Biomedical Engineering Reference
In-Depth Information
or equvialently (for signals represented in the frequency domain):
Z
∞
X
∗
(
e
i
2πξ
t
d
ξ
W
x
(
t
,
f
)=
X
(
f
+
ξ
/
2
)
f
−
ξ
/
2
)
(2.72)
−
∞
The
W
x
is implemented as
tfrwv
function in the Matlab toolbox
tftb
.
WVD has the following properties:
(
t
,
f
)
•
Energy conservation equation (2.68)
•
Conservation of marginal distributions equations: (2.69) and (2.70)
•
Conservation of time and frequency shifts:
y
(
t
)=
x
(
t
−
t
0
)
⇒
W
y
(
t
,
f
)=
W
x
(
t
−
t
0
,
f
)
(2.73)
e
i
2π
f
0
t
y
(
t
)=
x
(
t
)
⇒
W
y
(
t
,
f
)=
W
x
(
t
,
f
−
f
0
)
(2.74)
•
Conservation of scaling
√
kx
y
(
t
)=
(
kt
)
⇒
W
y
(
t
,
f
)=
W
x
(
kt
,
f
/
k
)
(2.75)
WVD is a quadratic representation, so an intrinsic problem occurs when the signal
contains more than one time-frequency component. Let's assume that we analyze a
signal
y
composed of two structures
x
1
and
x
2
:
y
(
t
)=
x
1
(
t
)+
x
2
(
t
)
(2.76)
Then, the WVD can be expressed as:
W
y
(
t
,
f
)=
W
x
1
(
t
,
f
)+
W
x
2
(
t
,
f
)+
2
Re
{
W
x
1
,
x
2
(
t
,
f
)
}
(2.77)
)=
R
∞
−
e
−
i
2π
f
τ
d
τ is called a cross-term.
WVD has many desired properties mentioned above and optimal time-frequency
resolution, but the presence of cross-terms in real applications can make the inter-
pretation of results difficult. The problem is illustrated in
Figure 2.13.
x
2
(
where
W
x
1
,
x
2
(
t
,
f
∞
x
1
(
t
+
τ
/
2
)
t
−
τ
/
2
)
2.4.2.1.2 Cohen class
The cross-terms oscillate in the time-frequency space with
relatively high frequency, as can be observed in Figure 2.13. The property can be
used to suppress the influence of the cross-terms simply by applying a low-pass
spatial filter on the WVD. A family of distributions obtained by filtering WVD is
called Cohen's class:
Z
∞
Z
∞
(
,
,
)=
(
−
,
−
)
(
,
)
C
x
t
f
Π
Π
s
t
ξ
f
W
x
s
ξ
dsd
ξ
(2.78)
−
∞
−
∞
where
Z
∞
Z
∞
e
−
i
2π
(
f
τ
+
ξ
t
)
d
τ
d
ξ
Π
(
t
,
f
)=
f
(
ξ
,
τ
)
(2.79)
−
∞
−
∞
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