BPSK probability of error

8/10/2019 BPSK probability of error

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ECE414 Wireless Communications, University of Waterloo, Winter 2012 1

Bandpass Transmission Techniques

for Wireless Communication

Chapter 3


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Outline

Introduction to Digital Communications

Signal (Vector) Space Representations

Digital Modulation Schemes (M-ASK, M-PSK, M-FSK)

Performance Measures for Modulation Schemes

- Bandwidth (spectral) efficiency

- Power efficiency

- Temporal characteristics (e.g., dynamic power range, peak/average ratio)

Power Spectral Density of Digital Modulation Schemes

Error Rate Performance of Digital Modulation Schemes

Comparison of Digital Modulation Schemes in terms of Spectral

Efficiency and Power Efficiency

Temporally Efficient Digital Modulation Schemes


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Original

message signal(analog)

Recovered

message

signal (analog)

A/D SourceEncoder

Modulator

Channel

De-

modulator

Channel

Decoder

Source

DecoderD/A

Block Diagram for a Digital Communication System

ChannelEncoder

Analog-to-Digital (A/D) Conversion:Analog (i.e., continuous-time

continuous-amplitude) message signal is converted into a discrete-time discrete-

amplitude digital signals by time-sampling and amplitude-quantization. The

resulting signals are then mapped to binary sequences.


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Source Encoding:Removes the redundant information embedded in the

message signal, therefore represents the message with as few binary digits as

possible, i.e., data compression

Channel Encoding: Introduces redundancy in a controlledmanner which can

be used at the receiver to overcome the effects of noise, interference and fading.

Provides noise immunityto transmitted information.

Source coding and channel coding will notbe studied in this course Modulation:Converts (maps) codewords to high-frequency analog waveforms.

A certain parameter of the carrier signal (i.e., modulated signal) is varied in

accordance with message signal (i.e. modulating signal) e.g. amplitude shift keying

(ASK), phase shift keying (PSK), frequency shift keying (FSK)

Receiver Blocks: Perform the inverse of the transmitter operations in order to

recover the original analog message (continuous-time continuous-amplitude) signal.

In a practical digital communication receiver, there are also additional sub-blocks

such as channel estimation, synchronization (frame/frequency/phase),

authentications, crypto, multiplexing, etc.

Block Diagram for a Digital Communication System (contd)


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Why is Modulation Required?

To achieve easy radiation:Dimensions of the transmit/receive antennas are

limited by the corresponding wavelength. The frequency conversion allows the

use of practical antenna lengths.

To accommodate for simultaneous transmission of several basebandsignals:Simultaneous transmission of different baseband signals which are

possibly overlapping can be facilitated by assigning slightly different frequency

carriers for each one.

Modulatio shifts the baseband signal to a higher frequency band, centered at the so-

calledcarrier frequency

.

Large bandwidths require high carrier frequencies: Practical requirements

in front-end filter design dictates the bandwidth-to-frequency carrier ratio (i.e.,

fractional bandwidth) be kept within a certain range.

1.001.0 cf

B

cf

B: Fractional bandwidth


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To (possibly) expand the bandwidth of the transmitted signal for

better transmission quality: When the bandwidth increases, the required

SNR (for fixed noise level, corresponding signal power) to achieve a specific

transmission rate decreases

Why is Modulation Required? (cont

d)

SNRBC 1log2 12 B

C

SNR

SNRBC 1log2

Channel capacity Bandwidth Signal-to-noise ratio

High-rate transmission requires larger bandwidths (therefore, higher

carrier frequencies): According to Shannon Theorem, channel capacity isdefined as the maximum achievable information rate that can be transmitted

over the channel. For the additive white Gaussian noise (AWGN) channel,


7/85ECE414 Wireless Communications, University of Waterloo, Winter 2012 7

Signal-Space Representations

Consider a modulation format where the transmitted signal waveforms belong to

the modulation set .

Each of the waveform can be represented as a point (vector) in anN-dimensional

signal space (sometimes called as vector space) defined by the orthonormal basis

functions

Mmm

ts1

Nnn

t1 MN

s

Tt0

jidttt jT

is

*

0

tstsdtttss nN

nnmn

T

mnm

s

1

,*

0,

Nmmmmm sssts ,2,1, ,...,,s

The Gram-Schmidt procedure (See Appendix A of the textbook) provides a

systematic approach to construct the set of orthonormal functions, which span the

signal space.



