Digital Signal Processing Reference
In-Depth Information
Finite-length signals are extremely convenient entities: their energy is
always and, as a consequence, no stability issues arise in processing. From
the computational point of view, they are not only a necessity but often the
cornerstone of very efficient algorithmic design (as we will see, for instance,
in the case of the FFT); one could say that all “practical” signal processing
lives in
l
g
r
,
y
i
d
.
,
©
,
L
s
N
. It would be extremely awkward, however, to develop the whole
theory of signal processing only in terms of finite-length signals; the asymp-
totic behavior of algorithms and transformations for infinite sequences is
also extremely valuable since a stability result proven for a general sequence
will hold for all finite-length signals too. Furthermore, the notational flexi-
bility which infinite sequences derive from their function-like definition is
extremely practical from the point of view of the notation. We can immedi-
ately recognize and understand the expression
x
[
n−k
]
as a
k
-point shift of
asequence
x
; but, in the case of finite-support signals, how are we to de-
fine such a shift? We would have to explicitly take into account the finiteness
of the signal and the associated “border effects”, i.e. the behavior of opera-
tions at the edges of the signal. For this reason, in most derivations which
involve finite-length signal, these signals will be
embedded
into proper se-
quences, as we will see shortly.
[
n
]
2.2.2
Infinite-Length Signals
Aperiodic Signals.
The most general type of discrete-time signal is rep-
resented by a generic infinite complex sequence. Although, as previously
mentioned, they lie beyond our processing and storage capabilities, they
are invaluably useful as a generalization in the limit. As such, they must
be handled with some care when it comes to their properties. We will see
shortly that two of the most important properties of infinite sequences con-
x
x
0
,
x
1
,
x
2
,...,
x
N−
2
,
x
N−
1
,
x
0
,
x
1
, ...
→
x
[
n
]=
...,
x
N−
2
,
x
N−
1
,
n
=
0
x
[
n
−
1
]=
...,
x
N−
3
,
x
N−
2
,
x
N−
1
,
x
0
,
x
1
,
x
2
,...,
x
N−
2
,
x
N−
1
,
x
0
, ...
x
Figure 2.6
Equivalence between a right shift by one of a periodized signal and the
circular shift of the original signal.
x
and
x
are the length-
N
original signal and its
right circular shift by one, respectively.
