Matched Filter

Matched Filters

By: Andy Wang

What is a matched filter? (1/1) A matched filter is a filter used in communications to

“match” a particular transit waveform. It passes all the signal frequency components while

suppressing any frequency components where there is only noise and allows to pass the maximum amount of signal power.

The purpose of the matched filter is to maximize the signal to noise ratio at the sampling point of a bit stream and to minimize the probability of undetected errors received from a signal.

To achieve the maximum SNR, we want to allow through all the signal frequency components, but to emphasize more on signal frequency components that are large and so contribute more to improving the overall SNR.

Deriving the matched filter (1/8) A basic problem that often arises in the study of communication systems is that

of detecting a pulse transmitted over a channel that is corrupted by channel noise (i.e. AWGN)

Let us consider a received model, involving a linear time-invariant (LTI) filter of impulse response h(t).

The filter input x(t) consists of a pulse signal g(t) corrupted by additive channel noise w(t) of zero mean and power spectral density No/2.

The resulting output y(t) is composed of go(t) and n(t), the signal and noise components of the input x(t), respectively.

)()()(

0),()()(

tntgty

Tttwtgtx

o

LTI filter of impulseresponse

h(t)∑

White noise w(t)

Signal

g(t)

y(t)x(t) y(T)

Sample at time t = T Linear receiver

Deriving the matched filter (2/8) Goal of the linear receiver

To optimize the design of the filter so as to minimize the effects of noise at the filter output and improve the detection of the pulse signal.

Signal to noise ratio is:

)(|)(||)(|

2

2

2

2

tnE

TgTgSNR o

n

o

where |go(T)|2 is the instantaneous power of the filtered signal, g(t) at point t = T, and σn

2 is the variance of the white gaussian zero mean filtered noise.

Deriving the matched filter (3/8)

We sampled at t = T because that gives you the max power of the filtered signal.

Examine go(t): Fourier transform

222

2

|)()(||)(|

)()()(

)()()(

dfefHfGtgthen

dfefHfGtgso

fHfGfG

ftjo

ftjo

o


Examine σn2:

dffSR

dfefSR

RtntnE

tnE

tnEtnE

nn

fjnn

n

n

n

10

0var2

2

22

22

but this is zero mean so

and recall that

autocorrelation is inverse Fourier transform of power spectral density

autocorrelation at 0

Deriving the matched filter (5/8) Recall:

dffHN

dfefGfHSNR

sodffHN

dffHN

RtnE

fHN

fS

o

fTj

oonn

on

2

2

2222

2

||2

|)()(|

||2

||2

0)(

||2

)(

H(f)

filter

SX(f) SX(f)|H(f)|2 = SY(f)

In this case, SX(f) is PSD of white gaussian noise,

Since Sn(f) is our output:2

)( oX

NfS


To maximize, use Schwartz Inequality.

dxx

dxx

22

21

||

|| Requirements: In this case, they must be finite signals.

dxxdxxdxxx 22

21

221 ||||||

This equality holds if φ1(x) =k φ2*(x).


We pick φ1(x)=H(f) and φ2(x)=G(f)ej2πfT and want to make the numerator of SNR to be large as possible

oo

oo

fT

fTjfT

N

dffG

N

dffGSNR

dffHN

dffGdffH

dffHN

dfefGfH

dfefGdffHdfefGfH

22

2

22

2

2

2222

|)(|2

2

|)(|

|)(|2

|)(||)(|

|)(|2

|)()(|

|)(||)(||)()(|

maximum SNR according to Schwarzinequality

Deriving the matched filter (8/8) Inverse transform

Assume g(t) is real. This means g(t)=g*(t) If

fGtgF

fGtgF

** )(

)(

then fGfG

fGfG

*

* for real signal g(t)

through duality

tTkgth

dfefGk

dfefGk

dfeefGkth

tTfj

tTfj

ftjfTj

2

2

22)(

Find h(t) (inverse transform of H(f))

h(t) is the time-reversed and delayed version of the input signal g(t).

It is “matched” to the input signal.

What is a correlation detector? (1/1) A practical realization of the

optimum receiver is the correlation detector.

The detector part of the receiver consists of a bank of M product-integrators or correlators, with a set of orthonormal basis functions, that operates on the received signal x(t) to produce the observation vector x.

The signal transmission decoder is modeled as a maximum-likelihood decoder that operates on the observation vector x to produce an estimate, .m̂

Detector

Signal Transmission Decoder

The equivalence of correlation and matched filter receivers (1/3) We can also use a corresponding set of matched filters

to build the detector. To demonstrate the equivalence of a correlator and a

matched filter, consider a LTI filter with impulse response hj(t).

With the received signal x(t) used as the filter output, the resulting filter output, yj(t), is defined by the convolution integral:

dthxty jj

The equivalence of correlation and matched filter receivers (2/3) From the definition of the

matched filter, we can incorporate the impulse hj(t) and the input signal φj(t) so that:

Then, the output becomes:

Sampling at t = T, we get:

tTth jj

dtTxty jj )(

dxty jj

The equivalence of correlation and matched filter receivers (3/3) So we can see that the

detector part of the receiver may be implemented using either matched filters or correlators. The output of each correlator is equivalent to the output of a corresponding matched filter when sampled at t = T.

Matched filters

Correlators

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Matched Filter