21

2,

0

2m

N

nnm

T

ms sdttsEs

s

sk

t( ),sl t( ) =1

Ts

sk

t( )0

Ts

sl t( )*

dt= sk,s

l = s

ks

l= s

k,ns*

l,n

n=1

N

Mmlk ,...2,1,

d2

sk

t( ),sl t( )( )= sk t( ) sl t( ) 2

dt

0

Ts

= d2

sk,s

l( )= sk,n sl,n2

n=1

N

Energy

Correlation

Euclidean

Distance

Signal-Space Representations (cont

d)

Nmmmm sssts ,2,1, ,...,, ms

Nkkkkk sssts ,2,1, ,...,, s

Nlllll sssts ,2,1, ,...,, s



M-ary Amplitude Shift Keying (M-ASK)

tfAts cmm 2cos sTt0 MmMmAm ...2,1,12

otherwise,0

0,2cos2 scs TttfTt

Basis Function(s) (Obtained through Gram-Schmidt procedure)

Signal-Space (Vector Space) Representation (Obtained through the use ofbasis functions)

2smmm TAts s

mlpmtfj

mcmm AtseAtfAts c ,

2Re2cos

Baseband (Equivalent Lowpass) Representation

1-dimensional

22

0

2sm

T

ms TAdttsEs

m Signal Energy


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M-ASK (cont

d)

Examples of M-ASK

Signal Constellations

M=4

Bandpass Modulation Signal

Equivalent Lowpass Signal

11 10 00 01


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M-ary Phase Shift Keying (M-PSK)

mcm tfAts 2cos sTt0 MmMmm ...2,1,12

otherwise,0

0,2cos21

scs TttfTt

otherwise,0

0,2sin22

scs TttfTt

Basis Functions

Signal-Space Representation

msmsmm TATAts sin2,cos2 s

mcm jlpmtfjj

mcm AetseAetfAts

,2

Re2cos


2-dimensional

220

2s

T

mss TAdttsEEs

m Signal Energy


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tfAts

tfAts

c

c

2cos0

2cos1

2

1

Example: Binary Phase Shift Keying (BPSK)

sEsE t1

t2 1 0 1 1 0

1 0 1 1 0

A

-A

A

-A

0,2 sEs


0,1 sEs

t

t

Bandpass Modulation Signal

Equivalent Lowpass Signal


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Example: Quadrature Phase Shift Keying (QPSK)

s

s

s

s

Es

Es

Es

Es

,0232cos11

0,2cos10

,022cos01

0,2cos00

44

33

22

11

tfAts

tfAts

tfAts

tfAts

c

c

c

c

sEsE t1

t2Signal-SpaceRepresentation

sE

sE


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Quadrature Amplitude Modulation (QAM)

tfAtfAts cimcrmm 2sin2cos ,,

Am,r,Am,i: Information-bearing signal amplitudes of the quadrature carriers

sTt0Mm ,...2,1

Alternatively, QAM can be considered as a combination of ASK and PSK.

2,

2, imrmm AAA rmimm AAarctg ,,

mcmm tfAts 2cos where sTt0Mm ,...2,1

Examples of QAM

Signal Constellations


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QAM (cont

d)

tfj

imrmcimcrmm

c

ejAAtfAtfAts

2

,,,, Re2sin2cos

mjarctgmimrmlpm eAjAAts ,,,

otherwise,00,2cos21 scs TttfTt

otherwise,00,2sin22 scs TttfTt

Basis Functions


2,2 ,, simsrmmm TATAts s 2-dimensional


22 ,2 ,0

2simrm

T

ms TAAdttsEs

m

Signal Energy


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M-ary Frequency Shift Keying (M-FSK)

tffAts mcm 2cos sTt0 Mmfmfm ...2,1,

tfjlpmtfjtfj

mcmmcm AetseAetffAts

2,

22Re2cos

Cross Correlation

flkTjT

ftlkj

slk e

flkT

flkTdte

T

s

sin1

0

2,

flkT

flkT

lk

2

2sin

,


0, lkFor andTf 21 lk

Tf 21Therefore, the minimum frequency separation between adjacent signals for

orthogonality of theMsignals is


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M-FSK (cont

d)

0......0011 sEts s

Tf 21Assuming frequency separation , the signal-space representationfor theM-FSK signals are given asN-dimensional vectors, whereN=M.

otherwise,0

0,2cos2 smcsm

TttffTt

0......0022 sEts s

sMM

Ets ......000 s

.

.

.

220

2s

T

mss TAdttsEEs

mwhere


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Performance Measures for Modulation Schemes

Bandwidth (spectral) efficiency:How much bandwidth is needed for a

given data rate?

zBits/sec/Hlog2

W

TM

W

R sss

: Bandwidth efficiency

: Data rate W: Bandwidth

The bandwidth depends on the modulation scheme and pulse shaping. Power

spectral density (PSD) is typically used to determine the bandwidth of the

transmitted signal. There are various definitions for bandwidth:

Main lobe (null-to-null) bandwidth: The width of the main spectral lobe.

Fractional power-containment bandwidth: The frequency interval that

contains (1-) of the total signal power, e.g. 99.9% of the total power.

