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?

  • $\begingroup$ 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 $|H(f)|^2$. $\endgroup$ Commented Aug 9, 2014 at 9:51

1 Answer 1


For a linear time-invariant filter, the output of the filter would be coloured noise.

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 $$ S_y(f) = |H(f)|^2 $$

  • $\begingroup$ Do you also know how these thing relates to auto correlation . $\endgroup$
    – Lakshmi
    Commented Aug 8, 2014 at 21:09
  • $\begingroup$ @Lakshmi See this answer $\endgroup$ Commented Aug 9, 2014 at 9:51
  • $\begingroup$ 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. $\endgroup$
    – SleuthEye
    Commented Aug 9, 2014 at 15:44

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