Digital Transmission S-72.1140 Transmission Methods in Telecommunication Systems (5 cr)

Digital Transmission

S-72.1140 Transmission Methods in Telecommunication Systems (5 cr)

2 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

I Baseband Digital Transmission

Why to Apply Digital Transmission? Digital Transmission Symbols and Bits

– M-level Pulse Amplitude Modulation (PAM)– Line codes (Binary PAM Formats)

Baseband Digital Transmission Link– Baseband Unipolar Binary Error Probability – Determining Decision Threshold – Error rate and Q-function – Baseband Binary Error Rate in Terms of Pulse Shape and

Pulse Shaping and Band-limited Transmission– Signaling With Cosine Roll-off Signals– Matched Filtering– Root-raised cos-filtering

Eye diagram


II Carrier Wave Digital Transmission

Waveforms of Digital Carrier Wave Communications Detection of Digital CW

– Coherent Detection• Error rate; General treatment

– Non-coherent Detection• Example of error rate determination (OOK)

Timing and Synchronization Error rate for M-PSK Error rate for M-QAM Comparison of digital CW methods


Why to Apply Digital Transmission? Digital communication withstands channel noise, interference

and distortion better than analog system. For instance in PSTN inter-exchange STP*-links NEXT (Near-End Cross-Talk) produces several interference. For analog systems interference must be below 50 dB whereas in digital system 20 dB is enough. With this respect digital systems can utilize lower quality cabling than analog systems

Regenerative repeaters are efficient. Note that cleaning of analog-signals by repeaters does not work as well

Digital HW/SW implementation is straightforward Circuits can be easily configured and programmed by DSP

techniques Digital signals can be coded to yield very low error rates Digital communication enables efficient exchange of SNR to

BW-> easy adaptation into different channels The cost of digital HW continues to halve every two or three

years

STP: Shielded twisted pair


DigitalTransmission

‘Baseband’ means that no carrier wave modulation is used for transmission

Information:- analog:BW & dynamic range- digital:bit rate

Information:- analog:BW & dynamic range- digital:bit rate

Maximization of information transferred

Maximization of information transferred

Transmitted power;bandpass/baseband signal BW

Transmitted power;bandpass/baseband signal BW

Message protection & channel adaptation;convolution, block coding

Message protection & channel adaptation;convolution, block coding

M-PSK/FSK/ASK..., depends on channel BW & characteristics

M-PSK/FSK/ASK..., depends on channel BW & characteristics

wireline/wirelessconstant/variablelinear/nonlinear

wireline/wirelessconstant/variablelinear/nonlinear

NoiseNoise

InterferenceInterference

ChannelChannel

ModulatorModulator

ChannelEncoder

ChannelEncoder

Source encoder

Source encoder

Channel decoder

Channel decoder

Source decoder

Source decoder

DemodulatorDemodulator

Information sink

Information sink

Information source

Information source

Message Message estimate

Received signal(may contain errors)

Transmitted signal

InterleavingInterleaving

Fights against burst errors

Fights against burst errors

DeinterleavingDeinterleaving

In baseband systemsthese blocks are missing


Symbols and Bits – M-ary PAM

1 1 00 1 11 110 1 0

bi ( 1/ ) ts/sb b bT r Tbitrate

( 1/ )D r Dsymbol rate baud

2nM

:

:

:

number of bits

: number of levels

Symbol duration

Bit duaration

b

n

M

D

T

2logn M

( ) ( ) k

ks t a p t kD

For M=2 (binary signalling):

For non-Inter-Symbolic Interference (ISI), p(t) mustsatisfy:

This means that at the instant of decision, received signal component is

( ) ( ) k b

ks t a p t kT

1, 0( )

0, , 2 ...

tp t

t D D

( ) ( ) ( )k K K K

ks t a p t kD a p t a

Generally: (a PAM* signal)

( )s t

*Pulse Amplitude Modulation


Binary PAM Formats (1)

Unipolar RZ and NRZ

Polar RZ and NRZ

Bipolar NRZ or alternate mark inversion (AMI)

Bit stream

Split-phase Manchester


Binary PAM Formats (2)

Unipolar RZ, NRZ:– DC component has no information, wastes power– Transformers and capacitors in route block DC– NRZ, more energy per bit, synchronization more difficult

Polar RZ, NRZ:– No DC term if ´0´and ´1´ are equally likely

Bipolar NRZ– No DC term

Split-phase Manchester – Zero DC term regardless of message sequence– Synchronization simpler– Requires larger bandwidth


Baseband Digital Transmission Link

( ) ( ) ( ) k d

ky t a p t t kD n t

( ) ( ) ( )

K k kk K

y t a a p KD kD n t

message reconstruction at yields K d

t KD t

message ISI Gaussian bandpass noise

Uni

pola

r P

AM

original message bits

decision instances

received wave y(t)

Dt


Baseband Unipolar Binary Error Probability

r.v. : ( ) ( ) k k k

Y y t a n t

The sample-and-hold circuit yields:

0

0

: 0,

( | ) ( )

k

Y N

H a Y n

p y H p y

Establish H0 and H1 hypothesis:

1

1

: 1,

( | ) ( )

k

Y N

H a Y A n

p y H p y A

and

pN(y): Noise probability density function (PDF) at the time

instance of sampling

Assume binary & unipolar x(t)


Determining Decision Threshold0

0

: 0,

( | ) ( )

k

Y N

H a Y n

p y H p y

1

1

: 1,

( | ) ( )

k

Y N

H a Y A n

p y H p y A

Choose Ho (ak=0) if Y<VChoose H1 (ak=1) if Y>V

The comparator implements decision rule:

1 1 1

0 0

( | ) ( | )

( | ) ( | )

V

e Y

Veo Y

p P Y V H p y H dy

p P Y V H p y H dy

Average error error probability: 0 0 1 1

e e eP PP PP

120 1 0 1

1/ 2 ( ) e e e

P P P P P

Channel noise is Gaussian with the pfd:2

2

1( ) exp

22N

xp x

Transmitted ‘0’but detected as ‘1’


Error rate and Q-function

21( ) exp

22 k dQ k

x m

0( )

Ve N

Vp p y dy Q

2

0 2

1exp

22Ve

xp dx

This can be expressed by using the Q-function

by

and also

0( )

Ve Np p y dy

1( )

V

e N

A VP p y A dy Q

m: mean2: variance

0ep

1ep


Assigment


Solution


Baseband Binary Error Rate in Terms of Pulse Shape

12 0 1 0 1( )

2e e e e e e

Ap p p p p p Q

for unipolar, rectangular NRZ [0,A] bits

setting V=A/2 yields then

2 2 21 1( ) (0) / 2

2 2RS A A

for polar, rectangular NRZ [-A/2,A/2] bits

2 2 21 1( / 2) ( / 2) / 4

2 2RS A A A

and hence

2 2

2

/(2 ),unipolar

/ ,polar2 4R R

R RRRN

S NA AS NN

probability of occurrence for bits ’0’ and ’1’


Assignment Determine average power for the following signals

T

T

A

-A

A

-A

A/2

-A/2


Solution

T

A

-A

A

-A

A/2

-A/2

2 2 21 1( ) ( )

2 2RS A A A

T

2 2 21 1 5( ) ( / 2)

2 2 8RS A A A


Pulse Shaping and Band-limited Transmission

In digital transmission signaling pulse shape is chosen to satisfy the following requirements:– yields maximum SNR at the time instance of decision

(matched filtering)– accommodates signal to channel bandwidth:

• rapid decrease of pulse energy outside the main lobe in frequency domain alleviates filter design

• lowers cross-talk in multiplexed systems


Signaling With Cosine Roll-off Signals

Maximum transmission rate can be obtained with sinc-pulses

However, they are not time-limited. A more practical choice is the cosine roll-off signaling:

( ) sinc sinc /

1( ) [ ( )]

p t rt t D

fP f F p t

r r

2

2( ) sinc

1 (4 )

cos tp t rt

t

2

/ 2

1( ) cos ( / 2 )

2

r

fP f f r

r r

for raised cos-pulses =r/2


Unipolar and Polar Error Rates in Terms of Eb/No

Eb/No is often indicated by

For sinc- pulse signalling the transmission BW is limited toand therefore noise before decision is limited to

and therefore

0 0/ /

b R bbNE S rN

0 0/ 2

R N bN N B N r

/ 2N b

B r

2

0 0

0 0

/(2 ) 2 /(2 ) ,unipolar

/( ) 2 / 2 ,polar2R R b b b b

R R b b b b

S N N r N rAS N N r N r

( ) ( )2 ,

2e e polar b e unipolar b

Ap Q p Q p Q


Matched Filtering

( ) ( )exp( ) ( ) ( )d d

H f KP f j t h t Kp t t ( ) ( )exp( ) ( ) ( )d d

H f KP f j t h t Kp t t

0

0

( ) ( )

( ) ( )exp( )R R

R R

x t A p t t

X f A P f j t

0

1[ ( ) ( )]

( ) ( )exp

dR

R d

t t tA F H f X f

H f P f j t dfA

2 22 ( ) ( ) ( )2n

H f G f df H f df

2

2

2

2

( ) ( )exp

( )2

d

R

H f P f j t dfAA

H f df

H(f)H(f)++( )

Rx t

( )2n

G f

( )D

y t

Should be maximized

Post filter noise

Peak amplitude to be maximized

Using Schwartz’s inequality2

2 2

( ) * ( ) ( ) ( )V f W f df W f df V f df


Assignment

What is the impulse response of the matched filter for the following signaling waveform?

How would you determine the respective output signal (after the matched filter)?

T

A


Avoiding ISI and enabling band-limiting inradio systems

Two goals to achieve: band limited transmission & matched filterreception

Hence at the transmitter and receiveralike root-raised cos-filtersmust be applied

TXfilt.

RXfilt.

Decisiondevice

noise

data

( )T f ( )R f

( ) ( ) ( ), raised-cos shaping

( ) *( ), matched filteringN

T f R f C f

T f R f

( ) ( ) ( )N

R f T f C f

raised cos-spectra CN(f)


Monitoring Transmission Quality by Eye Diagram

Required minimum bandwidth is

Nyqvist’s sampling theorem:

/ 2T

B r

Given an ideal LPF with thebandwidth B it is possible totransmit independent symbols at the rate:

/ 2 1/(2 )T b

B r T


Assignment

How many eye/openings you have in an M-level signaling?

S-72.1140 Transmission Methods in Telecommunication Systems (5 cr)

Digital Bandpass Transmission


Binary Waveforms in Carrier Wave Communications

ASK

FSK

PSK

DSB


Carrier Wave Communications

Carrier wave modulation is used to transmit messages over a distance by radio waves (air, copper or coaxial cable), by optical signals (fiber), or by sound waves (air, water, ground)

CW transmission allocates bandwidth around the applied carrier that depends on– message bandwidth and bit rate– number of encoded levels (word length) – source and channel encoding methods

Examples of transmission bandwidths for certain CW techniques:

MPSK, M-ASK Binary FSK (fd=rb/2)

MSK (CPFSK fd=rb/4), QAM:

2/ / log ( 2 )

T b b

nB r r n r M M T b

B r/ 2

T bB r

FSK: Frequency shift keyingCPFSK: Continuous phase FSK


Digital CW Detection

At the receiver, detection can be– coherent (carrier phase information used for detection)– noncoherent (no carrier phase used for detection)– differentially coherent (‘local oscillator’ synthesized from

received bits) CW systems characterized by bit or symbol error rate (number

of decoded errors(symbols)/total number of bits(symbols)) Number of allocated signaling levels determines constellation

diagram (=lowpass equivalent of the applied digital modulation format)


Coherent Detection by Integrate and Dump / Matched Filter Receiver Coherent detection utilizes carrier phase information and requires in-

phase replica of the carrier at the receiver (explicitly or implicitly) It is easy to show that these two techniques have the same

performance:

( )s t

( ) ( )h t s t

0( ) ( ) ( )v t s t y d

0

( ) ( ) ( )

