Digital Signal Processing Reference
In-Depth Information
Figure 3.1
Examples of various sampling point processes and their graphical depiction
t
k
. These signal readings obtained at discrete instants are usually
considered as signal sample values and the process of taking them is referred to
as sampling. The instants at which the samples are obtained form a stream of
uniform events, which can be depicted graphically as a sampling point process.
A few such point processes are shown in Figure 3.1. As demonstrated later, the
properties of the sampled signals depend to a considerable extent on the patterns
of the point processes generated and used for sampling.
When sampling is mentioned, it is usually assumed that the sampling process
considered is deterministic and periodic. The model of equidistant sampling,
according to which signal samples are separated by equal-length time intervals
T
, has been extensively studied, is now used almost exclusively and is actually
considered as unique. This is readily comprehensible because such a sampling
approach appears to be the most natural and obvious. It also has a number of
attractive advantages. However, periodic sampling, in reality, is just one among
many other possible sampling models.
It was established a relatively long time ago that application of periodic sam-
pling alone is not sufficient. The periodic sampling model is not applicable when
fluctuations in sampling instants cannot be ignored or when signal samples can be
obtained only at irregular or even random time intervals. In addition, studies have
revealed that randomness in sampling is not always harmful. It was discovered
that random irregularities in the sampling process sometimes might even be ben-
eficial. If properly introduced and exploited, these irregularities provide various
useful effects. Basically they help to preserve the structure of the original signals,
making it possible, for instance, to estimate frequencies of signal components
even when these frequencies exceed half of the sampling rate.
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k
=
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1
,
2
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