Bounded PSD bandwidth: The frequency interval where the PSD stays

above a prescribed certain threshold, e.g. sidelobes peaks 40 dB below its

maximum value

Roughly speaking, bandwidth of the modulation scheme is proportional to

the dimension number.

s

sR


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Power efficiency:How much power is needed for reliable transmission with a

specified fidelity?

The fidelity for a digital communication system is usually measured in terms of

symbol- or bit-error probability. For a given SNR, we aim to achieve a low error

probability (how low? it depends on the application).

Symbol error probability (SEP) is in general easier to evaluate. Bit error

probability (BEP) depends on the mapping of source bits onto modulation signals.

A bound on BEP is given as

Performance Measures for Modulation Schemes (cont

d)

ePeP

M

ePb

2log Two common mapping forms are natural mappingand Gray mapping.

In Gray mapping, the neighbour points differ in only one digit. It should be

noted that Gray mapping is not possible for every signal constellation.


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Temporal efficiency: How wide are the time variations of the transmitted signal?

Temporal efficiency=Peak power/Average power

The choice of amplifier depends on the temporal characteristics of the signal.

Other considerations:

Hardware/software implementation complexity & cost of implementation

Sensitivity to interference

Robustness to impairments encountered in a wireless channel

Performance Measures for Modulation Schemes (cont

d)

In most practical scenarios, these performance measures conflict with each

other. The communication system designer should be able to find the best trade-

offfor a given application under specific constraints.


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Comparison of Spectral Efficiency of Modulation Schemes

M-PSK and QAM

bits/seclog

rateData 2

T

M

Hz2

nulltonullBWT

s=Data rate

BW=

1

2log2M bits/sec/Hz[ ]

M-FSK

bits/seclograteData 2TM

Hz2

roughlyBWT

M

s=Data rate

BW=

2log2MM

bits/sec/Hz[ ]

M: Modulation order, Constellation size


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Power Spectral Density (PSD)

In practical, pulse shaping should be considered for a precise bandwidth

measurement and considered in the spectral efficiency calculations.

Power spectral density (PSD) describes the distribution of signal power in

the frequency domain. If the baseband equivalent of the transmitted signal

sequence is given as

ksk kTtpatg ka : Baseband modulation symbol

sT : Signal interval tp : Pulse shape

ffPT

f as

g 21

then the PSD ofg(t) is given as

Ra

n( )=1

2E a

ka

k+n

*

tpFfP

sfnTjn

aa enRf 2

where

See Ch.4 ofDigital Communications

by Proakis for the proof


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Example: PSD of BPSK with Rectangle Pulse Shaping

k

sk kTtpatg Aak

0,0

0,

0,

0, 2

*

2

*

n

nA

naEaE

naE

aaEnRnkk

k

nkka

22 AenRnRFf sfnTjn

aaa

p(t)

T/2 T

Autocorrelation of data sequence

Pulse shaping

t

p t( ) =t

Ts / 2

Ts

FT P f( ) =Tssinc fTs( ) e

j 2 f Ts 2( )

P f( )= Tssinc fTs( )

Baseband equivalent of BPSK sequence

Independent data symbols are assumed


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g f( )=P f( )

2

Ts a f( )=A

2Tssinc2 fTs( )

Example: PSD of BPSK with Rectangle Pulse Shaping (contd)

PSD of baseband BPSK sequence


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PSD of bandpass BPSK sequence

ccFT ffGffGfS *21 tfjcetgts 2Re

scsscs

cgcgs

TffTATffTA

fffff

2222

*

sinc4

1sinc

4

1

4

1

Example: PSD of BPSK with Rectangle Pulse Shaping (contd)

Bandpass BPSK sequence and its Fourier transform (spectral density)

Null-to-null bandwidth

SeeTutorial 1

See Ch.4 ofDigital Communications

by Proakis for the proof


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Example: PSD of QAM with Rectangle Pulse Shaping

kskk kTtpjbatg AAba kk 3,,

Baseband equivalent of QAM sequence

0,0

0,100,

0,22

n

nA

njbajbaE

njbaE

jbajbaEnR

nknkkk

kk

nknkkka

Autocorrelation of data sequence

PSD of baseband QAM sequence

fTTAfg 22 sinc10

PSD of bandpass QAM sequence

scsscss TffTATffTAf 2222 sinc

4

10sinc

4

10

Note that PSD of QAM has the

same general form as BPSK.


