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Matched Filters in the Frequency Domain

Module by: Behnaam Aazhang

Summary: An analysis of matched filters in the frequency domain.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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 τ h m T−τdτ2 N 0 2∫-∞∞| h m T−τ|2dτ SNR τ s m τ h m T τ 2 N 0 2 τ h m T τ 2 (1)
For the mm-th filter, h m h m can be expressed as
s m ˜ T=∫-∞∞ s m τ h m T−τdτ=ℱ-1 H m f S m f=∫-∞∞ H m f S m fⅇⅈ2πfTdf s m ˜ T τ s m τ h m T τ ℱ H m f S m f f H m f S m f 2 f T (2)
where the second equality is because s ~ m s ~ m is the filter output with input S m S m and filter H m H m and we can now define H m ^ f= H m f¯ⅇ-ⅈ2πfT H m ^ f H m f 2 f T , then
s m ˜ T=< S m f, H ^ m f> s m ˜ T S m f H ^ m f (3)

The denominator

∫-∞∞| h m T−τ|2dτ=∫-∞∞| h m τ|2dτ τ h m T τ 2 τ h m τ 2 (4)
h m * h m 0=∫-∞∞| H m f|2df=< H m f, H m f> h m h m 0 f H m f 2 H m f H m f (5)
h m * h m 0=∫-∞∞ H m fⅇⅈ2πfT H m f¯ⅇ-ⅈ2πfTdf=< 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 (6)
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> 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 (7)
with equality when
H m ^ f=α S m f H m ^ f α S m f (8)
or

Matched Filter in the frequency domain

H m f= S m f¯ⅇ-ⅈ2πfT H m f S m f 2 f T (9)

Figure 1
Matched Filter
Matched Filter (Figure4-31.png)

s m ˜ t=ℱ-1 s m f s m f¯=∫-∞∞| s m f|2ⅇⅈ2πftdf=∫-∞∞| s m f|2cos2πftdf s m ˜ t ℱ s m f s m f f s m f 2 2 f t f s m f 2 2 f t (10)
where ℱ-1 ℱ is the inverse Fourier Transform operator.

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