Digital Signal Processing Reference
In-Depth Information
For signals with rapid variations, this solution is thus not very satisfactory.
Historically, the Wigner-Ville distribution then appeared. This time-frequency tool
verifies a large number of desired properties of the time-frequency representations,
which explains its success. It is defined in the following way:
u
u
⎛ ⎞ ⎛ ⎞
( )
(
)
∫
Wtf
,
=
xt
+
x t
*
−
exp
−
j
2
π
fudu
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
2
2
It preserves the localizations in time and in frequency and allows an exact
description of the modulations of linear frequencies. On the other hand, it can take
negative values and its bilinear structure generates interferences between the
different components of a signal, sometimes making its interpretation difficult. In
order to reduce the interferential terms, smoothing is introduced. We obtain then the
smoothed pseudo Wigner-Ville distribution:
2
u
u
u
⎛⎞
⎛ ⎞ ⎛ ⎞
( )
(
)
(
)
W
t
,
f
=
∫∫
f
w
w
v
−
t
x
v
+
x
*
v
−
exp
−
j
2
π
fu dvdu
⎜⎟
⎜ ⎟ ⎜ ⎟
t
2
2
2
2
⎝⎠
⎝ ⎠ ⎝ ⎠
We can generalize this time-frequency representation by the formulation of a
class known as Cohen's class:
1
τ
τ
⎛ ⎞ ⎛ ⎞
(
)
( )
( )
(
)
∫∫∫
C t
,
f
=
Φ
θτ
,
x
u
+
x
*
u
−
exp
−
j
θ πτ θ
t
−
2
f
−
u
dud d
τ
−
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
3
2
π
2
2
Φ is an arbitrary kernel. The definition of the kernel is sufficient to determine a
representation. The desired properties of a time-frequency representation induce
certain conditions on the kernel. Based on this, the Wigner-Ville transform can be
derived but we can also introduce the distributions of Choi-Williams, Zao-Atlas-
Marx, Page, Rihaczek, etc. Thus, the user finds a set of time-frequency
representations at his disposal, each one having its own characteristics and favoring
a part of the desirable properties according to the case of interest. For example, if the
time localization of a frequency jump is a primordial criterion, the Zao-Atlas-Marx
distribution is without any doubt the most interesting to use.
Further details on these time-frequency representations can be found in
[HLA 05] and [FLA 93].
9.3. Parametric spectral estimation
Applying parametric models in a non-stationary context is interesting in so far as
that makes it possible to take advantage of the good frequency resolution of these
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