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Some Practical Pulse Shapes

Below are some pulse shapes commonly used in communication systems:

TtT

tAtp

0,sin

Half-Sinusoid Pulse

21sinc

21-sinc

2fTfTeATfP fTj

Full-Cosine Pulse

TtTtA

tp

0cos12

1sinc1sinc2sinc4

fTfTfTeAT

fP fTj


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Some Practical Pulse Shapes (cont

d)

Gaussian Pulse

22

22ln

2exp TtBAtp

2

2

2lnexp

2

2ln

B

fe

T

AfP fTj

where B is defined as the 3dB bandwidth of pulse

Raised Cosine Pulse

22241

cossin

Tt

Tt

Tt

Tttp

Tf

Tf

TTf

TTT

fT

fP

2

1,0

2

1

2

1,

2

1cos1

2

2

10,

10 : Roll-off factor


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Comparison of Pulse Shapes

Time-Domain

Gaussian

Half-sinusoid

Full-cosine

Square


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Comparison of Pulse Shapes (contd)

Frequency-Domain

Square

Gaussian

Half-sinusoid

Full-cosine

2/T

3/T

4/T

Square

BW=2/T

Half-sinusoid

BW=3/TFull-cosine

BW=4/T


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Comparison of Pulse Shapes (contd)

Raised Cosine

10

: Roll-off factor

TBWT

21

1/T

2/T

TBW

1


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For a given SNR (i.e. a given signal power for fixed noise power), we aim to

achieve a low error probability. To calculate error probability,first we need to

identify the receiver structure.

The receiver consists of a demodulator and a detector:

The demodulator converts the received waveform r(t) into aNdimensional

vector whereN is the dimension of the signal-space for the

given modulation type.

The detector decides which of the possibleMsignal waveforms was

transmitted based on r, whereMis the constellation size.

Optimum Receiver for AWGN

Nrrr ,..., 21r

Demodulator Detector tr r m

s tsm

tn nsr m tntstr m


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Optimum Receiver for AWGN (contd)

Correlation-type demodulator Matched-filter demodulator

For details, see ProakisDigital CommunicationsChapter 5


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We want to design a signal detector that makes a decision based on the

observation of the vector rsuch that the probability of a correct decision is

maximized. The optimal decision rule is based on the maximization of so-called

aposteriori probabilities

rsmp : The probability of choosing smm=1,2Mbased on the observation of r

This decision criterion is called theMaximum A Posteriori Probability (MAP) rule.

mMm

mmMm

mm

Mm

mMm

p

pp

p

pp

p

sr

ssr

r

ssr

rs

...2,1

...2,1

...2,1

...2,1

max

max

max

max

Bayes Theorem

rp : Common for all

Mp m 1s , i.e. Equally probablemessages

The conditional pdf is called the likelihood functionand the decision

criterion based on the maximization of over theM signals is called the

maximum likelihood(ML) criterion.

mp sr mp sr


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nsr m

0

2

02

2

2

2exp

1

2exp

2

1

02 N

n

N

nnf k

N

kk

For an AWGN channel, the components of the noise vector nare zero-mean Gaussian random variables with varianceN0/2

N

kkmkN

N

kkmk

N

kkmk

N

kkmkm

srNN

srNN

srfsrpp

1

2,

02

0

1

2,

00

1,

1,

1exp

1

1exp1

sr

The received signal will have a Gaussian conditional distribution


Nk ...2,1


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21

2, minminmax m

m

N

kkmk

mm

msrp srsr

The ML rule is then given as

The ML receiver decides in favor of the signal which is closest in Euclidean

distance to the received vector, r.


2222minmin mm

mm

mssrrsr

Expanding the decision rule,

where is the signal energy. Neglecting terms which do not affect

the decision and under the assumption that constant-energy modulation set

(e.g. PSK) is used

2mmE s

mm

mm

srsr maxmin2

Distancemetrics

Correlationmetrics


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Example: Error Probability for BPSK

11 2cos0 tfAtsb c 22 2cos1 tfAtsb c

01where

2where

tfTEtfAts cc 2cos22cos1

tfTEtfAts cc 2cos22cos2

tsts 12

Unlike otherM-PSK forM>2, we can represent this special form of BPSKsignal as 1-dimensional signal. The basis function is given as

otherwise,0

0,2cos21

TttfTt c

tr dt

T

0

.

Euclidean

Distance

Decoder

t1

Therefore, the optimal receiver has the following form of

r

i.e. antipodal signaling


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Example: Error Probability for BPSK (contd)

Assumes1(t) is sent. Under the assumption of AWGN, the received signal

twtstr 1

The output of demodulator

nEdtttwtsdtttrTT

0

110

1

where

2,0~ 00

1 NNdtttwnTdef

Assumes2(t) is sent. The output of demodulator is now

nEdtttrT

0

1


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Example: Error Probability for BPSK

Eb

Eb

1

0

0

0

1r

Here we have two possible alternatives, therefore we can use a zero threshold

detectoras an optimal detector.

