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Matched Filters

Module by: Behnaam Aazhang. E-mail the author

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−τd τ =∫−∞∞( s m ⁢τ+ N τ )⁢ h k ⁢t−τd τ =∫−∞∞ s m ⁢τ⁢ h k ⁢t−τd τ +∫−∞∞ N τ ⁢ h k ⁢t−τd τ 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−τd τ +∫−∞∞ N τ ⁢ h k ⁢T−τd τ 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−τd τ 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−τd τ =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−τd τ ⁢∫−∞∞ 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 2⁢δ⁢τ− τ ' ⁢ h k ⁢T−τ⁢ h k ⁢T− τ ' *d τ d τ ' = N 0 2⁢∫−∞∞| h k ⁢T−τ|2d 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
(7)

Signal Energy can be written as

∫−∞∞ s m ⁢τ⁢ h k ⁢T−τd τ 2 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−τd τ 2 N 0 2⁢∫−∞∞| h k ⁢T−τ|2d τ 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*d x 2≤∫−∞∞| g 1 ⁢x|2d x ⁢∫−∞∞| g 2 ⁢x|2d x 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−τd τ 2 N 0 2⁢∫−∞∞| h k ⁢T−τ|2d τ ≤2 N 0 ⁢∫−∞∞| s m ⁢τ|2d τ 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 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|2d τ =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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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks