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EE401: Advanced Communication Theory Professor A. Manikas Chair of Communications and Array Processing Imperial College London Principles of Decision and Detection Theory Prof. A. Manikas (Imperial College) EE.401: Detection Theory v.16 1 / 84

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Page 1: EE401: Advanced Communication Theory. ACT_M_ary_Detection_Beamer_201… · Basic Decision Theory Basic Decision Theory Detection Theory is concerned with determiningthe existence

EE401: Advanced Communication Theory

Professor A. ManikasChair of Communications and Array Processing

Imperial College London

Principles of Decision and Detection Theory

Prof. A. Manikas (Imperial College) EE.401: Detection Theory v.16 1 / 84

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Table of Contents1 Basic Decision Theory

Hypothesis Testing in an M-aryComm SystemTerminologyM-ary Decision CriteriaExamplesExample: Gaussian LikelihoodFunctions (LFs)Example: Signal + Gaussian NoiseExample: Rectangular andGaussian LFs

2 Basic Concepts on OptimumReceiversThe Concept of "ContinuousSampling"Optimum M-ary ReceiversGaussian Multivariable DistributionLikelihood Functions (ContinuousSampling)

3 Optimum Receivers usingMatched FiltersKnown Signals in White NoiseSignals and Matched Filters inFrequency Domain

4 Optimum Receivers in Non-WhiteNoiseKnown Signals in Non-WhiteNoiseA Special CaseOutput SNRA Special Case: Output SNR(Signal plus white Noise)Approximation To MatchedFilter Solution

Whitening Filter5 Optimum M-ary Rx based onSignal ConstellationIntroduction - OrthogonalSignalsThe “Approximation Theorem”of an Energy SignalComments on the "Approximation"Theorem

Gram-SchmidtOrthogonalizationM-ary Signals: Energy andCross-CorrelationSignal ConstellationDistance between two M-arysignalsExamples of SignalConstellation DiagramExamples of Plots of DecisionVariables (QPSK-Receiver)Optimum M-ary Receiver usingthe Signal-VectorsEncoding M-ary SignalsImportant RelationshipsBetween pe and pe ,cs

6 Appendices:Appendix-1: Proof ofEquation-26Appendix-2: Walsh-HadamardOrthogonal Sets and SignalsAppendix-3: IS95 - WalshFunction of order-64

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Basic Decision Theory

Basic Decision TheoryDetection Theory is concerned with determining the existence of asignal in the presence of noise - and can be classified as follows:

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Basic Decision Theory

N.B.:1 Adaptive or Learning :

F system parameters change as a function of input;F diffi cult to analyze;F mathematically complex

2 Non-Parametric :F fairly constant level of performance because these are based on generalassumptions on the input pdf;

F easier to implement.

3 Consider a parametric detector which has been designed for Gaussianpdf input.

F if input is actually Gaussian : then parametric’s performance issignificantly better;

F if input 6=Gaussian but still symmetric: then non-parametric detectorperformance may be much better than the performance of theparametric detector

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Basic Decision Theory Hypothesis Testing in an M-ary Comm System

Hypothesis Testing in an M-ary Comm System

A Hypothesis , a statement of a possible conditionIn an M-ary Comm. System we have M hypotheses:

r t =s t +n t( ) ( ) ( )iDecision rule=?

(To determine whichsignal is present)

Di

where si (t)= one of M signal (channel symbols)

hypotheses:H1 : s1(t) is

is being sent︷ ︸︸ ︷present with probability Pr(H1)

H2 : s2(t) is present with probability Pr(H2). . . ...HM : sM (t) is present with probability Pr(HM )

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Basic Decision Theory Hypothesis Testing in an M-ary Comm System

The statistics of the observed signal

r(t) = si (t) + n(t) (1)

are affected by the presence of s1(t), or s2(t), ..., or sM (t)

If si (t), ∀i are known

then their distributions are known

and the problem is translated to make a decision about one of theM distributions after having observed r(t)

This is called ‘Hypothesis Testing’

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Basic Decision Theory Terminology

Terminology

A priori probabilities :

Pr(H1),Pr(H2), ...,Pr(HM )

(these are calculated BEFORE the experiment is performed)

A posterior probabilities

Pr(H1/r),Pr(H2/r), ...,Pr(HM/r)

That is, if r =observation variablethen we have M Conditional Probabilities

Pr(Hi/r), ∀i ∈ [1, . . . ,M ]

known as a POSTERIOR PROBABILITIES(since these are calculated AFTER the experiment is performed).

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Basic Decision Theory Terminology

Pr(Hi/r) ∀i : diffi cult to find.A more natural approach is to find

Pr(r/Hi ), ∀i (2)

since in general pdfr/Hi , ∀iI are known orI can be found

Likelihood Functions (LF) :

pdfr/H1(r), pdfr/H2(r), ..., pdfr/HM (r) (3)

i.e. the M conditional probability density functions pdfr/Hi (r), ∀i , areknown as "Likelihood Functions"Likelihood Ratio (LR) : The ratio

pdfr/Hi (r)pdfr/Hj (r)

for i 6= j (4)

is known as "Likelihood Ratio"Prof. A. Manikas (Imperial College) EE.401: Detection Theory v.16 8 / 84

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Basic Decision Theory Terminology

