principles of -ary detection theory m - athanassios manikas 2 2005 - ln - trans... · advanced...

IMPERIAL COLLEGE LONDON,DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING.

COMPACT LECTURE NOTES on ADVANCED COMMUNICATION THEORY.Prof. Athanassios Manikas, Autumn 2005

Principles of -ary Detection TheoryM

Outline:

ì QHypothesis testing in -ary Communication SystemsìM-ary Decison CriteriaìBasic Concepts on Optimum ReceiversìOptimum Receivers using Matched FiltersìOptimum Receivers based on Signal Constellation (Orthogonal signals and the

"Approximation Theorem"; Gram-Schmidt Orthogona-lization; Energy & CrossCorrelation of -ary signals; Signal Constellation; Distance between two -ary signals)Q Q

ìì QBasic analysis of orthogonal and biorthogonal -ary communication systems.ìOrthogonal symbol sets: 64-ary Walsh-Hadamard.

Advanced Communication Theory Compact Lecture Notes

Principles of M-ary Detection Theory 3 A. Manikas

1. Introduction• A digital modulator is described by different channel symbols.Q

These channel symbols are ENERGY SIGNALS of duration .X-=

Digital Modulator:

Mapping binary digits to channel symbols

up conversion from baseband to bandpass

Digital Demodulator:

Mapping channel symbols to binary digits

down conversion from bandpass to baseband

Detector with a decision device

Note:It is common practice to ignore the up/down conversion and to workin 'baseband'.



• If Binary Digital Modulator Binary Communication SystemQ œ # Ê ÊIf M-ary Digital Modulator M-ary Communication SystemQ # Ê Ê



..01..101..00..

0..00 s t1( )=t

0..01 s t2( )=t

1..11 s tM( )= t

ts t( )=

Tcs

Tcs

Tcs

Tcs

: H ,Pr(H )1 1

: H ,Pr(H )2 2

: H , Pr(H )M M

..00..101..10..

0..00 s t1( )=t

0..01 s t2( )=t

1..11 s tM( )= tt

r t s t t( )= ( )+n( )

Tcs

Tcs

Tcs

Tcs

: D , Pr(D )1 1

: D , Pr(D )2 2

: D , Pr(D )M M

Channel

s t( )

r t( )=Detectorwith a

DecisionDevice

(DecisionRule)

D



• In this topic we will focus on the following block of a digital communication system:

Tcs

Tcs

Tcs

Tcs


DecisionDevice

(DecisionRule)

D

This is the 'heart' of a communication system andsome elements of detection/decision theory will beemployed for its investigation.



• Detection Theory:is concerned with determining the of a signal inexistencethe presence of noise - and can be classified as follows:



N.B.:i) Adaptive or Learning:

ì system as a function of input;parameters changeì to analyze;difficultì mathematically complex

ii) Non-Parametric: ì fairly because these are based onconstant level of performance

general assumptions on the input pdf;ì .easier to implement

iii) Consider a parametric detector which has been designed for Gaussian pdf input.

ì if : then parametric's performance isinput is actually Gaussiansignificantly better;

ì if but : then non-parametricinput=Gaussian still symmetric/detector performance may be much better than the performance ofthe parametric detector



2. Hypothesis Testing in an -ary Comm SystemQ• A Hypothesis ´ a statement of a possible condition• In an ary Comm. System we have hypotheses:Q - Q

CHANNEL

-B +B f

1r(t)

n (t)i

Decision rule=?(To determine which

signal is present)= >3( )Å

one out of M signals

D 3

ì hypotheses:

: is present with probability : is present with probability

... ...............

ÚÝÝÝÛÝÝÝÜ

çH H1 1 1= Ð>Ñ Ð Ñ

is being sent

PrH H2 2= Ð>Ñ Ð ÑPr #

.....: is present with probability H HQ Q Q= Ð>Ñ Ð ÑPr



• The of the observed signalstatistics =<Ð>Ñ = Ð>Ñ3 8Ð>Ñare affected by the presence of or or= Ð>Ñ = Ð>Ñ ÞÞÞ = Ð>Ñ" # Q

If are = Ð>Ñß3 a3 known

then are their distributions known

and the problem to make a decisionis translated about afterone of the distributionsQ having observed .<Ð>Ñ

This is called 'Hypothesis Testing'



3. Terminology

• H ..., a priori probabilitiesPrÐ ÑßH1 PrÐ Ñß À# PrÐ ÑHQ

• Let .r = observation variable

Then we have Q Conditional Probabilities PrÐ Ña3 − Ò"ß ÞÞßQÓH3/rknown as PROBABILITIESa POSTERIOR

Ðsince they are calculated AFTER the experiment isperformed .Ñ



• : difficult to find PrÐ Ñ a3H3/r .A more natural approach is to find

H

since in general pdf are known or can be found.

PrÐ Ñ a3

a3

r/ 3 ,

Å

r/H3

• Let us change the notation ÚÛÜ

:3Ð Ñ

Ð Ñ

r

p r

=

=

pdf

pdf

r/

r

H3a3

• The functions are known asQ :3Ð Ñr a3 Likelihood Functions

• Note: The ratio is known as the : Ð Ñ: Ð Ñ3

4

rr Likelihood Ratio



4. 'Max. A POSTERIOR PROB' Decision Criterion (MAP)

• DECISION choose hypothesis : H3 i e. DÞ 3

iff Pr PrÐ Ñ Ð Ña4 À 4 Á 3H H3 4/r /r

i e. :i e. :

... ....................

... ....................

ÚÝÝÝÝÛÝÝÝÝÜ

H H HH

1 Þ HÞ H

"

#

iff H iff H

Pr PrPr PrÐ Ñ Ð Ña3 Á "Ð Ñ Ð Ña3 Á #

" 4

4

/r /r/r /r2 2

H HQ Qi e. :Þ HQ iff Pr PrÐ Ñ Ð Ña3 Á Q/r /rH4

This decision is known as criterionMAP

• The observation space is partitioned into regions Q ß ß ÞÞÞßR R R" # Q

Then is chosen iff a given observation H R3 3r −



• By using Bayes rule PrÐ ÑH3/r = p r .p r

3Ð ÑÐ ÑPrÐ ÑH3

in conjunction with the MAP Criterion D3: iff Pr PrÐ Ñ Ð Ña4 À 4 Á 3H H3 4/r /rwe have

p r .p r p r

p r .3 4Ð ÑÐ Ñ Ð Ñ

Ð ÑPr PrÐ Ñ Ð ÑH H3 4 a4 À 4 Á 3

i.e. MAP: : iff D3 p r . p r .3 4Ð Ñ Ð ÑPr PrÐ Ñ Ð Ñ a4 À 4 Á 3H H3 4>

or equivalentlyi.e. MAP: D3 3: K œ max

4a bp r .4 4Ð Ñ PrÐ Ñß a4H



5. M-ary Decision Criteria• the sets of parameters P1 and P2 where:Consider

P1 denotes the set of parameters H , H , ..., HPr Pr PrÐ Ñ Ð Ñ Ð Ñ1 2 Q

P2 represents the set of costs weights Î G ß a3ß 434

associated with the transition probabilities Pr(D H3 4l Ñ i.e. one weight for every element of the transition matrix )Ð …

