dc02 detection theory

8/9/2019 DC02 Detection Theory

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DETECTION THEORY


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Gram-Schmidt Procedure

Permits the representation of any set ofMenergy signals, {si(t)}, as linear

combinations ofNorthonormal basisfunctions { }, whereN M)(tj


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Represent the given set of real-valued energysignals1(t), s2(t), ,sM(t), each of duration Tseconds, in the form

N

j

jiji tsts1

)()( Mi

Tt

,....,2,1

0

T

jiij dtttss0

)()( Nj

Mi

,....,2,1

,....,2,1


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The real-value basis functions , .are orthonormal

Each basis function is normalized to have unit energy

The basis functions are orthogonal with respect toeach other over the interval 0 t T

T

ji dttt0

01)()(

jiji

)(1 t )(2 t


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Example


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Stage 1: We have to establish whether or not the given

set of signals1(t), s2(t), , sM(t) is linearly

independent Lets1(t), s2(t), , sN(t) denote subset of linearly

independent signals, whereN M eachmember of the original set of signals s1(t), s2(t),

, sM(t) may be expressed as a linearcombination of this subset of Nsignals


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Stage 2:

Construct a set of Northonormal basisfunctions

1(t),

2(t), ,

N(t) from the linearly

independent signalss1(t), s2(t), , sN(t)


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Geometric Interpretations ofSignals

With orthonormal basis functions {i(t)},j = 1,2, , N, each real-valued energy signals1(t),

s2(t), ,sM(t), can be expanded as

N

j

jiji tsts1

)()( Mi

Tt

,....,2,1

0

T

jiij dtttss0

)()( Nj

Mi

,....,2,1

,....,2,1


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Example


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Solution


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Each signal in the set {si(t)} is completely determined

by the vector of its coefficients

Si is called the signal vector

iN

i

i

i

s

ss

2

1

s Mi ,....,2,1


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Set of signal vector {si}, i = 1, 2, , Masdefining a corresponding set of Mpoints inaN-dimensional Euclidean space

N-dimensional Euclidean space is called

signal space


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In anN-dimensional Euclidean space,

define lengths of vectors and anglesbetween vectors


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Square-length of any signal vector si is define tobe the inner product or dot product of si

The cosine of the angle between the vectors siand sj equals the ratio

N

j

ijiii s1

22 ),( sss

ji

ji

ss

ss ),(


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Energy of a signalsi(t) of duration T seconds is

equal to

Euclidean distance between the pointsrepresented by the signal vector si and sk

T N

jijii sdttsE 0 1

22

)(

N

j

T

kikjijki dttstsss1 0

22 )]()([ss


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Model of a DigitalCommunication System


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At the transmitter input, we have amessage source that emits one symbolevery T seconds, denote by m

1

, m2

,, mM

Assume that all symbols are equally likely

( )i iP P m 1

M


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The output of message source is presentedto a vector transmitter, producing a vectorof real numbers

1

2

.

.

.

i

i

i

i N

s

s

s

s


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The channel is assumed to have twocharacteristics:

The channel is linear, with a bandwidth that islarge enough to accommodate the transmissionof thesi(t) without distortion

The transmitted signalsi(t) is perturbed by an

additive, zero-mean, stationary, white Gaussiannoise process, denote by W(t)


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We expressed the received randomprocess, X(t)

x(t) is referred as the received signal

( ) ( ) ( )iX t S t W t 01,2,...,t T

i M


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The requirements is to design the vectorreceiver so as to minimize the averageprobability of symbol error defined as

mi is the transmitted symbol and is estimateproduced by the vector receiver

( )e iP P m m

m


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If the receiver is phase locked to thetransmitter, it is referred to coherentreceiver

On the other hand, the receiver is referredto non-coherent receiver


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Response of Bank of Correlatorsto Noisy Input

Suppose that the input of the bank of N

product-integrator or correlators israndom processX(t)

W(t) is a white Gaussian noise process of zero

mean and power spectral densityN0/2

( ) ( ) ( )iX t S t W t 0

1, 2,...,

t T

i M

f k f C l


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R f B k f C l


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The output of each correlator is

0

( ) ( )T

i jX X t t dt ij js W 1, 2,...,j N

R f B k f C l t


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The first component, sij, is a deterministicquantity contributed by the transmitted signalsi(t)

