Digital Signal Processing Reference
In-Depth Information
10.1.2 Arrangements for Sample Value Processing
Basically there are two different sources of errors distorting the results of process-
ing nonuniformly sampled signals. The first type of signal parameter estimation
error is linked to incompletely suppressed aliasing. The second kind of error is
related to sampling irregularities. It should be realized that the hybrid sampling
technique provides for elimination of errors due to aliasing and that benefit is due
to the fact that hybrid sampling is based on superposition of partial sparse periodic
sampling pulse streams. This sampling approach is much better for elimination of
aliases than the more conventional nonuniform sampling techniques. However,
it does not automatically suppress the errors caused by random skipping of the
signal sample value taking. These error components need to be diminished at the
stage of signal processing in some other way. For instance, the signal process-
ing methods adapted to sampling nonuniformities, discussed in Chapter 18, are
effective for that.
As shown in Chapter 8, variable-phase periodic sampling, if performed cor-
rectly, provides the conditions necessary for suppression of aliasing. The consid-
ered method for hybrid sampling exploits the dependencies described there. To
explain how aliases present in the digital signal obtained as a result of hybrid sam-
pling can be eliminated, the signal model will be used again, according to which
a signal
x
(
t
) is composed of a multitude of sinusoidal components at arbitrary
frequencies in a limited frequency band. Suppose that such a signal is sampled
periodically and that it is done according to superposition of a number of phase-
shifted periodic sampling point processes. Consider the case where the frequency
range is extended four times. In that case, the highest frequency of the signal
might exceed two times the frequency of the periodic sampling point processes
used. The sampling point processes are periodic with the period
T
=
1
/
f
s
an
d the
signal sample values are taken at sampling instants
t
[
m
]
k
t
[0]
k
=
+
m
δ,
m
=
0
,
3
,
where
m
4 is the phase angle of the
m
th sampling point process. Su-
perposition of the particular sampling point processes apparently has to be per-
formed so that the minimal distance between adjacent sampling instants in the
resulting sampling process should not be shorter than the smallest sampling in-
terval allowed for that ADC. Under the specified conditions, each of the signal
basic frequencies
f
0
within the frequency range (0
δ
=
mT
/
1
4
−
f
u
) has three potentially
aliasing frequencies:
f
1
f
3
. The task is to recover the signal components at
all indicated frequencies. At least four periodic sampling point processes are
used.
To compose equations describing the sampling process under these conditions,
account is taken of the fact that all aliasing frequencies represented by sample
,
f
2
,