• the likelihood functionsEstimate identifyÎ ,: Ð<Ñ : Ð<Ñß ÞÞÞß : Ð<Ñ" # Q

where : Ð<Ñ3 =˜ pdf<lH3ß a3



• : choose the hypothesis (i.e. with the Decision H Di 3Ñ maximum G3Ð<Ñ

where depends on the chosen criterion as followsK Ð<Ñ3 :

C1) CriterionBAYES C2) ( ) CriterionMinimum Probability of Error pmin( ) e

C3) CriterionMAP C4) CriterionMINIMAX C5) ( ) CriterionNewman-Pearson N-P

C6) ( ) CriterionMaximum Likelihood ML



C if then Criterion. That is:P1=knownP2=known1) œ BAYES

choose Hypothesis with the max G4Ð<Ñ Ð Ñ ‚œ̃ ‚weight H pdf4 4 <lPr H4

C ,C3)# if then Criteria. That is:P1=knownP2= knownœ ?8

Ð Ñmin p or MAP e

choose Hypothesis with the max G 4Ð<Ñ Ð Ñ ‚œ̃ Pr H pdf4 <lH4

C4) if then Criterion. That is:P1= knownP2=known œ ?8 MINIMAX

choose Hypothesis with the max G 4Ð<Ñ ‚œ̃ weight pdf4 <lH4



C5) if then Criterion:P1= knownP2= knownœ ?8

?8Newman-Pearson (N-P)

choose Hypothesis with the max G constraint4Ð<Ñ by solving a optimisation problem

C6) if an is required then any informationapproximate/initial solution about the sets of parameters P1 and/or P2 can be ignored.

In this case the ( ) Criterion should beMaximum Likelihood ML used. That is:

choose Hypothesis with the maximum K < œ4 <l( ) pdf˜ H4



Note-1: weight4 œ̃

Á

G G! a b3œ"

Q

34 44

i j

or, simply, weight4 œ !a b3œ"

Q

34 44G G

(since the term for is equal to zero)3 œ 4

Note-2: Sometimes, for convenience, will be used (i.e. ( ))K K ´ K <4 4 4



Example-1:

• By solving ( ) the decision threshold K < œ K Ð<Ñ <4 3 >2ß43

can be estimated• area-A = Pr(D |H ) and area-B = Pr(D |H )4 3 3 4

• Pr(D H ) = symbol error probability : œ ß

3 Á 4

/ß-= 4 34œ" 3œ"

Q Q! !



Example-2:





Example-3:Advanced Communication Theory Compact Lecture Notes




..01..101..00..

0..00 s t1( )=t

0..01 s t2( )=t

1..11 s tM( )= t

ts t( )=

Tcs

Tcs

Tcs

Tcs

: H ,Pr(H )1 1

: H ,Pr(H )2 2

: H , Pr(H )M M

..00..101..10..

0..00 s t1( )=t

0..01 s t2( )=t

1..11 s tM( )= tt

r t s t t( )= ( )+n( )

Tcs

Tcs

Tcs

Tcs

: D , Pr(D )1 1

: D , Pr(D )2 2

: D , Pr(D )M M

Channel

s t( )


DecisionDevice

(DecisionRule)

D



6. BASIC CONCEPTS on OPTIMUM RECEIVERSConsider a -ary communication model in which one of Q Qsignals is received in the times t3Ð Ñß for 3 œ "ß #ß ÞÞQßinterval in the presence of white noiseÐ Ñ!,T-=

i.e.

+ n t<Ð Ñt œ Ð Ñ ! Ÿ > Ÿ X

Ú ÞÝ áÝ áÛ ßÝ áÝ áÜ às t"Ð Ñs t#Ð ÑÞÞÞÞÞÞs tQÐ Ñ

-=

where denotes the received (observable) signal. <Ð Ñt



= Ð>Ñ À3

8Ð>Ñ À



<Ð>Ñ œ = Ð>Ñ 8Ð>Ñ3



6.1. How to get a probabilistic description of a continuoussignal (Continuous Sampling):

1 assume that amplitude samples of the received signalÑ P with sampling frequency )Ð "

>?

<Ð Ñt ß ! Ÿ > Ÿ X ß-=

are available i.e. < < <" # P, , ..., ,

where ,..., L<5 5œ "

Ú ÞÝ áÝ áÛ ßÝ áÝ áÜ às"5s#5ÞÞÞÞÞÞsQ5

n ; 5 œ



or, in more compact form

+ < œ

Ú ÞÝ áÝ áÛ ßÝ áÝ áÜ às"s#ÞÞÞÞÞÞsQ

n

where < œ < ß < ß ÞÞÞß <c d" # PT

8 œ 8 ß 8 ß ÞÞÞß 8c d" # PT

=3 3" 3# 3Pœ = ß = ß ÞÞÞß =c dT

2 take the limit as Ñ P > PÞ > XÄ _ß Ä !ß Ä? ? -=



< = 8œ 3 ( )P samplesAdvanced Communication Theory Compact Lecture Notes


6.2. Optimum -ary ReceiversQObjective: to design a receiver which operates on and<Ð Ñtchooses one of the following two hypotheses:

or

+ +

+ +

+

Ú ÚÝ ÝÝ ÝÝ ÝÝ ÝÛ ÛÝ ÝÝ ÝÝ ÝÝ ÝÜ Ü

H H" " " ": :

: :

:

<Ð Ñ Ð Ñ

<Ð Ñ Ð Ñ

<Ð Ñ Ð Ñ

t n t

t n t

t n t

œ œ

œ œ

œ

s tÐ Ñ

H H2 2s t# #Ð ÑÞÞÞÞÞ ÞÞÞÞÞÞ ÞÞÞÞÞ ÞÞÞÞHQ s tQÐ Ñ

<

<

s n

nsÞÞ

HQ : < œ sQ + n

Optimum Decision Rule: depends on the likelihoodfunctions : Ð Ñ3 r

amplitude pdf of ? <Ð>Ñ œ



Optimum Receiver - Initial Conceptual Structure:CHANNEL

Compute likelihoodfunctions

M

-B +B f

1

OPTIMUM RECEIVER

n (t)i

s (t)i ∆t Compare variables and get max

Di?r(t)

Noise: PSD PSD rectn nN N f

iÐ Ñ œ Ð Ñ œf f ! !

