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Chapter 6Chapter 6Matched FiltersMatched Filters

Huseyin BilgekulEEE 461 Communication Systems II

Department 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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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 Filterh(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

2o

out o

s tS

N n t

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

maximizing the signal.

• s(t) is the input signal and s0(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 be known.

• Design the filter such that instantaneous output signal power is maximized at a sampling instant t0, compared with the average output noise power:

2

2o

out o

s tS

N n t

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

s(t)

T T

2

2o

out o

s tS

N n t

h(t)H(f)

Threshold Detector

Samplert = 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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Matched FilterMatched Filter• The matched filter does 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

a transfer function given by:

• where S(f) = F[s(t)] of duration T sec. • t0 is the sampling time• K is 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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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 if A(f) = K B*(f)

1 oj to os t t F S f H f S f H f e df

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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

nnn

KS f e KS f eH f P f H f

P fP f

oj t

n

S f eH f K

P f

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

• Here Es is the energy of the input signal. The filter H(f ) is:

• The output SNR depends on the signal energy Es and not on the particular shape that is used.

• Impulse response is the known signal wave shape played “Backwards” and shifted by to.

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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 for T seconds, then there are 3 cases: (to< T, to= T and to> T)

– to< T gives a NONCAUSAL input response

– to> T gives a DELAY in deciding what was sent

– to= T gives the MINIMUM DELAY for a decision plus it is REALIZABLE.

2 2o

Fj t

oo o

K Kh t s t t H f S f e

N N

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Impulse Response of Matched Impulse Response of Matched FilterFilter

• Thus, s(t) and h(t) have duration T.• The delay is also T

• The output has duration 2T because s0(t) = s(t)*h(t).

• Note that the peak value is at T.

2T

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

F

j Th t Cs T t H f CS f e

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Impulse Response of Matched Impulse Response of Matched FilterFilter

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

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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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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)

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Integrate and Dump (Matched) Integrate and Dump (Matched) FilterFilter

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

Input Signal

Backward Signal

Matched Filter Impulse Response

Matched Filter Output Signal

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Integrate and Dump Realization of Matched Integrate and Dump Realization of Matched FilterFilter

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Correlation ProcessingCorrelation Processing

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

realized by correlating the input with s(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 o t Tr t r t s t dt

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Correlation (Matched Filter) Detection of Correlation (Matched Filter) Detection of BPSKBPSK

cos If 1

cos If 0

( 1)

c

c

A ts t

A t

nT t n T