The second component, Wj, is a random variablethat arises because of the presence of noise atthe input

0( ) ( )Tij i js s t t dt

0( ) ( )

T

j jW W t t dt

R f B k f C l t


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Consider next a new random process,X(t)

1'( ) ( ) ( )

N

j jJX t X t X t

1

'( ) ( ) ( ) ( ) ( )N

i ij j j

j

X t s t W t s w t

1

( ) ( )

'( )

N

j j

j

W t W t

W t

R f B k f C l t


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We may express the received randomprocess as

1

( ) ( ) '( )

N

j j

j

X t X t X t

1

( ) '( )N

j j

j

X t W t

R f B k f C l t


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The noise process W(t) has zero mean,hence the random varialbe Wj extractedfrom W(t) has zero mean

The mean value of jth correlator output Xjdepends only onsij

jx j

m E X

ij jE s w ij j ijs E W s

Response of Bank of Co elato s


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The variance ofXj

2

j jx Var X

2

j ijE X s

2

jE W

Response of Bank of Correlators


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2

0 0( ) ( ) ( ) ( )

j

T T

X j jE W t t dt W u u du

0 0 ( ) ( ) ( ) ( )

T T

j jE t u W t W u dtdu

0 0 ( ) ( ) ( , )T T

j j Wt u R t u dtdu

0 0

( ) ( ) ( ) ( )T T

j t j u E W t W u dtdu



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Since

Then

0( , ) ( )2

W

NR t u t u

2 0

0 0( ) ( ) ( )

2jT T

X j j

Nt u t u dtdu

20

0

( )2

T

j

Nt dt

0

2

N



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Similarly, we find thatXj are mutually uncorrelated

j kj k j X k X

Cov X X E X m X m

0 0

0 0

0

0 0

0

0

( ) ( ) ( ) ( )

( ) ( ) ( , )

( ) ( ) ( )2

( ) ( ) 02

j ij k ik

j k

T T

j k

T T

j k W

T T

j k

T

j k

E X s X s

E W W

E W t t dt W u u du

t u R t u dtdu

Nt u t u dtdu

Nt t dt



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Define the vector of N random variables atthe correlator outputs as

1

2

.

.

.

N

X

X

X

X



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Since the elements of the vector X arestatistically independent, we may express theconditional probability density function of thevector X, given that the signal si(t) or

corresponding the symbol mi was transmitted

This is called likelihood functions

1

| |j

N

i X j i

j

f m f x m

X x 1, 2,...,i M



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Since eachXj is a Gaussian random variablewith meansij and varianceN0/2, we have

2

00

1 1| exp

jX j i j ijf x m x s

NN



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Likelihood functions of AWGN channel aredefined by

2

2

0

10

1| exp

N N

i j ij

jf m N x sN

x x

1, 2,...,i M


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Maximum Likelihood Detector

Suppose that, when the observation vectorhas the value x, we make decision

The average probability of symbol error inthis decision, denoted byPe(mi,x), is

mm

( , ) not sent|e i iP m x P m x

1 sent|iP m x


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The optimum decision rule is

set if

for all k i

Where k = 1, 2, , M

This decision rule is referred to as maximum aposteriori probability (MAP)

im m

sent| sent|i kp m p mx x


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Applying Bayes rule, we may restate thisdecision rule as

set if

is maximum for k =i

i

m m

|k kp f mf

x

x

x


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SincefX(x) is independent of the transmittedsignal, and the a priori probability pk= pi when allthe signals are transmitted with equal probability,we may simplify the decision rule as

set if

is maximum for k =i

The decision rule is referred to as maximum likelihood, and thedevice for its implementation is referred as maximum-likelihooddetector

im m

| kf mx x


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Since likelihood function is alwaysnonnegative, we may restate the decisionrule

set if

is maximum for k = i

im m

ln | kf m x x


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Let Z denote the N-dimensional space of allpossibly observed vector x. we refer to this space

as observation space.

