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4.9 Matched filters in the frequency domain

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An analysis of matched filters in the frequency domain.

The time domain analysis and implementation of matched filters can be found in Matched Filters .

A frequency domain interpretation of matched filters is very useful

SNR ∞ ∞ s m h m T 2 N 0 2 ∞ ∞ h m T 2
For the m -th filter, h m can be expressed as
s m T ∞ ∞ s m h m T H m f S m f f ∞ ∞ H m f S m f 2 f T
where the second equality is because s ~ m is the filter output with input S m and filter H m and we can now define H m f H m f 2 f T , then
s m T S m f H m f

The denominator

∞ ∞ h m T 2 ∞ ∞ h m 2
h m h m 0 f ∞ ∞ H m f 2 H m f H m f
h m h m 0 f ∞ ∞ H m f 2 f T H m f 2 f T H m f H m f
Therefore,
SNR S m f H m f 2 N 0 2 H m f H m f 2 N 0 S m f S m f
with equality when
H m f S m f
or

Matched filter in the frequency domain

H m f S m f 2 f T

Matched filter

s m t s m f s m f f ∞ ∞ s m f 2 2 f t f ∞ ∞ s m f 2 2 f t
where is the inverse Fourier Transform operator.
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Source:  OpenStax, Digital communication systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10134/1.3
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