Nonstationary processes - Statistical Signal Processing of Complex-Valued Data - page 223

9 Nonstationary processes

Wide-sense stationary (WSS) processes admit a spectral representation (see Result 8.1 )
in terms of the Fourier basis, which allows a frequency interpretation. The transform-
domain description of a WSS signal x ( t ) is a spectral process ξ ( f ) with orthogonal

increments d ( f ). For nonstationary signals, we have to sacrifice either the Fourier

basis, and thus its frequency interpretation, or the orthogonality of the transform-domain
representation. We will discuss both possibilities.
The Karhunen-Loeve (KL) expansion uses an orthonormal basis other than the Fourier
basis but retains the orthogonality of the transform-domain description. The KL expan-
sion is applied to a continuous-time signal of finite duration, which means that its
transform-domain description is a countably infinite number of orthogonal random
coefficients. This is analogous to the Fourier series, which produces a countably infinite
number of Fourier coefficients, as opposed to the Fourier transform, which is applied
to an infinite-duration continuous-time signal. The KL expansion presented in Section
9.1 takes into account the complementary covariance of an improper signal. It can be
considered the continuous-time equivalent of the eigenvalue decomposition of improper
random vectors discussed in Section 3.1 .
An alternative approach is the Cramer-Loeve (CL) spectral representation, which
retains the Fourier basis and its frequency interpretation but sacrifices the orthogonality
of the increments d

ξ

( f ) of the spectral
process of an improper signal can have nonzero Hermitian correlation and complemen-
tary correlation between different frequencies. Starting from the CL representation, we
introduce energy and power spectral densities for nonstationary signals. We then discuss
the CL representation for analytic signals and discrete-time signals.
Yet another description, which allows deep insights into the time-varying nature of
nonstationary signals, is possible in the joint time-frequency domain. In Section 9.3 ,we
focus our attention on the Rihaczek distribution, which is a member of Cohen's class of
bilinear time-frequency distributions. The Rihaczek distribution is not as widely used
as, for instance, the Wigner-Ville distribution, but it possesses a compelling property:
it is an inner product between the spectral increments and the time-domain process at a
given point in the time-frequency plane. This property leads to an evocative geometrical
interpretation. It is also the basis for extending the rotary-component and polarization
analysis presented in the previous chapter to nonstationary signals. This is discussed in
Section 9.4 . Finally, Section 9.5 presents a short exposition of higher-order statistics for
nonstationary signals.

ξ ( f ). As discussed in Section 9.2 , the increments d ξ