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White Noise



Definition: To say that $ v(n)$ is a white noise means merely that successive samples are uncorrelated:

$\displaystyle E\{v(n)v(n+m)\} = \left\{\begin{array}{ll} \sigma_v^2, & m=0 \\ [5pt] 0, & m\neq 0 \\ \end{array} \right. \isdef \sigma_v^2 \delta(m) \protect$ (C.26)

where $ E\{f(v)\}$ denotes the expected value of $ f(v)$ (a function of the random variables $ v(n)$ ).

In other words, the autocorrelation function of white noise is an impulse at lag 0. Since the power spectral density is the Fourier transform of the autocorrelation function, the PSD of white noise is a constant. Therefore, all frequency components are equally present--hence the name ``white'' in analogy with white light (which consists of all colors in equal amounts).



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