been forced to schedule use of the telemetry spectrum by the hour to accommodate all of the required users. Under these conditions, it has become

imperative that all programs be as efficient and flexible as possible to ensure that they are able to secure the use of a sufficiently large portion of the spectrum to complete

their required testing.

10

100

1,000

10,000

100,000

1970 1975 1980 1985 1990 1995 2000 2005 2010Year

Bit

Rat

e (K

bps)

199

To be spectrally efficient means that data rates must be kept at the absolute minimum bandwidth, so proper transmitter pre-modulation filtering techniques must be

used, and efficient modulation schemes must be employed. Currently the most common modulation scheme is PCM/FM; this is a very robust modulation technique that

is fairly efficient. To be efficient these systems must have proper pre-modulation filtering and be properly aligned. As can be seen from figure below, improper selection

of pre-modulation filtering, and over deviation of the transmitter can increase the required channel bandwidth by over 50%. Figure shows ideal spectrum with proper

modulation filtering of .7 x Fb (Bit frequency), improper pre-modulation filtering at 1.4 x Fb & 2 x Fb, and excessive transmitter deviation of 1.2 x Fb (.35 x Fb is ideal).

figure 154

• Ideal spectrum with proper modulation filtering of 0.7 x Fb (Bit frequency),

• improper pre-modulation filtering at 2 x Fb,

• excessive transmitter deviation of 1.2 x Fb (0.35 x Fb is ideal).

Improvement in pre-modulation filter selection and tighter controls on alignment of systems will provide a great improvement in spectrum efficiency. New modulation

techniques such as GMSK and FQPSK have the potential to double spectrum utilization. As can be seen from figure below, these new modulation techniques are much more efficient than PCM/FM. Once the equipment can be developed, we can

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

1450.5 1452.5 1454.5 1456.5 1458.5 1460.5

Over deviation 1.2 x Fb

IdealPCM/FM

ImproperPre-modulation Filter

2 x Fb

200

expect to see vast improvements in spectrum efficiency. In the meantime efforts should be concentrated on insuring that currently deployed telemetry systems have been

properly constructed and aligned.

figure 155. New Modulation Techniques vs PCM/FM.

201

21 CORRELATED DETECTION

Let us consider a linear system (filter) through which passes a signal when a white noise is present. The aim of filter is to extract the signal with the maximum S/N ratio at the output. We want to find the linear filter transfer function H(ω) that maximize the S/N ratio at the output. Calling f(t) the signal applied to filter and ε, the noise spectral power

density which is present within the filter bandwidth.

figure 156

• Considering first the signal f(t):

The Fourier transform of the input signal f(t) is

∫+∞

∞−

−= dtetfF tjωω )()( eq 288

The filter output signal voltage is therefore:

)()()( ωωω HFG = eq 289

and

[ ] ωωωπ

ωω ω deHFGFFtg tj∫+∞

∞−

− == )()(21)()()( 1 eq 290

Its absolute value at time τ is

[ ] ωωωπ

ωωτ ω deHFGFFg tj∫+∞

∞−

− == )()(21)()()( 1 eq 291

Linear filter H(ω) f(t) g(t)

ε

202

let us consider now the noise

The filter input noise power spectral density is

2ε

=NoisePSD (W/Hz) eq 292

Where the spectrum is extended from positive to negative frequencies, then the filter output noise power spectral density will be

2)(2

)( ωεω HGN = eq 293

Where the filter Power spectrum is: 2)(ωH

The filter output noise power is therefore

∫+∞

∞−

= ωωεπ

dHPN2)(

221

eq 294

We want find H(ω) in order to maximize the output signal-to-noise ratio S/N:

⎟⎟⎠

⎞⎜⎜⎝

⎛

NPg )(max

2 τ where powersignal=)(2 τg

Ones the input signal is defined, it’s also defined its energy which is a constant given by

∫ ∫+∞

∞−

+∞

∞−

== ωωπ

dFdttfE 22 )(21)( eq 295

The signal to noise ratio, we want to maximize, can be divided by the energy E without altering the maximum S/N ratio determination. Therefore

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⋅=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

