Tài liệu Modulation and coding course- lecture 5 doc

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

Period 3 - 2007 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

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

), ,

,( i1 i2 iN

s

), ,

,(z1 z2 z N

=

z

) , ,

, (n1 n2 n N

=

n

n s

z = i +

2/

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)

is simplified to:

or equivalently:

which is known as maximum likelihood

i k m

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

Detection rule (ML)…

„ It can be simplified to:

or equivalently:

i k

insidelies

Vector

s z

z

allfor maximum

is

,21

ifregion

insidelies

Vector

1

i k E

a z

Z

k N

j

kj j

Maximum likelihood detector block

Schematic example of ML decision regions

)(

1 t

ψ

)(

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(

)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

i M

| (

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

/)

()

(

0

2 1

P m

Union bound

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

„ Applying Union bounds yields

The probability of a finite union of events is upper bounded

by the sum of the probabilities of the individual events

),()

i k i

P

1

2( , ) )

i k k

i k

Union bound:

Example of union bound

1 ψ

)

| ( )

Z Z Z

Upper bound based on minimum distance

2

/)

exp(

1

sent)is

when ,

than closer to

isPr(

),(

0 0

2 0

2

N

d Q

du N

u N

P

ik d

i i

k i

k

s s

s z

s s

1 (

) , (

1 )

M

P M

M

P

M

i M

i k k

i k

k i

ik

d = s −s

ik k i k

Example of upper bound on av Symbol

error prob based on union bound

)(

1 t

ψ

)(

d

4 , 3

d

3 , 2

d

4 , 1

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:

„ 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

b b

R

W N

S W

N

ST N

: Bit rate : Bandwidth

Example of Symbol error prob For PAM

signals

) (

1 t

ψ

T

1