The Periodogram

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}

In the limit as goes to infinity, the expected value of the periodogram equals the true power spectral density of the noise process . This is expressed by writing

(7.24) |

where denotes the power spectral density (PSD) of . (``Expected value'' is defined in Appendix C on page .)

In terms of the sample PSD defined in §6.7, we have

(7.25) |

That is, the periodogram is equal to the smoothed sample PSD. In the time domain, the autocorrelation function corresponding to the periodogram is Bartlett windowed.

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 the Fourier transform, it
has not been averaged in any way that reduces randomness, 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.

In principle, we should be able to recover from
a
*filter*
which, when used to filter white noise,
creates a noise indistinguishable statistically from the observed
sequence
. However, the DTFT is evidently useless for this
purpose. How do we proceed?

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.

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