Digital Signal Processing Reference
In-Depth Information
x
1
(t)
x
2
(t)
0
0
t
t
(
a
)
(
b
)
Figure 1.1
Two samples of continuous-time signals.
real- or complex-valued function of time. We can also define a continuous-time
signal as a mapping of the set of all values of time to a set of corresponding
values of the functions that are subject to certain properties. Since the function is
well defined for all values of time in
−∞
to
∞
, it is differentiable at all values
of the independent variable
t
(except perhaps at a finite number of values). Two
examples of continuous-time functions are shown in Figure 1.1.
A
discrete-time signal
is a function that is defined only at discrete instants of
time and undefined at all other values of time. Although a discrete-time function
may be defined at arbitrary values of time in the interval
−∞
to
∞
, we will
consider only a function defined at equal intervals of time and defined at
t
nT
,
where
T
is a fixed interval in seconds known as the
sampling period
and
n
is an integer variable defined over
−∞
to
∞
. If we choose to sample
f(t)
at
equal intervals of
T
seconds, we generate
f(nT)
=
|
t
=
nT
as a sequence of
numbers. Since
T
is fixed,
f(nT)
is a function of only the integer variable
n
and
hence can be considered as a function of
n
or expressed as
f(n)
. The continuous-
time function
f(t)
and the discrete-time function
f(n)
are plotted in Figure 1.2.
In this topic, we will denote a discrete-time (DT) function as a DT sequence,
DT signal, or a DT series. So a DT function is a mapping of a set of all integers
to a set of values of the functions that may be real-valued or complex-valued.
Values of both
f(t)
and
f(n)
are assumed to be continuous, taking any value
in a continuous range; hence can have a value even with an infinite number of
digits, for example,
f(
3
)
=
f(t)
0
.
4
√
2inFigure1.2.
A zero-order hold (ZOH) circuit is used to sample a continuous signal
f(t)
with a sampling period
T
and hold the sampled values for one period before the
next sampling takes place. The DT signal so generated by the ZOH is shown in
Figure 1.3, in which the value of the sample value during each period of sam-
pling is a constant; the sample can assume any continuous value. The signals of
this type are known as
sampled-data signals
, and they are used extensively in
sampled-data control systems and switched-capacitor filters. However, the dura-
tion of time over which the samples are held constant may be a very small
fraction of the sampling period in these systems. When the value of a sample
=
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