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
if input noise is white noise , its Power spectral density (Psd) is .
then H(f) becomes
From the properties of Fourier Transforms , by Conjugate Symmetry property
Equation (I) becomes
From Time-shifting property of Fourier Transforms
From Time-Reversal Property
By Shifting the signal by T Seconds in positive direction(time) ,the Fourier Transform is given by
Now the inverse Fourier Transform of the signal from the Equation(II) is
Let the constant is set to 1, then the impulse response of Matched Filter will become .