Digital Signal Processing Reference
In-Depth Information
cross terms. For instance, the WD of the sum of two signals
x
(
t
)
+
y
(
t
)
WD
x
+
y
(
t, f
)
=
WD
x
(
t, f
)
+
2
e
(
WD
x, y
(
t, f
))
+
WD
y
(
t, f
)
(3.14)
(
(
))
has a cross-term 2
e
WD
x, y
t, f
in addition to the two autocomponents,
where the cross WD is defined as
x
t
y
∗
t
e
−
j
2
π
f
τ
d
+∞
+
2
−
2
(
)
=
τ.
WD
x, y
t, f
(3.15)
−∞
The cross or interference terms are often a serious problem in practical ap-
plications, especially if a WD outcome is to be visually analyzed by a human
analyst. Cross terms lie between two autocomponents and are oscillatory,
with their frequencies increasing with increasing distance in time-frequency
between the two autocomponents.
14
,
19
,
21
For real-valued bandpass signals,
the WD of the analytic signal is generally used because removal of the nega-
tive frequency components also eliminates cross terms between positive and
negative frequency components. Although helpful in eliminating cross terms,
cross terms between multiple components in the analytic signal still make in-
terpretation difficult. Because cross terms have oscillations of relatively high
frequency, they can be attenuated by means of a smoothing operation that
corresponds to the convolution of the WD with a 2D smoothing kernel. For
most methods, the smoothing tends to produce the following effects:
1. A (desired) partial attenuation of the interference terms
2. An (undesired) broadening of signal terms, i.e., a loss of time-
frequency concentration
3. A (sometimes undesired) loss of some of the mathematical proper-
ties of the WD
The interference terms (ITs) of the WD are a consequence of the WD bilinear
(or quadratic) structure. That is, they occur in the case of a multicomponent
signal and can be identified mathematically with quadratic cross terms.
15
,
16
,
19
According to the quadratic superposition law, the WD of an
N
-component
signal consists of
N
signal terms and [
N
2ITs . Each signal component
generates a signal term, and each pair of signal components generates an IT.
While the number of signal terms grows linearly with the number
N
of signal
components, the number of ITs grows quadratically with
N
.
19
(
N
−
1
)
]
/
This is shown
in Figure 3.12.
The goal of filtering in the WD plane is thus to reduce, or eliminate, the
cross terms while preserving the autoterms. To achieve this result, a center
affine filter can be applied to the WD. To take advantage of the structure of the
filtering problem, the following modifications can be made to the standard
center affine filtering operation:
14
1. The absolute values of the samples,
, rather than their actual
values, are utilized to calculate the respective affinities. Accordingly,
|
W
m
1
,m
2
|
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