MAP CriterionDECISION−choose hypothesisHi i.e. Di :

iff Pr(Hi/r) > Pr(Hj/r), ∀j : j 6= i (5)D1 : iff Pr(H1/r) > Pr(Hj/r), ∀j 6= 1D2 : iff Pr(H2/r) > Pr(Hj/r), ∀j 6= 2· · · · · ·DM : iff Pr(HM/r) > Pr(Hj/r), ∀j 6= M

This detection is known as max a posterior probability (MAP)criterion

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Basic Decision Theory Terminology

Equivalent MAP expressions:

Di : iff Pr(Hi )×pdfr/Hi (r) > Pr(Hj )×pdfr/Hj (r); ∀j : j 6= im

Di = maxj ,∀j(Pr(Hj )× pdfr/Hj (r)︸ ︷︷ ︸

,Gj

)

Note:I The above can be easily proven using the Bayes rule

Pr(Hi/r) =pdfr/Hi (r).Pr(Hi )

pdfr (r)(6)

I Gj is known as "decision variable"I In this topic the symbol G will be used to represent a "decisionvariable".

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Basic Decision Theory Terminology

Correlation between two analogue energy signals r(t) and s(t) ofduration Tcs :

corr ,∫ Tcs

0r(t).s(t).dt (7)

Correlation between the discretised versions r and s of the signalsr(t) and s(t):

corr , rT s (8)

wherer = [r1, r2, .... rL]

T (9)

ands = [s1, s2, .... sL]

T (10)

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Basic Decision Theory M-ary Decision Criteria

M-ary Decision Criteria

Consider the sets of parameters P1 and P2 where:P1 denotes the set of parameters Pr(H1),Pr(H2), ..., Pr(HM )P2 represents the set of costs/weights Cij , ∀i , j

associated with the transition probabilities Pr(Di |Hj )(i.e. one weight for every element of the channel transition matrix F -see EE303, Topic on "Comm Channels")

Estimate/identify the likelihood functions

pdfr |H1(r), pdfr |H2(r), ..., pdfr |HM (r)

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Basic Decision Theory M-ary Decision Criteria

Decision: choose the hypothesis Hi (i.e. Di )with the maximum Gi (r)where Gi (r) depends on the chosen criterion as follows:

F BAYES CriterionF Minimum Probability of Error (min(pe)) CriterionF MAP CriterionF MINIMAX CriterionF Newman-Pearson (N-P ) CriterionF Maximum Likelihood (ML ) Criterion

P1 P2 choose Hypothesis with max(Gj (r))

Bayes known known Gj (r) ,weight j × Pr(Hj )×pdfr |Hjmin(pe ) known unknown Gj (r) , Pr(Hj )× pdfr |Hjor MAP

Minimax unknown known Gj (r) ,weight j× pdfr |HjN-P unknown unknown by solving a constraint optim. problem

ML don’t care don’t care Gj (r) , pdfr |Hj

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Basic Decision Theory M-ary Decision Criteria

Notes:I Note-1:if an approximate/initial solution is required then any informationabout the sets of parameters P1 and/or P2 can be ignored. In thiscase the Maximum Likelihood (ML) Criterion should be used.

I Note-2:

weight j ,M

∑i=1i 6=j

(Cij − Cjj

)(11)

or (since the term for i = j is equal to zero) simply,

weight j =M

∑i=1

(Cij − Cjj

)(12)

I Note-3:Sometimes, for convenience, Gj will be used (i.e. Gj , Gj (r)) - i.e. theargument will be ignored.

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Basic Decision Theory M-ary Decision Criteria

Decision Criteria: Mathematical Architecture

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Basic Decision Theory Examples

ExamplesExample: Gaussian LFs

A Brth,ji

r0 1Dj Di

pdf ( )r/Hj r pdf ( )r/Hi r

(Volts)

By solving Gj (r) = Gi (r) the decision threshold rth,ji can beestimated

area-A = Pr(Dj |Hi ) and area-B = Pr(Di |Hj )

pe ,cs =M

∑j=1

M

∑i=1i 6=j

Pr(Dj ,Hi ) = symbol error probability/rate (SER)

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Basic Decision Theory Examples

Example: Signal + Gaussian Noise

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Basic Decision Theory Examples

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Basic Decision Theory Examples

Example: Rectangular and Gaussian LFs

A discrete channel employs two equally probable symbols and thelikelihood functions are given by the following expressions

pdfr |H0 =12rect

{ r2

}pdfr |H1 = N

(0,σ =

12

)If the detector employed has been designed in an optimum way, findthe decision rule and model the above discrete channel.

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Basic Decision Theory Examples

(Volts)

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Basic Concepts on Optimum Receivers

Basic Concepts on Optimum Receivers

Receiver = Detector + Decision Device

Consider a M-ary communication model in which one of M signalssi (t), for i = 1, 2, ..M, is received in the time interval (0,Tcs) in thepresence of white noisei.e.

r(t) =

s1(t)

or s2(t)· · ·

or sM (t)

+ n(t), 0 ≤ t ≤ Tcs (13)

where r(t) denotes the received (observable) signal.

