Digital Signal Processing Reference
In-Depth Information
cross-interference coefficients could be converted and given as
A
i
C
m
=
D
c
+
V
c
,
B
i
C
m
=
D
s
+
V
s
,
(18.8)
A
i
S
m
=
V
s
−
D
s
,
B
i
S
m
=
D
c
−
V
c
,
where
N
−
1
N
−
1
1
N
1
N
D
c
=
cos 2
π
(
f
i
−
f
m
)
t
k
,
V
c
=
cos 2
π
(
f
i
+
f
m
)
t
k
,
k
=
0
k
=
0
(18.9)
N
−
1
N
−
1
1
N
1
N
D
s
=
sin 2
π
(
f
i
−
f
m
)
t
k
,
V
s
=
sin 2
π
(
f
i
+
f
m
)
t
k
.
k
=
0
k
=
0
Note that all the equations of (18.9) describe the mean values of nonuniformly
sampled sinusoid limited realizations [0,
Θ
] either for frequency (
f
i
−
f
m
)or
ϕ
=
π
/
(
f
i
0. This is a significant fact. It means
that the impact of random sampling irregularities on the essential properties of
randomly sampled composite signals can be revealed by studying estimation
specifics of the mean value of a sinusoid. The estimate
m
(
f
) of the mean value
and the expected value of the squared estimate of the mean value
E
[
m
2
(
f
)],
derived by Bilinskis and Mikelsons in 1992 for various types of random sam-
pling point processes, prove to be very informative and convenient for describing
effects caused by random sampling irregularities including those related to the
cross-intereference. The values of
D
c
+
f
m
) and phase angles
2or
ϕ
=
V
c
and
V
s
characterize deviations from
the mean values of sine waves at frequency (
f
i
,
D
s
,
f
m
), which are due
to the nonuniformity of the used sampling point process. This means that the con-
sidered cross-interference coefficients are related in this way to these deviations
and are tied to specific realizations of a sampling point process.
−
f
m
)or(
f
i
+
18.1.3 Approximation
Consider the estimates first in a generalized form not related to specific sampling
conditions. Assume that an analog signal
x
(
t
)
=
sin(2
π
ft
+
ϕ
) is randomly sam-
pled. It can then be represented in the following form:
x
(
t
k
)
=
sin(2
π
ft
k
+
ϕ
)
.
(18.10)
The estimate of the mean value of this signal is given by
N
1
N
m
(
f
)
=
sin(2
π
ft
k
+
ϕ
)
.
(18.11)
k
=
1
