Digital Communication I: Modulation and Coding Course-Lecture 5 potx

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 spac

Digital Communications I: Modulation and Coding Course

Term 3 - 2008 Catharina Logothetis

Lecture 5

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

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

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.

mˆi

Statistics of the observation Vector

 AWGN channel model:

Gaussian random variables with zero-mean and

variance The noise vector pdf is

independent Gaussian random variables Its pdf is

), ,

,(z1 z2 z N



z

) , , ,

(n1 n2 n N



n

n s

/ 0

exp

1)

(

N N

/ 0

exp

1)

|

(

N N



allfor ,)

|sentPr(

)

|sentPr(

ifˆ

Set

M k

i k m

m

m m

k i

i

k p

m

p p

m m

k k

i





allfor maximum

is

,)

(

)

|(

ifˆ

Set

z

z z z

Detection …

 Partition the signal space into M decision

regions, such that Z , ,1 ZM

i

k k

i

m m

i

k p

m

p p

.all

for maximum

is

],)

(

)

|

(ln[

ifregion

insidelies

Vector

z z z

z z

Detection (ML rule)

 For equal probable symbols, the optimum

decision rule (maximum posteriori probability)

p

m m

is ),

|(

ifˆ

Set

z z

i k m

p

m m

is )],

|(ln[

ifˆ

Set

z z

i k m

p

Z



ˆ

meansThat

allfor maximum

is )],

|(ln[

ifregion

insidelies

Vector

z z z

Z k

i



ifregion

insidelies

Vector

s z

z

)

(of

energy the

iswhere

allfor maximum

is

,21

ifregion

insidelies

Vector

1

t s E

i k E

a z

Z k

N

j

kj j

Maximum likelihood detector block

Schematic example of the ML decision regions

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

or equivalently

sent)inside

lienot does

sent)Pr(

Pr(

)ˆ

sent)inside

liessent)Pr(

Pr(

)ˆ

ˆPr(

)

Av prob of symbol error …

 Average probability of symbol error :

 For equally probable symbols:

) ˆ

( Pr )

| (

1 1

) (

1 1

) (

1 )

(

1

1 1

M

i

i e

E

i

d m

p M

m

P M

m

P M

M P

z z

z

Example for binary PAM

2

Q P

2

/

2

/)

()

(

0

2 1

P m

Union bound

the symbol vector than , when is transmitted

The probability of a finite union of events is upper bounded

by the sum of the probabilities of the individual events

),()

i k i

i k

| ( )

Z Z Z

1 2

Upper bound based on minimum distance

2

/)

exp(

1

sent)is

when ,

than closer to

isPr(

),(

0 0

du N

u N

P

ik d

i i

k i

k

ik 

s s

s z

s s

1 (

) , (

1 )

M

P M

i k

k i

ik

d  s  s

ik k

Example of upper bound on av Symbol

error prob based on union bound

d

4 , 3

d

3 , 2

d

4 , 1

d E s s

i

E d

k

i

E d

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:

and hence, are energy signal (zero power)

different DCSs transmitting different number of bits per symbol

b

b b

R

W N

S W

N

ST N

: Bit rate : Bandwidth

Example of Symbol error prob For PAM

signals

) (

1 t



T

1