Because we have assumed that the decision rulemust say , where i = 1, 2, , M, the totalobservation space Z is correspondinglypartitioned intoMdecision regions, denoted by

Z1, Z2, , ZM

imm


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We may restate the decision rule as follow:

vector x lies inside region Zi if

is maximum for k = i ln | kf m x x


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We may formulate the maximum likelihooddecision rule for AWGN channel as


is maximum for k = i 2

10

1 N

j kjj

x sN


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Equivalently, we may restate


is minimum for k = i2

1

( )N

j kj

j

x s


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Since

We may rewrite the decision rule


is minimum for k = i

22

1

( )N

j kj kj

x s

x s

2

kx s


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This show that maximum-likelihooddecision rule is simply to choose themessage point closest to the received

signal point


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Probability of Error

The average probability of symbol errorequals

i1 ( sent) ( does not lie inside Z | sent)

M

e i iiP P m P m

x

i

1

11 ( lies inside Z | sent)

M

i

i

P mM

x


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Probability of Error

i( lies inside Z | sent) ( | )

i

i i

Z

P m f m d xx x x

1

11 ( | )

i

i

M

ei Z

P f m dM

x x x


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Correlation Receiver

The optimum receiver consists of two subsystems The detector part of the receiver consists of bank of M

product-integrators or correlators supplied with acorresponding set of coherent reference signals or

orthonormal basis functions The second part of the receiver, namely the vector

receiver, is implemented in the form of a maximum-likelihood detector that operates on the observationvector x to produce an estimate of the transmitted

symbol mi, i = 1, 2, , M, in a way that would minimizethe average probability of symbol error

m


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Correlator Detector


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Correlator Receiver


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Matched Filter Receiver

Since each of the orthonormal basis functions1(t), 2(t), , N(t) is assumed to be zero outsidethe interval 0t Tmay be avoided because

analog multipliers are usually hard to build Consider a linear filter with impulse response hj(t).

With the receiver signalx(t) used as the filterinput, the resulting outputyj(t)

dthxty jj )()()(


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Suppose

The resulting filter output is

)()( tTth jj

dtTxty jj )()()(


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Sampling the output at time t = T, we get

Since j(t) is zero outside the interval 0t T,finally we get

Note thatyj(t) = xj, wherexj is thejth

correlator outputproduced by the received signalx(t)

dxTy jj )()()(

T

jj dxTy0

)()()(


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The detector part of the optimumreceiver may also be implemented asbank of matched filter

The optimum receiver based on thisdetector is referred as the matched filter

receiver


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Matched Filter

Consider a linear filter of impulse response h(t),with an input that consist of a known signal, (t),and an additive noise component, w(t)

T is the observation instant

w(t) is the sample function of a white Gaussian noise

process of zero mean and power spectral density N0/2

)()()( twttx Tt0


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Matched Filter

The resulting output,y(t), may be expressed as

0(t) and n(t) are prodeuced by the signal and noise

components of the input x(t), respectively

)()()( 0 tntty

h d l


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Matched Filter

The output signal component 0(t) be considerablygreater than the output noise component n(t) is to

have the filter make the instantaneous power in theoutput signal 0(t), measured at time t = T, as large

as possible compared with the average poser of theoutput noise n(t)

This is equivalent to maximizing the output signal-to-noise ratio

)()(

)(2

20

tnE

TSNR O

M h d Fil


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Matched Filter

(f) denote the Fourier transform fo the knownsignal (t), andH(f) denote the transfer function

of the filter

When the filter output is sampled at time t = T

dfftjffHt

2exp)()()(0

22

0 2exp)()()( dffTjffHT

M t h d Filt


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Matched Filter

The power spectral density SN(f) of the outputnoise n(t)

The average power of the output noise n(t)

2)(

2

)( 0

fHN

fSN

dffStnE N )()(

2

dffHN 20 )(2

M t h d Filt


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Matched Filter

The output signal-to-noise ratio

2

2)(

2

2exp)()(

0

dffHN

dffTjffH

SNR O

M t h d Filt


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Matched Filter

According to Schwarzs inequality

The output signal-to-noise ratio

dffdffHdffTjffH222

)()(2exp)()(

dff

NSNR O

2

0

)(2

M t h d Filt


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Matched Filter

The signal energy given

The noise power spectral densityN0/2

dtfdtt22

)()(

M t h d Filt


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Matched Filter

Consequently, the output signal-to-noise ratio willbe a maximum whenH(f) is chosen so that the

equality holds

The optimum value of this transfer function isdefined by

dffNSNRO

2

0

max, )(2

fTjffHop t 2exp)()( *

Matched Filter


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Matched Filter

The impulse response of matched filter

Since for real-valued signal (t), we have*(f) = (-f)

dftTfjfthop t )(2exp)()(

*

dftTfjfthopt )(2exp)()( *

tT

Matched Filter


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Matched Filter

Properties of Matched filter

The spectrum of the output signal of a matchedfilter with the matched signal as input is, except

for a time delay factor, proportional to the energyspectral density of the input signal