∫ ∫

∫

∫ ∫

∫∞+

∞−

∞+

∞−

∞+

∞−∞+

∞−

∞+

∞−

∞+

∞−

ωωεωωπ

ωωω

ωωεπ

ωωπ

ωωωπτ

ωω

22

2

22

2

2

)(2

)(21

)()(max

)(22

1)(21

)()(21

max)(maxHdF

deHF

HdF

deHF

PEg

tjtj

N

eq 296

203

In order to maximize the preceding ratio we can use the Schwarz theorem for complex function integral:

∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

⋅≤ ωωωωωωω dYdXdYX 222

)()()()( eq 297

The “=” sign is applied only when

)()( ωω ∗= kYX eq 298

so we have

∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

⋅= ωωωωωωω dYdkYdYkY 22*

2

* )()()()( eq 299

Where Y*(.) is the conjugate complex value of Y(.)

Therefore calling:

tjeFYHX

ωωω

ωω

)()()()(

=

= eq 300

We can write

tj

tj

ekFHeiekFkYHX

ω

ω

ωω

ωωωω−

−

=

===

)()(..)()()()(

*

**

eq 301

Then substituting on the S/N ratio we can obtain the maximum value given by:

πε

ωωεππ

ωωπ

ωωεπ

ωωπ

ωωωωπ

ωωεπ

ωωπ

ωωωπτ

ω

4

1

)(22

121

)(21

max

)(22

1)(21

)()(21

max)(

221)(

21

)()(21

max)(max

22

22

22

22

22

2

2

=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⋅

⋅=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

∫

∫

∫ ∫

∫ ∫

∫ ∫

∫

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−∞+

∞−

∞+

∞−

∞+

∞−

dFk

dFk

HdF

dHdF

HdF

deHF

PEg

tj

N

204

Therefore this maximum S/N output correspond to a filter response given by

kEdFkg == ∫+∞

∞−

ωωπ

2max )(

2 eq 302

This is the optimum filter function to reveal the signal when white noise is present. If we get k=1 then by Parseval theorem we can rewrite:

EdttfdFgT

=== ∫∫+∞

∞−

)()(21 22

max ωωπ

eq 303

Where the second integral it has been extended to the duration T of the input signal.

Therefore, the optimum correlator receiver must to perform an integration, over the signal period T, of the product between the input signal and the replica of the input

signal locally available. The term correlator follows by the correlation existing between input signal and the locally available reference signal.

The reference signal (replica of input signal) can be obtained by PLL circuits. These circuits have a bandwidth which can be reduced to a small bandwidth around the

carrier, thus the carrier can be extracted and can be used as a reference signal. Also PLL circuits use the correlation.

From a general point of view we can say that the correlation detection is a way to use as better as possible the known characteristics of the input signal (frequency. Shape, ..).

The equation for the correlation detection is:

dttstvtgT R∫= )()()( eq 304

Where

v(t)=s(t)+n(t) is the signal corrupted by noise

sR(t) is the reference replica signal obtained by use of PLL circuits

T= signal period

A blocks scheme of the correlator receiver is shown in figure below

205

figure 157

The Modulation process must be conserve some power on the carrier (other to that on the side band) in order to allow carrier extraction by the PLL circuits.

PLL

x ∫T

dt(..)

v(t)=s(t)+n(t)

sR(t)

g(t)

correlator

206

P - Power T – Time F - Frequency

T

P

T

F

P

T

FDMA

TDMA FDMA

CDMA FDD

F

22 INTRODUCTION TO CDMA

22.1 Multiple Access A cell in a cellular radio network could be seen as a multi-user communication system, in which a large amount of users share a common physical resource to transmit and receive information. The resource is the frequency band in the radio spectrum. There are several different radio access techniques in which multiple users could send the information through the common channel to the receiver:

figure 158

In FDMA and TDMA the common channel is partitioned into orthogonal single user subchannls. A problem arises if the data from the users accesing the network is bursty in nature. A single user who has reserved a channel may transmit data irregularly so that silent periods are even longer than transmission periods. For example, a speech signal may contain long pauses. In such cases TDMA or FDMA tends to be inefficient because a certain percentage of frequency or of the time slots allocated to the user

caries no information. An inefficiently designed multiple access system limits the number of simultaneous users of the common communication channel.

