Fuzzy Methods in Nonlinear Signal Processing: Part II—Applications - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 86

(S)

W 1

(t,f)

(t , f )
11

direction of oscillation

1/τ 12

(S)
2

W (t,f)

(t 12 , f 12 )

(I)

W 12 (t,f)

1/ν 12

(t , f 2

2

FIGURE 3.12
Interference geometry of Wigner distribution. (From G.R. Arce and S.R. Hasan, IEEE Trans. Signal
Process. , 48, 2321-2331, Aug. 2000. c 2000 IEEE. With permission.)

the reference point of the affinity function is set as the absolute value
of the center-sample, .

2. The affinity spread parameter

| W c |

is made proportional to the local
variance at the observation window location. Thus, a higher value
of

γ

is obtained, in general, for an observation window centered
in a cross-term region than for a window centered in an autoterm
region.

γ

While modification 1 may not affect the affinities of samples in the autoterm
region (as most of the samples are already positive), it drastically changes the
affinities of samples in the cross-term region. Specifically, both positive and
negative samples have higher affinities, resulting in more samples contribut-
ing to the estimate and trending the filter to have linear lowpass characteris-
tics. Condition 2 further ensures that more samples (including both positive
and negative samples) receive higher affinities in the cross-term region while
assigning very low affinities to most of the samples in the autoterm region.
This scenario is depicted in Figure 3.13.
To evaluate the cross-term reduction, we consider the special case of a signal
composed of the sum of time-frequency-shifted signals x

( t ) = c 1 x 1 ( t ) + c 2 x 2 ( t )

e j 2 π f i t , for i

where x i ( is a

Gaussian atom. Interest in a Gaussian-shaped atom derives from the fact that
it has good locality (energy concentration) in the joint t

t ) = x 0 ( t − t i ) = 1 , 2. Consider the case where x 0 ( t )

f domain, and its
WD is nonnegative. 22 Figure 3.14 demonstrates the center affine filtered WD
of the signal composed of two Gaussian atoms.
To demonstrate the relative performance of the center affine filtered WD, a
comparison of the experimental results provided by several time-frequency
representations is shown. Consider first a 128-point test signal comprised of

−