Digital Signal Processing Reference
In-Depth Information
suring times. This “look and write down” operation is what is normally re-
ferred to as
sampling
. There are real-world phenomena which lend them-
selves very naturally and very intuitively to a discrete-time representation:
the daily Dow-Jones index, for example, solar spots, yearly floods of the Nile,
etc. There seems to be no irrecoverable loss in this neglect of intermediate
values. But what about music, or radio waves? At this point it is not alto-
gether clear from an intuitive point of view how a sampled measurement of
these phenomena entail no loss of information. The mathematical proof of
this will be shown in detail when we study the sampling theorem; for the
time being let us say that “the proof of the cake is in the eating”: just listen
to your favorite CD!
The important point to make here is that, once a real-world signal is
converted to a discrete-time representation, the underlying notion of “time
betweenmeasurements” becomes completely abstract. All we are left with is
a sequence of numbers, and all signal processing manipulations, with their
intended results, are independent of the way the discrete-time signal is ob-
tained. The power and the beauty of digital signal processing lies in part
with its invariance with respect to the underlying physical reality. This is in
stark contrast with the world of analog circuits and systems, which have to
be realized in a version specific to the physical nature of the input signals.
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2.1.2
Basic Signals
The following sequences are fundamental building blocks for the theory of
signal processing.
Impulse.
The discrete-time impulse (or discrete-time
delta function
)ispo-
tentially the simplest discrete-time signal; it is shown in Figure 2.5(a) and is
defined as
1
n
=
0
δ
[
n
]=
(2.5)
0
n
=
0
Unit Step.
The discrete-time unit step is shown in Figure 2.5(b) and is de-
fined by the following expression:
1
n≥
0
0
n
u
[
n
]=
(2.6)
<
0
The unit step can be obtained via a discrete-time integration of the impulse
(see eq. (2.16)).
