Spectral Characteristics of Noise

As we know, the spectrum of a time series has both a magnitude and a phase . The phase of the spectrum gives information about when the signal occurred in time. For example, if the phase is predominantly linear with slope , then the signal must have a prominent pulse, onset, or other transient, at time 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 , we will always eliminate phase information by working with in the frequency domain (the squared-magnitude Fourier transform), where .

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.