Many signals are either partly or wholly stochastic, or random.
Important examples include human speech, vibration in machines,
and CDMA communication signals.
Given the ever-present noise in electronic systems, it can be argued
that almost all signals are at least partly stochastic.
Such signals may have a distinct average spectral
structure that reveals important information (such as for speech
recognition or early detection of damage in machinery).
Spectrum analysis of any single block of data using
window-based
deterministic spectrum analysis, however, produces
a random spectrum that may be difficult to interpret.
For such situations, the classical statistical spectrum estimation
methods described in this module can be used.
The goal in classical statistical spectrum analysis is to estimate
E|Xω|2
X
ω
2
,
the power spectral density (PSD) across frequency
of the stochastic signal.
That is, the goal is to find the expected (mean, or average)
energy density of the signal as a function of frequency.
(For zero-mean signals, this equals the variance of each frequency sample.)
Since the spectrum of each block of signal samples is itself random,
we must average the squared spectral magnitudes over a number of blocks
of data to find the mean.
There are two main classical approaches,
the periodogram
and auto-correlation methods.
The periodogram method divides the signal into a number of shorter
(and often overlapped) blocks of data, computes the squared magnitude
of the windowed
(and usually zero-padded)
DFT,
X
i
ω
k
X
i
ω
k
,
of each block,
and averages them to estimate the power spectral density.
The squared magnitudes of the DFTs of LL possibly overlapped
length-NN windowed blocks
of signal (each probably with zero-padding) are averaged to estimate the
power spectral density:
X
ω
k
̂=1L∑i=1L|
X
i
ω
k
|2
X
ω
k
1
L
i
L
1
X
i
ω
k
2
For a fixed total number of samples,
this introduces a tradeoff: Larger individual data blocks provides
better frequency resolution due to the use of a longer window,
but it means there are less blocks to average, so the estimate
has higher variance and appears more noisy.
The best tradeoff depends on the application.
Overlapping blocks by a factor of two to four increases the number
of averages and reduces the variance, but since the same data is being
reused, still more overlapping does not further reduce the variance.
As with any window-based spectrum estimation procedure, the window
function introduces broadening and sidelobes into the power spectrum
estimate.
That is, the periodogram produces an estimate of the windowed spectrum
Xω
̂=E|Xω*
W
M
|2
X
ω
X
ω
W
M
2
, not of
E|Xω|2
X
ω
2
.
Figure 1 shows the non-negative frequencies of the DFT
(zero-padded to 1024 total samples) of 64 samples of a
real-valued stochastic signal.
With no averaging, the power spectrum is very noisy and difficult
to interpret other than noting a significant reduction in spectral energy
above about half the Nyquist frequency.
Various peaks and valleys appear in the lower frequencies,
but it is impossible to say from this figure whether they
represent actual structure in the power spectral density (PSD)
or simply random variation in this single realization.
Figure 2 shows the same frequencies of a length-1024 DFT of a
length-1024 signal. While the frequency resolution has improved,
there is still no averaging, so it remains difficult to
understand the power spectral density of this signal.
Certain small peaks in frequency might represent narrowband
components in the spectrum, or may just be random noise peaks.
In
Figure 3, a power spectral density computed from averaging
the squared magnitudes of length-1024 zero-padded DFTs of 508 length-64
blocks of data (overlapped by a factor of four, or a 16-sample
step between blocks) are shown.
While the frequency resolution corresponds
to that of a length-64 truncation window, the averaging greatly
reduces the variance of the spectral estimate and allows the user to
reliably conclude that the signal consists of lowpass broadband noise
with a flat power spectrum up to half the Nyquist frequency, with
a stronger narrowband frequency component at around 0.65 radians.
The averaging necessary to estimate a power spectral density
can be performed in the discrete-time domain, rather than in frequency,
using the auto-correlation method.
The squared magnitude of the frequency response,
from the DTFT multiplication and conjugation properties,
corresponds in the discrete-time domain to the signal convolved
with the time-reverse of itself,
|Xω|2=Xω
X
*
ω↔xn
x
*
-n=rn
↔
X
ω
2
X
ω
X
*
ω
x
n
x
*
n
r
n
or its auto-correlation
rn=∑xk
x
*
n+k
r
n
k
x
k
x
*
n
k
We can thus compute the squared magnitude of
the spectrum of a signal by computing
the DFT of its auto-correlation.
For stochastic signals, the power spectral density
is an expectation, or average, and by linearity of
expectation can be found by transforming the
average of the auto-correlation.
For a finite block of NN
signal samples, the average of the autocorrelation values,
rn
r
n
,
is
rn=1N−n∑k=0N−(1−n)xk
x
*
n+k
r
n
1
N
n
k
N
1
n
0
x
k
x
*
n
k
Note that with increasing lag,
nn,
fewer values are averaged, so they introduce
more noise into the estimated power spectrum.
By
windowing the auto-correlation before
transforming it to the frequency domain, a
less noisy power spectrum is obtained, at the
expense of less resolution.
The multiplication property of the DTFT shows
that the windowing smooths the resulting power
spectrum via convolution with the DTFT of the window:
Xω
̂=∑n=-MMrnwnⅇ-ⅈωn=E|Xω|2*Wω
X
ω
n
M
M
r
n
w
n
ω
n
X
ω
2
W
ω
This yields another important interpretation of how the auto-correlation method works:
it estimates the power spectral density by
averaging the power spectrum over nearby frequencies,
through convolution with the window function's transform,
to reduce variance.
Just as with the periodogram approach, there is always a
variance vs. resolution tradeoff.
The periodogram and the auto-correlation method give
similar results for a similar amount of averaging; the user should
simply note that in the periodogram case, the window introduces smoothing
of the spectrum via frequency convolution before squaring the magnitude,
whereas the periodogram convolves the squared magnitude with
Wω
W
ω
.
The Discrete Fourier Transform
