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White Noise

White noise may be defined as a sequence of uncorrelated random values, where correlation is defined in Appendix C and discussed further below. Perceptually, white noise is a wideband ``hiss'' in which all frequencies are equally likely. In Matlab or Octave, band-limited white noise can be generated using the rand or randn functions:

y = randn(1,100); % 100 samples of Gaussian white noise
                  % with zero mean and unit variance

x = rand(1,100);  % 100 white noise samples, 
                  %   uniform between 0 and 1.
xn = 2*(x-0.5);   % Make it uniform between -1 and +1
True white noise is obtained in the limit as the sampling rate goes to infinity and as time goes to plus and minus infinity. In other words, we never work with true white noise, but rather a finite time-segment from a white noise which has been band-limited to less than half the sampling rate and sampled.

In making white noise, it doesn't matter how the amplitude values are distributed probabilistically (although that amplitude-distribution must be the same for each sample--otherwise the noise sequence would not be stationary, i.e., its statistics would be time-varying, which we exclude here). In other words, the relative probability of different amplitudes at any single sample instant does not affect whiteness, provided there is some zero-mean distribution of amplitude. It only matters that successive samples of the sequence are uncorrelated. Further discussion regarding white noise appears in §C.3.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University