Matched Filters 2.14 2001/06/14 2005/09/20 10:27:21.915 GMT-5 Behnaam Aazhang aaz@ece.rice.edu Dinesh Rajan dinesh@ece.rice.edu Mohammad Jaber Borran mohammad@ece.rice.edu Roy Ha rha@rice.edu Shawn Stewart mrshawn@alumni.rice.edu Behnaam Aazhang aaz@ece.rice.edu Cauchy-Schwarz inequality correlator CSI linear matched filter signal-to-noise ratio SNR time-invariant A description of matched filters, which is a demodulation technique with LTI filters which achieves maximum SNR. Signal to Noise Ratio (SNR) at the output of the demodulator is a measure of the quality of the demodulator. SNR signal energy noise energy In the correlator described earlier, E s s m 2 and σ η n 2 N 0 2 . Is it possible to design a demodulator based on linear time-invariant filters with maximum signal-to-noise ratio?

If s m t is the transmitted signal, then the output of the kth filter is given as y k t τ r τ h k t τ τ s m τ N τ h k t τ τ s m τ h k t τ τ N τ h k t τ Sampling the output at time T yields y k T τ s m τ h k T τ τ N τ h k T τ The noise contribution: ν k τ N τ h k T τ The expected value of the noise component is ν k τ N τ h k T τ 0 The variance of the noise component is the second moment since the mean is zero and is given as ν k ν k 2 τ N τ h k T τ τ ' N τ ' h k T τ ' ν k 2 τ ' τ N 0 2 δ τ τ ' h k T τ h k T τ ' N 0 2 τ h k T τ 2 Signal Energy can be written as τ s m τ h k T τ 2 and the signal-to-noise ratio (SNR) as SNR τ s m τ h k T τ 2 N 0 2 τ h k T τ 2 The signal-to-noise ratio, can be maximized considering the well-known Cauchy-Schwarz Inequality x g 1 x g 2 x 2 x g 1 x 2 x g 2 x 2 with equality when g 1 x α g 2 x . Applying the inequality directly yields an upper bound on SNR τ s m τ h k T τ 2 N 0 2 τ h k T τ 2 2 N 0 τ s m τ 2 with equality τ h k opt T τ α s m τ . Therefore, the filter to examine signal m should be Matched Filter τ τ h m opt τ s m T τ The constant factor is not relevant when one considers the signal to noise ratio. The maximum SNR is unchanged when both the numerator and denominator are scaled. 2 N 0 τ s m τ 2 2 E s N 0 Examples involving matched filter receivers can be found here. An analysis in the frequency domain is contained in Matched Filters in the Frequency Domain. Another type of receiver system is the correlation receiver. A performance analysis of both matched filters and correlator-type receivers can be found in Performance Analysis.