Digital Signal Processing Reference
In-Depth Information
TABLE 9.1
Equivalent Operations
Continuous Time
Discrete Time
t
n
1.
x(t)dt
x[k]
a
L
-
q
k=-
q
dx(t)
dt
2.
3.
x[n] - x[n - 1]
x(t)d(t) = x(0)d(t)
x[n]d[n] = x[0]d[n]
du(t)
dt
4.
d(t) =
d[n] = u[n] - u[n - 1]
t
n
5.
u(t) =
L
d(t)dt
u[n] =
d[k]
a
-
q
k=-
q
Equivalent operations with impulse functions are given by
x(t)d(t) = x(0)d(t) 3 x[n]d[n] = x[0]d[n]
(9.17)
[see (9.11)], where is the continuous-time unit impulse function. Equivalent op-
erations with impulse and step functions are given by
d(t)
du(t)
dt
3 d[n] = u[n] - u[n - 1]
d(t) =
(9.18)
[see (9.12)], and
t
n
u(t) =
L
d(t)dt 3 u[n] =
d[k].
(9.19)
a
-
q
k=-
q
As an example of this summation, let
n = 3:
3
d[k] =
Á
+ d[-1] + d[0] + d[1] + d[2] + d[3]
u[3] =
a
k=-
q
= d[0] = 1.
Recall that
d[0] = 1
and
d[n] = 0
for
n Z 0.
These equivalent operations are sum-
marized in Table 9.1.
In this section, we have introduced discrete-time signals and systems. In addition,
a difference equation, which models an integrator, was solved. A general method for
solving linear difference equations with constant coefficients is given in Chapter 10.
9.2
TRANSFORMATIONS OF DISCRETE-TIME SIGNALS
In this section, we consider six transformations on a discrete-time signal
x
[
n
]. Three
transformations are on the independent variable
n
and the other three on the de-
pendent variable x[
#
].