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Basic Concepts on Optimum Receivers

Tcs

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Basic Concepts on Optimum Receivers The Concept of "Continuous Sampling"

The Concept of "Continuous Sampling"Continuous Sampling helps to get a probabilistic description of acontinuous signal:

1. assume that L amplitude samples of the received signal

r(t), 0 ≤ t ≤ Tcsare available, i.e. r1, r2, . . . , rL,(with sampling frequency 1

4t )

where rk =

s1k

or s2k. . .

or sMk

+ nk ; k = 1, . . . , L

or, in more compact form,

r =

s1

or s2· · ·

or sM

+ n where

r =

[r1, r2, . . . , rL

]Tn =

[n1, n2, · · · , nL

]Ts i =

[si1, si2, · · · , siL

]TProf. A. Manikas (Imperial College) EE.401: Detection Theory v.16 23 / 84

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Basic Concepts on Optimum Receivers The Concept of "Continuous Sampling"

i.e. r = s i + n (L samples)

2. take the limit as L→ ∞, ∆t → 0, L× ∆t → Tcs

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Basic Concepts on Optimum Receivers Optimum M-ary Receivers

Optimum M-ary Receivers

Objective : to design a receiver which operates on r(t) and choosesone of the following M hypotheses:

H1 : r(t) = s1(t) + n(t)H2 : r(t) = s2(t) + n(t)· · · · · ·HM : r(t) = sM (t) + n(t)

or

H1 : r = s1 + nH2 : r = s2 + n· · · · · ·HM : r = sM + n

Optimum Decision Rule:depends on the likelihood functions pdfr |Hi|(r)

Optimum Receiver - Initial Conceptual Structure:

Computelikelihoodfunctions

M

OPTIMUM RECEIVER

∆t Comparevariables and get max

Di?r t =s t +n t( ) ( ) ( )i

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Basic Concepts on Optimum Receivers Optimum M-ary Receivers

Assumptions :I Noise: PSDni (f ) =

N02 ⇒PSDn(f ) =

N02 rect

{f2B

}Therefore, the received signal is sampled at intervals

∆t =12B

(14)

I Sampling: the samples are uncorrelated and statistically independent

Example - ML Receiver with "Cont. Sampling" :

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Basic Concepts on Optimum Receivers Optimum M-ary Receivers

Gaussian Multivariable Distribution

Consider the Gaussian random vector: r = [r1, r2, ..., rL]T

Then

pdf(r) =1√

(2π)L det(Rrr )exp

{−(r − µ

r)TR−1rr (r − µ

r)

2

}(15)

where

mean = µr= E{r} = [E{r1}, E{r2}, ..., E{rL}]T (16)

cov(r) = Rrr = E{(r − µ

r

) (r − µ

r

)}T(17)

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Basic Concepts on Optimum Receivers Optimum M-ary Receivers

If r = s + n then the mean µrand covariance Rrr are given as follows:

µr= s and Rrr = Rnn (18)

proof :

E{r} = µr, E{s + n} = E{s}+ E{n} = E{s} = s (19)

cov{r} , Rrr = E{(r − E{r})(r − E{r})T

}= E

{(r − s)(r − s)T

}= ε{nnT }

⇒ cov{r} = Rnn = σ2nIL = N0BIL (20)

Note that:

If Rnn = σ2nIL then det(Rnn) =(σ2n)L

(21)

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Basic Concepts on Optimum Receivers Optimum M-ary Receivers

Likelihood Functions (Continuous Sampling)

pdfr/Hi(r): Based on the above (for AWGN) and for “ContinuousSampling”we have

pdfr/Hi (r) =

(1√2πσ2n

)L· exp

{− (r − s i )

T (r − s i )2σ2n

}(22)

However{

σ2n = N0B4t = 1

2B

}=⇒ σ2n =

N02·4t

Therefore

pdfr/Hi (r) =

(√∆t

πN0

)L· exp

{− (r − s i )

T (r − s i ) · ∆tN0

}and for “Continuous Sampling”, i.e. L −→ ∞,4t −→ 0, L×4t −→ Tcswe have

pdfr/Hi (r(t)) = const · exp{− 1N0.∫ Tcs

0(r(t)− si(t))2dt

}(23)

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Basic Concepts on Optimum Receivers Optimum M-ary Receivers

ML APPROACH using "Continuous Sampling":Rule : choose the hypothesis Hi with the maximum likelihoodpdfr/Hi(r(t)) where

pdfr/Hi(r(t)) = const · exp{− 1N0.∫ Tcs

0(r(t)− si (t))2dt

}(24)

for i = 1, . . . ,M

MAP APPROACH using "Continuous Sampling":This is a more general approach than the Maximum LikelihoodRule : choose the hypothesis Hi with themaximum Pr(Hi )× pdfr/Hi (r(t)) where

pdfr/Hi (r(t)) = const. exp{− 1N0.∫ Tcs

0(r(t)− si (t))2dt

}(25)

for i = 1, . . . ,M

i.e. max{Pr(Hi )× pdfr/Hi (r(t))

}= max

{∫ Tcs

0r(t)s∗i (t)dt +

N02ln (Pr (Hi ))−

12Ei

}(26)

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Basic Concepts on Optimum Receivers Optimum M-ary Receivers

Equation-26 (for proof see Appendix 2) suggests that an M-aryreceiver 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  M­ary RECEIVER:  CORREL. RECEIVER

where DCi ≡ N02 ln(Pr(Hi ))−

12Ei with i = 1, 2, . . . ,M

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Basic Concepts on Optimum Receivers Optimum M-ary Receivers

Note that if the a priori probabilities are equal,i.e.

Pr(H1) = Pr(H2) = · · · = Pr(Hi ) (27)

and the signals have the same energy,i.e.

