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Spectral Characteristics of Noise

As we know, the spectrum $ X(\omega)$ of a time series $ x(n)$ has both a magnitude $ \left\vert X(\omega)\right\vert$ and a phase $ \angle{X(\omega)}$ . The phase of the spectrum gives information about when the signal occurred in time. For example, if the phase is predominantly linear with slope $ -\tau$ , then the signal must have a prominent pulse, onset, or other transient, at time $ \tau$ in the time domain.

For stationary noise signals, the spectral phase is simply random, and therefore devoid of information. This happens because stationary noise signals, by definition, cannot have special ``events'' at certain times (other than their usual random fluctuations). Thus, an important difference between the spectra of deterministic signals (like sinusoids) and noise signals is that the concept of phase is meaningless for noise signals. Therefore, when we Fourier analyze a noise sequence $ x(n)$ , we will always eliminate phase information by working with $ \left\vert X(\omega)\right\vert^2$ in the frequency domain (the squared-magnitude Fourier transform), where $ X(\omega) \isdeftext \hbox{\sc DTFT}_\omega(x)$ .


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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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