)()()(0 ffHf opt

fTjf

fTjff

2exp

2exp)()(

2

*

Matched Filter


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Matched Filter


The output signal of a matched filter isproportional to a shifted version of the

autocorrelation function of the input signal towhich the filter is matched

TtRt )(0

ERT )0()(0

Matched Filter


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Matched Filter


The output signal-to-noise ratio of a matched filterdepends only on the ratio of the signal energy to

the power spectral density of the white noise atfilter input

ENdffNtnE2

)(2

)( 02

02

00

2

max,2

2 NE

ENESNR O

Matched Filter


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Matched Filter


The matched filter operation may be separatedinto two matching conditions; namely, spectral

phase matching that produces the desired outputpeak at time T, and spectral amplitude matchingthat gives this peak value its optimum signal-to-noise density ratio

)()( ffH fTjfjfHfH 2)(exp)()(

Detection of Signals WithUnknown Phase In Noise


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Unknown Phase In Noise

Up to this point in our discussion, we have assumed thatthe information bearing signal is completely known at thereceiver.

In practice, however, it is often found that in addition tothe uncertainty due to the additive noise of a receiver,there is an additional uncertainty due to the randomnessof certain signal parameters

The usual cause of this uncertainty is distortion in thetransmission medium. The most common random signalparameter is the phase, which is especially true fornarrow-band signals.



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Synchronization with the phase of thetransmitted carrier may be too costly, andthe designer may simply choose to

disregard the phase information in thereceived signal at the expense of somedegradation in the noise performance ofthe system



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Consider a digital communication system inwhich the transmitted signal equals

Eis the signal energy

T is the duration of the signaling interval

fi is an integral multiple of 1/2T

2

( ) cos 2i i

E

s t f tT

0 t T



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When no provision is made to phase synchronize thereceiver with the transmitter, the received signal will, foran AWGN channel, be of the form

w(t) is the sample function of a white Gaussian noise process ofzero mean and power spectral density N0/2

The phase is unknown, and is usually considered to be the

sample value of a random variable uniformly distributed between0 and 2radians

2

( ) cos 2 iE

x t f t w tT

0 t T



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Using a well-known trigonometric identity, we may write

Suppose that the received signal x(t) is applied to a pair ofcorrelators; we assume that one correlator is suppliedwith the reference signal and the other issupplied with the reference signal

For Both correlators, the observation interval is 0 t T

2 2

( ) cos cos 2 sin sin 2i iE E

x t f t f t w tT T

0 t T

2 cos 2 iT f t 2 sin 2 iT f t



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In the absence of noise, we find that the firstcorrelator output equals and the secondcorrelator output equals

The dependence on unknown phase may be

removed by summing the squares of the twocorrelator outputs, and then taking the squareroot of the sum

When the noise w(t) is zero, the result of these

operations is simply , which is independent ofthe unknown phase

cosE

sinE

E



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The suggests that for the detection of asinusodal signal of arbitrary phase, andwhich is corrupted by an additive white

Gaussian noise, we may use the so-calledquadrature receiver

This receiver is optimum in the sense that

it realizes this detection with the minimumprobability of error



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To obtain the second equivalent form of thequadrature receiver, suppose we have a filter thatis matched to

The envelope of the matched filter output isobviously unaffected by the value of the phase

2 cos 2 is t f tT 0 t T



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The output of such a filter in response tothe received signalx(t) is given

0

2cos 2

T

i

y t x f T t dT

0

0

2cos 2 cos 2

2 sin 2 sin 2

T

i i

T

i i

f T t x f dT

f T x f dT



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The envelop of matched filter output, evaluatedat time t = T, will be

This is just the output of the quadrature receiver.Therefore, the output at time T) of a filter matched to the

signal , of arbitrary phase ,followed by an envelope detector is the same as thecorresponding ouput of the quadrature receiver

1/2 2

0 0

2 2cos 2 sin 2

T T

i i il x f d x f d

T T

2 cos 2 is t T f t



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The combination of matched filter andenvelope detector is called a noncoherentmatched filter



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dc02 detection theory

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