207

One way of overcoming this problem is to allow more than one users to share the channel or sub-channel by the use of spread spectrum signals. In this method each user is assigned a unique code sequence or signature sequence that allow the user’s signal

to be spread on the common channel.

By designing these code sequence with relatively little cross correlation, the crosstalk inherent in the demodulation signals received from multiple transmitters is minimized. This multiple access method is called CODE DIVISION MULTIPLE ACCESS (CDMA)

22.2 Spread spectrum modulation The general concept of spread spectrum modulation is presented in figure below:

figure 159: spread spectrum system concept

Formally the operation of both transmitter and receiver can be partitioned into two steps. At the transmitter site:

• the first step is modulation in which the narrowband signal Sn, which occupies frequency band Wi, is formed. In the modulation process, bit sequences of length n are mapped to 2n different narrowband symbols constituting the narrowband signal Sn.

• in the second step the spreading is carried out, in which the narrowband signal Sn is spread in a large frequency band Wc. The spread signal is denoted by Sw, and the spreading functions expressed as ε(). An example for spreading is reported in figure below:

Sn Sn ε() ε()=ε-1()

Sn

Trasmitter

Channel

Receiver

n(t) i(t)

208

figure 160: spreading process

where:

RATE. CHIP

TR

RATE; BIT T

R

cc

bb

1

1

=

=

Tc<<Tb and Tb=N⋅Tc, eq 305

N i.e. the Rc/Rb, is defined as spreading factor, in the example above the spreading factor is SF=8.

c

b

b

c

TT

RRSF == eq 306

Data 1

-1

Spreading/expansion code

1

-1

Expanded signal = Data * Expansion code

1

-1

Chip period Tc

Bit period Tb

209

A general CDMA modulating process can be represented as shown in figure below:

figure 161

At the receiver side the first step is dispreading, which can be formally represented by the inversion function ε-1()=ε(). In dispreading, the wideband signal Sw, is converted back to a narrowband signal Sn which can then be demodulated using digital demodulation schemes. The primary reason for going to the process of spreading and dispreading is to enable the CDMA multiple access method, but due to the signal spreading and resulting enlarged bandwidth, spread spectrum signals have many other interesting properties that differ from those of narrowband signals. The most important are discussed in the following section.

Source of information Coding

Antenna

Spreading sequence O.L.

Modulator

RF signal

B

210

22.3 Tolerance to Narrowband Interference A spread spectrum system is tolerant to narrowband interference, as shown in figure below:

figure 162: dispreading process in the presence of interference

Assume that a signal Sw is received in the presence of a narrowband interference signal in. The despreading process can be presented as follows:

( ) ( )[ ] [ ] wnnnnw iSiSiS +=+=+ −−− 111 εεεε eq 307

The dispreading operation converts the input signal into a sum of the useful narrowband signal Sn and an interfering wideband signal iw.

Sw iW

in

Sn P[W/Hz]

f [Hz] f [Hz] f0 fi f0 fi

Wc Wc

Despreading

Wi

iwr

P[W/Hz]

211

After the dispreading operation a narrowband filtering (operation F()) is applied, with the bandpass filter of bandwidth Bn equal to the bandwidth Wi of Sn , this result in:

wrnwnwn iSiFSiSF +=+=+ )()( eq 308

Only a small proportion of the interfering signal energy passes the filter and remains as residual interference, because the bandwidth Wc of iw is much larger than Wi. The ratio between the transmitted modulation bandwidth and the information signal bandwidth is called processing gain, Gp:

i

cp W

WG = eq 309

To prevent any filter-or modulation-specific properties, from this point: Wc=chip rate Wì=bit rate In WCDMA system the value of Wc is 3.84 Mcps which, owing to spectral sidel lobes, results in 5 MHz carrier raster. Its important to note that processing gain process is composed of the spreading part and coding part.

