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Matched Filters

Module by: Behnaam Aazhang

Summary: 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 energynoise energy SNR signal energy noise energy (1)
In the correlator described earlier, E s =| s m |2 E s s m 2 and σ η n 2= N 0 2 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?

Figure 1
Figure 1 (Figure4-24.png)

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

y k t=∫-∞∞ r τ h k t-τdτ=∫-∞∞ s m τ+ N τ h k t-τdτ=∫-∞∞ s m τ h k t-τdτ+∫-∞∞ N τ h k t-τdτ 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 (2)
Sampling the output at time TT yields
y k T=∫-∞∞ s m τ h k T-τdτ+∫-∞∞ N τ h k T-τdτ y k T τ s m τ h k T τ τ N τ h k T τ (3)
The noise contribution:
ν k =∫-∞∞ N τ h k T-τdτ ν k τ N τ h k T τ (4)
The expected value of the noise component is
E ν k =E∫-∞∞ N τ h k T-τdτ=0 ν k τ N τ h k T τ 0 (5)
The variance of the noise component is the second moment since the mean is zero and is given as
σ ν k 2=E ν k 2=E∫-∞∞ N τ h k T-τdτ∫-∞∞ N τ ' ¯ h k T- τ ' ¯d τ ' ν k ν k 2 τ N τ h k T τ τ ' N τ ' h k T τ ' (6)
E ν k 2=∫-∞∞∫-∞∞ N 0 2δτ- τ ' h k T-τ h k T- τ ' ¯dτd τ ' = N 0 2∫-∞∞| h k T-τ|2dτ 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 (7)

Signal Energy can be written as

∫-∞∞ s m τ h k T-τdτ2 τ s m τ h k T τ 2 (8)

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

SNR=∫-∞∞ s m τ h k T-τdτ2 N 0 2∫-∞∞| h k T-τ|2dτ SNR τ s m τ h k T τ 2 N 0 2 τ h k T τ 2 (9)

The signal-to-noise ratio, can be maximized considering the well-known Cauchy-Schwarz Inequality

∫-∞∞ g 1 x g 2 x¯dx2≤∫-∞∞| g 1 x|2dx∫-∞∞| g 2 x|2dx x g 1 x g 2 x 2 x g 1 x 2 x g 2 x 2 (10)
with equality when g 1 x=α g 2 x g 1 x α g 2 x . Applying the inequality directly yields an upper bound on SNR
∫-∞∞ s m τ h k T-τdτ2 N 0 2∫-∞∞| h k T-τ|2dτ≤2 N 0 ∫-∞∞| s m τ|2dτ 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 (11)
with equality ∀τ: h k opt T-τ=α s m τ¯ τ h k opt T τ α s m τ . Therefore, the filter to examine signal m m should be

Matched Filter

∀τ: h m opt τ= s m T-τ¯ τ τ h m opt τ s m T τ (12)
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 ∫-∞∞| s m τ|2dτ=2 E s N 0 2 N 0 τ s m τ 2 2 E s N 0 (13)
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.

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