0011100,11,0

bPbbPbPbbP

bbPbbPeP

2/110 bPbP

0110 bbPbbP Due to symmetryEqually probable messages 10 bbPeP

Under the assumption that b=1 is sent zEr

drbrfbrPbbP

0

11010

LetP(e) denote the error probability

EE

Decision regions

0 b1

b

Bayes Theorem


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40

Example: Error Probability for BPSK (contd)

0

2NEQ

drbrfbbP

0

110

drNEr

N

0 0

2

0

exp1

20N

Ery

dyy

NE

02

2

2exp

2

1

where Q-function is defined as dyexQx

y

22

2

1

E E


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Example: Error Probability for QPSK

22cos11

232cos10

22cos01

2cos00

4

3

2

1

tfAts

tfAts

tfAts

tfAts

c

c

c

c

tr

dtT

0

. Detector

t1

dt

T

0 .

t2

otherwise,0

0,2cos21

scs TttfTt

otherwise,0

0,2sin22

scs TttfTt

2

4,3,2,1min

msrs

m


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Example: Error Probability for QPSK (contd)

Under the assumption of AWGN (which exhibits symmetry), rotating and

moving the signal constellation does not change the error probability. Therefore,

we can rotate/move our signal constellation in such a way that the resulting

constellation allows easy mathematical derivation.

Here, we move our constellation as the targetsignal is located on the origin.

If there is no symmetry in the signal constellation, this should be repeated for

each signal.Decision

regions

First, we calculate P(c), i.e. the probability of making a correct decision. Then,

probability of error is simply found as P(e)=1-P(c).


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Assume that the signal located at the origin has been transmitted. If the

received signal is in the shaded area, this means we will make a correct decision.

2

0

2

0

1

1

22

2,2

N

EQ

N

EQ

dnPdnP

dndnPscP

s

s

QI

QI


sEd 2

d

2d

2d 2,0~ 0

01 NNdtttwn

Tdef

I

2,0~ 00

2 NNdtttwnTdef

Q

QnP

2,~ Nn

QQ 1


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Due to symmetry,

4321 scPscPscPscPcP

2

00

2

2

00

222

21

NEQ

NEQ

N

EQ

N

EQcPeP

bb

EE

ss

bs



45/85


Example: Error Probability for BFSK

tT

fAts c2

12cos0 1

tr

dtT

0

. Detector

t1

dtT

0

.

t2

2

4,3,2,1min

msrs

m

tT

fAts c1

2cos1 2

otherwise,00,212cos2

1scs TttTfTt

otherwise,0

0,12cos22

scs TttTfTt


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sE t1

t2

sE

Example: Error Probability for BFSK (contd)

sEd 2

0N

EQeP s

By rotation, it can be easily shown that

Now, we will study the same problem without rotation:

Assume was sent. The received signal is 0,11 sEts s QIs nnE ,r

Decision is based on mm

mm

srsr maxmin2

0

121

N

EQEnPEnnPPeP sssIQsrsrs

2,0~, 0NNnn IQ 0,0~ NNnnn IQdef

Due to symmetry,

01

N

EQePeP ss


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A Union Bound on Error Probability

2mmd sr

In most cases, probability of error can not be obtained in closed form.

Therefore, one needs to find some bounds or approximations which can work for

any signal constellation.

We have already shown that the optimal decoder for any signal constellation

over AWGN is given by the Euclidean distance decoder, i.e.

M

mmm

M

mm eP

M

PePeP11

1sss meP s : Probability of making a

decision error when smwas sent

M

lmlm

M

lmml

M

lmmlm

ll

ll

ml

P

ddP

ddPeP

1

1

1

sss

s

ss

i

i

i

i APAP

Union Bound (U-B)

lmP ss : The probability of choosing slinstead of the originally transmitted sm


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A Union Bound on Error Probability (contd)

M

l

N

dBUB

M

l

mlBU

N

M

l

mlM

l

mlmm

ml

ml

ml

mlml

e

N

dQ

dQPeP

1

4

1 0

,

21

,

1

0

2

,

02

2

2

ssss

U-B: Union Bound

M

m

M

l

mlM

mm

ml

N

dQ

MeP

MeP

1 1 0

,

1 2

11s

Assuming equal-probable message signals, the probability of error is

UB-B: Union-

Bhattacharyya Bound

2, mlmld ss where

2/2xexQ


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The U-B requires the computation of all distances dl,mamong signals in

the constellation. A looser bound can be obtained as follows

0

min

1 0

,

21

2 N

dQM

N

dQeP

M

l

mlBU

m

ml

s

0

min

21

N

dQM

P(e) is dominated by the minimum Euclidean distance of the signal

constellation.

A Union Bound on Error Probability (contd)

M

m

M

l

mlM

mm

ml

N

dQ

MeP

MeP

1 10

,

1 2

11s

Then the probability of error is found as

mlml

dd ,,

min minwhere is the minimum Euclidean distance of the constellation.

