Digital Signal Processing Reference
In-Depth Information
Ce
t
/
C
Figure 2.14
Signal illustrating the time
0
t
constant.
The time constant of an exponential signal is illustrated in Figure 2.14. The de-
rivative of
x(t)
in (2.25) at
t = 0
is given by
dx(t)
dt
C
t
e
-t/t
C
t
.
t = 0
=-
t = 0
=-
If the signal continued to decay from at this rate, it would be zero at Ac-
tually, the value of the signal at is equal to 0.368
C
; that is, the signal has de-
cayed to 36.8 percent of its amplitude after
t = 0
t = t.
t = t
t
seconds. This result is general; the
x(t) = Ce
-t/t
0.368Ce
-t
1
/t
,
signal
at time
t
1
+ t
is equal to
or
0.368x(t
1
).
The inter-
ested reader can show this.
Table 2.2 illustrates the decay of an exponential as a function of the time con-
stant While infinite time is required for an exponential to decay to zero, the ex-
ponential decays to less than 2 percent of its amplitude in
t.
4t
units of time and to less
than 1 percent in units of time.
In most applications, in a practical sense the exponential signal can be ignored
after four or five time constants. Recall that the models of physical phenomena are
never exact. Hence, in many circumstances, high accuracy is unnecessary either in
the parameters of a system model or in the amplitudes of signals.
5t
CASE 2
C
Complex,
a
Imaginary
Next we consider the case that
C
is complex and
a
is imaginary—namely,
x(t) = Ce
at
; C = Ae
jf
= A∠f,
a = jv
0
,
(2.26)
TABLE 2.2
Exponential Decay
e
-
t/
t
t
0
1.0
0.3679
0.1353
0.0498
0.0183
0.0067
t
2t
3t
4t
5t
