tele3113 wk11tue

6 Oct. 2009 TELE3113 1

TELE3113 Analogue and Digital Communications –Detection Theory

Wei [email protected]

School of Electrical Engineering and TelecommunicationsThe University of New South Wales

6 Oct. 2009 TELE3113 2

Digital Signal Detection

Receive filter

n(t)AWGN

si(t)r(t)

Decision device

Sampled at t=kTs

1 if y(kTs)>λ

y(t) y(kTs)

Thresholdλ

0 if y(kTs)<λ

At the receiving end of the digital communication system:

Noise power spectral density: Sn(ω)=η/2

≤≤−=≤≤+=

=0for 0 )(1for 0 )(

)(2

1

TtAtsTtAts

tsi

Polar NRZ Signaling

6 Oct. 2009 TELE3113 3

Suppose there are M possible signal symbols: {si} for i=1,…,M

We can represent these symbols in vector form

Similarly the noise vectors, and the received signal vectors,

Thus


rrnr

nsr iirrr

+=

ϕ1

ϕ2

ϕ3

1sr

3sr 2sr

4sr1rr

nr

nr

nr

nr

isr

6 Oct. 2009 TELE3113 4

Digital Signal DetectionIn each time interval, the signal detector makes a decision based on the observation of the vector so that the probability of correct decision is maximized.

rr

Consider a decision rule based on the posterior probabilities

)|21for ) vector received| ted transmit was signal(

rsP( ,M,, i rsP

i

irr

Krr

==

The decision is based on selecting the signal corresponding to the maximum set of posterior probabilities.

)(

)()|()|

rfsPsrf

rsP( iii r

rrrrr=

where is probability of si being transmitted and is the pdf function of the received signal vector .

This kind of decision is called maximum a posteriori probability (MAP) criterion

)rf(r

rr

Choose to maximize:

∑=

=M

1mms|rf()rf( where )() msP rrrrisr

)( isP r

6 Oct. 2009 TELE3113 5

If the M symbols are equally probable; i.e. for all i, the decision rule based on finding the signal that maximizes is equivalent to finding the signal that maximizes .


The decision based on the maximum of over M signal symbols is called the maximum-likelihood (ML) criterion

)| rsP( irr

)s|rf( irr

∑=

==M

1mms|rf()rf( where )()

)()()|(

)| mii

i sPrf

sPsrfrsP( rrrr

r

rrrrr

MAP criterion

)s|rf( irr

where is called the likelihood function.)s|rf( irr

MsP i /1)( =r

6 Oct. 2009 TELE3113 6

Thus {rk} are statistically independent Gaussian variables

where N is number of base vectors

and

Digital Signal Detectionnsr irrr

+=Recall:

For AWGN, the noise {nk} components are uncorrelated Gaussian variables which are statistically independent

ikkikkk s]nE[s]E[r , ]E[n =+== mean) (zero 0

( )22

][][ 22222 ησσησ ==→=−= nrn nEnEVariance

)s|f(r)s|rf( ikki

N

k 1=Π=

rr

η

σ

πη

σπ

/)(

)2/()(

2

22

ikk

nikk

sr

sr

n

e

e

−−

−−

=

=

1

21)s|f(r ikk

Take natural logarithm on both sides, gives

( ) ( )∑=

−−−

=N

kikk srN

1

21ln2

lnη

πη)s|rf( irr

6 Oct. 2009 TELE3113 7

Digital Signal Detection( ) ( )∑

=

−−−

=N

kikk srN

1

21ln2

lnη

πη)s|rf( irr

With ln(•) is a monotonic function, the maximum of over is equivalent to finding the signal that minimizes the Euclidean distance:

)s|rf( irr

isr

isr

∑=

−=N

kikki srsrD

1

2)(),( rr

So, the ML decision criterion (maximize over M signal symbols, i.e. i=1,…M) reduces to finding the signal that is the closest in distance to the received signal vector .

isr)s|rf( i

rr

rr

Example: 3 signal symbols.

Note the decision regions formed by the perpendicular bisectors of any two signal symbols.

6 Oct. 2009 TELE3113 8

Digital Signal DetectionDetection error will occur when the received signal vector falls into the decision region of other signal symbols. This is due to the presence of strong random noise.

Consider there are two signal symbols s1 and s2 , which are spaced d apart. The decision boundary is their perpendicular bisector.

