Digital Signal Processing Reference
In-Depth Information
Example 3.2
The upper frequency
f
n
of a signal spectrum is equal to 0.25
f
s
. The demands
of the sampling theorem are obviously met. The estimate
E
[
] is calculated
on the basis of the commonly used algorithm given below. The question is whether
or not the estimate will be corrupted by errors due to aliasing. To find the answer,
it is sufficient to consider the estimation of the mean absolute value of only one
sinusoidal signal component at frequency
f
n
.
|
x
{
nT
}|
It can be shown that
2
c
π
.
On the other hand, when this signal is sampled and digitally processed as required,
the expected estimate
|
=
c
sin 2
π
f
n
t
|
=
E
[
x
(
t
)
]
=
N
1
N
c
sin
π
k
2
c
2
.
E
[
|
c
sin 2
π
f
n
kT
|
=
lim
N
]
⇒∞
k
=
1
Comparison of the true and the estimated values reveals that the latter is biased
and that aliasing therefore occurs. To understand why this is so, the Fourier
series expansion of
should be considered. It can be shown that it
contains components at frequencies 2
f
n
,4
f
n
,6
f
n
,....Under the given conditions,
4
f
n
|
c
sin 2
π
f
n
t
|
=
2
f
s
and 8
f
n
=
2
f
s
, .... The components at these frequencies cause the
bias errors.
The given examples show how misleading a superficial interpretation of the sam-
pling theorem can be. They are given in order to draw attention to the fact that
conditions for alias-free signal processing, in the cases where signals are to be
functionally converted, differ from those stated by the sampling theorem. It is not
sufficient to know the upper frequency of the signal spectrum to determine how
high the sampling frequency should be to be sure that there will be no aliasing
induced errors. The subsequent specific processing of the digitized signal should
also be taken into account.
Note that functional conversions, performed either before or after the sampling
operation, lead to the same additional restrictions on the spectra of the correspond-
ing signals. If the original signal
x
(
t
) is functionally converted before digitizing, as
shown in Figure 3.3(a), then, naturally, the required sampling frequency is deter-
mined by the spectrum of the converted signal
F
[
x
(
t
)]
When the equivalent con-
version is performed in the course of processing the digital signal
x
(
kT
)
.
as shown
in Figure 3.3(b), then the signal is sampled at the same rate as in the first case.
This rule is applied to the cases illustrated by Examples 3.1 and 3.2. It can
easily be established that the permitted bandwidth of the signal
x
(
t
) is less than
1
,
x
3
(
t
). In the case when
/
3
f
s
in the case of the functional conversion
F
[
x
(
t
)]
=