Minimum Euclidean

distancebound


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An Approximation for Error Probability

As an alternative, we can also the following approximate upper bound

0

min,

~

1 0

,

2

2 min N

dQN

N

dQeP md

M

l

mlBU

m

ml

s

Approximate upper bound

mdN min, : Number of signals at distance dminfrom sm

0

min

2min N

dQNd

M

m

mdd N

M

N

1

,minmin

1

M

mmeP

MeP

1

1s

M

mmd

N

dQN

M 1 0

min,

2

1~min


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Error Probability for M-PSK

sE

MMEMEd bs 2

2

22

min sinlog4sin4

2,2

2,1

min M

MNd

2,sinlog2

2

2,2

22

0

0

MM

MN

EQ

MN

EQ

eP

b

b

Replacing and into the formula on p.50, we obtain2mind mindN


52/85


Error Probability for M-PSK (cont

d)

Error rate degrades asMincreases.

Recall that spectral efficiency increases asMincreases.


53/85


Error Probability for QAM

tfAtfAts cimcrmm 2sin2cos ,, 2,2 ,, simsrmm TATA s

AAAA imrm 3,, ,,

sssss TAEEEE29

151230

sssss TAEEEE2

10965

sssss

ssss

TAEEEE

EEEE

25

1413117

8421

s

M

m

ss TAE

M

Emavg

2

1

51

avg

bs

avg b

EE

ss EETAd5

8

5

222

4

min

0s 1s 2s 3s

4s 5s 6s 7s

8s 9s 10s 11s

12s 13s 14s 15s

mind


54/85


Error Probability for QAM (cont

d)

0s 1s 2s 3s

4s 5s 6s 7s

8s 9s 10s 11s

12s 13s 14s 15s

2 neighbours

3 neighbours

4 neighbours

151230 ,,, ssss

10965 ,,, ssss

11714138421 ,,,,,,, ssssssss

2 neighbours

4 neighbours

3 neighbours

00

min

5

43

2

~min N

EQ

N

dQNeP

avgb

d

Using the result from p.50, we obtain an approximate upper bound

31

1min,min

M

mdd m

NM

N

3 neighbours


55/85


Error Probability for QAM (cont

d)

Power efficiency decreases with increasingM, but not early as fast asM-PSK.

Recall that spectral efficiency increases asMincreases.


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tffAts mcm 2cos sTt0 MmTmfm ...2,1,2

Error Probability for M-FSK

Each signal occupies its own

dimension. Therefore, each

signal hasM-1neighbours,

separated from each other by

sEd 2min

0

2

0

log11N

MEQMNEQMeP bs


57/85


Error Probability for M-FSK (cont

d)

AsMincreases, power efficiency improves (i.e. lessEbis

required).

Recall that spectral efficiency decreases asMincreases.

ForM=2, BFSK requires 3dB more energy/bit to achieve the sameP(e)asBPSK.

In other words, BPSK is 3dB more power efficient that BFSK.

0

2

N

EQeP b

BPSK

0NEQeP b

BFSK


58/85


Comparison of Power Efficiency of Modulation Schemes

We will use BPSK/QPSK as a benchmark with which to compare the

power efficiency of other modulation schemes.

BPSK/QPSK has . Now, define the power efficiency of a

modulation scheme (relative to BPSK/QPSK) asbEd 4

2min

b

P

E

d

4

log102min

10


59/85


Differential Phase Shift Keying (DPSK)

So far, we assumed that coherent demodulation is performed, i.e. that the

carrier phase is perfectly known at the receiver. This normally requires carrier

phase estimation.

An alternative is differentially encoding, where the data is encoded in

phase difference from one symbol to the next. Assuming binary signalling,

kkkkk baabd 11,11,0

dk 0 1 1 1 0 1

bk +1 -1 -1 -1 +1 -1

ak +1 +1 -1 +1 -1 -1 +1

This diagram mightcorrespond to either

PSK or DPSK!


60/85


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Error Probability of DPSK

tAats ckcos kTtTk 1

kk

j

TkT

j

Tk

NaATedttNdtaAedttyy TNNNk 02,0~

tNAea jk

211 kkdef

yy 212 kkdef

yy Defining the decision variable

can be written as

2221* 1Resgn kkk yyz

110Re11 21*

1 kkkkkk bPbyyPbbPeP

tjytyty QI


62/85


We need to find statistical properties of and :1

2

22/ 111 kkkkj NNaaATe

22/ 112 kkkkj NNaaATe

2/11 kkj aaATeE

2/12 kkj aaATeE

TNNNNNEEEVar kkkk 0*112111 41 TNNNNNEEEVar kkkk 0*112222

4

1

Error Probability of DPSK (cont

d)

211 kkdef

yy 212 kkdef

yy

First, we recall the definitions

E P b bili f DPSK ( d)


63/85


kkk baa 1 Encoding scheme:

