The periodogram is based on the definition of the power
spectral density (PSD) (see Appendix C). Let
denote a windowed segment of samples from a random process
,
where the window function
(classically the rectangular window)
contains
nonzero samples. Then the periodogram is defined as the
squared-magnitude DTFT of
divided by
[120, p. 65]:^{7.7}
(7.24) |
In terms of the sample PSD defined in §6.7, we have
(7.25) |
In practice, we of course compute a sampled periodogram , , replacing the DTFT with the length FFT. Essentially, the steps of §6.9 include computation of the periodogram.
As mentioned in §6.9, a problem with the periodogram of noise signals is that it too is random for most purposes. That is, while the noise has been split into bands by a Fourier transform, it has not been averaged in any way that reduces randomness within each band, and each band produces a nearly independent random value. In fact, it can be shown [120] that is a random variable whose standard deviation (square root of its variance) is comparable to its mean. This is not a measure of noise level in each band.
The trick to noise spectrum analysis is that many sample power spectra (squared-magnitude FFTs) must be averaged to obtain a ``stable'' statistical estimate of the noise spectral envelope. This is the essence of Welch's method for spectrum analysis of stochastic processes, as elaborated in §6.12 below. The right column of Fig.6.1 illustrates the effect of this averaging for white noise.