E1 = E2 = · · · = EMthen Equation-26 is simplified as follows:

max{Pr(Hi )× pdfr/Hi (r(t))

}= max{pdfr/Hi (r(t))}

= max{∫ Tcs

0r(t)s∗i (t) · dt

}(28)

N.B. - ML receiver: similar structure to MAP with DCi = 12Ei

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Optimum Receivers using Matched Filters Known Signals in White Noise

Optimum M-ary Receivers using Matched FiltersKnown Signals in White Noise

Consider the output of one branch of the correlation receiver:

CORRELATOR

s (t)i

0

Tcs 1r t =s t +n t( ) ( ) ( )i

at point-1 = output =∫ Tcs

0r(t) · si (t) · dt

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Optimum Receivers using Matched Filters Known Signals in White Noise

In this section we will try to replace the correlator of a correlationreceiver with a linear filter (known as Matched Filter ).

MATCHED FILTER

Linearfilterh (t)i

Tcs1 2r t =s t +n t( ) ( ) ( )i

at point-1 =∫ t

0r(u) · hi (t − u) · du

at point-2 = output =∫ Tcs

0r(u) · hi (Tcs − u) · du

N.B.:compare correlator’s o/p (point-1) with matched filter’s o/p (point-2).

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Optimum Receivers using Matched Filters Known Signals in White Noise

If we choose the impulse response of the linear filter as

hi (t) = si (Tcs − t) 0 ≤ t ≤ Tcs (29)

then that linear filter, defined by the above equation, is calledMatched Filter.

at point-2 =∫ Tcs

0r(u) · si (u) · du

=⇒ at point-2 =∫ Tcs

0r(t) · si (t) · dt

N.B.:Output Of Correlator =

↑only at time t=Tcs

Output Of Matched Filter

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Optimum Receivers using Matched Filters Known Signals in White Noise

example:

Reversal inTime

Tcs

s (t)i

h (t)i

0

t

t

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Optimum Receivers using Matched Filters Known Signals in White Noise

CORRELATION RECEIVER:

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  M­ary RECEIVER:  CORREL. RECEIVER

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Optimum Receivers using Matched Filters Known Signals in White Noise

or, MATCHED FILTER RECEIVER:

+DC1

+DC2

+DCM

G1

G2

GM

CompareDecisionVariables

and

choosemaximum

Di

If Gi=maxthen

Gi

Di

r t( )

t

Tcs

Tcs

Tcs

Tcs

OPTIMUM  M­ary 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

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Optimum Receivers using Matched Filters Signals and Matched Filters in Frequency Domain

Signals and Matched Filters in Freq. Domain

Let h(t) ={s(Tcs − t) 0 ≤ t ≤ Tcs0 elsewhere

Time DomainFT7−→ Freq. Domain

s(t)FT7−→ S(f ) =

∫ Tcs0 s(t)e−j2πftdt

h(t)FT7−→ H(f ) =

∫ ∞−∞ h(t)e

−j2πftdt

=∫ Tcso s(Tcs − t)e−j2πftdt

=∫ Tcs0 s(u)e−j2πf (Tcs−u)du

= e−j2πfTcs∫ Tcs0 s(u)e+j2πfudu

= e−j2πfTcs · S∗(f )

thereforeH(f ) = e−j2πfTcs · S∗(f ) (30)

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Optimum Receivers in Non-White Noise Known Signals in Non-White Noise

Known Signals in Non-White NoiseNow let us assume that the noise n(t) is described as follows:

n(t) ={zero meanRnn(τ) = not necessarily white or Gaussian

(31)

Letr(t) = s(t)︸︷︷︸

completely known

+ n(t) (32)

objective: design a linear filter h(t) such that

SNRoutat t=Tcs

= max (33)

MATCHED FILTER

Linearfilterh(t)

Tcsr t =s t +n t( ) ( ) ( )

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Optimum Receivers in Non-White Noise Known Signals in Non-White Noise

output =∫ Tcs

0h(τ) ·

s(Tcs−τ)+n(Tcs=τ)

⇓︷ ︸︸ ︷r(Tcs − τ) ·dt

=∫ Tcs

0h(τ) · s(Tcs − τ)dt︸ ︷︷ ︸

=s(Tcs )

+∫ Tcs

0h(τ) · n(Tcs − τ)dτ︸ ︷︷ ︸

n(Tcs )

∴ SNRout =E{s(Tcs )2}E{n(Tcs )2}

=s(Tcs )2

E{n(Tcs )2}(34)

∴ maxh(t)

(SNRout ) ={minh(t)E{n(Tcs )2} with s(Tcs ) = constant

}(35)

∴ minh(t)

ξ where ξ = E{n(Tcs )2} − λ · s(Tcs ) (36)

i.e

optimum impulse response , hopt = argminh(t)

ξ (37)

where ξ = E{n(Tcs )2} − λ·s(Tcs )

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Optimum Receivers in Non-White Noise Known Signals in Non-White Noise

The solution of Equation 37 is the solution of the following

FREDHOLM INTEGRAL EQUATION of 1st KIND:

∫ Tcs

0hopt (z).Rnn(τ − z).dz = s(Tcs − τ); 0 ≤ τ ≤ Tcs (38)

GENERAL EXPRESSION FOR MATCHED FILTERS

Proof of the above result (Equation 38) is not important. What isreally important is that the proof does not involve any assumptionabout the pdf or PSD of the noise (i.e. that the noise is whiteGaussian).

