Digital Signal Processing Reference
In-Depth Information
9
DISCRETE-TIME SIGNALS
AND SYSTEMS
In this chapter, we first consider
discrete-time signals,
or, more simply,
discrete sig-
nals
. A discrete-time signal is defined only at discrete instants of time. We denote a
discrete-time signal as
x
[
n
], where the independent variable
n
may assume only
integer values. As a second topic in this chapter, we consider
discrete-time systems,
or, simply,
discrete systems
. A discrete-time system is defined as one in which all sig-
nals are discrete time. This chapter follows closely the outline of Chapter 2.
As stated, a discrete signal is defined at only discrete instants of time. For ex-
ample, suppose that a continuous-time signal is to be processed by a digital com-
puter. [This operation is called
digital signal processing
(DSP).] Because a computer
can operate only on a number, the continuous-time signal must first be converted to
a sequence of numbers. This conversion process is called
sampling
. If the signal is
sampled at regular increments of time
T
, the number sequence
results. The time increment
T
is called the
sampling period
.
(Since there is little danger of confusion in this and following chapters, the symbol
T
is used to denote the sampling period, instead of as in Chapters 5 and 6.) The
sampling process is illustrated in Figure 9.1(a), where each sample value is repre-
sented by a dot at the end of a vertical line.
The hardware normally used in sampling is represented in Figure 9.1(b). As
described in Chapter 1, an analog-to-digital converter (A/D or ADC) is an elec-
tronic circuit that samples a voltage signal and converts each sample into a binary
number; the binary numbers can then be transmitted to a digital computer for pro-
cessing or for storage. Hence, an A/D is used to generate and transmit the number
sequence to the computer. The instants that samples are taken are deter-
mined by timing pulses from the computer.
A word is in order concerning notation. The notation indicates a continuous-
time signal. The notation indicates the value of at The notation
f
[
n
] denotes a
discrete-time signal
that is defined only for
n
an integer. Parentheses
indicate continuous time; brackets indicate discrete time. However, this notation is
f(t)
f(nT), n =Á,
-2, -1, 0, 1, 2, Á ,
T
S
f(nT)
f(t)
f(nT)
f(t)
t = nT.
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