# # #F Ê š ›Therefore, the received signal is sampled at intervals ?t œ "

#F

Sampling: the samples are uncorrelated and statisticallyindependent



Example - MAP Receiver:



Likelihood functions: Gaussian multivariable distribution

Mean and Variance:X X{r n× Ö ×œ œ s s3 3 +

VAR { { {r r r r r× × ×œ X X X ŸŠ ‹Š ‹X

œ œXÖ ×88X #8 œ 5 ˆ ˆL LN B!



Likelihood function : Ð Ñ3 r : (based on the above)

: Ð Ñ œ3 r . "

#

#É 15 5

8#

#8

L

exp Ÿ Ð Ñ Ð Ñr rs s3 3T

However N .B

t

Ú ÞÛ ßÜ à5

?5

#!

"#

##

n

B

nN. t

œ

œœÊ !

?



Therefore

: Ð Ñ œ3 r É ?1

tN

L

! ! . exp Ÿ Ð Ñ Ð Ñr r

Rs s3 3

T . t?

and for "Continuous Sampling"i.e. L Ä _ß Ä !ß Ä? ?> Þ > XL -=

we have

: Ð<Ð ÑÑ œ <Ð Ñ3 t . . t dtconst s texp ŸŠ ‹ "

!

X #

N!

-=

3Ð Ñ



ìMAXIMUM LIKELIHOOD APPROACH using the“CONTINUOUS SAMPLING" concept:

Rule: choose the hypothesis with the H3 maximumlikelihood p r tiÐ Ð ÑÑ

where

: Ð<Ð ÑÑ œ <Ð Ñ3 t . . t dtconst s texp ŸŠ ‹ "

!

X #

N!

-=

3Ð Ñ

1for 3 œ " # Q, , ..., Ð Ñ



ìMAP APPROACH using the “CONTINUOUSSAMPLING" concept: This is a more general approachthan the Maximum Likelihood

Rule maximum: choose the hypothesis with the Hi PrÐ ÑH3 ‚ p r tiÐ Ð ÑÑ

where

: Ð<Ð ÑÑ œ <Ð Ñ3 t . . t dtconst exp ŸŠ ‹ "

!

X #

N!

-=

s t3Ð Ñ Ð Ñ2

for 3 œ " # Q, , ...,



The above rule can be rewritten as follows:

maxe f a b a bPr PrÐ Ñ Ð ÑH H3 3‚ p r t3Ð Ð ÑÑ œ ÞmaxÖln ln -98=>

" #N0' X-=

0 a br tÐ Ñ s t3Ð Ñ .dt×

œ œmax lnš ›a b a bPrÐ ÑH3" #

N0' X-=

0 r tÐ Ñ s t3Ð Ñ .dt

œ max lnÖ a bPrÐ ÑH3" "

N N0 0' 'X X-= -=

0 0r t#Ð ÑÞ Þdt dts t#3 Ð Ñ

dt Þ#N0'

0X-=r tÐ Ñs t‡

3 Ð Ñ ×



œ I I max lnš ›a bPrÐ ÑH3

" #N N N

10 0 0< 3

X'0

-=r t dtÐ Ñ Þs t‡3 Ð Ñ

œ I œmax lnš ›a bR# #

"3

X! -=PrÐ ÑH3 '0 r t dtÐ Ñ Þs t‡

3 Ð Ñ

œ Imax lnš ›a b' X R# #

"3

-= !

0 r t dtÐ Ñ Þ s t‡3 Ð Ñ PrÐ ÑH3

i.e.max maxe f 'PrÐ ÑH3 ‚ Þp r t r t3

‡3Ð Ð ÑÑ Ð Ñœ Ö X-=

0 s tÐ Ñ dt

R#! lna bPrÐ ÑH3 Ð Ñ"

# 3I × 3



ì QEquation-0 suggests that an -ary receiver will have the following form:

s (t)1

0

Tcs

+DC1

s (t)2

0

Tcs

+DC2

s (t)M

0

Tcs

+DCM

G1

G2

GM

CompareDecisionVariables

and

choosemaximum

Di

If Gi=maxthen

Gi

Di

r t( )

t

Tcs

Tcs

Tcs

Tcs

OPTIMUM M-ary RECEIVER: CORREL. RECEIVER

where DC ln with =1,2, ...,3 3"´ N!

# # Ð ÐL ÑÑ 3 QPr I3



ìNote that if the a priori probabilities are equal,i.e. H H HPr Pr PrÐ Ñ œ Ð Ñ œ Þ œ Ð Ñ" # 3.. . ,

and the signals have the same energy,i.e. I œ I œ ÞÞÞÞ œ I" # Q

then Equation-0 is simplified as follows:

Hmax Pr maxe f e fÐ Ñ ‚ Ð Ð ÑÑ Ð Ð ÑÑ3 p r t p r ti iœ

4œ Ð Ñmaxš ›' X-=

0 r t s t dtÐ Ñ Ð ÑÞ‡i

ìN.B. - ML receiver: similar structure with MAP with DC3"´ #I3



7. Optimum -ary Receivers using Matched FiltersQ

7.1. KNOWN SIGNALS IN WHITE NOISEConsider the output of one branch of the correlationreceiver:

CHANNEL

-B +B f

1

CORRELATOR

s (t)i

0

Tcs

r(t)

n (t)i

1s (t)i

output .at point-1 œ œ '

!

X-=<Ð Ñt . dts t3Ð Ñ



In this section we will try to replace the correlator of acorrelation receiver with a linear filter known as MatchedÐFilter .Ñ

CHANNEL

-B +B f

1

MATCHED FILTER

r(t)

n (t)i

Linearfilterh (t)i

Tcs1 2s (t)i

at point-1 œ 0t' <Ð Ñu . .du2 3Ð Ñt u

outputat point-2 œ œ 0

T' -=<Ð Ñu . .du2 Ð Ñ3 -=T u

N.B.: compare o/p at point-1 of correlator with at point-2 of matched filter



Reversal in Time

Tcs

s (t)i

h (t)i

0

t

t

e.g.

If we choose the impulse response of the linear filter as 2 Ð Ñ Ð Ñ3 t s T tœ 3 cs 0 t TŸ Ÿ cs

then that linear filter, defined by the above equation, is calledthe MATCHED FILTER. at point-2 œ 0

T' -= <Ð Ñu . .dus u3Ð Ñ

Ê œat point-2 0T' -= t<Ð Ñ. .dts t3Ð Ñ



N.B.:

Output Of Correlator Output Of Matched Filterœ

only at time > œ X-=



CORRELATION RECEIVER:

s (t)1

0

Tcs

+DC1

s (t)2

0

Tcs

+DC2

s (t)M

0

Tcs

+DCM

G1

G2

GM


and

choosemaximum

Di

If Gi=maxthen

Gi

Di

r t( )

t

Tcs

Tcs

Tcs

Tcs


or,



MATCHED FILTER RECEIVER:

+DC1

+DC2

+DCM

G1

G2

GM


and

choosemaximum

Di

If Gi=maxthen

Gi

Di

r t( )

t

Tcs

Tcs

Tcs

Tcs

OPTIMUM M-ary RECEIVER: MATCHED FILTER. RECEIVER

Tcs

h t =s T -tM M cs ( ) ( )

Tcs

h t =s T -t2 2( ) ( )cs

Tcs

h t =s T -t1 1( ) ( )cs



7.2. SIGNALS AND MATCHED FILTER IN FREQ. DOMAIN

• Let 0 h t s T t 0 t TelsewhereÐ Ñ œ

Ð Ñœ -= -= Ÿ Ÿ

• Time Domain Freq. DomainFTÈ

s t f s tÐ Ñ Ð Ñ Ð Ñ ÈFT e dtS œ o

T j2 ft' -= 1

h t f h tÐ Ñ Ð Ñ Ð Ñ ÈFT e dtH œ _

_ ' j2 ft1

œ oT j2 ft' -=s T tÐ Ñ-= e dt 1

œ e duoT j2 f T u' -= -=s uÐ Ñ 1 Ð Ñ

e e duœ j2 fT +j2 fuo

T1 1-= -=' s uÐ Ñ

œ e .j2 fT *1 -= S Ð Ñftherefore HÐ Ñf e . œ j2 fT *1 -= S Ð Ñf Ð Ñ5



7.3. KNOWN SIGNALS IN NON-WHITE NOISE

noise œ š zero mean

not necessarily white or GaussianRnnÐ Ñœ7

ìLet <Ð Ñ Ð Ñt n tœ Ð Ñs t + Å

Æ

signal : completely known

ì Ð Ñobjective: design a linear filter such thath t

SNR outat t Tœ -=

œ max



CHANNEL

-B +B f

1

MATCHED FILTER

r(t)

n (t)i

s(t) Linearfilterh(t)

Tcs

output . .d

s T

œÌ

Ð Ñ

oT' -=h T

n T

Ð Ñ <Ð Ñ

Ð Ñ

7 7

7

-=

-=

-= 7

7

.s T d + dœ Ð Ñðóóóóóóóóóñóóóóóóóóóò ðóóóóóóóóóóñóóóóóóóóóóòo oT T' '-= -=h h .n TÐ Ñ Ð Ñ Ð Ñ7 7 7-=

œ œ

7 7 7

s Tµ Ð Ñ-=

-=

n Tµ Ð Ñ-=



.. . SNRout œ œ XX Xe fe f e fs T s Tµ µÐ Ñ Ð Ñ-= -=

#

# #-=n TµÐ Ñ-=

#

n TµÐ Ñ

.. with constant. max minh t h t

Ð Ñ Ð Ñ

œa b ŸSNRout œ Xe fn Tµ Ð Ñ-=# s Tµ Ð Ñ-=

.. where . minh t

.Ð Ñ

œ .Xe fn Tµ Ð Ñ-=# s Tµ Ð Ñ-=

i.e. h t

.

optimum impulse response ´

h9pt œÐ Ñ

œ

arg min

.where Xe fn Tµ Ð Ñ

Ð Ñ

-=# s Tµ Ð Ñ-=

6



The solution of the above Equation is the solution of thefollowing FREDHOLM INTEGRAL EQUATION of 1stKIND:

oT' -=hoptÐ Ñz . .dz T 0 T RnnÐ Ñ7 z œ Ð Ñ às -= -= Ÿ Ÿ7 7

GENERAL EXPRESSION FOR MATCHED FILTERS

7Ð Ñ



N.B.:1. Proof of the above result is not important. What is realy

important is that the proof does not involve anyassumption about the pdf or PSD of the noise i.e. thatÐthe noise is white Gaussian .Ñ

# Ð. In the case of white noise the above Equation Equation-7 can be solved easily as follows:Ñ

. .dz ToT' -=hoptÐ Ñz ðóóñóóòN

2! . z

z

$ 7

7

Ð Ñ

Ð Ñ

œ

Rnn

œ Ð Ñs -= 7

Ê Ê œ h hopt optÐ Ñ Ð Ñ7 7. T . TN2 N!

!œ Ð Ñ Ð Ñs s-= -=7 7#

Ð Ñ8Be careful - is not influenced by the factor SNR 2

N!



ìConsider next you have got the optimum impulse responseh t0Ð Ñ Ð Ñ by using Equation-7 . This impulse responseprovides the maximum output- which can be estimatedSNRas follows:

CHANNEL

-B +B f

1

MATCHED FILTER

r(t)

n (t)i

s(t) Linearfilterh (t)opt

Tcs

SNRout œ ?



output œ Ð Ñ <Ð ÑoT

opt' -=h . T .d7 7 7-= Å

s T + n TÐ Ñ Ð Ñ-= -= 7 7

h .s T d + h . n T d

s T n T

œ Ð Ñ Ð Ñ Ð Ñ Ð Ñ

Ð Ñ Ð Ñ

ðóóóóóóóóóóñóóóóóóóóóóò' '! !

X X-= -=

-= -=

-= -=opt opt7 7 7 7 7 7

œ œµ µðóóóóóóóóóóóñóóóóóóóóóóóò

Therefore, SNRoutmax

œ .......XX Xe fe f e fs T

n T n Ts Tµ

µ µµÐ Ñ

Ð Ñ Ð ÑÐ Ñ-=

#

-= -=# #

-=#

œ

..... for you .....Ð Ñ



..... for you .....Ð Ñ

. T z .dz i.e. SNRoutmaxs Tn T o

Tœ œ Ð Ñ

µµ -=Ð ÑÐ Ñ

-=#

-=#

-=

Xe f ' h zoptÐ Ñ s

Ð Ñ9



IMPORTANT NOTE:For white noise we have seen that: h z . s T zopt

2NÐ Ñ œ Ð Ñ!

