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The next result for harmonizable almost CS processes, which follows immediately from
the CL spectral representation in Result
9.2
, connects the cyclic PSD and C-PSD to the
second-order properties of the increments of the spectral process
ξ
(
f
).
ν
n
,
Result 10.1.
The cyclic PSD P
xx
(
f
)
can be expressed as
ξ
∗
(
f
P
xx
(
ν
n
,
f
)d
f
=
E
[d
ξ
(
f
)d
−
ν
n
)]
,
(10.12)
and the cyclic C-PSD P
xx
(˜
ν
n
,
f
)
as
P
xx
(˜
ν
n
,
f
)d
f
=
E
[d
ξ
(
f
)d
ξ
(˜
ν
n
−
f
)]
.
(10.13)
Moreover, the spectral correlation
S
xx
(
ν,
f
) and the complementary spectral corre-
lation
S
xx
(
ν,
δ
f
), defined in Result
9.2
, contain the cyclic PSD and C-PSD on
-ridges
parallel to the stationary manifold
ν
=
0:
S
xx
(
ν,
f
)
=
P
xx
(
ν
n
,
f
)
δ
(
ν
−
ν
n
)
,
(10.14)
n
S
xx
(
P
xx
(˜
ν,
f
)
=
ν
n
,
f
)
δ
(
ν
−
ν
n
)
˜
.
(10.15)
n
On the stationary manifold
ν
=
0, the cyclic PSD is the usual PSD, i.e.,
P
xx
(0
,
f
)
=
=
P
xx
(
f
). It is clear from
the bounds in Result
9.3
that it is not possible for the spectral correlation
S
xx
(
P
xx
(
f
), and the cyclic C-PSD the usual C-PSD, i.e.,
P
xx
(0
,
f
)
ν,
f
)to
have only
0. CS processes must always have a WSS component, i.e.,
they must have a PSD
P
xx
(
f
)
δ
-ridges for
ν
=
≡
0. However, it is possible that a process has cyclic PSD
0 but cyclic C-PSD
P
xx
(˜
P
xx
(
ν,
f
)
≡
0 for all
ν
=
ν
n
,
f
)
≡
0forsome ˜
ν
n
=
0.
Example 10.1.
Let
u
(
t
) be a real WSS random process with PSD
P
uu
(
f
). The complex-
modulated process
x
(
t
)
u
(
t
)e
j2
π
f
0
t
=
has PSD
P
xx
(
f
)
=
P
uu
(
f
−
f
0
) and cyclic PSD
0. However,
x
(
t
) is CS rather than WSS because the cyclic
C-PSD is nonzero outside the stationary manifold for
P
xx
(
ν,
f
)
≡
0 for all
ν
=
2
f
0
:
P
xx
(2
f
0
,
ν
=
f
)
=
P
uu
(
f
−
f
0
).
Unfortunately, there is not universal agreement on the nomenclature for CS processes.
It is also quite common to call the cyclic PSD
P
xx
(
ν
n
,
f
) the “spectral correlation,” a
term that we use for
S
xx
(
f
). In the CS literature, one can often find the statement
“only (almost) CS processes can exhibit spectral correlation.” This can thus be translated
as “only (almost) CS processes can have
P
xx
(
ν,
ν
n
,
f
)
≡
0for
ν
n
=
0,” which means that
S
xx
(
ν,
f
) has
δ
-ridges outside the stationary manifold
ν
=
0. However, there also exist
processes whose
S
xx
(
ν,
f
)have
δ
-ridges that are curves (not necessarily straight lines)
in the (
f
)-plane. These processes have been termed “generalized almost CS” and
were first investigated by
Izzo and Napolitano (1998
).
ν,
10.1.2
Cyclic spectral coherence
The cyclic spectral coherence quantifies how much the frequency-shifted spectral pro-
cess
ξ
(
f
−
ν
n
) tells us about the spectral process
ξ
(
f
). It is defined as the cyclic
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