22.4 Direct Sequence Spread Spectrum System There are a number of techniques for spreading the information-bearing signal by use of the code signals:

• Direct Sequence • Frequency Hopping • Time Hopping

The most common technique used in cellular radio networks is DS (Direct Sequence Spread Spectrum). In this system the signal spreading is achieved by modulating the data-modulated signal a second time by a wideband spreading signal. The wideband spreading signal has to be approximated closely to a random signal with uniform distribution of the symbols. Typical representatives of such signal in digital form are pseudonoise (PN) sequence over a finite alphabet.

212

Since the WCDMA system has to maximize system capacity during the spreading, the operation is done in two phases:

• In the first the user signal is spread by the channelisation code. This is called the Orthogonal Variable Spreading Factor (OVSF) Code, its construction is based on the Hadmard Matrix. The code has the property that two different codes from the family are perfectly orthogonal if in phase. Thus, its use guarantees maximum capacity, measured by the number of active users. The channelization operation, transforms every data symbol into a number of chips, thus increasing the bandwidth of the signal. The number of chips per data symbol is called the Spreading Factor (SF).

• In the second, all the spread users’signals are scrambled by the cell specific scrambling sequence, which has the statistical properties of random sequence. The second operation is the scrambling operation, where a scrambling code is applied to the spread signal.

figure 163

22.4.1 Channelization operation

In 3GPP the OVSF codes used for different symbol rates are uniquely described as Cch,SF,k, where SF is the Spreading Factor of the code and k is the code number (0 ≤ k ≤ SF - 1). Each level of the code tree defines the channelization codes of length SF, where SF is the spreading factor of the codes. The channelization codes have orthogonal properties and are used for separating the information transmitted from a single source, i.e.

• different connections within one cell in the downlink (thus reducing the own interference),

• dedicated physical data channels from one UE in the uplink. In the downlink the OVSF codes are a limited resources and need to be managed by the radio network controller, whereas in the uplink such a problem does not exist. The OVSF codes are effective only when the channels are perfectly synchronized at chip level (the loss in crosscorrelation, e.g. due to multipath, is compensated for by the additional scrambling operation).

Data Bit rate Chip rate Chip rate

Channelisation code

Scrambling code

213

22.4.1.1 Channelization codes (OVSF codes) generation The channelization codes are OVSF (Orthogonal Variable Spreading Factor) codes that preserve the orthogonality between a user’s different physical channels. The OVSF codes can be defined using the code tree of figure below:

figure 164

Each level in the code tree defines channelization codes of length SF, corresponding to a spreading factor SF. The channelization codes are Welsh codes. Their generation method, which is based on the Hadamard matrix is shown in figure below:

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡=

...111111111111

111111

111C

1,2,1,2,

1,2,1,2,

0,2,0,2,

0,2,0,2,

3,3,

2,3,

1,3,

0,3,

0,1,

0,1,

0,1,

0,1,

1,2,

0,2,

ch,1,0

chch

chch

chch

chch

ch

ch

ch

ch

ch

ch

ch

ch

ch

ch

CCCCCC

CC

CCCC

CC

CC

CC

figure 165

It is easy to verify that all the codes with the same SF are orthogonal, i.e. they fulfil the orthogonality condition specified by equation:

Cch, 1, 0 = (1)

Cch, 2, 0 = (1, 1)

Cch, 2, 1 = (1, -1)

Cch, 4, 1 = (1, 1, -1, -1)

Cch, 4, 0 = (1, 1, 1, 1)

Cch, 4, 3 = (1, -1, -1, 1)

Cch, 4, 2 = (1, -1, 1, -1)

Cch, 8, 0 = (1, 1, 1, 1, 1, 1, 1, 1)

Cch, 8, 1 = (1, 1, 1, 1, -1, -1, -1, -1)

Cch, 8, 2 = (1, 1, -1, -1, 1, 1, -1, -1)

Cch, 8, 3 = (1, 1, -1, -1, -1, -1, 1, 1)

Cch, 8, 4 = (1, -1, 1, -1, 1, -1, 1, -1)

Cch, 8, 5 = (1, -1, 1, -1, -1, 1, -1, 1)

Cch, 8, 6 = (1, -1, -1, 1, 1, -1, -1, 1)

Cch, 8, 7 = (1, -1, -1, 1, -1, 1, 1, -1)

SF = 8SF = 4SF = 2SF = 1

( Cch, 2, 0 , Cch, 2, 0 )

( Cch, 2, 0 , - Cch, 2, 0 )

( Cch, 2, 1 , - Cch, 2, 1 )

( Cch, 2, 1 , Cch, 2, 1 )

214

01

0,2,1 =⋅∑

−

=

SF

kkk SS eq 310

This orthogonality, however, is not guaranteed between a generic code and the codes belonging to its subtree. As a consequence, the selection of one code, blocks the corresponding subtree.