-=

Then Equation-9 becomes:

SNRoutmax oT

œ Ð Ñ Ð Ñ œ

œ

' -=ðóóóñóóóò2N!

. T z . T z .dz s s-= -=

hopt

. T z .dzœ

œ

2N! ðóóóóóóóñóóóóóóóòo

T' -=sE

#-=Ð Ñ œ

Ê # SNRoutmaxœ Ð ÑI

R! for white noise 10

Provided that the filter is matched to the signal, it is obvious from Equation-3 that signal waveformÀ SNR outmaxÁ f{ }

signal bandwidthÁ f{ } peak powerÁ f{ } time durationÁ f{ }



7.3.1. Approximation To Matched Filter SolutionìWe have seen that the matched filter can be found by solving

the following integral equation:

MATCHED FILTER GENERAL EQUATIONo

T' -=hoptÐ Ñz . .dz TT

RnnÐ Ñ7 z œ Ð Ñs -=-=

! Ÿ Ÿ

77

a Fredholm integral of the 1st kind.Ð Ñ

ìHowever, it is very difficult to solve the above equation in ageneral case.

N.B.: when solution easy noise whiteœ œ



ìIn order to find an approximation to matched filter solutionÐ Ñi.e. we need to relax the conditionw z!Ð Ñ ¶ h zoptÐ Ñ 0 Ÿ Ÿ7 T-= .to _ Ÿ Ÿ _7

Then _

_' w z T! !Ð ÑÅ Æ Å Æ

. .dzRnnÐ Ñ7 z œ Ð Ñs 7

maximizes SNR at some time

However the above integral is a convolution integral

.. . –!Ð Ñf . .PSDnÐ Ñf œ ’* j2 fÐ Ñf e 1 T!

.. 11. –!Ð Ñf œ Ð Ñ .f

’* j2 fÐ Ñf e 1 T!

PSDnÐ Ñ



7.3.2. Whitening Filter non-white

ì œ + output<Ð Ñ Ð Ñt n ts tÐ ÑÅ

Æ

+ known s tÐ Ñ n t3Ð ÑÅ

whiteìwhitening filter:

ˆ Ä regions in Frequ Domain inattenuates which NOISE LARGEœ

ˆ Ä regions in Frequ Domain inaccentuates which NOISE LOWœ



ìcombined - transfer function:Whitening Filter Matched Filter

<Ð Ñ

Ð Ñ

t output

f‹

i.e. combined transfer function ´ œ‹Ð Ñf f .e’* j fÐ Ñ #1 T!

PSDnÐ Ñf



8. Optimum -ary Receivers based on Signal ConstellationQ8.1. Introduction - Orthogonal SignalsTwo signals and are called orthogonal signals- Ð>Ñ - Ð>Ñ3 4a bi.e. in the interval [ , ] iff- Ð>Ñ ¼ - Ð>Ñ À3 4 a b

= 12for for

' œab- Ð>ÑÞ- Ð>Ñ .> Ð Ñ

I 3 œ 4! 3 Á 43 4

‡ -3

i.e.c ti( )

c tj( ) a

b = for

for œI 3 œ 4! 3 Á 4

-3



Note:

ˆ I "if = then the signals are called orthonormal-3

ˆ - Ð>ÑÞ- Ð>Ñ .> œ I'ab

3 -3‡

3

and this represents the energy of the signal - Ð>Ñ3

in the interval [ , ].a b



8.2. THE "APPROXIMATION THEOREM" of an Energy SignalIt states that any energy signal can be approximated by a linear combination of a set of othogonal signals.

Theorem:Consider an energy signal Furthermore consider a set of D orthogonal signals . Then=Ð>ÑÞ - Ð>Ñß - Ð>Ñß ÞÞÞß - Ð>Ñe f" # D

ì =Ð>Ñ - Ð>Ñ The signal can be approximated by a linear combination of the orthogonal signals as follows:3

=Ð>Ñ œ A Þ- Ð>Ñ œ Ð>Ñs Ð Ñ!3œ"

‡3 =

Di A -H 13

where (coefficients or weights) c dc dA

-

= " # H

" # H

œ A ß A ß ÞÞÞß A

Ð>Ñ œ - Ð>Ñß - Ð>Ñß ÞÞÞß - Ð>Ñ

T

T

ì The best approximation is the one which uses the following coefficients

optimum weightsA -=" ‡œ =Ð>ÑÞ Ð>ÑÞ.> œI-

'ab

Ð Ñ14

where I- - - -œ I ß I ß ÞÞÞß Ic d" # H

T

The above coefficients are optimum in the sence that they minimize the energy of the error signalI%

%Ð>Ñ œ =Ð>Ñ =Ð>ÑÞs

N.B.: denotes Hadamard multiplication



c t1( ) c t2( ) c tD( ) c( )ts t( )

s t( )

w1w2 wD

ws

Ec1

1

%(t)

s t( )



N.B.:ìThe energy of the error signal is minimum when the%Ð>Ñ

set of signals is ane f%Ð>Ñß - Ð>Ñß - Ð>Ñß ÞÞÞß - Ð>Ñ" # H

orthogonal set.

i.e., if then or

Þßà

ÚÛÜ%Ð>Ñ ¼ - Ð>Ñ a3

=Ð>Ñ Ð>Ñ Þ Ð>ÑÞ.> œ

3

=‡

,

' a bab

A - - !H

I œ% minimum

ìIn general However if then the set=Ð>Ñ ¸ =Ð>ÑÞ =Ð>Ñ œ =Ð>Ñs sof orthogonal signals is known ase f- Ð>Ñß - Ð>Ñß ÞÞÞß - Ð>Ñ" # H

a of orthogonal signals.COMPLETE SET



8.3. GRAM-SCHMIDT Orthogonalization





Example



8.4. -ary Signals: Energy and Cross-CorrelationQìBinary Com. Systems: use 2 possible waveforms Ö= Ð>Ñß = Ð>Ñ×! "

or Ö= Ð>Ñß = Ð>Ñ×1 2Q Ö= Ð>Ñß ÞÞÞß = Ð>Ñ×-ary Com. Systems: use possible waveformsQ 1 Q

Note: these waveforms are energy signals of duration X-=

ìThe signals (or channel symbols) are characterized byQ

their energy I3

. 15I œ = Ð>Ñ .> à a3 Ð Ñ3#3

'0X-=

Furthermore their similarity (or dissimilarity) ischaracterized by their cross-correlation . . 16334 3

"

I I4œ = Ð>Ñ = Ð>Ñ .> Ð ÑÉ 3 4

'0X-= *



The signals may be also expressed, by using the Approx.QTheorem described in the previous section, as a linearcombination of a set of orthogonal functions {H - Ð>Ñ×5

17= Ð>Ñ œ à a3 Ð Ñ3 A -=3H Ð>Ñ

where 18-Ð>Ñ œ Ð Ñ- Ð>Ñß - Ð>Ñß ÞÞÞß - Ð>Ñc d" # HT

c t1( ) c t2( )s ti( )

w1iw2i

Ec1

1

%(t)

c tD( ) c( )t

wDiwsi

s ti( ) s ti( )

???



