# Matched Filter, impulse response h(t)

Matched Filter can be considered as a special case of Optimum Filter. Optimum Filter can be treated as Matched Filter when the noise at the input of the receiver is White Gaussian Noise.

Transfer Function of Matched Filter:-

Transfer Function of Optimum filter is $H(f)=\frac{k&space;X^{*}(f)e^{-j2\pi&space;fT}}{S_{ni}(f)}$

if input noise is white noise , its Power spectral density (Psd) is $S_{ni}(f)=\frac{N_{o}}{2}$.

then H(f) becomes $H(f)=\frac{k&space;X^{*}(f)e^{-j2\pi&space;fT}}{\frac{N_{o}}{2}}$

$H(f)=\frac{2k}{N_{o}}X^{*}(f)e^{-j2\pi&space;fT}-----Equation(I)$

From the properties of Fourier Transforms , by Conjugate Symmetry property  $X^{*}(f)&space;=&space;X(-f)$

Equation (I) becomes

$H(f)=\frac{2k}{N_{o}}X(-f)e^{-j2\pi&space;fT}------Equation(II)$

From Time-shifting property of Fourier Transforms

$x(t)\leftrightarrow&space;X(f)$

From Time-Reversal Property  $x(-t)\leftrightarrow&space;X(-f)$

By Shifting the signal $x(-t)$ by T Seconds in positive direction(time) ,the Fourier Transform is given by  $x(T-t)\leftrightarrow&space;X(-f)e^{-j2\pi&space;ft}$

Now the inverse Fourier Transform of the signal from the Equation(II) is

$F^{-1}[H(f)]=F^{-1}[\frac{2k}{N_{o}}X(-f)e^{-j2\pi&space;fT}]$

$h(t)=\frac{2k}{N_{o}}x(T-t)$

Let the constant $\frac{2k}{N_{o}}$ is set to 1, then the impulse response of Matched Filter will become $h(t)&space;=&space;x(T-t)$.

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