( ) ( )

v t s t y t

s t y d

( )s t

( )y t

( )y t ( )v T

( )v T


Non-coherent Detection

2-ASK

2-FSK

Base on filtering signal energy on allocated spectra and using envelope detectors

Has performance degradation of about 1-3 dB when compared to coherent detection (depending on Eb/N0)

Examples:


Coherent (Optimum) Binary Detection

Received signal consists of bandpass filtered signal and noise that is sampled at the decision time instants tk yielding decision variable:

Quadrature presentation of the signaling waveform is

Assuming that the BPF has the impulse response h(t), signal component at the sampling instants is then expressed by

( )k m

Y y t z n

( 1)

0

( ) ( ) ( ) ( )

( ) ( )

b

b

b

m m b m k

m b

bk

k T

kT

T

t tz s t kT h t s h t d

s h d

kT

T

( ( ) x( ) ( ) )A

x y t y t d

0,1m

( ) ( )cos( ) ( )sin( )m C k i C k q C

s t A I p t t Q p t t


Optimum Binary Detection - Error Rate Assuming ‘0’ and ‘1’ reception is equally likely, error happens

when H0 (‘0’ transmitted) signal hits the dashed region or for H1 error hits the left-hand side of the decision threshold that is at

1 0( ) / 2optV z z

2 020 0

0 1 1 0

1exp / 2

2

( ) / 2 / 2

opt

opte

e e e

V

V zp z d Q

p p p Q z z

Errors for ‘0’ or/and ‘1’ are equal alike, for instance for ‘0’:

For optimum performancewe have the maximized

SNR that is obtainedby matched filtering/

integrate and dump receiver

2

1 0 / 2z z

xQ


Optimum Binary Detection (cont.)

Express energy / bit embedded in signaling waveforms by

Therefore, for coherent CW we have the SNR and error rate

2

1 0

2

0 0 1

2

1

0 10

0 0

1

0 0

( ) ( )

( ) 2 ( ) ( )

( )b b

b b

T T

T T

E

E E

s s d d

s d s s d

s

Note that the signaling waveform correlation greatly influences the SNR!

2

1 0 1 0 10 1 0 10 10

2

2 / 2

2 2

4 2 2b

e

z z E E E E E E E Ep Q Q

1 0 / 2ep Q z z 1 0 / 2ep Q z z

2max / 2

/ 2b b

o

o

E ESNR N

N


Example: Coherent Binary On-off Keying (OOK)

For on-off keying (OOK) the signaling waveforms are

and the optimum coherent receiver can be sketched by

1 0( ) ( )cos , ( ) 0

C Tb Cs t A p t t s t

2 2

1 1 0 10 1 00 0

( ) / 2, 0, ( ) ( ) 0b b

C b

T T

E s d A T E E s s d 2

0 1( ) / 2 / 4

b C bE E E A T 10b b

e b

E E Ep Q Q Q

1/( )R R R b b

b

bW T

S S S T E


Timing and Synchronization Performance of coherent detection is greatly

dependent on how successful local carrier recovery is Consider the bandpass signal s(t) with width Tb rectangular

pulses pTb(t), that is applied to the matched filter h(t): ( ) ( )cos( ),

( ) ( ) ( )cos( )C Tb C

b Tb b C

s t A p t t

h t Ks T t Kp T t t

( )s t( )h t

( )z t

( ) ( ) ( )

cos cos 2b

c c c

b

z s t h t

TKA t

T

2

0

2

( )

/ 2

bT

C b

E s t dt

A T

2

0

2

( )