Let us define the following matrixÐH ‚HÑ

‘-- œ -Ð>ÑÞ-Ð>Ñ .> œ'0X-= H.

=

Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÕ Ø

' ' '' ' '0 0 0

0 0 0

X X X

X X X

-= -= -=

-= -= -=

- Ð>Ñ.>ß - Ð>Ñ- Ð>Ñ.>ß ÞÞÞß Ð>Ñ- Ð>ÑÞ.>

- Ð>ÑÞ- Ð>ÑÞ.>ß - Ð>ÑÞ.> ÞÞÞß - Ð>Ñ- Ð>ÑÞ.>

Þ

#" " "# H

# #" H##

* *

* *

-

ÞÞß ÞÞÞß ÞÞÞß ÞÞÞ

- Ð>ÑÞ- Ð>ÑÞ.>ß - Ð>ÑÞ- Ð>ÑÞ.>ß ÞÞÞß - Ð>ÑÞ.>' ' '0 0 0X X X-= -= -=

H H" ##H

* *

i.e.

‘ ˆ-- Hœ -Ð>ÑÞ-Ð>Ñ .> œ œ Ð Ñ

" ! ÞÞÞ !! ÞÞÞ !ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ! ! ÞÞÞ "

'0X-= H. 191

Ô ×Ö ÙÖ ÙÕ Ø



Then by using Equation-15 in conjunction with Equation-17, the energyßI = Ð>Ñ a33 3 of the signal , , can be expressed as follows:

. .

. .

..

I œ A Þ-Ð>Ñ Þ -Ð>Ñ A .>

œ = Ð>Ñ œ = Ð>Ñ

œ A Þ -Ð>ÑÞ-Ð>Ñ .> A

œ A Þ A

œ A Þ A

œ A A

œ A

3 = =

3‡3

= =

X

= =--

= =H

= =

=#

(0

X-= ðñò ðóñóò

¼ ¼

3 3

3 3

-=

3 3

3 3

3 3

3

H H

H H

0H

H

H

(‘

ˆ



Similarly, by using Equation-16 in conjunction with Equation-17 the cross-correlationcoef. becomes334 a34

. .

. .

.

.

.

3

‘

ˆ

343 4

X

= =

3 4= =

X

3 4= =--

3 4= =H

3 4= =

œ A Þ-Ð>ÑÞ-Ð>Ñ A .>"

I I

œ ÞA Þ -Ð>ÑÞ-Ð>Ñ .> A"

I I

œ A Þ A"

I I

œ A Þ A"

I I

œ A A"

I I

ÈÈ ÈÈÈ

((

-=

3 4

3 4

-=

3 4

3 4

3

0

H H

T H

0

H

H

H4

3 4œA A= =

H.¼ ¼ ½ ½A Þ A= =3 4



i.e. 20I œ A A œ A Ð Ñ3 = = =

#

3 3 3

H ¼ ¼

. 21334"I I = =

A A

A Þ Aœ A A œ Ð ÑÈ ½ ½ ½ ½3 4 3 4

= =3 4

= =3 4

H .H



8.5. Signal Constellation

c t1( ) c t2( )s ti( )

w1iw2i

Ec1

1

%(t)

c tD( ) c( )t

wDiwsi

s ti( ) s ti( )

ìWith reference the figure:

if the error signal %Ð>Ñ œ !then the knowledge of the vector is as good asA=

knowing the transmitted signal ,=Ð>Ñor, in the case of signals ( -ary system),Q Q knowledge of knowledge of 22œ œ = Ð>Ñ œ ß a3 Ð Ñ3 =A

3



ìTherefore, we may represent the signal = Ð>Ñ3

by a in a -dimensional Euclidean spacepoint Ð Ñ HA=3

with H Ÿ QÞ

ìThe set of points (vectors) specified by the columns of thematrix

– œ ß ß ÞÞÞßc dA A A= = =" # Q

is known as "signal constellation".



8.6. Distance between two -ary signalsMì The distance between two signals and is the Euclidean distance= Ð>Ñ = Ð>Ñ3 4

between their associate vectors and A A= =3 4

wsj

wsi

dij

Origin

D-dimensional space

i.e.

Re( )

2

. œ A A

œ A A A A

œ A A A A # A A

œ I I I I

34 = =

= = = =

= = = = = =

3 4 34 3 4

½ ½ÊŠ ‹ Š ‹ÉÉ È

3 4

3 4 3 4

3 3 4 4 3 4

H

H H H

3

2 23Ê . œ I I I I Ð Ñ#34 3 4 34 3 43 È



ìIt is clear from the above that the Euclidean distance .34associated with two signals and indicates, like= Ð>Ñ = Ð>Ñ3 4

the cross-correlation coefficient, the similarity ordissimilarity of the signals.

ìAn involving Important Bound . À34

ˆIf denotes the prob. of error associated with:/ß- Ð= Ð>ÑÑs 3

the channel symbol , it can be found that (see= Ð>Ñ3

S.Haykin p.498)

24: Ÿ Ð Ñ/ß- Ð= Ð>ÑÑ4œ"4Á3

Q.

#Rs 3

34

!

!T{ }È



Then, the symbol error probability is bounded as:/ß=-follows:

25: Ÿ Ð Ñ/ß-"Q

3œ"4œ"

Q Q

4Á3

.