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Optimum Receivers in Non-White Noise A Special Case

A Special Case

In the case of white noise the above Equation (Equation-38) can besolved easily as follows:∫ Tcs

0hopt (z) · N02 · δ(τ − z)︸ ︷︷ ︸

=Rnn(τ−z )

· dz = s(Tcs − τ) (39)

=⇒ hopt (τ) · N02 = s(Tcs − τ) =⇒

hopt = 2N0· s(Tcs − τ) (40)

Be careful - SNR is not influenced by the factor 2N0

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Optimum Receivers in Non-White Noise Output SNR

Output SNR

Consider next you have got the optimum impulse response h0(t) (byusing Equation-38). This impulse response provides the maximumoutput-SNR which can be estimated as follows:

MATCHED FILTER

Linearfilterh (t)opt

Tcsr t =s t +n t( ) ( ) ( )

Important Question: SNRout =?we can answer this question as follows: ↘

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Optimum Receivers in Non-White Noise Output SNR

o/p =∫ Tcs

0hopt (τ). r(Tcs − τ)

↑s(Tcs−τ)+n(Tcs−τ)

=∫ Tcs

0hopt (τ).s(Tcs − τ)dτ︸ ︷︷ ︸

=s(Tcs )

+∫ Tcs

0hopt (τ).n(Tcs − τ)dτ︸ ︷︷ ︸

=n(Tcs )

(41)

therefore,

SNRoutmax

=E{s(Tcs )2}E{n(Tcs )2}

=s(Tcs )2

E{n(Tcs )2}= (42)

= · · · · · ·

. . . (for you). . .

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Optimum Receivers in Non-White Noise Output SNR

. . . (for you). . .

i.e. SNR outmax

= s(Tcs )2

E{n(Tcs )2} =∫ Tcs0 hopt (z) · s(Tcs − z)dz (43)

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Optimum Receivers in Non-White Noise A Special Case: Output SNR (Signal plus white Noise)

A Special Case: Output SNR (Signal plus white Noise)For white noise we have seen that: hopt (z) =

2N0.s(Tcs − z)

Then Equation-43 becomes:

SNR outmax

=∫ Tcs

0

2N0· s(Tcs − z)︸ ︷︷ ︸=hopt

· s(Tcs − z) · dz

= 2N0

∫ Tcs

0s2(Tcs − z) · dz =

=⇒ SNR outmax

= 2 EN0 for white noise (44)

Provided that the filter is matched to the signal, it is obvious fromEquation-44 that

SNIR outmax

6=f{signal waveform}6=f{signal bandwidth}6=f{peak power}6=f{time duration}

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Optimum Receivers in Non-White Noise Approximation To Matched Filter Solution

Approximation To Matched Filter Solution

We have seen that the matched filter can be found by solving thefollowing integral equation:∫ Tcs

0hopt (z) · Rnn(τ − z) · dz = s(Tcs − τ); 0 ≤ τ ≤ Tcs

MATCHED FILTER GENERAL EQUATION(a Fredholm Integral of the 1st kind.)

However, it is very diffi cult to solve the above equation in a generalcase.

N.B.: solution=easy when noise=white

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Optimum Receivers in Non-White Noise Approximation To Matched Filter Solution

In order to find an approximation to matched filter solution, i.e.

w0(z) ' hopt(z)

we need to relax the condition 0 ≤ τ ≤ Tcs to −∞ ≤ τ ≤ ∞

0 ≤ τ ≤ Tcs to −∞ ≤ τ ≤ ∞

.Then ∫ ∞

−∞w0(z)↘

Rnn(τ − z) · dz = s(T0↗

Maximizes SNR at some time

− τ) (45)

However the above integral is a convolution integral

∴ W0(f ) · PSDn(f ) = S∗(f ).e−j2πfT0 (46)

=⇒ W0(f ) =S∗(f ).e−j2πfT0

PSDn(f )(47)

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Optimum Receivers in Non-White Noise Whitening Filter

Whitening Filter

r(t) = s(t)↑

known

+

non-white↓n(t) : i/p o/p: s(t) + ni (t)

↑white

whitening filter:I attenuates−→regions in Frequ Domain in which NOISE=LARGEI accentuates−→regions in Frequ Domain in which NOISE=LOW

combined Whitening Filter-Matched Filter transfer function:

r(t) output

combined transfer function≡ L(f ) = S∗(f ).e−j2πfT0

PSDn(f )

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Optimum M-ary Rx based on Signal Constellation Introduction - Orthogonal Signals

Optimum M-ary Rx based on Signal ConstellationIntroduction - Orthogonal Signals

Two signals ci (t) and cj (t) are called orthogonal signals(i.e. ci (t) ⊥ cj (t)) in the interval [a, b] iff:∫ b

aci (t) · c∗j (t)dt =

{Eci for i = j0 for i 6= j (48)

c ti( )

c tj( ) a

b

=

{Eci for i = j0 for i 6= j

Note:I if Eci = 1 then the signals are called orthonormalI∫ ba ci (t).c

∗i (t) dt = Eci = the energy of ci (t) in the interval [a, b].

