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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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Correlation ProcessingCorrelation Processing
C l ti P iC l ti P i
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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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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
+=
< +