Digital Signal Processing Reference
In-Depth Information
x
[
n
]
3
H
2
H
H
0
H
2
H
3
H H
Figure 9.5
Summation yielding
approximate integration.
We now compare certain operations on discrete-time signals with equivalent opera-
tions on continuous-time signals. First,
integration
in continuous time is considered
to be equivalent to
summation
in discrete time. This is illustrated in Figure 9.5,
where the discrete-time signal is assumed to be generated by sampling a continuous-
time signal. By Euler's rule, we see that
t
n
x(t)dt 3 H
a
x[k],
(9.14)
L
-
q
k=-
q
where
In a like manner, we can approximate the slope of a continuous-time signal
with the samples
x
[
n
], by the relation
t = nH.
x(t)
dx(t)
dt
x[k]
-
x[k
-
1]
H
`
t =kH
L
.
(9.15)
This relation is illustrated in Figure 9.6. The numerator in the right side of (9.15) is
called
the first difference,
which is considered to be the equivalent operation to the
first derivative of a continuous-time signal:
dx(t)
dt
3 x[n] - x[n - 1].
(9.16)
Approximation
for derivative
x
[
n
]
0
k
1
k
n
Figure 9.6
Approximate differentiation.