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Optimum M-ary Rx based on Signal Constellation The “Approximation Theorem” of an Energy Signal

The “Approximation Theorem”of an Energy Signal

Theorem : Consider an energy signal s(t).Furthermore consider aset of D orthogonal signals

{c1(t), c2(t), ..., cD (t)}

ThenI The signal s(t) can be approximated by a linear combination of theorthogonal signals ci (t) as follows:

s(t) =D

∑i=1

w∗i ci (t) = wHs c(t) (49)

where{w s = [w1,w2, . . . ,wD ]T

c(t) = [c1(t), c2(t), . . . , cD (t)]TI w s is a vector of coeffi cients or weights

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Optimum M-ary Rx based on Signal Constellation The “Approximation Theorem” of an Energy Signal

I The best approximation is the one which uses the following coeffi cients

w s =1E c�∫ bas(t) · c∗(t)dt = optimum weights (50)

whereE c = [Ec1 ,Ec2 , . . . ,EcD ]

T (51)

and � denotes Hadamard multiplication

I The above coeffi cients are optimum in the sense that they minimizethe energy Eε of the error signal

ε(t) = s(t)− s(t) (52)

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Optimum M-ary Rx based on Signal Constellation The “Approximation Theorem” of an Energy Signal

Comments on the "Approximation" Theorem

The above theorem states that any energy signal can beapproximated by a linear combination of a set of orthogonal signalsand can be equivalent described by the following diagram:

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Optimum M-ary Rx based on Signal Constellation The “Approximation Theorem” of an Energy Signal

The energy of the error signal ε(t) is minimum when the set ofsignals

{ε(t), c1(t), c2(t), ..., cD (t)} = is an orthogonal set

i.e if

ε(t) ⊥ ci (t), ∀i

or∫ ba

(s(t)− wHs c(t)

).c∗(t).dt = 0

then Eε = min

(53)

In generals(t) ≈ s(t) (54)

However, ifs(t) = s(t) (55)

then {c1(t), c2(t), ..., cD (t)} , a complete set of orthog. signals.

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Optimum M-ary Rx based on Signal Constellation Gram-Schmidt Orthogonalization

Gram-Schmidt Orthogonalization

Suppose that we have a set of energy signals

{s1(t), s2(t), ..., sM (t)} = non-orthogonal

and we wish to construct a set of orthogonal signals

{c1(t), c2(t), ..., cD (t)} = orthogonal

One popular approach to construct this orthogonal set is theGRAM-SCHMIDT orthogonalisation , described as follows:

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Optimum M-ary Rx based on Signal Constellation Gram-Schmidt Orthogonalization

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Optimum M-ary Rx based on Signal Constellation Gram-Schmidt Orthogonalization

Example

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Optimum M-ary Rx based on Signal Constellation M-ary Signals: Energy and Cross-Correlation

M-ary Signals: Energy and Cross-Correlation

Binary Com. Systems : use 2 possible waveforms {s0(t), s1(t)}or {s1(t), s2(t)}

M-ary Com. Systems : use M possiblewaveforms { s1(t), ..., sM (t)}Note: these waveforms are energy signals of duration TcsThe M signals (or channel symbols) are characterized by their energyEi

Ei =∫ Tcs

0s2i (t)dt ∀i (56)

Furthermore their similarity (or dissimilarity) is characterized by theircross-correlation

ρij =1√EiEj

∫ Tcs

0si (t) · s∗j (t)dt (57)

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Optimum M-ary Rx based on Signal Constellation M-ary Signals: Energy and Cross-Correlation

The M signals may be also expressed, by using the Approx. Theoremdescribed in the previous section, as a linear combination of a set ofD orthogonal signals {ck (t)}

si (t) = wHsi c(t); ∀i (58)

where c(t) = [c1(t), c2(t), . . . , cD (t)]T (59)

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Optimum M-ary Rx based on Signal Constellation M-ary Signals: Energy and Cross-Correlation

Let us define the following (D ×D)matrix

Rcc =∫ Tcs

0c(t).c(t)H .dt

=

∫ Tcs0 c21 (t)dt,

∫ Tcs0 c1(t)c*2 (t)dt, ...,

∫ Tcs0 c1(t)c*D (t).dt∫ Tcs

0 c2(t).c*1 (t).dt,∫ Tcs0 c22 (t).dt ...,

∫ Tcs0 c2(t)c*D (t).dt

..., ..., ..., ...∫ Tcs0 cD (t).c*1 (t).dt,

∫ Tcs0 cD (t).c*2 (t).dt, ...,

∫ Tcs0 c2D (t).dt

i.e.

Rcc =∫ Tcs

0c(t).c(t)H .dt =

1 0 ... 00 1 ... 0... ... ... ...0 0 ... 1

= ID (60)

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Optimum M-ary Rx based on Signal Constellation M-ary Signals: Energy and Cross-Correlation

Then, by using Equation-56 in conjunction with Equation-58, theenergy Ei of the signal si (t), ∀i , can be expressed as follows:

Ei =∫ Tcs

0wHsi c(t)︸ ︷︷ ︸=si (t)

· c(t)Hw si︸ ︷︷ ︸=s∗i (t)

· dt

= wHsi

(∫ Tcs

0c(t)c(t)Hdt

)w si

= wHsi ·Rcc · w si= wHsi · ID · w si= wHsi w si=

∥∥w si∥∥2 (61)

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Optimum M-ary Rx based on Signal Constellation M-ary Signals: Energy and Cross-Correlation

Similarly, by using Equation-57 in conjunction with Equation-58 thecross-correlation coef. ρij ∀ij becomes

ρij =1√EiEj

∫ Tcs

0wHsi · c(t) · c(t)

