Nonstationary processes - Statistical Signal Processing of Complex-Valued Data - page 242

on the other hand, depends on the Hermitian correlation only:

{ V xx [ k 1 ,θ 1 ) , V xx [ k 2 ,θ 2 ) }

cov

µ 2 φ

φ ∗ [ m 2 ,µ 2 ]e − j( µ 1 θ 1 − µ 2 θ 2 ) = [ m 1 ,µ 1 ]

m 1 m 2 µ 1

×{ r xx [ k 1 + m 1 + µ 1 , k 2 − k 1 + m 2 − m 1 + µ 2 − µ 1 ]

×

r xx [ k 1 + m 1 , k 2 − k 1 + m 2 −

+

r xx [ k 1 + m 1 , k 2 − k 1 + m 2 − m 1 + µ 2 ] m 1 ]

×

r xx [ k 1 + m 1 + µ 1 , k 2 − k 1 + m 2 − m 1 − µ 1 ]

} .

We can derive the following approximate expression for the variance of the HR-
TFD estimator, assuming analytic and quasi-stationary signals whose time duration of
stationarity is much greater than the duration of correlation:

π
0 |

{ V xx [ k

2 2

,θ }≈ − ,θ − ω | | ,ω |

var ) [ m k ) V xx [ k )

m

d ω

| V xx [ k ∗ [ m

2

+ [ m − k ,θ − ω ) − k ,θ + ω ) ,ω ) |

π .
(9.105)

2

However, the same simplifying approximation of large duration of stationarity leads to a
vanishing CR-TFD estimator because stationary analytic signals have zero complemen-
tary correlation. This result should not be taken as an indication that the CR-TFD is not
important for analytic signals. Rather it shows that the assumption of quasi-stationarity
with large duration of stationarity is a rather crude approximation to the general class of
nonstationary signals.

9.4 Rotary-component and polarization analysis

In Section 9.3.1 , we determined the LMMSE estimator of the time-domain signal at a
fixed time instant t from the frequency-domain representation at a fixed frequency f .
This showed us how well the signal can be approximated from clockwise or counter-
clockwise turning circles (phasors). It thus seems natural to ask how well the signal can
be approximated from ellipses , which combine the contributions from clockwise and
counterclockwise circles. This framework will allow us to extend rotary-component and
polarization analysis from the stationary case, described in Section 8.4 , to a nonstation-
ary setting. 12 We thus construct a WLMMSE estimator of x ( t ), for fixed t , from the
frequency-domain representation at + f and − f , where throughout this entire section

f shall denote a fixed nonnegative frequency:

( f )e j2 π ft ξ ∗ ( f )e − j2 π ft x f ( t ) = W 1 ( t , f )d ξ + W 2 ( t , − f )d

f )e − j2 π ft ξ ∗ ( f )e j2 π ft

+ W 1 ( t , − f )d ξ ( − + W 2 ( t , f )d − . (9.106)