#Rs ! !T{ }34

!È

ˆN.B.: the following expression was used to go fromEqu.24 to Equ.25

(see S.Haykin p.498): œ :/ß"Q

3œ"

Q

/ß- Ð= Ð>ÑÑcs s!3

ì definition of the "minimum distance": . œ .a3ß 4

min mine f34



8.7. Examples of Signal Constellation DiagramConsider an M-ary System having the following signals

Ö ×= Ð>Ñ = Ð>Ñ = Ð>Ñ1 2, , ...., Q with ! Ÿ > Ÿ X-=

ì M-ary ASKˆ channel symbols: = Ð>Ñ E3 3œ Þ Ð J >Ñcos 2 1 - where E œ #3 "Q3 given= (say)ˆ œdimensionality of signal space H œ "ˆ if thenQ œ %

s t1( )0100 11 10s t2( ) s t3( ) s t4( )

Es1 Es2 Es3 Es4Origin

ˆ if then Q œ )

s t1( )000

Es1

001s t2( )

Es2

011s t3( )

Es3

010s t4( )

Es4Origin

s t5( )110

Es5

s t6( )

Es6

101s t7( )

Es7

100s t8( )

Es8



ì M-ary PSKˆ channel symbols:= Ð>Ñ Þ 3 "3

#Qœ EÞ # J > 3 œ "ß #ß ÞÞßcosˆ ‰1 -1 a b 9

ˆ œdimensionality of signal-space H œ 2

ˆ if & then if & thenQ œ % œ ! Q œ % œ9 9° 45°

s t1( )

01

0011

s t2( )

s t3( )

Es1

Es2

Es3 Origin

10s t4( )

Es4

s t1( )00Es1

01s t2( )

Es2

11s t3( )

Es3

Origin

10s t4( )

Es4

450



ˆ if & thenQ œ œ !8 ° 9

Origin

s t1( )000 Es1

001s t2( )

Es2

011s t3( )

Es3

010s t4( )

Es4

s t5( )110Es5

s t6( )

Es6

101s t7( )

Es7

100s t8( )

Es8



ì M-ary FSK Àvery difficult to be represented using constellation diagram



Examples of Decision Variables (QPSK- Receiver) Plots

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Re

Im

-8 -6 -4 -2 0 2 4 6 8

x 106

-8

-6

-4

-2

0

2

4

6

8x 106

Re

Im

'good' 'bad'



8.8. Optimum M-ary Receiver using the Signal-VectorsìLet us consider again the general M-ary problem:

CHANNEL

-B +B f

1r(t)

n (t)i

Decision rule=?(To determine which

signal is present)= >3( )Å

one out of M signals

D 3

ì hypotheses:

: is present with probability : is present with probability

... ....................

... ....

ÚÝÝÝÝÛÝÝÝÝÜ

H H1 1 1= Ð>Ñ Ð ÑPrH H2 2= Ð>Ñ Ð ÑPr #

................: is present with probability H HQ Q Q= Ð>Ñ Ð ÑPr



ìwe have seen that the MAP criterion, using the “CONTINUOUSSAMPLING" concept, is described by the following rule:

Rule: choose the hypothesis with the maximumH3 HPrÐ Ñ3 ‚ p r t3Ð Ð ÑÑ

where maxe fPrÐ ÑH3 ‚ p r t3Ð Ð ÑÑ œ

= max lnÖ ×' a bX R#

-= !

0 r tÐ Ñs t‡3 Ð Ñ IÞ dt PrÐ ÑH3 3 "

#



This rule can also be described by the following two equivalent receivers:



s (t)1

0

Tcs

+DC1

s (t)2

0

Tcs

+DC2

s (t)M

0

Tcs

+DCM

G1

G2

GM


and

choosemaximum

Di

If Gi=maxthen

Gi

Di

r t( )

t

Tcs

Tcs

Tcs

Tcs


where DC ln with =1,2, ...,3 3"´ N!

# # Ð ÐL ÑÑ 3 QPr I3



ìNow let us use the Approx. Theorem of energy signals

to represent the received signals <Ð>Ñ

as well as the -ary signal , ,Q = Ð>Ñ a33

i.e.

ÚÝÛÝÜˆ ‰<Ð>Ñ œ

œ œ

œ

A<H

H H

H

-

- -

-

Ð>Ñ

Ð>Ñ Ê Ð>Ñ

Ð>Ñ a3

= Ð>Ñ = Ð>Ñ3 = 3 =‡

=

A A

A3 3

3

‡



In this case the MAP receiver (Equ-0) becomes

maxe fPrÐ ÑH3 ‚ p r tiÐ Ð ÑÑ œ

œ max lnš ›a b' X R# #

"-= !

0 r tÐ Ñs t dt‡i Ð ÑÞ PrÐ ÑH3 I3

œ Þ max lnš ›a b' X R

# #"-= !

0H HA< - -Ð>ÑÞ Ð>Ñ A= 33

dt PrÐ ÑH I3

œ max lnš ›a bA<H H

0Š ‹' X R# #

"-= !- -Ð>ÑÞ Ð>Ñ .> A= 33PrÐ ÑH I3

œ max ln˜ ™a bA<HA= 33

R# #

"! PrÐ ÑH I3



i.e. max max lne f a b˜ ™Pr PrÐ Ñ Ð ÑH H3 = 3‚ p r tiÐ Ð ÑÑ œ A<

HA3

R# #

"! I3

Ð Ñ26The above equation may be implemented as follows:



orßAdvanced Communication Theory Compact Lecture Notes


8.9. Encoding -ary SignalsQì The performance of -ary systems is evaluated by means of the average Q probability of

symbol error probability of bit , which for is different than the average : ß Q #ß/ß-=

error Bit-Error-Rate BER (or ), .:/

That is 27for for œ : Á : Q #

: œ : Q œ #Ð Ñ/ß-= /

/ß-= /

However, because we transmit binary data, the probability of bit error is a more:/natural parameter for performance evaluation than .:/ß-=Although, these two probabilities are related i.e. 28: œ : Ð Ñ/ /ß-=f{ }

their relationship depends on the encoding approach which is employed by the digitalmodulator for mapping binary digits to -ary signals (channel symbols)Q

bits channel1st bit 2nd bit ....//... -th bit

-th word of bitsÄ Ä3

È =

Digital Modulator

èëëëëëëëëëëëëëëëëëëëéëëëëëëëëëëëëëëëëëëëê#

#

cs

-=

3

symbols

where #cs 2œ QÞlog



ì Two important relationships between and are the following:: :/ /ß-=

1. If the encoder provides a mapping where adjacent symbols differ in one binary digit then it can be proved (see Haykin p500) that

29"/ß-= / /ß-=#-=

Þ: Ÿ : Ÿ : Ð Ñ

e.g. -ary PSK for Gray code, otherwise difficult): Q Ð

BER p œ œ Þ:e"

/ß-=#-=Ð Ñ30

N.B.: Note that Gray encoder provides a mapping where adjacent symbols differ inone binary digit. This property is very important because the most likely errors dueÐto channel noise effects are related with an erroneous selection (wrong decision) ofÑan adjacent symbol.

