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problems because of the following. One would think that the minimum requirements
for a distribution to have an interpretation as a distribution of energy or power are
that
it is a bilinear function of the signal (so that it has the right physical units),
it is covariant with respect to shifts in time and frequency,
it has the correct time- and frequency-marginals (instantaneous power and energy
spectral density, respectively), and
it is nonnegative.
Yet Wigner's theorem
6
says that such a distribution does not exist. The Rihaczek dis-
tribution satisfies the first three properties but it is complex-valued. The Wigner-Ville
distribution,
7
which is more popular than the Rihaczek distribution, also satisfies the
first three properties but it can take on negative values. Thus, we will not attempt an
energy/power-distribution interpretation. Instead we argue for the Rihaczek distribution
as a distribution of correlation and then present an evocative geometrical interpreta-
tion. This geometrical interpretation is the main reason why we prefer the Rihaczek
distribution over the Wigner-Ville distribution.
9.3.1
Interpretation
A key insight is that the HR/CR-TFDs are inner products in the Hilbert space of second-
order random variables, between the time-domain signal
at a fixed time instant t
and the
frequency-domain representation
at a fixed frequency f
:
=
x
(
t
)
(
f
)e
j2
π
ft
,
E
[
x
∗
(
t
)d
(
f
)e
j2
π
ft
]
V
xx
(
t
,
f
)d
f
=
ξ
,
d
ξ
(9.89)
=
x
∗
(
t
)
(
f
)e
j2
π
ft
.
V
xx
(
t
(
f
)e
j2
π
ft
]
,
f
)d
f
=
E
[
x
(
t
)d
ξ
,
d
ξ
(9.90)
How should these inner products be interpreted? Let's construct a linear minimum
mean-squared error (LMMSE) estimator of the random variable
x
(
t
), at fixed
t
,from
the random variable d
(
f
)e
j2
π
ft
,atfixed
f
:
8
ξ
E
x
(
t
)d
ξ
∗
(
f
)e
−
j2
π
ft
(
f
)e
j2
π
ft
x
f
(
t
)
=
E
d
ξ
∗
(
f
)e
−
j2
π
ft
d
ξ
(
f
)e
j2
π
ft
d
ξ
V
xx
(
t
,
f
)
d
ξ
(
f
)
d
f
e
j2
π
ft
=
.
(9.91)
S
xx
(0
,
f
)
The MMSE is thus
2
2
)
E
|
x
f
(
t
)
−
x
(
t
)
|
=
r
xx
(
t
,
0)(1
−|
ρ
xx
(
t
,
f
)
|
,
(9.92)
where
2
|
V
xx
(
t
,
f
)
|
2
|
ρ
xx
(
t
,
f
)
|
=
(9.93)
r
xx
(
t
,
0)
S
xx
(0
,
f
)
is the magnitude-squared
rotational
time-frequency coherence
9
between the time-
and frequency-domain descriptions. Similarly, we can linearly estimate
x
(
t
) from
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