Assuming known stationary signal and noise spectra and additive noise, the Wiener filter is a filter used in signal processing to provide an estimate of a desired or target random process through linear time-invariant (LTI) filtering of an observed noisy process. The mean square error between the intended process and the estimated random process is reduced by the Wiener filter. Fig: Block diagram view of the FIR Wiener filter for discrete series. An input signal x[n] is convolved with the Wiener filter g[n] and the result is compared to a reference signal s[n] to obtain the filtering error e[n]. In the big picture, the signal is attenuated and added with noise, then the signal is passed through a Wiener filter. And the function of the Wiener filter is to minimize the mean square error between the filter output of the received signal and the reference signal by adjusting the Wiener filter tap coefficient.   Description...

Fourier Transform of a delta function If x(t) = \( \delta(t) \), then Fourier transform, X(\( \omega \)) = \( \int_{-\infty}^{\infty} \delta(t)e^{-j\omega t} dt \) = \( \int_{-\epsilon}^{\epsilon} \delta(t)e^{-j\omega t} dt \) = \( \int_{-\epsilon}^{\epsilon} \delta(t)e^{-j\omega 0} dt \) = 1 Thus, the Fourier transform of a delta/impulse is a constant equal to 1, independent of frequency. Remember that derivation used the shifting property of the impulse to eliminate the integral.       Fourier transform of a unit step function   \( \gamma(t) \) = 0 for t < 0                            ...