Communication Systems, 5e - Western Michigan Universitybazuinb/ECE4600/Ch11_3.pdf · Communication Systems, 5e ... Hill Companies, Inc. Permission required for reproduction or

Communication Systems, 5e

Chapter 11: Baseband Digital Transmission

A. Bruce CarlsonPaul B. Crilly

© 2010 The McGraw-Hill Companies

Chapter 11: Baseband Digital Transmission

• Digital signals and systems• Noise and errors• Bandlimited digital PAM systems• Synchronization Techniques

© 2010 The McGraw-Hill Companies

3

Digital Formatting and Transmission

EncodeTransmitPulse

modulateSample Quantize

Demodulate/Detect

Channel

ReceiveLow-pass

filter Decode

PulsewaveformsBit stream

Format

Format

Digital info.

Textual info.

Analog info.

Textual info.

Analog info.

Digital info.

source

sink

4

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

(a) unipolar RZ & NRZ (b) polar RZ & NRZ (c) bipolar NRZ (d) split-phase Manchester (e) polar quaternary NRZ: Figure 11.1-1

ABC Binary PAM formats

5

ABC PCM Definitions

• Unipolar– The signal can be considered on and off, with signal

levels of 0 and A in amplitude.

• Polar– The signal has opposite polarity components, so that the

signal has a zero (0) DC component if 0’s and 1’s are equally likely.

• Bipolar– The signal also has opposite polarity components, but

also inlcudes zero (0) as a pseudo-trinary format.

6


Figure 11.2-1

Baseband Binary Receiver

• Synchronous Time sampling of maximum filter output

thtnthkTtpaty ink

k

kkk tnaty

k

k kTtpatx

Symbol Detection Keys

• Filter– Appropriate bandwidth to trade-off signal power and

noise power– A matched filter is optimal!

• Detection Threshold– What threshold is used to detect a binary symbol?– Minimize the probability of a bit error

• Based on hypothesis testing

7

8

Polar Binary Error Probability

• Hypothesis Testing using a voltage threshold– Hypothesis 0– Hypothesis 1

• For Noise that is AWGN

1e10e0error PHPPHPP

2| 0AypHyp kNkY

2A-| 1 kNkY ypHyp

2

2

20 22exp

21

2|

AyAypHyp NY

x

2

d2

exp21xQ

22

21AQ

VAQdyAypVYPP

V

Ne

22

20AQ

VAQdyAypVYPP

VNe

0Vfor

9

Relationship to signal power

• Defining the average received signal power– Unipolar

– Polar

– Bipolar

• In terms of SNR

AASR ,0,21 2

2,

2,

41 2 AAASR

Polarfor

NS

UnipolarforNS

21

N4A

2A

R

R

R

22

AAASR ,,2

BipolarforNS

NAA

RR

22

2

2

21limT

TcTR dttx

TES

10

SNR to Eb/No

• For the Signal to Noise Ratio – SNR relates the average signal power and average noise

power (Tb is bit period, W is filter bandwidth)

– Eb/No relates the energy per bit to the noise energy(a multiple of the time-bandwidth product)

WR

NE

WT1

NE

WNT1E

NS b

0

b

b0

b

0

bb

WTNS

RW

NS

NE

bb0

b

11

Relationship to Eb/No

• Defining the energy per bit to noise power ratiofor a time-bandwidth product of

– Unipolar

– Polar

– Bipolar

0

b

RR

22

NE

NS

21

N4A

2A

0

b

RR

22

NE2

NS

N4A

2A

0

b

RR

22

NE2

NS

NAA

21T

2RTW b

bb

12

Relationship to Bit Error Probability

• Defining the binary bit error probabilityfor a time-bandwidth product

– Unipolar

– Polar

– Bipolar

0

berror N

EQ2AQP

0

berror N

E2QAQP

0

berror N

E2Q2AQP

13

Bit Error Rate Plot

10-3 10-2 10-1 100 1010

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5B

it E

rror R

ate

Eb/No

Classical Bit Error Rates

OrthogonalAntipodal

EbNo=(0:10000)'/1000;

% Q(x)=0.5*erfc(x/sqrt(2))