H · w sj · dt

=1√EiEj

· wTsi ·(∫ Tcs

0c(t) · c(t)H · dt

)· w sj

=1√EiEj

wHsi ·Rcc · w sj

=1√EiEj

wHsi · ID · w sj

=1√EiEj

wHsi w sj =wHsi w sj∥∥w si∥∥ ∥∥∥w sj∥∥∥

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Optimum M-ary Rx based on Signal Constellation M-ary Signals: Energy and Cross-Correlation

i.e.Ei = wHsi w si =

∥∥w si∥∥2 (62)

ρij =1√EiEj

wHsi · w sj =wHsi w sj∥∥w si∥∥ ∥∥∥w sj∥∥∥ (63)

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Optimum M-ary Rx based on Signal Constellation Signal Constellation

Signal ConstellationWith reference to the figure below

If the error signal ε(t) = 0

then the knowledge of the vector w s is as good asknowing the transmitted signal s(t)

or, in the case of M signals (M-ary system){knowledge of si (t)}={knowledge of w si }, ∀i

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Optimum M-ary Rx based on Signal Constellation Signal Constellation

Therefore, we may represent the signal si (t)by a point (w si ) in a D-dimensional Euclidean space with

D ≤ M (64)

The set of points (vectors) specified by the columns of the matrix

W =[w s1 , w s2 , ..., w sM

](65)

is known as “signal constellation” .

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Optimum M-ary Rx based on Signal Constellation Distance between two M-ary signals

Distance between two M-ary signals

The distance between two signals si (t) and sj (t) is the Euclideandistance between their associate vectors w si and w sj

wsj

wsi

dij

Origin

D­dimensional spacei.e. dij =

∥∥∥w si − w sj∥∥∥=√(w si − w sj )H (w si − w sj )

=√wHsi w si + w

Hsj w sj − 2Re(wHsi w sj )

=√Ei + Ej − 2ρij

√EiEj

d2ij = Ei + Ej − 2ρij√EiEj (66)

It is clear from the above that the Euclidean distance dij of twosignals indicates, like the cross-correlation coeffi cient, the similarity ordissimilarity of the signals.

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Optimum M-ary Rx based on Signal Constellation Distance between two M-ary signals

An Important Bound involving dij :I If pe ,cs(si (t)) denotes the prob. of error associated with the channelsymbol si (t), it can be found that (see S. Haykin p.498)

pe ,cs(si (t)) ≤M∑j=1j 6=i

T{

dij√2N0

}(67)

Then, the symbol error probability pe ,sc is bounded as follows:

pe ,sc ≤ 1M

M∑i=1

M∑j=1j 6=i

T{

dij√2N0

}(68)

I N.B.: the following expression was used to go from Equ.67 to Equ.68

pe ,sc =1M

M∑i=1

pe ,cs(si (t)) (see S.Haykin p.498) (69)

“minimum distance”: dmin , min∀i ,j{dij}

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Optimum M-ary Rx based on Signal Constellation Examples of Signal Constellation Diagram

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

{s1(t), s2(t), . . . , sM (t)} with 0 ≤ t ≤ Tcs

M-ary ASKI channel symbols: si (t) = Ai · cos(2πFc t) whereAi =given= 2i − 1−M (say)

I dimensionality of signal space = D = 1I if M = 4 then

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

Es1 Es2 Es3 Es4Origin

I if M = 8 then

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

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Optimum M-ary Rx based on Signal Constellation Examples of Signal Constellation Diagram

M-ary PSK

I channel symbols: si (t) = A. cos(2πFc t + 2π

M . (i − 1) + φ)

for i = 1, 2, ..,I dimensionality of signal-space = D = 2

M = 4 & φ = 0◦ M = 4 & φ = 45◦ M = 8 & φ = 0◦

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

450Origin

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

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Optimum M-ary Rx based on Signal Constellation Examples of Signal Constellation Diagram

M-ary FSK:very diffi cult to be represented using constellation diagram

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Optimum M-ary Rx based on Signal Constellation Examples of Plots of Decision Variables (QPSK-Receiver)

Examples of Plots of Decision Variables (QPSK- Receiver)

­1.5 ­1 ­0.5 0 0.5 1 1.5­1.5

­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"

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Optimum M-ary Rx based on Signal Constellation Optimum M-ary Receiver using the Signal-Vectors

Optimum M-ary Receiver using the Signal-Vectors

Let us consider again the general M-ary problem:

r t =s t +n t( ) ( ) ( )iDecision rule=?

(To determine whichsignal is present)

Di

where si (t) = one of M signals (channel symbols)

Hypotheses:H1 : s1(t) is present with probability Pr(H1)H2 : s2(t) is present with probability Pr(H2)· · · · · · · · · · · ·HM : sM (t) is present with probability Pr(HM )

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Optimum M-ary Rx based on Signal Constellation Optimum M-ary Receiver using the Signal-Vectors

we have seen that the MAP criterion, using the “CONTINUOUSSAMPLING" concept, is described by the following rule:Rule: choose the hypothesis Hi with the maximum

Pr(Hi )× pdfr/Hi (r(t))

where

max{Pr(Hi )× pdfr/Hi (r(t))

}= max

{∫ Tcs

0r(t)s∗i (t)dt +

N02ln(Pr(Hi ))−

12Ei

}(70)

Remember that this rule can also be described by the following twoequivalent receivers:

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Optimum M-ary Rx based on Signal Constellation Optimum M-ary Receiver using the Signal-Vectors

