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Matched Filters in the Frequency Domain

Module by: Behnaam Aazhang

Summary: 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 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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