As, , the uncertainty of the received signal vector is mainly contributed by the random noise ( ), which is Gaussian-distributed, around the signal symbol.

rr

nsr irrr

+= rr

nr

s2s1

decision boundary

d

noise distribution

( )isr rr−=

6 Oct. 2009 TELE3113 9


s2s1

decision boundary

d

noise distribution

Assume s1 is sent,

at the receiver, the probability of detection error is:

( )

∫

∫∞

−

∞

=

=

−>−=

2/

/

21

121

12

21

)(

)(

d

n

d/

dne

dns|nf

ssrsrP

ssP

η

πη

sent is detected is

rr

rrrrr

rr/η) n (

/η) sr (

e)s|nf()s)|s-rf(() s|rf(

e)s|rf(

2

21

1

1With

1111

1

r

rr

rrrrrrr

rr

−

−−

===

=

πη

πη

∫∞

−=x

y dyexQ 2/2

21)(πUsing and let η

ny 2=

=

= ∫∞

−

η

πη

2

21)sent is detected is (

2/

2/12

2

dQ

dyessPd

yrr

6 Oct. 2009 TELE3113 10

Digital Signal DetectionFor a signal symbol set: Misi ,...1for }{ =

r

Detection error probability is

][2

][][

11

1

i

M

i

M

kki

ik

M

iiie

sPss

Q

sPsPP

rrr

rr

sent | detection erroneous

∑∑

∑

==≠

=

−≤

=

η

If all signal symbols are equally probable, i.e. MsP i /1][ =r

sent | detection erroneous

∑∑

∑

==≠

=

−≤

=

M

i

M

kki

ik

M

iiie

ssQ

M

sPsPP

11

1

21

][][

η

rr

rr

6 Oct. 2009 TELE3113 11

Digital Signal DetectionCalculation of error probabilities:

(a) Antipodal signaling: ( ) ( )0, ;0, 21 EsEs −=+=

E

s

+

1

E

s

−

2

[ ]

=

+

=

+

≤

+=

η

η

ηη

EQ

sPsPEQ

sPEQsPEQ

sPssPsPssPPe

2

)()(2

)(2

2)(2

2

)()|()()|(

21

21

221112

detectedisdetected is

Example: for NRZ signaling which takes amplitude either +A or 0. For bit interval Tb, the energy per bit Eb=A2Tb.

signal symbol energy=E

=

=

ηηb

eEQTAQP

2

6 Oct. 2009 TELE3113 12

2 sE+

[ ]

=+

=

+

≤

+=

ηη

ηη

EQsPsPEQ

sPEQsPEQ

sPssPsPssPPe

)()(

)(22)(

22

)()|()()|(

21

21

221112

detectedisdetected is

Digital Signal Detection(b) Orthogonal signaling: ( ) ( )EsEs +=+= ,0 ;0, 21

(c) Square signaling:( ) ( ) ( ) ( )EEsEEsEEsEEs −−=+−=++=−+= , ;, ;, ;, 4321

E

s

+

1

E+

E+E−

E− 1s

3s

4s

2s

+

=

+

+

≤

=

∑

∑

=

=

ηη

ηηη

EQEQ

EQEQEQsP

sPssPP

ii

iiiie

222

2

2222

22)(

)()|detectednot is (

4

1

4

1

6 Oct. 2009 TELE3113 13

Digital Signal DetectionIntegrate-and-Dump detector

r(t)=si(t)+n(t)

≤≤−=≤≤+=

=0for 0 )(1for 0 )(

)(2

1

TtAtsTtAts

tsi

[ ]

++

=+= ∫+

0for )(1for )(

)()()(2

10

0 o

oTt

ti nta

ntadttntstzOutput of the integrator:

where

( )

∫

∫

∫

+

+

+

=

−=−=

==

Tt

to

Tt

t

Tt

t

dttnn

ATdtAa

ATAdta

0

0

0

0

0

0

)(

2

1

6 Oct. 2009 TELE3113 14

Digital Signal Detectionno is a zero-mean Gaussian random variable.

{ } { }∫∫++

==

=Tt

t

Tt

to dttnEdttnEnE

0

0

0

0

0)()(

{ } { }

{ }

( )

22

2

)()(

)(

0

0

0

0

0

0

0

0

0

0

2

22

Td

dtdt

dtdntnE

dttnEnEnVar

Tt

t

Tt

t

Tt

t

Tt

t

Tt

t

Tt

toon

o

o

o

ηεη

εεδη

εε

σ

==

−=

=

===

∫

∫ ∫

∫ ∫

∫

+

+ +

+ +

+

( ) )/()2/( 222 12

1 T

nn e

Tef on

o

o

ηασα

πησπα −− ==pdf of no:

6 Oct. 2009 TELE3113 15


≤≤−=≤≤+=

=0for 1for

TtAtsTtAts

tsi 0)(0)(

)(0

1

AT

s

+

1

AT

s

−

0 As

0We choose the decision threshold to be 0.

Two cases of detection error:

(a) +A is transmitted but (AT+no)<0 no<-AT

(b) -A is transmitted but (-AT+no)>0 no>+AT

Error probability:

[ ]

Tudue

APAPdT

e

dT

eAPdT

eAP

APAATnPAPAATnPP

TA

u

AT

T

AT

TAT T

ooe

ηα

π

απη

απη

απη

η

ηα

ηαηα

2 2

)()(

)()(

)()|()()|(

2

2

2

22

2

2/

)/(

)/()/(

==

−+=

−+=

−>+−<=

∫

∫

∫∫

∞ −

∞ −

∞ −−

∞−

−

Q

( )

∫

∫

=

=

=

=

∞ −

T

bb

x

u

e

dtAEEQ

duexQTAQP

0

2

2/2

2

2 2

2

Q

Q

η

πηThus,

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