Matched Filters in the Frequency Domain 2.11 2001/06/27 2005/09/20 11:12:12.105 GMT-5 Behnaam Aazhang aaz@ece.rice.edu Roy Ha rha@rice.edu Behnaam Aazhang aaz@ece.rice.edu Cauchy-Schwarz inequality correlator CSI linear matched filter signal-to-noise ratio SNR time-invariant 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 τ s m τ h m T τ 2 N 0 2 τ h m T τ 2 For the m-th filter, h m can be expressed as 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 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 τ h m T τ 2 τ h m τ 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 α S m f or Matched Filter in the frequency domain H m f S m f 2 f T

Matched Filter

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 where ℱ is the inverse Fourier Transform operator.