Digital Signal Processing Reference
In-Depth Information
a
n
,
to the discrete exponential signal
provided that the discrete signal is based on a
sampling process.
Time constant of a discrete exponential signal
EXAMPLE 9.7
x[n] = (0.8)
n
,
For the discrete-time signal
from (9.49),
t
T
=
-1
ln
0.8
= 4.48 Q t = 4.48T.
Hence, there are 4.48 samples per time constant. Assuming that an exponential decays to a
negligible amplitude after four time constants (see Section 2.3), this signal can be ignored for
nT 7 4t = 4(4.48T) L 18T
or for
n 7 18
samples.
■
We now generalize the exponential signal
x[n] = Ca
n
(9.50)
by considering the cases that both parameters
C
and
a
can be complex. Of course,
complex signals cannot appear in nature. However, as is the case for differential
equations, the solutions of many difference equations are simplified under the
assumption that complex signals can appear both as excitations and as solutions.
Then, in translating the results back to the physical world, only the real parts or the
imaginary parts of complex functions are used. We now consider three cases of the
discrete complex exponential signal.
CASE 1
C
and
a
Real
x[n] = Ca
n
,
For the first case, consider the signal
with both
C
and
a
real. This signal is plot-
ted in Figure 9.18 for both
C
and
a
positive. For
a 7 1,
the signal increases exponentially with
increasing
n
. For
0 6 a 6 1,
the signal decreases exponentially with increasing
n
. For
, and the signal is constant.
a = 1, x[n] = C(1)
n
= C
Ca
n
Ca
n
Ca
n
x
[
n
]
x
[
n
]
x
[
n
]
C
C
C
n
n
n
a
1
0
a
1
a
1
Figure 9.18
Discrete-time exponential signals.
