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4.7 Matched filters

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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 signalenergy noiseenergy
In the correlator described earlier, E s s m 2 and n 2 N 0 2 . Is it possible to design a demodulator based on linear time-invariantfilters with maximum signal-to-noise ratio?

If s m t is the transmitted signal, then the output of the k th filter is given as

y k t ∞ ∞ r h k t ∞ ∞ s m N h k t ∞ ∞ s m h k t ∞ ∞ N h k t
Sampling the output at time T yields
y k T ∞ ∞ s m h k T ∞ ∞ N h k T
The noise contribution:
k ∞ ∞ N h k T
The expected value of the noise component is
k ∞ ∞ N h k T 0
The variance of the noise component is the second moment since the mean is zero and is given as
k k 2 ∞ ∞ N h k T ' ∞ ∞ N ' h k T '
k 2 ' ∞ ∞ ∞ ∞ N 0 2 ' h k T h k T ' N 0 2 ∞ ∞ h k T 2

Signal Energy can be written as

∞ ∞ s m h k T 2

and the signal-to-noise ratio (SNR) as

SNR ∞ ∞ s m h k T 2 N 0 2 ∞ ∞ h k T 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 g 2 x . Applying the inequality directly yields an upper bound on SNR
∞ ∞ s m h k T 2 N 0 2 ∞ ∞ h k T 2 2 N 0 ∞ ∞ s m 2
with equality h k opt T s m . Therefore, the filter to examine signal m should be

Matched filter

h m opt s m T
The constant factor is not relevant when one considers the signal to noise ratio. The maximum SNR is unchanged when boththe numerator and denominator are scaled.
2 N 0 ∞ ∞ s m 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 receiverscan be found in Performance Analysis .
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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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