eee461lect11(matched filters)

8/2/2019 EEE461Lect11(Matched Filters)

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EEE 461 1

Chapter 6Chapter 6Matched FiltersMatched Filters

Huseyin Bilgekul

EEE 461 Communication Systems IIDepartment of Electrical and Electronic Engineering

Eastern Mediterranean University

Matched Filters Matched filters for white noise Integrate and Dump matched filter Correlation processing


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EEE 461 2

Matched FilterMatched Filter

The Matched Filter is the linear filter that maximizes:

Recall( ) ( ) ( ) ( ) ( ) ( )y t h t x t Y f H f X f= =

Matched Filter

h(t)

H(f)

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

R(f)

ro(t)=so(t)+no(t)

Ro(f)

( ) ( ) ( )2

y xS f H f S f =

( )

( )

2

2

o

out o

s tS

N n t

=


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EEE 461 3

Matched FilterMatched Filter Design a linear filter to minimize the effect of noise while

maximizing the signal. s(t) is the input signal ands0(t) is the output signal.

The signal is assumed to be known and absolutely time limited and

zero otherwise.

The PSD,Pn(f) of the additive input noise is also assumed to beknown.

Design the filter such that instantaneous output signal power is

maximized at a sampling instant t0, compared with the average

output noise power:( )

( )

2

2

o

out o

s tS

N n t

=


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EEE 461 4

Matched FilterMatched Filter The goal is maximize (S/N)out

s(t)

T T

( )

( )

2

2

o

out o

s tS

N n t

=

h(t)

H(f) ThresholdDetector

Sampler

t= tor(t)=s(t)+n(t)

R(f)

ro(t)=so(t)+no(t)

Ro(f)

so(t)

r(t)=s(t)+n(t)ro(t)=so(t)+no(t)


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EEE 461 5

Matched FilterMatched Filter The matched filterdoes not preserve the input signal shape.

The objective is to maximize the output signal-to-noise ratio.

The matched filter is the linear filter that maximizes (S/N)out and has atransfer function given by:

where S(f) =F[s(t)] of duration Tsec.

t0 is the sampling time Kis an arbitrary, real, nonzero constant.

The filter may not be realizable.

( )( )

( )

oj t

n

S f eH f K

P f

=


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EEE 461 6

Signal and Noise CalculationSignal and Noise Calculation Signal output:

Output noise power or variance

Putting the pieces together gives:

Simplify Using Schwartz Inequality.

Equality occurs only ifA(f) =KB*(f)

( ) ( ) ( ){ } ( ) ( )2 oj to os t t F S f H f S f H f e df

= = =


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EEE 461 7

Signal and Noise CalculationSignal and Noise Calculation

Apply the Schwartz Inequality:

Then we obtain:

Maximum (S/N)out is attained when equality occurs if we

choose:

( ) ( ) ( ) ( ) ( ) ( ), oj tn nA f H f P f B f S f e P f= =

( ) ( ) ( ) ( ) ( )

( ) ( )

or

o oj t j t

n

nn

KS f e KS f e

H f P f H f P fP f

= =

( )( )

( )

oj t

n

S f eH f K

P f

=


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EEE 461 8

Matched Filter for White NoiseMatched Filter for White Noise For a white noise channel,Pn(f) =No/2

HereEs is the energy of the input signal. The filterH(f) is:

The output SNR depends on the signal energyEs and not on the

particular shape that is used.

Impulse response is the known signal wave shape played

Backwards and shifted by to.

h l f h


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EEE 461 9

Matched Filter for White NoiseMatched Filter for White Noise Increase in the time-bandwidth product does not change the output

SNR.

If a symbol lasts forTseconds, then there are 3 cases: (to< T, to= T and

to> T)

t< T ives aNONCAUSAL in ut res onse

( ) ( ) ( ) ( )2 2

o

Fj t

o

o o

K Kh t s t t H f S f e

N N

= =


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EEE 461 10

Impulse Response of Matched FilterImpulse Response of Matched Filter

Thus,s(t) and h(t) have duration T.

The delay is also T

The output has duration 2Tbecause s0(t) = s(t)*h(t). Note that the peak value is at T.

2T

s(t)+n(t)so(t)

( ) ( ) ( ) ( )

Fj Th t Cs T t H f CS f e = =


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EEE 461 11

Impulse Response of Matched FilterImpulse Response of Matched Filter

The output is obtained by performing convolution s0(t) = s(t)*h(t).


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EEE 461 12

MF Example for White NoiseMF Example for White Noise

Consider the set of signals:

Draw the matched filter for each signal and

sketch the filter responses to each input

T/2 T

s1(t)

T/2 T

s2(t)


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EEE 461 13

T/2 T

h1(t)

T/2 T

s1(t)

T/2 T

s2(t)

MF Example for White NoiseMF Example for White Noise

T/2 T

h2(t)

T/2 T

y11(t)=s1(t)*h1(t)

T/2 T

y21(t)=s2(t)*h1(t)

d D (M h d) F lI d D (M h d) Fil


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EEE 461 14

Integrate and Dump (Matched) FilterIntegrate and Dump (Matched) Filter

d ( h d) F lI d D (M h d) Fil


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EEE 461 15

Integrate and Dump (Matched) FilterIntegrate and Dump (Matched) Filter

Input Signal

Backward Signal

Matched Filter Impulse Response

Matched Filter Output Signal

I d D R li i f M h d Fil


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EEE 461 16

Integrate and Dump Realization of Matched FilterIntegrate and Dump Realization of Matched Filter

C l i P iC l ti P i


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EEE 461 17

Correlation ProcessingCorrelation Processing

C l ti P iC l ti P i


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EEE 461 18

Correlation ProcessingCorrelation Processing Theorem: For the case of white noise, the matched filter can be realized

by correlating the input withs(t) where r(t) is the received signal and

s(t) is the known signal wave shape.

Correlation is often used as a matched filter for Band pass signals.

( ) ( ) ( )o

o

t

o ot T

r t r t s t dt

=

C l ti (M t h d Filt ) D t ti f BPSKC l ti (M t h d Filt ) D t ti f BPSK


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EEE 461 19

Correlation (Matched Filter) Detection of BPSKCorrelation (Matched Filter) Detection of BPSK

( )

cos If 2

cos If 2

( )2

c

c

A t

s t A t

nT t n T

+=

< +

eee461lect11(matched filters)

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