Digital Signal Processing Reference
In-Depth Information
the signal is not really a sequence, although it can be arbitrarily extended to
one. More importantly (and more often), the finiteness of a discrete-time
signal is explicitly imposed by design since we are interested in concentrat-
ing our processing efforts on a small portion of an otherwise longer signal; in
a speech recognition system, for instance, the practice is to cut up a speech
signal into small segments and try to identify the phonemes associated to
each one of them.
(5)
A special case is that of periodic signals; even though
these are bona-fide infinite sequences, it is clear that all information about
them is contained in just one period. By describing one period (graphically
or otherwise), we are, in fact, providing a full description of the sequence.
The complete taxonomy of the discrete-time signals used in the topic is the
subject of the next Sections ans is summarized in Table 2.1.
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2.2.1
Finite-Length Signals
As we just mentioned, a finite-length discrete-time signal of length
N
are
just a collection of
N
complex values. To introduce a point that will reappear
throughout the topic, a finite-length signal of length
N
is entirely equivalent
to a vector in
N
. This equivalence is of immense import since all the tools
of linear algebra become readily available for describing and manipulating
finite-length signals. We can represent an
N
-point finite-length signal using
the standard vector notation
=
x
0
...
x
N−
1
T
x
x
1
Note the transpose operator, which declares
x
as a
column
vector; this is
the customary practice in the case of complex-valued vectors. Alternatively,
we can (and often will) use a notation that mimics the one used for proper
sequences:
x
[
n
]
,
n
=
0, ...,
N
−
1
[
]
[
]
Here we
must
remember that, although we use the notation
x
n
,
x
n
is
not defined
for values outside its support, i.e. for
n
N
.Note
that we can always obtain a finite-length signal from an infinite sequence
by simply dropping the sequence values outside the indices of interest. Vec-
tor and sequence notations are equivalent and will be used interchangeably
according to convenience; in general, the vector notation is useful when we
want to stress the algorithmic or geometric nature of certain signal process-
ing operations. The sequence notation is useful in stressing the algebraic
structure of signal processing.
<
0orfor
n
≥
(5)
Note that, in the end, phonemes are pasted together into words and words into sen-
tences; therefore, for a complete speech recognition system, long-range dependencies
become important again.