Ortho=0.5*erfc(sqrt(EbNo)/sqrt(2));Antipodal=0.5*erfc(sqrt(2*EbNo)/sqrt(2));

semilogx(EbNo,[Ortho Antipodal])ylabel('Bit Error Rate')xlabel('Eb/No')title('Classical Bit Error Rates')legend('Orthogonal','Antipodal')

14

BER Performance, Classical Curveslog-log plot

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1510

-7

10-6

10-5

10-4

10-3

10-2

10-1

100B

it E

rror R

ate

Eb/No

Classical Bit Error Rates

OrthogonalAntipodal

15

Antipodal and Orthogonal Signals

• Antipodal– Distance is twice “signal voltage”– Only works for one-dimensional signals

• Orthogonal– Orthogonal symbol set– Works for 2 to N dimensional signals

bE2d

jiforjifor

dttstsE

zT

jiij 111

0

jifor0jifor1

dttstsE1z

T

0jiijbE2d

16

M-ary Signals

• Symbol represents k bits at a time– Symbol selected based on k bits– M waveforms may be transmitted

• Allow for the tradeoff of error probability for bandwidth efficiency

• Orthogonality of k-bit symbols– Number of bits that agree=Number of bits that disagree

k2M

jifor0jifor1

K

bbsumbbsumz

N

1k

jk

ik

K

1k

jk

ik

ij

Q Function

• Another defined function that is related to the Gaussian (and used) is the Q-function.:

• The Q-function is the complement of the normal function, :

• Therefore note that:

17

Q Function Table p. 858

duuxQxu

2

exp21 2

xxQ 1

xQxQ 1

XxQxFX 1

Using MATLAB• Another way to find values for the Gaussian

– The error function

18

duuxerfx

u

0

2exp2

21

21 xerfxQ

22

121

21

211 XxerfXxerfxFX

The error function (Y = ERF(X)) is built-in to MATLAB From MATLAB:ERF Error function.

Y = ERF(X) is the error function for each element of X. X must bereal. The error function is defined as:

erf(x) = 2/sqrt(pi) * integral from 0 to x of exp(-t^2) dt.See also erfc, erfcx, erfinv.

Reference page in Help browserdoc erf

Using MATLAB (2)

• The complementary error function

19

xerf1xerfc

2xerfc

21xQ

The error function (Y = ERFC(X)) is built-in to MATLAB. . From MATLAB:ERFC Complementary error function.

Y = ERFC(X) is the complementary error function for each elementof X. X must be real. The complementary error function isdefined as:

erfc(x) = 2/sqrt(pi) * integral from x to inf of exp(-t^2) dt.= 1 - erf(x).

Class support for input X:float: double, single

See also erf, erfcx, erfinv.Reference page in Help browser

doc erfc

Qfn and Qfninv

• These function are now in the Misc_Matlab zip file on the web site

function [Qout]=Qfn(x)% Qfn(x) = 0.5 * erfc(x/sqrt(2));Qout = 0.5 * erfc(x/sqrt(2));

function [x]=Qfninv(Pe)% For Qfn(x) = 0.5 * erfc(x/sqrt(2));% The inverse can be found asx=sqrt(2)*erfcinv(2*Pe);

20

21

Matched Filter PCM (1)

• A matched filter is the time reversed version of the transmitted signal p(t)

TkTtpatx kk 2

tTpKth

2

dTpKTtpatTpKTtpaty2222 100

'''0 22

dpTTtpKaty

'''00

dptpKatRKaTty pp

• By definition, the maximum of the autocorrelation occurs at t=0. Therefore, the output is maximized at y(T).

22


• If we define the constant K as

dp

dptpa

RtR

aTtypp

pp

200 0

• The maximum at t=0 becomes

• And subsequent samples at time k x T are

dpRK ppeq

201

0aTy

1 kaTky

23


• The noise power for a matched filter is

dTpKtnthtntnr 2

tTpKth

2

dTpKtndTpKtnEtnE r 222

ddTpTptntnKEtnE r 2222

ddTpTptntnEKtnE r 22

22

ddTpTpNKtnE r 222

022

24


• The noise power output becomes

– Using the definition of K

• For eqT=1/req, this is the minimum bandwidth; therefore, the output achieves the maximum SNR

tTptTpKth

eq 21

2

dTpKNtnE r

2202

22

2

12

22 0

02

2

022 eq

eqeqr

rNN

dTpNtnE

25


(a) received pulse (b) impulse response (c) output pulse: Figure 11.2-6

Matched filtering with rectangular pulses

26

Matched Filter for NRZ

• An integrate and dump circuit can provide the optimal matched filter output at times k x T

+Vdc

-Vdc

Vout

R1Vin

C2

V-

V+

Sync

27

Properties of Matched and Wiener Filters

• See ECE3800 Notes– Review from Chapter 9

Matlab PulseDetect.m

• See simulation

28

29

Defining a Filter for Pulses

• We want to minimize or zero inter-symbol interference (ISI)

• We want a frequency band limited filter

– Allowable signal rates with as the excess bandwidth

k

dk Tkttpaty

,2,0

01TTt

ttp

fBfP 0

TBBandrwithrBwhere 2

0,2

BrBforBr 2,2

Tr 1

30

Defining a Filter for Pulses

• Possible solutions

,2,0

01TTt

ttp

fBfP 0

trtptp sinc

fforfPtp 0

10

dffPp

• Therefore we select

rf

rfPfP rect1

These are considered the Nyquist conditions for the filter

Tr 1

31

Cosine Spectral Shaping

• A candidate filter is (with with as the excess BW)

2rect

42cos

4fffP

From Chap 2Raised cosine

pulse

Raised cosine pulse. (a) Waveform (b) Derivatives(c) Amplitude spectrumFigure 2.5-7

Convolving

• Raised Cosine Convolution with Bandlimited Spectrum

32

rf

rfPfP rect1

TBBandrwithrBwhere 2

0,2

20

22242cos1

21

2

rf

rfrrfr

rfr

fP

trt

ttp

sinc412cos

2

• Transforming to the time domain filter

33

Nyquist/Raised Cosine Pulse Shaping

GNU FDL:Oli Filth, Raised Cosine Filter, Impulse Response, en.wikipedia.org, 3 November 2005, Oli Filth

GNU FDL:Oli Filth, Raised Cosine Filter Response , en.wikipedia.org, 3 November 2005, Oli Filth

Tr

rABC

12

Tr

rABC

12

http://en.wikipedia.org/wiki/Raised-cosine_filter

Nyquist Filter

34

trt

ttp

sinc412cos

2

% function hnyq=nyquistfilt(alpha,M)% or% function hnyq=nyquistfilt(alpha,fsymbol,fsample,k)%% alpha roll-off% fsample rate% fsymbol rate% M = fsample/fsymbol (an integer value)% k is 1/2 the number of symbols in the filter% The filter length is euqal to 2*ceil(k*M)+1%% A discrete time cosine taperd Nyquist filter% Based on frederic harris, Multirate Signal Processing for Communications% Prentice-Hall, PTR, 2004. p. 89

MknMkforMn

MnMn

np

,sinc21

cos

2

r 2

sfnt

10 2

0 r

Mfr s

trtrtrtp

sinc21

cos2

Mntr

MATLAB Raised Cosine Filters

• rcosine– [NUM, DEN] = RCOSINE(Fd, Fs, ‘fir’, R)– FIR raised cosine filter to filter a digital signal with the digital

transfer sampling frequency Fd. The filter sampling frequency is Fs. Fs/Fd must be a positive integer. R specifies the rolloff factor which is a real number in the range [0, 1].

• rcosfir– B = RCOSFIR(R, N_T, RATE, T)– Raised cosine FIR filter. T is the input signal sampling period, in

seconds. RATE is the oversampling rate for the filter (or the number of output samples per input sample). The rolloff factor, R, determines the width of the transition band. N_T is a scalar or a vector of length 2. If N_T is specified as a scalar, then the filter length is 2 * N_T + 1 input samples.

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Communication Systems, 5e - Western Michigan Universitybazuinb/ECE4600/Ch11_3.pdf · Communication Systems, 5e ... Hill Companies, Inc. Permission required for reproduction or