22.4.2 Scrambling operation The second operation is the scrambling operation, where a scrambling code is applied on top of the spread signal, so it does not change the signal bandwidth but only makes the signals from different sources separable from each other. With the scrambling, it would not matter if the actual spreading were performed with identical codes for several transmitters. With the scrambling operation the real (I) part imaginary (Q) parts of the spread signal are further multiplied by a complex-valued scrambling code. The scrambling codes are used to

• separate different cells in the downlink, • separate different terminals in the uplink.

The scrambling codes have good correlation properties (interference averaging) and are always used on top of spreading codes, thus not affecting the transmission bandwidth.

22.5 Orthogonal sequences reminder

Let us consider two finit energy signals ( )ts1 and ( )ts2 defined in the interval [a, b]. The AUTOCORRELATION ( )τiiR of the finite energy signal ( )tsi (i = 1, 2) is defined as

( ) ( ) ( ) dttstsR i

b

aiii

∗∫ += ττ ˆ eq 311

The CROSS CORRELATIONS ( )τ21R and ( )τ12R between the finite energy signals ( )ts1 and ( )ts2 are defined as

( ) ( ) ( ) dttstsRb

a

∗∫ += 2121 ˆ ττ eq 312

and

( ) ( ) ( ) dttstsRb

a

∗∫ += 1212 ˆ ττ eq 313

215

Definition:

The finite energy signals ( )ts1 and ( )ts2 are ORTHOGONAL ⇔ ( ) 0021 =R ⇔ ( ) 0012 =R .

In other words, the finite energy signals ( )ts1 and ( )ts2 are ORTHOGONAL ⇔

( ) ( ) 021 =∗∫ dttstsb

a

⇔ ( ) ( ) 012 =∗∫ dttstsb

a

eq 314

Let us now suppose that ( )ts1 and ( )ts2 are two channelisation sequences with the same SF, as shown in figure below. We will show that

the sequences ( )ts1 and ( )ts2 are ORTHOGONAL ⇔ 01

0,2,1 =⋅∑

−

=

SF

kkk SS

Note:

The term ∑−

=

⋅1

0,2,1

SF

kkk SS is also called the cross-product between the two

sequences. If we indicate

[ ][ ]1,22,21,22

1,12,11,11

,,ˆ

,,ˆ

−

−

=

=

SF

SF

SSSS

SSSS

L

L eq 315

the cross-product between the two sequences can also be indicated as

TSS 21 ⋅ eq 316

Proof:

The sequences ( )ts1 and ( )ts2 are ORTHOGONAL ⇔ ( ) ( ) 021 =∗∫ dttstsbT

⇔

⇔ 01

0,2,1 =⋅⋅∑

−

=

SF

kckk TSS ⇔ 0

1

0,2,1 =⋅∑

−

=

SF

kkk SS eq 317

216

figure 166

22.6 Modulation and Tolerance to Wideband Interference Figure below depict the basic operations of spreading and dispreading for DS-CDMA system. User data is here assumed to be a BPSK-modulated bit sequence of rate R, the user data bit assuming the value of ±1. The spreading operation, in this example, is the multiplication of each user data bit with a sequence of 8 code bits, called chips. The resulting spread is at rate of 8xR, and has the same random (pseudo-noise like) appearance as the spreading code. During despreading we multiply the spread user data/chip sequence, bit by bit duration, with the very same 8 code chips as we used during the spreading of this bits. As shown the original user bit sequence has been recovered perfectly, provided we have also perfect synchronization between the spread user signal and the despreading code. The correlation receiver integrates the resulting products (data x code) for each user bit.

