Digital Signal Processing Reference
In-Depth Information
In summary, a discrete-time signal is an ordered sequence of numbers. The se-
quence is usually expressed as
5
f[n]
6
where this notation denotes the sequence
We usually consider
,
Á , f[-2], f[-1], f[0], f[1], f[2], Á .
f[n],
for
n
a noninteger,
to be undefined.
Some of the reasons that engineers are interested in discrete-time signals are
as follows:
1.
Sampling is required if we are to use digital signal processing, which is
much more versatile than analog signal processing.
2.
Many communication systems are designed on the basis of the transmission
of discrete-time signals, for a variety of reasons.
3.
Sampling a signal allows us to store the signal in discrete memory.
4.
The outputs of certain sensors that measure physical variables are discrete-
time signals.
5.
Complex strategies for automatically controlling physical systems require
digital-computer implementation. The controlling signals from the computer
are discrete time.
6.
Many consumer products such as CDs, DVDs, digital cameras, and MP3
players use digital signals.
9.1
DISCRETE-TIME SIGNALS AND SYSTEMS
In this section, we introduce by example discrete-time signals. We use numerical in-
tegration as the example. Suppose that we wish to integrate a voltage signal,
using a digital computer. Integration by a digital computer requires that we use a
numerical algorithm. In general, numerical algorithms are based on approximating
a signal with an unknown integral with a signal that has a known integral. Hence, all
integration algorithms are approximate in nature.
We use Euler's rule (discussed in Section 1.3), which is depicted in Figure 9.2.
Euler's rule approximates the area under the curve by the sum of the rectangu-
lar areas shown. In this figure, the step size
H
(the width of each rectangle) is called
the
numerical-integration increment
. The implementation of this algorithm requires
x(t),
x(t)
x
(
t
)
•••
•••
0
(
n
1)
H
nH
(
n
1)
H
t
Figure 9.2
Euler integration.
