Digital Signal Processing Reference
In-Depth Information
Figure 9.6
Typical differing decompositions of an additive sampling point process: (a) de-
composition into eight components; (b) decomposition into five components
decompositions play an equally important role in secondary aliasing. The decom-
position into
n
components is special. The question is: why it is so special?
To answer this question, look at the features characterizing the considered
decompositions a little more carefully. Two typical decompositions of an additive
sampling point process for the cases where
n
=
μ/δ
are illustrated
in Figures 9.6(a) and (b) respectively. The decomposed additive sampling point
process is characterized by
=
μ/δ
and
n
=
μ/δ
μ
=
δ
μ
+
τ
8
, the sampling intervals are equal to
k
τ
,
±
δ
and the random variable
) with equal probability. It
can be seen that the sampling points drift across the decomposition levels in the
first case where
n
k
assumes the values (0
=
μ/δ
and jump chaotically between these levels in the second
case where
n
. That leads to the characteristic decomposition distributions
illustrated in Figure 9.6.
In general, the distributions of the sampling points between the constituents
represented by various phase-shifted particular periodic processes with random
skips are shown in the cases where
n
=
μ/δ
are closer to uniform (histograms a,
b, c and e in Figure 9.7). The differences in the quantities of the sampling points
belonging to these randomly decimated periodic processes with particular phase
=
μ/δ