Tc

Tb (One data symbol, SF chips)

Tc 2Tc 0 (SF-1)Tc

( )ts1

( )ts2

S1, 0 S1, 1 S1, 2 S1, SF-1

S2, 0 S2, 1 S2, 2 S2, SF-1

217

figure 167

Figure below show the effect of despreading operation when applied to the CDMA signal of another user whose signal is assumed to have been spread with a different spreading code. The result of multiplying the interfering signal with the own code and integrating the resulting products leads to interfering signal values lingering around 0. The basic idea is that the receiver works as a correlation receiver, which means that it correlates a known (reference) code with an incoming signal that is composed of several different CDMA signal (from different users or channels), with general interference (from other RF systems), and with a noise (of thermal nature). The output from the receiver is in the form of the autocorrelation function of the wanted signal.

Spreding signal=Data x code 11-11

Spreding code 11-11

Data 11-11

Correlation result

88

-8

Spreading

DeSpreading

Spreding code 11-11

Data=spreading code x spreading signal

11-11

No interference scenario

218

figure 168

As can be seen, the amplitude of the own signal (correlation result in graphics) increases on average by a factor of 8 relative to that of the user of the other interfering system, i.e. the correlation detection has raised the desired user signal by the spreading factor, here 8, from the interference presenting the CDMA system. This effect is termed “processing gain” and is a fundamental aspect of all CDMA systems, and in general of all spread spectrum system. Processing gain is what gives CDMA systems the robustness against-interference that is necessary in order to reuse the available 5 MHz carrier frequencies over geographically close distances. Let’s take an example with real WCDMA parameters. Speech service with a bit rate of 12.2 kbps has a processing gain of:

dBLogRRLog

rateBitrateChipLogPg

b

c 25102.121084.31010

__10 3

6

=⎥⎦

⎤⎢⎣

⎡⋅⋅

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡= eq 318

After despreading, the signal power needs to be typically a few decibel above the interference and noise power. The required power density over the interference power density after despreading is designated as Eb/N0, where Eb is the energy, or power density, per user bit and N0 is the interference and noise power density. For speech service the ratio Eb/N0 is typically in the order of 5.0 dB, and the required wideband signal to interference ratio C/I is therefore 5.0 dB minus the processing gain i.e.

dBdBdBdBIC

wdeband 20255)( −=−= eq 319

Spreding code 11-11

Other interfering signal 11-11

Data after multiplication 11-11

Correlation result 88

-88

Interference example

219

In other words, the signal power can be 20 dB under the interference or thermal noise power, and the WCDMA receiver can still detect the signal. Due to spreading and despreading, C/I can be lower in WCDMA than, for example, in GSM. A good quality speech connection in GSM requires C/I=9-10 dB. Since the wideband signal can be around the thermal noise level, its detection is difficult without knowledge of the spreading sequence. For this reason spread spectrum systems were originated in military applications where the wideband nature of the signal allowed have to be hidden below omnipresent thermal noise. It should be observed that the more is the bit rate Rb, the lower is Pg(dB), therefore higher bit-rate service require an increasing C/I.

Rb (Kbit/sec) Pg Pg(dB)

12.2 314.7541 25.0 64 60 17.8 384 10 10.0 2000 1.92 2.8

table 8

It should be also observed that the processing gain comes at the price of an increased transmission bandwidth (by the amount of the processing gain).

22.7 Uplink Modulation In the uplink direction there are basically two additional terminal-oriented criteria that need to be taken into account in the definition of the modulation and spreading methods. The uplink modulation should be designed so that the terminal amplifier efficiency is maximized and/or the audible interference from the terminal is minimized. Discontinuosus uplink transmission can cause audible interference to audio equipment that is very close to the terminal; this is a completely separate issue from the interference in the air interface. With GSM operation we are familiar with the occasional audible interference with audio equipment that is not properly protected. The interference from GSM has a frequency of 217 Hz, which is determined, by the GSM frame frequency. With WCDMA system, the same issues arise when discontinuous uplink transmission is used, for example with a speech service. During the silent period no information bits need to be transmitted, only the information for link maintenance purposes, such as power control with a 1.5 KHz command rate. With such a rate the transmission of the pilot and the power control symbols with time multiplexing in the uplink direction would cause audible interference in the middle of the telephony voice