2. If all -ary signals are equally likely, then it can be proved (see Haykin p.500) that Q

pe csœ Þ:## " /ß#

#

-="

-= Ð Ñ31

(which, for large is simplified to #cs pe csœ Þ: Ñ"# /ß

e.g. -ary FSK: 32Q Ð ÑBER pœ œ Þ:e cs## " /ß#

#

-="

-=



9. Performance of ary Communication SystemsM-A. Orthogonal ary SignalsM-

If the -ary signals are equipowered with cross-correlation coeff. =0,ì M 3

s t .s t .dt ' X34

-=

0 3 4 34Ð Ñ Ð Ñ œ IÞ"!

$ where ifif$ œ

3 œ 43 Á 4œ

then the signals are orthogonal and the symbol error probability can be estimated byp /ß-=

using the following approach:

ìÄÄ

Let us assume: H is true. Then, if with no error is madeothetwise an error is made3

4 3 G G œ a4 4 Á 3

By using the above, the probability of symbol error can be found as follows:

p = /ß-= " Ð ÑPr no error is made Ê

no-error 33

pdf

Ê Ð Ñp = | G . . . dG

G

/ß-= 3_

_"

#

ÐK Ñ#

3

" Ð Ñ

œ Ð Ñ

' Pr ðóóóóóóóóóóñóóóóóóóóóóò È 15

.5#

3 3#

#

3

exp

G

i



However no-errorPr PrÐ Ñ œ Ð Ñ| G G <G , G <G ,...,G <G ,G <G , ...,G <G =3 " 3 # 3 3" 3 3 " 3 3+ M

œ Ð Ñ Ð Ñ Þ Ð Ñ Ð Ñ Þ Ð Ñ G <G . G <G . .. G <G . G <G . .. G <GPr Pr Pr Pr Pr" 3 # 3 3" 3 3 " 3 3+ M

œ Ð Ñ 4 Á 3 G < G Š ‹Pr 4 3

"M

œ .dzŠ ‹Š ‹' K3

_"

#

Ð Ñ#

Q"

È 15

.5#

4#

#exp z

œ .dzŒ Š ‹' K 3 4.

5

_ # #

Q"1 zÈ 1

exp #

œ Š ‹1 34 Ð ÑT{ }K 3 4.5

Q"

From the above expressions it is clear that the variance and the mean and should5 . .#4 3

be estimated.



By using 35G = r t . s t .dt

s t +n t

4 4

3

' X-=

0 Ð Ñ Ð ÑÅ

Ð Ñ Ð Ñ

Ð Ñ

it can be proved that this is for you!!!Ð Ñ

= G | = ,5#4 3 #var H ;š › N!E a3 4

= G | = ; . X4 4 3š ›H ! 3 Á 4

= G | = energy

. X3 3 3š ›H EÅ

ìNote that in order to prove the above Equations, the following relations should be usedÐ Ñand proved!! G G | = Xš ›4 5 3H ! 4 Á 5

G | =

Xš › #4 3

# R#

#

HE 4 œ 3

4 Á 3

!E

EN0



ìBy using the above expressions it is easy to show that Equation-33 can be written asfollows:

p + . . . . d/ß-=_

_I BR ##

œ " " B # B Ð Ñ' ’ “É T{ }!

#Q"1È 1

exp 36

ìExample:

Qœ œ

II

œ œ

-ary FSKEUE where denotes the average energy per bit

is the average symbol energyBUE ?

ÚÝÛÝÜœI

R R QI

F<

,

! ! #

,

log,



B. Biorthogonal ary SignalsM-

ì If the -ary signals are biorthogonal, i.e. there is a set of orthogonal signals { } andM s t3Ð Ñthe set of their negative counterpart { }, then the probability of error can be found Ð Ñs t3

to be:

37p . + . . . . de,cs .

œ " " # # B Ð Ñ'

#

_ I " BR #

"

#É ÈIR!

!

Q# #’ “É Tš ›B

1exp

(for proof see Appendix-1)

ìExamples: BPSK, 4-PSK

ìNote:

orˆ

ß334 œ

" ß!œ

ˆ" !Þ&

ÚÛÜ

BPSK EUE

4 PSK EUE EUE

p =

- p = p

/ß-=

/ß-= /ß-=#

T

T . T

š ›š › Š ‹š ›

Èa b È È#

" " œ # # #BPSK

where the above expression for -PSK follows from the statistical independence of%the noise on the quadrature carriers



10. Walsh-Hadamard Orthogonal Sets and Signals = 1ì‡1 c d matrix‡# œ ‚

" "" "” • Ð# #Ñ

6-times

‡ ‡ ‡ ‡ ‡ ‡ ‡64 œ Œ Œ Œ Œ Œðóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóò# # # # # #

where denotes the Kronecker product of two matricesŒ

e.g. for two matrices and

if then

œ Œ œE E E EE E E E” • ” •"" "# "" "#

#" ## #" ##



Properties:ì

ˆ œ '%‡ ‡ ˆ64T

64 '%

ˆ 3Let denote the -th column of h3 ‡64

i.e. ‡64 œ ß ß ÞÞÞßc dh h h" # '%

Then the following example illustrates an application of "WalshHadamard" in a mobile comm. environment (say) where each user inthe same cell and same frequency channel is assigned one column ofWalsh matrix,



ðóóóóóóóóóóóóñóóóóóóóóóóóóò ðóóóóóóóóóóóóóóóóóóñóó.... , ...., , ....input binary sequence

Walshnd column7 ß7 7 ß 7 Ä Ä1 # $ 5

Transmitter

#h#

óóóóóóóóóóóóóóóóò.... , ...., , ....output binary sequence

7 ß7 7 ß 71T T T Th h h h# # # ## $ 5

ðóóóóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóóóóò

ÚÝÝÝÝÝÝÝÝÝÝÛÝÝÝÝÝÝÝÝÝÝÜ

.... , ...., ....input binary sequence

7 ß7 7 ß 7 Ä

Ä

1T T T Th h h h# # # ## $ 5

Receiver-A

Receiver-B

Walshnd column

/64

.... , ...., , ....output binary sequence

Walsh3rd colum

#h#

Ä 7 ß7 7 ß 7

Ä

ðóóóóóóóóóóóóñóóóóóóóóóóóóò1 # $ 5

n/64

.... 0 0, 0 ...., 0, ....output binary sequenceh3

Ä ß ßðóóóóóóóóóñóóóóóóóóóò



11. Appendix-1: Proof of Equation-37



principles of -ary detection theory m - athanassios manikas 2 2005 - ln - trans... · advanced...

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