Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

8.10 WHITE NOISE

Noise appears to some degree in nearly all real signals and systems. Of particular interest is noise that has a flat PSD.

Definition: White-Noise Process Wide-sense stationary random process X(t) with flat PSD

(8.179)

for is called white noise. It has zero mean, its autocorrelation function is

(8.180)

and thus CXX(τ) = RXX(τ).

The units of No are watts/Hz. It is called white noise because (i) the waveforms have a noisy appearance (and sound like noise in the audible frequency range) and (ii) it includes all frequencies equally (a flat spectrum), similar to white light in the visible range of the electromagnetic spectrum (which is flat across wavelength). This definition does not state anything about the distribution of white noise, which can be of any type, though a Gaussian pdf is often assumed.

Theorem 8.5 A white-noise random process with the autocorrelation function in (8.180) must have zero mean.

Proof. This result is a consequence of the defining property in (8.179). Let Y(t) = X(t)+μ Y, where X(t) has zero mean and the autocorrelation function in (8.180). Assume that μ Y is nonzero. The autocorrelation function of Y(t) is

(8.181)

which has PSD

(8.182) ...