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Wiener-Khinchin Theorem
In the case of periodic deterministic signals, the signals may
bedescribed either in terms of periodicity (in the time domain) or
in termsof their line spectra (in the frequency domain). No new
information isgleaned by moving from one domain to the other.
Practicalconsiderations usually determine which representation is
the mostuseful in specific applications.
In dealing with stochastic signals it was seen that the
appropriatemeasures by which they may be summarized are the
correlation &spectral density functions. Wiener & Khinchin
first drew attention tothe fact that correlation & spectral
density functions form Fouriertransform pairs. This observation is
observed in the case of
autocorrelation function & the power spectral density
function asshown below:
The definitions of the autocorrelation & the power
spectraldensity functions over samples 2T long gives:
Rxx()= lim Tinf 1/2T -T+T x(t).x(t+ )dt
Sxx()= lim Tinf 1/2T {X().X*()}
and taking the Fourier transform (FT) of the autocorrelation
functiongives
FT {Rxx()}= -T+T{lim Tinf 1/2T -T+T x(t).x(t+ )dt} e-j .d
Substituting for (t+ ) & noting that d becomes d, then
FT {Rxx()}= -T+T{lim Tinf 1/2T -T+T x(t).x()dt} e-j(-t) .d
results which, after expanding the exponential term &
separating the 2integrals, becomes
FT {Rxx()}= lim Tinf 1/2T {-T+T x(t). ejt .dt -inf+inf x().e-j
.d
It is recognized that the 2 integrals represent the Fourier
transform ofx(t), i.e. X() & its conjugate complex X*(), and
thus:
FT {Rxx()}= lim Tinf 1/2T {X().X*()} = Sxx()
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By identical reasoning it can be shown that the
cross-correlationfunction & the cross spectral density function
also form Fouriertransform pairs. The Wiener-Khinchin are usually
written formally as:
Sxx()= -inf +inf Rxx(). e-j .d
Rxx()= 1/2 -inf+inf Sxx(). ej .d
These 2 equations exhibit the fact that there is no new
informationgenerated by moving from the time domain to the
frequency domainor vice-versa. However, in some applications of
industrial diagnostics itis easier to interpret the underlying
phenomena in one domain or theother.
One use of the first of the 2 equations is to use it to
calculate thepower spectral density from a known estimate of the
autocorrelationfunction. It is now more common to use the Fast
Fourier Transform
(FFT) to calculate the power spectrum directly from the raw time
seriesassociated with the signal under investigation.