220

frequency band. Therefore in WCDMA uplink the two dedicated physical channel are not time multiplexed but I-Q/code multiplexing is used. The continuous transmission achieved with an I-Q/code multiplexed control channel is shown in figure below.

figure 169 : parallel transmission of DPDCH and DPCCH when data is present/absent (DTX)

Now, as the pilot and the power control signaling are maintained on a separate continuous channel, non pulsed transmission occurs. The only pulse occurs when the data channel DPDCH is switched on and off, but such switching has happens quite seldom. For the best possible power amplifier efficiency, the terminal transmission should have as low-peak-to-average ratio (PAR) as possible to allow terminal to operate with a minimal amplifier back-off requirement. With the I-Q/code multiplexing, also called dual-channel QPSK modulation, the power level of the DPDCH and DPCCH are typically different, especially as data rates increase, and would lead in extreme case to a BPSK-type transmission when transmitting the branches independently. This has been avoided by using a complex-valued scrambling operation after the spreading with channelisation codes. The signal constellation of the I-Q/code multiplexing before complex scrambling is shown in figure below, the same constellation is obtained after descrambling in the receiver for data detection.

G=0.5

I

Q

I

Q G=1

figure 170: Constellation of I-Q/code multiplexing before complex scrambling, G denotes the relative gain factor between DPCCH and DPDCH

Data (DPDCH)

Dedicate Phisical Data Channel

User Data (DPDCH)

Dedicate Phisical Data Channel

Phisical Layer Control Information (DPCCH)

Dedicate Phisical Control Channel

DTX Period

221

Figure below illustrates the principle of the uplink spreading of DPCCH and DPDCHs. The binary DPCCH and DPDCHs to be spread are represented by real-valued sequences, i.e. the binary value "0" is mapped to the real value +1, while the binary value "1" is mapped to the real value –1. The DPCCH is spread to the chip rate by the channelization code cc, while the n-th DPDCH called DPDCHn is spread to the chip rate by the channelization code cd,n. One DPCCH and up to six parallel DPDCHs can be transmitted simultaneously, i.e. 1 ≤ n ≤ 6.

figure 171: Spreading for uplink DPCCH and DPDCHs

After channelization, the real-valued spread signals are weighted by gain factors, βc for DPCCH and βd for all DPDCHs. At every instant in time, at least one of the values βc and βd has the amplitude 1.0. The β-values are quantized into 4 bit words. The quantization steps are given in table below.

I

Σ

j

cd,1 βd

Sdpch,n

I+jQ

DPDCH1

Q

cd,3 βd

DPDCH3

cd,5 βd

DPDCH5

cd,2 βd

DPDCH2

cd,4 βd

DPDCH4

cd,6 βd

DPDCH6

cc βc

DPCCH

Σ

S

222

Signalling values for βc and βd

Quantized amplitude ratios βc and βd

15 1.0 14 14/15 13 13/15 12 12/15 11 11/15 10 10/15 9 9/15 8 8/15 7 7/15 6 6/15 5 5/15 4 4/15 3 3/15 2 2/15 1 1/15 0 Switch off

table 9: The quantization of the gain parameters

After the weighting, the stream of real-valued chips on the I- and Q-branches are then summed and treated as a complex-valued stream of chips. This complex-valued signal is then scrambled by the complex-valued scrambling code Sdpch,n. The scrambling code is applied aligned with the radio frames, i.e. the first scrambling chip corresponds to the beginning of a radio frame. In the uplink, the complex-valued chip sequence generated by the spreading process ( spreding phase + scrambling phase) is QPSK modulated as shown in Figure below:

figure 172: Uplink Modulation

S Im{S}

Re{S}

cos(ωt)

Complex-valued chip sequence from spreading operations

-sin(ωt)

Split real & imag

Pulse- shaping

Pulse- shaping

223

22.7.1 One UL parallel channel Let us start by considering only one of two parallel channels,

figure 173: Only the DPCCH channel is active

Supposing the gain factor parameter equal to one: βc=1; then the signal can assume only the values ±1 along the Q axes i.e. ±1j, in other word we are obtaining a BPSK constellation.

