Databases Reference
In-Depth Information
x
(
t
)
Figure 9.5
The dark solid line shows the time-domain signal
x
(
t
) as it moves through the
complex plane. At
t
=
t
, the ellipse is a perfect approximation.
with
E
x
(
t
)d
Ξ
H
(
t
f
)
V
x
(
t
,
f
)d
f
=
,
V
xx
(
t
f
)
d
f
V
xx
(
t
V
xx
(
t
=
V
xx
(
t
(9.112)
,
f
)
,
−
f
)
,
−
f
)
,
and
E
d
Ξ
(
t
f
)
f
)(d
f
)
2
f
)d
Ξ
H
(
t
K
(
t
,
=
,
,
.
(9.113)
)
†
denotes the pseudo-inverse, which is necessary because
K
(
t
In (
9.111
), (
f
) can
be singular. Some background on the pseudo-inverse is given in Section
A1.3.2
of
Appendix 1. Finding a closed-form solution for the ellipse
·
,
ε
t
,
f
(
t
) is tedious because it
involves the pseudo-inverse of the 4
×
4matrix
K
(
t
,
f
). However, in special cases
(i.e., proper, WSS, or analytic signals),
K
(
t
f
) has many zero entries and the com-
putations simplify accordingly. In particular, in the WSS case,
,
ε
t
,
f
(
t
) simplifies to
ε
f
(
t
)in(
9.109
). The analytic case is discussed in some detail
the stationary ellipse
below.
9.4.1
Ellipse properties
ε
t
,
f
(
t
) approximates
x
(
t
)at
t
=
In order to measure how well the local ellipse
t
,we
may consult the magnitude-squared
time-frequency coherence
13
f
)
K
†
(
t
f
)
V
x
(
t
V
x
(
t
,
,
,
f
)
2
|
ρ
xx
(
t
¯
,
f
)
|
=
,
(9.114)
r
xx
(
t
,
0)
which is closely related to the approximation error at
t
=
t
:
2
2
)
E
|
ε
t
,
f
(
t
)
−
x
(
t
)
|
=
r
xx
(
t
,
0)(1
−|
ρ
xx
(
t
¯
,
f
)
|
.
(9.115)
2
The magnitude-squared time-frequency coherence satisfies 0
≤|
ρ
xx
(
t
¯
,
f
)
|
≤
1. If
2
ε
t
,
f
(
t
) is a perfect approximation of
x
(
t
)at
t
=
|
ρ
xx
(
t
¯
,
f
)
|
=
1, the ellipse
t
.
2
ε
t
,
f
(
t
) has zero
This is illustrated in Fig.
9.5
.If
|
ρ
xx
(
t
¯
,
f
)
|
=
0, the best-fit ellipse
Search WWH ::
Custom Search