Spectral Analysis - Signal Processing of Random Physiological Signals - page 166

The left-hand side of (15.5), is defined as the total energy in the signal v ( t ).
A periodic signal contains real and imaginary components of power. However,
determination of even symmetry (real power only) and odd symmetry (imaginary power
only) simplifies calculations because b n and a n cancel for each case, respectively. Fourier
Series Analysis is a very powerful tool, but it is not applicable to random signals.

15.2.1 Power Spectra Estimation Via the Autocorrelation Function
The classic method used to efficiently obtain the estimated Power Spectra from random
processes before the advent of the Fast Fourier Transform was by taking the finite Fourier
transform of the correlation function. The process required one to calculate the autocor-
relation function (or cross-correlation) first, and then to take the Fourier Transform of
the resulting correlation function.
The autocorrelation function ( R xx ) and cross-correlation function ( R xy ) will not
be discussed in this section since the material was presented in Chapter 9. Hence, let
us assume that the correlation function exists and that its absolute value is finite. The
Fourier Transform (FT) is defined as shown in (15.6) for the autocorrelation and (15.7)
for the cross-correlation:

∞

) e − j ωτ d S xx ( ω ) = R xx ( τ τ (15.6)

−∞

∞

) e − j ωτ d

S xy ( ω ) = R xy ( τ τ (15.7)

−∞

where S xx ( ω ) is the autospectral density function for an input function x ( t ), and S xy ( ω )

is the cross-spectral density function for the input x ( t ) and output y ( t ).
The cross-spectral density function is estimated by taking the cross-correlation
of two different functions in the time domain, that is, the input signal to a system and
the output signal from the same system, and then taking the Fourier transform of the
resulting cross-correlation function, S xy . The Inverse Fourier Transforms are then

∞

1
2

) e j ω t d R xx ( τ ) = S xx ( ω ω (15.8)

π

−∞