## 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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White Noise

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Introduction to Noise