Nonstationary processes - Statistical Signal Processing of Complex-Valued Data - page 240

measures how well the time-domain signal x ( t ) can be approximated by counterclock-
wise rotating phasors. Similarly, the magnitude of the CR-TFD, normalized by the
time- and frequency-marginals of the HR-TFD, measures how well the time-domain
signal x ( t ) can be approximated by clockwise rotating phasors. This suggests that the
distribution of coherence (rather than energy or power) over time and frequency is a
fundamental descriptor of a nonstationary signal.
Finally we point out that, if x ( t ) is WSS, the estimator x f ( t ) simplifies to

( f )e j2 π ft x f ( t ) = d ξ (9.98)

for all t , which has already been shown in Section 8.4.2 . Furthermore, since WSS
analytic signals are proper, the conjugate-linear estimator is identically zero: x f ( t )

0
for all t . Hence, WSS analytic signals can be estimated only from counterclockwise
rotating phasors, whereas nonstationary improper analytic signals may also be estimated
from clockwise rotating phasors.

=

9.3.2 Kernel estimators

The Rihaczek distribution is a mathematical expectation. As a practical matter, it must
be estimated. Because such an estimator is likely to be implemented digitally, we present
it for a time series x [ k ]. We require that our estimator be a bilinear function of x [ k ] that
is covariant with respect to shifts in time and frequency. Estimators that satisfy these
properties constitute Cohen's class. 10 (Since we want to perform time- and frequency-
smoothing, we do not expect our estimator to have the correct time- and frequency-
marginals.) The discrete-time version of Cohen's class is

V xx [ k

] x ∗ [ k m ]e − j µθ , ,θ ) = x [ k + m + µ ] φ [ m ,µ + (9.99)

m µ

where m is a global and µ a local time variable. The choice of the dual-time kernel

] determines the properties of the HR-TFD estimator V xx [ k

φ ). Omitting the

conjugation in ( 9.99 ) gives an estimator of the CR-TFD. It is our objective to design
a suitable kernel

[ m ,µ ,θ

φ [ m ,µ ]. A factored kernel that preserves the spirit of the Rihaczek

distribution is

w 3 [ m ] φ [ m ,µ ] = w 1 [ m + µ ] w 2 [ µ ] . (9.100)

In practice, the three windows w 1 , w 2 , and w 3 might be chosen real and even. By

inserting ( 9.100 )into( 9.99 ), we obtain

m ] π
− π

m w 3 [ m ] x ∗ [ k

)e j m ( θ − ω ) d

ω

V xx [ k

,θ ) = + W 2 ( ω ) F 1 [ k ,θ − ω

π .

(9.101)

2

Here W 2 ( ω ) is the discrete-time Fourier transform (DTFT) of w 2 [ µ ] and F 1 is the

short-time Fourier transform (STFT) of x [ k ] using window w 1 , defined as

n w 1 [ n ] x [ n

k ]e − j n θ . F 1 [ k ,θ ) = + (9.102)