I

Q

figure 174: BPSK modulation constellation

jβc Cc Chanelisation

Code

SC=Scrambing Code

I+jQ

DPCCH

DPDCH OFF

Q

I

S

224

In order to minimize the zero crossing by the vector constellation, we are using the scrambling codes which are formed in such a way that the rotations between consecutive chips within one symbol period are limited to ±90°. The full 180° rotation can happen only between consecutive symbols. This method further reduces the peak-to-peak-average ratio of the transmitted signal from the normal QPSK transmission. The scrambling codes, during the chip time, can assume only the values:

(1+j); (1-j); (-1+j); (-1-j); which once multiplicand by the expanded code signal yield:

( )( )( )( ) jjj

jjjjjj

jjj

−=−−⋅−−=+−⋅

+=−⋅+−=+⋅

1111

1111

( )( )( )( ) jjj

jjjjjj

jjj

−=−−⋅−+=+−⋅−−−=−⋅−

−=+⋅−

111111

11

eq 320

As a consequence the vector constellation is no longer a BPSK constellation but a balanced QPSK constellation.

figure 175: balanced QPSK constellation

I

Q

1-j -1-j

1+j 1-j

1+j

1-j

-1+j

-1-j

225

22.7.2 Two UL parallel channel As a second analysis case let us consider the transmission of two parallel channels, DPDCH (dedicated physical data channel) and DPCCH (dedicated physical control channel):

figure 176

Consider the peak-to-average power ratio (PAR=crest factor) defined as the ratio between the peak and average power. The amplifier efficiency is maximized when the peak-to-average power ratio (PAR=crest factor) is minimized. This mean that the mobile phone amplifier can operate with a minimum back-off, in other word with a minimum output power reduction with respect to the maximum output power allowed. The peak-to-average power ratio (PAR=crest factor) is affected and degraded by power unbalance of I and Q channels, notably in phase and quadrature component of the signal. To minimize peak-to-average power ratio (PAR=crest factor) is required a balanced QPSK constellation. The power ratio between Q and I component is called G=βc/βd The multicode transmission of more channels increases the peak-to-average power ratio (PAR=crest factor) and, as in the case of one channel, the Scrambling code application at the end, transform the Unbalanceed-QPSK into a Balanced-QPSK. In such a way the efficiency of the power amplifier remains constant irrespective of the power difference G=βc/βd between DPCCH and DPDCH. This can be explained by figure below, which show the signal constellation for I-Q/code multiplexed control channel with complex spreading. There are three case study in which we use/not use scrambling operation:

• one with G=βc/βd=0 dB i.e. βd=βc=1 • one with G=βc/βd=-6 dB i.e. βd=1 βc=1/2 • one with G=βc/βd=-10 dB

jβc cc

cd βd

I+jQ

DPCCH

DPDCH

Q

ISc

S

226

22.7.2.1 No scrambling code operation When no scrambling code are used, we obtain an unbalance QPSK which is poorly efficient from the amplifier point of view; this is shown in simulation below :

In this condition especially for G= -6 dB the peak-to-average power ratio (PAR=crest factor) is very poor

22.7.2.2 Scrambling code operation When scrambling code are used, then even when G=βc/βd≠0 dB we obtain always a balance rotated QPSK. Thus, the signal envelope variations with complex spreading, are very similar to a balanced QPSK transmission for all value of G=βc/βd:

• G=βc/βd=0 dB i.e. βd=βc=1 Multiplication of signal and scrambling code yield:

No scrambling code are used Unbalance QPSK G=βc/βd=-6 dB

No scrambling code are used G=βc/βd=0 dB

I

Q

1 2

I

Q

227

( ) ( )( ) ( )( ) ( )( ) ( ) jjj

jjjj

jjj

211211

211211

−=−−⋅+−=+−⋅+

=−⋅+=+⋅+

( ) ( )( ) ( )( ) ( )( ) ( ) 211

211211

211