+1 +1 +1 +1 0

-1 +1 -1 0 -1

-1 -1 +1 -1 0

+1 -1 -1 0 1

ka 1ka kb 21 kk aa 21 kk aa

01 E

jATeE 2

TNVar 01

TNVar 02

211 ,0~, NIR

22 ,cos~ ATNR 2

2 ,sin~ ATNI

Rician:Rayleigh,: 21

Under the assumption that is sent1kb

202 TNwhere

IR j 111

IR j 222

Complex Gaussian

Complex Gaussian


d)


64/85

E P b bilit f DPSK ( t d)


65/85


dxmx

Imxxm

dxx

Ixx

dIeP

20

2

22

0 22

2

202

22

02

202

22

0

22

2

2exp

2exp

2

1

22exp

2

1

2

exp

2

exp

2def

x

Variable change

2def

m

=1

0

0

2

exp21

2exp

2

1

NE

N

TA

222 TAdttsET


d)

202 TN

AT

E P b bilit f DPSK ( t d)


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0 2 4 6 8 10 1210

-6

10-5

10-4

10-3

10-2

10-1

100

Coherent vs. Differential PSK

SNR [dB]

BER

CoherentDifferential


d)

There is some performance degradation due to differential detection,but now a less complex receiver can be used (i.e. no need for phase

tracking).


67/85


Temporal Characteristics of Modulation Schemes

So far, we have considered pulse shapes which are strictly limited in the

symbol interval. By using a pulse shape to spill overinto adjacent symbol

intervals, better spectral efficiency can be achieved, however this also results in

intersymbol interference (ISI).

The following block diagram is commonly used for studying ISI. Assuming

matched filter type implementation for the demodulator,

tw

thT thC thR

slTtDetector

ksk kTtpatg

ActualChannel

EquivalentChannel

sseqk

kl lTnTklhaz where thththh RCTeq thtwtn R

T l Ch t i ti f M d l ti S h ( t d)


68/85


Temporal Characteristics of Modulation Schemes (cont

d)

Here, we use in pulse shapes which spill over adjacent symbols. This will

bring ISI terms:

otherwise,0

0,1 nnTh seq

constant

sleq T

l

fH

This condition is known as Nyquist pulse-shaping criterionor Nyquist

condition for zero ISI.

sseqlk

keqlsseqk

kl lTnTklhahalTnTklhaz

0

ISI terms The condition for no ISI is

In frequency domain, this requires

See proof Proakis Digital CommunicationsChapter 9

T l Ch t i ti f M d l ti S h ( t d)


69/85



d)

WTs

2

1

WTs

2

1

For this case, there is no choice forHeqto satisfy Nyquist criterion.

sleq

T

lfH

sleq

T

lfH

otherwise,0

, WfTfH

seq

For this case, there is only one solution:

W: Bandwidth of equivalent ch.

In the following, we consider three distinct cases:


70/85

Temporal Characteristics of Modulation Schemes (cont d)


71/85



d)

Under the matched-filter assumption (i.e. which maximizes the output

signal-to-noise ratio), the transmit and receive filters satisfy

fHfH RT

Under the ideal channel assumption , i.e.

fHfHfH eqRT

1fHC

For raised-cosineequivalent channel response, we can divide it into two

root-raised-cosine(RRC) filters.

s

sss

ss

ss

RCRRC

Tf

Tf

TTfTT

TfT

fHfH

2

1,0

2

1

2

1,2

1cos2

2

1,

Temporal Characteristics of BPSK


72/85


Temporal Characteristics of BPSK

sk k kTtpatg

Consider the baseband BPSK modulated signal with RRC pulse shape

1ka

Eye patternis a sketch ofg(t) for all possible combinations of ,...,, 321 aaa

Minimum instantaneous

power=0

Maximum instantaneous

power=(1.6)2=4.1 [dB]

Dynamic range=4.1 [dB]

Average power=1=0 [dB] dB1.4powerAvg.powerPeak

For this example, we observe large dynamic range of instantaneous power

and large peak/average ratio. These make the design of TX power amplifier

difficult.

Temporal Characteristics of QPSK


73/85


The QPSK signal with pulse shaping can be written as

tkTtpbtkTtpatg ck skck sk sincos

1, kk ba

The instantaneous power of the QPSK signal is

22

ksk

ksk kTtpbkTtpa

Hence, a QPSK signal suffers similar time-domain problems as a BPSK

signal. Now assume, different pulses are used for I&Q channels. If Q channel

pulse is delayed by 1/2 symbol relative to I channel pulse, i.e.

the instantaneous power is

2sTtptq

22

2 kssk

ksk TkTtpbkTtpa

Both terms can not pass through zero simultaneously, hence significantly

increasing the minimum instantaneous power and reducing dynamic range of the

signal. PSD and BER remain unchanged. This is known as Offset QPSK (OQPSK).

