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Derive the expression for frequency response of the matched filter with non-white noise.
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Solution:

Matched filter with nonwhite noise: In the derivation of the matched-filter characteristic, the the spectrum of the noise accompanying the signal was assumed to be white; that is, it was independent of frequency.

If this assumption were not true, the filter which maximizes the output signal-to-noise ratio would not be the same as the matched filter.

It has been shown that if the input power spectrum of the interfering noise is given by [Ni (f)]2 , the frequency response the function of the filter which maximizes the output signal-to-noise ratio is ,

$H(f)=\frac{G_a S^*(f) \exp \left(-j 2 \pi f t_1\right)}{\left[N_i(f)\right]^2}$

When the noise is nonwhite, the filter which maximizes the output signal-to-noise ratio is called the NWN (nonwhite noise) matched filter.

For white noise [Ni(f)]2 = constant and the NWN matched-filler frequency-response function of Eq. above reduces to that of Eq. discussed earlier in white noise. The equation above can be written as,

$H(f)=\frac{1}{N_i(f)} \times G_a\left(\frac{S(f)}{N_i(f)}\right)^* \exp \left(-j 2 \pi f t_1\right)$

This indicates that the NWN matched filter can be considered as the cascade of two filters.

The first filter, with frequency-response function l/Ni (f), acts to make the noise spectrum uniform, or white.

It is sometimes called the whitening filter. The second is the matched filter when the input is white noise and a signal whose spectrum is S(f )/Ni(f ).