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Digital CommunicationSystems

Dr. Shurjeel WyneLecture 6

Baseband Demodulation andDetection

Today we are going to talkabout: Signal detection in AWGN channels

MAP and Maximum likelihood detector Minimum distance detector

Average probability of symbol error Union bound on error probability Upper bound on error probability based onthe minimum distance

Detection of Binary Signal in Gaussian Noise

For any binary modulation, the transmitted signal over asymbol interval (O,T) is:

{s o( t) O

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Receiver FunctionalityThe recovery of signa l a t the receiver consist of two parts:1. Waveform-to-sample transformationo Demodulator followed by a samplero At the end of each symbol duration T, the output of the sampleryields a sample z{T), called test~Wtii~iic

z(T) = a,(T) + no(T) = J, 2 (3.3)Where a,{T) is the desired signal component,and no{T) is the noisecomponent

2. Detection of symbolo Assume that input noise is a zero-mean Gaussian random

process and receiving filter is linear, then no(1) is a Gaussianrandom variable

1 l - 2 1( " n o o J 2Jp C n o ) ~ r;- ex pCY ov2;r (3.4)

Find Filter Transfer Function Ho(f) 1 i i i i : O~jedilfe: To maximizes (SINh :: Expressing signal a,(t) at filter output in terms of filter transf function H(f)

a iel) ~ [H(f) S(f)where S(f) is the Four ier t ransform of input s igna l s(t )

Output noise power can be expressed as:, , ~ ~ . ! ! .. . . 2 .. _ f o o IH(! ) 1 2 df2 -00

Expressing (SINh as:1 [H(f) S(f) e j2HjT df 1 2

~ o [IH(f)12 df

ma x [ ~ L 2E

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Probabilities Review P[sol, P[s,l --+ a priori probabilities

These probabi li ties are known before transmission P[zl

probabilit y of the received sample p(z lso) , P(ZIS,)

condit ional pdf of received sample z, conditioned on the eventthat s, was transmit ted ( i = 0,1) . A lso cal led like lihood of s,

P[solz]' P[s, lz l --+ a posteriori probabilities After examining the sample z, we make a refinement of our

previous knowledge about the transmitted symbol, s, P[s,lso l, P[SOIS, l

Probabilit y of a wrong decision (error) P[slls,l, P[solsol

Probabilit y of a correc t decision

MAP detect ion rule:

I

L(Z ) 11 p(zl s ,) : ' pe so ) I = l ikel ihood rat io tes t (LRT)- p(zlso) : 0 P(s , ) 1

Maximum likelihood (ML) detect ion ruleWhen the two signals, so(t) and s,(t), are equally likely, i.e., peso! =PIs,) = 0.5, then the maximum a posterior i (MAP) decision rulesimplifies to:

L(z) = p(zls , ) : ' 1p( z : 0

Hence:

Taking the natural log of both sides will giveHI

A=ln {L(z)}= ze a l ~ a o ) _ (a ,' -~5) > 00"0 20 -0 z(o,~oo) > (0,2_0;) (0,+00)(0,-00)0"0 < 2 0 "~ 2 0 "~

HO

How to Choose the decision threshold? :::: Optimum decision rule based on Maximum a posteriori probabili / .:: If

P[solz] > P[s, lz ]- -> Ho else

P[s, lz ] > P[so lz]- -> H,

Proble is that a posteriori probabilities are not known. Solut io: Use Baye's theorem to rep lace a poster ior i p robab il it ies:

P ( '. 1 z) - p ( z l S i) P ( S i )b, Z - p(z)

Substituting the pdfs l [ J 2 JI I z- a op( z Iso ) =---exp -- --(J"o& 2 (J"ol ( J ' JI I z - a ,p(zls,)=---exp -- --(J"o& 2 (J"o

HI HI_I exp[__ I (z- a l' ]L(z)= p(zls ,) > I = : > er o& 2e ro '

p(zlso) < _1_exp[ __ I_(z-aol ']HO ero& 2e ro

> O"~(a,+a,,)(~ -a,,) > (a , + a o) '"

z 2 Z < --2---ro< 20"0 (a , -a,,)n ; H o

'III = (a, + 0,)12 isthe optimum threshold level for minimizing the probabilityof ma king a n inc or rec t d ec ision in thi s binary ca se.. Example: For antipodal Signal, s,(t) = - So (t) a, = - aoH I>z 0