⇐⇒

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  M­ary RECEIVER:  CORREL. RECEIVER

where

DCi =N02ln(Pr(Hi ))−

12Ei (71)

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Optimum M-ary Rx based on Signal Constellation Optimum M-ary Receiver using the Signal-Vectors

Now let us use the Approx. Theorem of energy signals to representthe received signals r(t) as well as the M-ary signal si (t), ∀i ,

i.e.

r(t) = wHr c(t)si (t) = wHsi c(t) =⇒ s∗i (t) = (w

Hsi c(t))

= c(t)Hw si ∀iIn this case the MAP receiver (Equ-70) becomes

max{Pr (Hi)× pdfr/Hi(r(t))

}= max

{∫ Tcs

0r(t)s∗i (t)dt +

N02ln(Pr(Hi ))−

12Ei

}= max

{∫ Tcs

0wHr c(t) · c(t)Hw sidt +

N02ln(Pr(Hi ))−

12Ei

}= max

{wHr (

∫ Tcs

0c(t) · c(t)Hdt)w si +

N02ln(Pr(Hi ))−

12Ei

}= max

{wHr w si +

N02ln(Pr(Hi ))−

12Ei

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Optimum M-ary Rx based on Signal Constellation Optimum M-ary Receiver using the Signal-Vectors

i.e. max{Pr(Hi )× pdfr/Hi(r(t))

}= max

{wHr w si +

N02ln(Pr(Hi ))−

12Ei

}(72)

The above equation may be implemented as follows:

or

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Optimum M-ary Rx based on Signal Constellation Encoding M-ary Signals

Encoding M-ary SignalsThe performance of M-ary systems is evaluated by means of theaverage probability of symbol error pe ,cs , which, for M > 2, isdifferent than the average probability of bit error (or Bit-Error-RateBER), pe .

That is{pe ,cs 6= pe for M > 2pe ,cs = pe for M = 2

(73)

However, because we transmit binary data, the probability of bit errorpe is a more natural parameter for performance evaluation than pe ,cs .Although, these two probabilities are related, i.e. pe =f{pe ,cs} theirrelationship depends on the encoding approach which is employed bythe digital modulator for mapping binary digits to M-ary signals(channel symbols)

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Optimum M-ary Rx based on Signal Constellation Important Relationships Between pe and pe ,cs

Important Relationships Between BER and SER

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

1γcs· pe ,cs ≤ pe ≤ pe ,cs (74)

e.g. M-ary PSK (for Gray code, otherwise diffi cult):

BER = pe =1

γcs· pe ,cs (75)

I N.B.: Note that Gray encoder provides a mapping where adjacentsymbols differ in one binary digit. This property is very importantbecause the most likely errors (due to channel noise effects) are relatedwith an erroneous selection (wrong decision) of an adjacent symbol.

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Optimum M-ary Rx based on Signal Constellation Important Relationships Between pe and pe ,cs

If all M-ary signals are equally likely, then it can be proved (seeHaykin p.500) that

pe =2γcs−1

2γcs − 1 · pe ,cs (76)

which, for large γcs is simplified to

pe =12· pe ,cs (77)

Example:

M-ary FSK: BER = pe =2γcs−1

2γcs − 1 · pe ,cs (78)

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Appendices: Appendix-1: Proof of Equation-26

Appendix-1: Proof of Equation-26The above rule can be rewritten as follows:

max{Pr(Hi )× pdfr/Hi(r(t))

}= max

{ln (Pr(Hi )) + ln (const.)−

1N0

∫ Tcs

0(r(t)− si (t))2dt

}= max

{ln (Pr(Hi )) −

1N0

∫ Tcs

0(r(t)− si (t))2dt

}= max

{ln (Pr(Hi )) −

1N0

∫ Tcs

0r2(t)dt − 1

N0

∫ Tcs

0s2i (t)dt +

2N0

∫ Tcs

0r(t)s∗i (t)dt

}= max

{ln (Pr(Hi ))−

1N0Er −

1N0Ei +

2N0

∫ Tcs

0r(t)s∗i (t)dt

}= max

{N02ln (Pr(Hi ))−

12Ei +

∫ Tcs

0r(t)s∗i (t)dt

}= max

{∫ Tcs

0r(t)s∗i (t)dt +

N02ln (Pr (Hi ))−

12Ei

}

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Appendices: Appendix-2: Walsh-Hadamard Orthogonal Sets and Signals

Appendix-2: Walsh-Hadamard Orthogonal Sets and Signals

H1 = [1]

H2 =

[1 11 −1

](2× 2) matrix

H64 = H2 ⊗H2 ⊗H2 ⊗H2 ⊗H2 ⊗H2︸ ︷︷ ︸6−times

where ⊗ denotes the Kronecker product of two matricese.g. for two matrices A and B

if A =

[A11 A12A21 A22

]then A⊗B =

[A11B A12BA21B A22B

]Properties:I HT

64 H64 = 64 I64I Let hi denote the i-th column of H64i.e. H64 =

[h1, h2, . . . , h64

]

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Appendices: Appendix-2: Walsh-Hadamard Orthogonal Sets and Signals

The following example illustrates an application of “Walsh Hadamard”in a mobile communications where (each user in the same cell andsame frequency channel is assigned one column of Walsh matrix,

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Appendices: Appendix-3: IS95 - Walsh Function of order-64

Appendix-3: IS95 - Walsh Function of order-64

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