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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.

Testing for White Noise

To test whether a set of samples can be well modeled as white noise, we may compute its sample autocorrelation and verify that it approaches an impulse in the limit as the number of samples becomes large; this is another way of saying that successive noise samples are uncorrelated. Equivalently, we may break the set of samples into successive blocks across time, take an FFT of each block, and average their squared magnitudes; if the resulting average magnitude spectrum is flat, then the set of samples looks like white noise. In the following sections, we will describe these steps in further detail, culminating in Welch's method for noise spectrum analysis, summarized in §6.9.

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Sample Autocorrelation
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Spectral Characteristics of Noise