Digital Communications I: Modulation and Coding Course

Spring - 2013Jeffrey N. Denenberg

Lecture 3c: Signal Detection in AWGN

Lecture 5 2

Last time we talked about:

Receiver structure Impact of AWGN and ISI on the

transmitted signal Optimum filter to maximize SNR

Matched filter and correlator receiver Signal space used for detection

Orthogonal N-dimensional space Signal to waveform transformation and

vice versa

Lecture 5 3

Today we are going to talk about:

Signal detection in AWGN channels Minimum distance detector Maximum likelihood

Average probability of symbol error Union bound on error probability Upper bound on error probability

based on the minimum distance

Lecture 5 4

Detection of signal in AWGN

Detection problem: Given the observation vector ,

perform a mapping from to an estimate of the transmitted symbol, , such that the average probability of error in the decision is minimized.

m̂im

Modulator Decision rule m̂imzis

n

zz

Lecture 5 5

Statistics of the observation Vector

AWGN channel model: Signal vector is deterministic. Elements of noise vector are i.i.d

Gaussian random variables with zero-mean and variance . The noise vector pdf is

The elements of observed vector are independent Gaussian random variables. Its pdf is

),...,,( 21 iNiii aaas

),...,,( 21 Nzzzz

),...,,( 21 Nnnnn

nsz i

2/0N

0

2

2/0

exp1

)(NN

p N

nnn

0

2

2/0

exp1

)|(NN

p iNi

szszz

Lecture 5 6

Detection

Optimum decision rule (maximum a posteriori probability):

Applying Bayes’ rule gives:

.,...,1 where

allfor ,)|sent Pr()|sent Pr(

if ˆSet

Mk

ikmm

mm

ki

i

zz

ikp

mpp

mm

kk

i

allfor maximum is ,)(

)|(

if ˆSet

z

z

z

z

Lecture 5 7

Detection …

Partition the signal space into M decision regions, such that MZZ ,...,1

i

kk

i

mm

ikp

mpp

Z

ˆ

meansThat

. allfor maximum is ],)(

)|(ln[

if region inside lies Vector

z

z

z

z

z

Lecture 5 8

Detection (ML rule)

For equal probable symbols, the optimum decision rule (maximum posteriori probability) is simplified to:

or equivalently:

which is known as maximum likelihood.

ikmp

mm

k

i

allfor maximum is ),|(

if ˆSet

zz

ikmp

mm

k

i

allfor maximum is )],|(ln[

if ˆSet

zz

Lecture 5 9

Detection (ML)…

Partition the signal space into M decision regions, .

Restate the maximum likelihood decision rule as follows:

MZZ ,...,1

i

k

i

mm

ikmp

Z

ˆ

meansThat

allfor maximum is )],|(ln[

if region inside lies Vector

z

z

z

Lecture 5 10

Detection rule (ML)…

It can be simplified to:

or equivalently:

ik

Z

k

i

allfor minimum is ,

if region inside lies Vector

sz

z

).( ofenergy theis where

allfor maximum is ,2

1

if region inside lies Vector

1

tsE

ikEaz

Z

kk

k

N

jkjj

i

r

Lecture 5 11

Maximum likelihood detector block diagram

1,s

Ms,

12

1E

ME2

1

Choose

the largestz m̂

Lecture 5 12

Schematic example of the ML decision regions

)(1 t

)(2 t

1s3s

2s

4s

1Z

2Z

3Z

4Z

Lecture 5 13

Average probability of symbol error

Erroneous decision: For the transmitted symbol or equivalently signal vector , an error in decision occurs if the observation vector does not fall inside region . Probability of erroneous decision for a transmitted symbol

or equivalently

Probability of correct decision for a transmitted symbol

sent) inside lienot does sent)Pr( Pr()ˆPr( iiii mZmmm z

sent) inside liessent)Pr( Pr()ˆPr( iiii mZmmm z

iZ

iiiic dmpmZmP zz z z )|(sent) inside liesPr()(

)(1)( icie mPmP

imis

z iZ

sent) and ˆPr()( iiie mmmmP

Lecture 5 14

Av. prob. of symbol error …

Average probability of symbol error :

For equally probable symbols:

)ˆ(Pr)(1

i

M

iE mmMP

)|(1

1

)(1

1)(1

)(

1

11

M

i Z

i

M

iic

M

iieE

i

dmpM

mPM

mPM

MP

zzz

Lecture 5 15

Example for binary PAM

)(1 tbEbE 0

1s2s

)|( 1mp zz)|( 2mp zz

2

)2(0

N

EQPP b

EB

2/

2/)()(

0

2121

NQmPmP ee

ss

Lecture 5 16

Union bound

Let denote that the observation vector is closer to the symbol vector than , when is transmitted.

depends only on and . Applying Union bounds yields

zisks

is

kiA

Union bound

The probability of a finite union of events is upper bounded by the sum of the probabilities of the individual events.

),()Pr( 2 ikki PA ssis

ks

M

ikk

ikie PmP1

2 ),()( ss

M

i

M

ikk

ikE PM

MP1 1

2 ),(1

)( ss

Lecture 5 17

Union bound:

Example of union bound

1

1s

4s

2s

3s

1Z

4Z3Z

2Z r 2

1

1s

4s

2s

3s

2A r

1

1s

4s

2s

3s

3A

r

1

1s

4s

2s

3s

4A

r2 22

432

)|()( 11

ZZZ

e dmpmP rrr

2

)|(),( 1122

A

dmpP rrss r 3

)|(),( 1132

A

dmpP rrss r 4

)|(),( 1142

A

dmpP rrss r

4

2121 ),()(

kke PmP ss

Lecture 5 18

Upper bound based on minimum distance

2/

2/)exp(

1

sent) is when , than closer to is Pr(),(

00

2

0

2

N

dQdu

N

u

N

P

ik

d

iikik

ik

ssszss

2/

2/)1(),(

1)(

0

min

1 12

N

dQMP

MMP

M

i

M

ikk

ikE ss

kiikd ss

ik

kikidd

,

min minMinimum distance in the signal space:

Lecture 5 19

Example of upper bound on av. Symbol error prob. based on union

bound

)(1 t

)(2 t

1s3s

2s

4s

2,1d

4,3d

3,2d

4,1dsEsE

sE

sE

4,...,1 , iEE siis

s

ski

Ed

ki

Ed

2

2

min

,

Lecture 5 20

Eb/No figure of merit in digital communications

SNR or S/N is the average signal power to the average noise power. SNR should be modified in terms of bit-energy in DCS, because: Signals are transmitted within a symbol

duration and hence, are energy signal (zero power).

A merit at bit-level facilitates comparison of different DCSs transmitting different number of bits per symbol.

b

bb

R

W

N

S

WN

ST

N

E

/0

bR

W: Bit rate

: Bandwidth

Lecture 5 21

Example of Symbol error prob. For PAM signals

)(1 t0

1s2s

bEbE

Binary PAM

)(1 t0

2s3s

52 bE

56 bE

56 bE

52 bE

4s 1s4-ary PAM

T t

)(1 t

T1

0