Digital Signal Processing Reference
In-Depth Information
The expected value of
m
(
f
) is obtained by averaging the right-hand side of
Equation (18.11) over the sampling instants
{
t
k
}
. Then
∞
N
1
N
E
[
m
(
f
)]
=
sin(2
π
ft
+
ϕ
)
p
k
(
t
)d
t
(18.12)
−∞
k
=
1
where
p
k
(
t
) is the probability density function of the time intervals [0
t
k
].
Thus the expected value of
m
(
f
) depends both on the sinusoidal signal param-
eters and on the sampling point process used when sampling this signal. In the
case of periodic sampling with jitter,
,
1
/
T
for
t
∈
[(
k
−
1)
T
,
kT
]
,
p
k
(
t
)
=
0
for
t
∈
[(
k
−
1)
T
,
kT
]
.
Substituting this function into Equation (18.12) leads to
∞
N
1
Θ
ω
=
ω
+
ϕ
E
[
m
(
)]
sin(
t
) d
t
(
k
−
1)
T
k
=
1
Θ
1
Θ
=
ω
+
ϕ
sin(
t
)d
t
0
sin
ω
2
sin
ωΘ/
2
=
+
ϕ
,
(18.13)
ωΘ/
2
2
π
f
and
where
ω
=
Θ
is the time interval during which the signal is observed.
In this particular case,
NT
. This expected value of
E
[
m
(
f
)] of a sine-wave
signal sampled periodically with jitter is shown in Figure 18.1 as a function of the
normalized frequency
Θ
=
ν
=
f
Θ
=
fNT
. The two diagrams (a) and (b) illustrate
ϕ
=
π
/
the cases where
0 respectively.
This function is directly tied to the expected values of
D
c
,
2 and
ϕ
=
V
c
and
V
s
and
therefore also to the expected values of the cross-interference coefficients. While
in the wide frequency range the expected values of these coefficients are mean-
ingless as they are in a wide frequency range close to zero, the function describing
the expected values of them in the low-frequency range is quite useful. The point
is that the expected value of
E
[
m
(
f
)] represents a good approximation of the
cross-interference function there.
Basically there are two reasons why the mean value of a sinusoid might differ
from zero. Firstly, it happens if such a signal is observed and the mean value is
calculated for a time interval not equal to an integer number of its periods and,
secondly, the mentioned deviations occur as a result of nonuniform sampling.
Both of these factors often act simultaneously, causing the cross-interference
D
s
,
