# white noise filtering

If I have a white noise fed into a filter what would be the output of the filter - what would you expect to see? I know that white noise has a unit power spectral density (PSD), how these thing relate to each other?

• The definition of a white noise process is that when applied to the input of a LTI filter with transfer function $H(f)$, the output process has PSD proportional to $|H(f)|^2$. Aug 9, 2014 at 9:51

Again assuming linear time-invariant filter, the effect of the filter on the output can be further characterized in terms of its power spectrum density (PSD). Specifically, for an input $X(f)$ with power spectrum density $S_x(f)$ to a linear time-invariant filter with frequency response $H(f)$, the power-spectrum density $S_y(f)$ of the output of the filter is given by:
$$S_y(f) = |H(f)|^2 S_x(f)$$
So, for a white noise input with unit power spectrum density $S_x(f)=1$, the corresponding PSD of the output would be $$S_y(f) = |H(f)|^2$$
• Given the autocorrelation $R_h(\tau)$ of the (again linear-time-invariant) filter's impulse response, it can be shown that the Fourier transform $\mathcal{F}\{R_h(\tau)\} = |H(f)|^2$. Also, for WSS process (see @dilip answer link) $\mathcal{F}\{R_y(\tau)\} = S_y(f)$. So in this case, the autocorrelation of the filtered noise corresponds to the autocorrelation of the filter's impulse response. Aug 9, 2014 at 15:44