Temporal Characteristics of QPSK

Temporal Characteristics of QPSK (cont d)


74/85


+ =

jjkk

ee

jba

,2 434 , jj

kk

ee

jba

kee

kee

jba

jj

jj

kk

oddfor,,

evenfor,,

2

434

Another variant of QPSK is /4-QPSK. This modulation scheme is a

superposition of two QPSK signal constellations offset by /4 relative to each

other.

PSD and BER of /4-QPSK are the same as QPSK.

Temporal Characteristics of QPSK (cont

d)

Temporal Characteristics of QPSK (cont d)


75/85


In QPSK, transitions between opposite points in the signal constellation

cause the instantaneous power to zero, leading to a large dynamic range.

The special structure of /4-QPSK avoids transitions which pass the origin,

reducing dynamic range and peak-to-average power ratio.

Temporal Characteristics of QPSK (cont

d)

Continuous FSK


76/85


Continuous FSK

We can get perfect temporal properties by using continuous FSK (CFSK)

sk k kTtpatg 1ka

k

skc

t

c kTtqahtfdghtfts 22cos22cos

where

dkTptqt

s

def

Instantaneous power= constant

Dynamic range=0dB

Peak-to-average power ratio=0dB

There is no abrupt switching from one phase to another, avoiding phase

discontinuities.

h: Modulation index

Continuous FSK (cont d)


77/85


1/2Ts

t

1/2

t

p(t): Frequency pulse

Ts

Ts

Here, we assume a rectangle pulse shape forp(t).

q(t): Phase pulse

ns

sn

kk

ksk

aT

nTthah

kTtqaht

22

2;

1

0

a

-3h

-2h

-h

h2h

3h

0

Ts 2Ts 3Ts

+1

-1

+1

-1

+1

-1

The shaded path illustrates the phases for

the input sequence {+1,+1,-1}

ss TntnT 1

Phase Tree

Continuous FSK (cont

d)

n=0,1,..


78/85

Continuous FSK (cont d)


79/85


We have already introduced MSK as a special case of modulation family of

CFSK.

An MSK signal can be also considered as a special form of OQPSK where

the rectangular pulses are replaced with half-sinusoidal pulses.

tfTkTtpatfkTtpats csskkcs

kk 2sin22cos2

evenodd

otherwise,0

20,2

cos ss

TtT

t

tp

The transmission rate on the two orthogonal carriers is 1/2Tsbits/sec so thatthe combined transmission rate is 1/Tsbits/sec.

Continuous FSK (cont

d)

Comparison of MSK QPSK and OQPSK


80/85


Comparison of MSK, QPSK and OQPSK

Continuous phase is assured in MSK while 90 and 180 phase changes are

observable for OQPSK and QPSK respectively.



81/85


In terms of temporal

efficiency, MSK obviouslyoutperforms QPSK and

OQPSK.

The main lobe of MSK is

wider than that of QPSK and

OQPSK and, in terms of null-to-null bandwidth MSK is less

spectral efficient.

MSK has lower sidelobes

than QPSK and OQPSKLess

adjacent channel interference

MSK, QPSK and OQPSK

have the same power efficiency.


(contd)

Gaussian MSK


82/85


Gaussian MSK

The spectral efficiency of MSK can be further improved by prefiltering.

sk

k kTtpatg GaussianLPF

MSK

Modulator

The frequency response function of Gaussian LPF filter is given as

2ln

2exp

2ln

2

2

2lnexp

2222 tBBth

B

ffH

whereBis 3dB-bandwidth of the filter.

We are interested in how a rectangle pulse passed through a Gaussian

LPF will look like.

Gaussian MSK (cont d)


83/85


T

Tt

T

TtQtf

2/2/

Frequency pulse

2ln2 BT

2

1

2exp

2

111

2 x

x

xxQx

tq

Phase pulse

T

Tt

x

2/

1

T

Tt

x

2/

2

Phase pulse corresponding to rectangular

pulse shaping (i.e. no filtering) is also

included in the figure.

Gaussian MSK (cont

d)

BT: Normalized

3dB-Bandwidth



84/85


ForBT, the pulse shape takes its original unfilteredform , i.e. rectangle

pulse. GMSK

MSK The frequency pulse has a duration of 2Tsalthough signaling rate is 1/Ts. Such a

LPF will result in intersymbol interference which requires sequence estimation for

optimal detection.

Gaussian MSK (cont

d)

BT: Normalized

3dB-Bandwidthof Gaussian filter



85/85

BTshould be chosen as to find a good

compromise between spectral efficiencyand ISI.

AsBTdecreases, the spectral

efficiency improves (i.e. less bandwith).

Also sidelobes fall off very rapidly (i.e.

less adjacent channel interference).

However, reducingBTresults in ISI

and error rate performance degrades (i.e.

observation of an irreducible error floor

due to ISI) In practical application,BTis typically

chosen as (0.2, 0.5). GSM systems use

GMSK withBT=0.35.

Gaussian MSK (cont

d)

BPSK probability of error

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