/ 2

bT

C b

E s t dt

A T

k

t

2 2

1 0 0 0 1

2

1

0 10

0 0 0 0

1

( ) ( ) ( ) 2 ( ) ( )( )b b b bT T T T

E E E

s s d d s d s s ds 2 2

1 0 0 0 1

2

1

0 10

0 0 0 0

1

( ) ( ) ( ) 2 ( ) ( )( )b b b bT T T T

E E E

s s d d s d s s ds

cT

cosb

c

b

TKA

T

( )c

yelding after filtering:

nominal point of inspection at Tb


Analyzing phase error by Mathcad

210 cosb

e

c

E Ep Q

Therefore, due to phase mismatch at the receiver, the error rate is degraded to


Example Assume data rate is 2 kbaud/s and carrier is 100 kHz for an BPSK

system. Hence the symbol duration and carrier period are

therefore the symbol duration is in radians

Assume carrier phase error is 0.3 % of the symbol duration. Then the resulting carrier phase error is

and the error rate for instance for is

that should be compared to the error rate without any phase errors or

Hence, phase synchronization is a very important point to remember in coherent detection

1/ 2kbaud/s = 0.5msS

T 31/ 1/100 10 10C C

T f s

10 2314.2rad

0.5ms xs

x

o0.003 0.94 rad 54x

8 9dB 2 2 2( 2 cos ( 16cos 54) 10

ep Q Q

5( 16) 3 10e

p Q

(or carrier cycles)


Error rate for M-PSK

In general,PSK error rate can be expressed by

where d is the distance between constellation points (or a=d/2 is the distance from constellation point to the decision region border) and is the average number of constellation points in the immediate neighborhood. Therefore

Note that for matched filter reception

2e n n

d ap n Q n Q

nn

qn

decision region

000

001

011111

101

100

110

010

d

2 sin( / 2)2 2 2 sin( / )

2 2e

d A Ap Q Q Q M

2

0

2, log ( )

b b

A EE nE M E

N

2M

2nM

/ 2sin( / 2)

d

A

A


Error rate for M-QAM, example 16-QAM

2e n n

d ap n Q n Q

2 2 22 2

4 4 8 3 4 23

4 8 4

4 2 8 10 4 1810

16

nn

a a aA a

2

20

23 3 3

10 10e

a A Ep Q Q Q

N

symbol error rate

Constellation follows from 4-bit words and therefore

0 04

43 2 3

4 10 4 5b

bb

E E

EEp Q Q

N N

2

0

log ,

( ) /

/ 2 /

b

e

n M E nE

p p E n

A E N

2

2

2 3

18

a

a

2 2

2

3

10

a a

a

22a

2a


Non-coherent Detection


Example: Non-coherent On-off Keying (OOK)

Bandpass filter is matched to the signaling waveform (not to carrier phase), in addition fc>>fm, and therefore the energy for ‘1’ is simply

Envelopes follow Rice and Rayleigh distributions for ‘1’ and ‘0’ respectively:

2

1( / 2)

b CE T A

distribution for "1"

distribution for ”0”"


Noncoherent OOK Error Rate

The optimum decision threshold is at the intersection of Rice and Rayleigh distributions (areas of error probability are the same on both sides of decision threshold)

Usually high SNR is assumed and hence the threshold is approximately at the half way and the error rate is the average of '0' and '1' reception probabilities

Therefore, error rate for noncoherent OOK equals

0 11

2e e eP P P

2 2

0 0

1 10

/2

/2

( | ) exp /8 exp / 2

( | ) ( / 2 ) ( )

C

C

e Y C b

e Y C b

A

A

P p Y H dy A

P p Y H dy Q A Q

probability to detect "0" in error

probability to detect "1" in error

1 1exp( / 2) ( ) exp( / 2), 12 2e b b b bP Q


Comparison


Error Rate Comparison

a: Coherent BPSKb: DPSKc:Coherent OOKd: Noncoherent FSKe: noncoherent OOK

a: Coherent BPSKb: DPSKc:Coherent OOKd: Noncoherent FSKe: noncoherent OOK


Comparison of Quadrature Modulation Methods

Note that still the performance is good, envelope is not constant. APK (or M-QASK) is used for instance in modems

APK=MQASK

(pe=10-4)

M-APK: Amplitude Phase Shift Keying

(pe=10-4)

PRK BPSK

APK M QASK M QAM

Digital Transmission S-72.1140 Transmission Methods in Telecommunication Systems (5 cr)

Documents

digital transmissionsymbols

cr digital transmission

respect digital systems

communications laboratory

bw dynamic range digital

analog systems interference

decision threshold error

synchronizationerror