
The asymptotic approximation for H(e
jvT
) in Equation 8.87 also predicts a delay of vT=2 rad in
x
A
(n) relative to the continuous-time output x(t). The time delay can be estimated by zooming in on
the respon ses in the top half of Figure 8.17.
Alternatively, the time delay can be determined by finding the time between occurrences of equal
values of x
A
(n) and x(t). For example, at t ¼4 s, the discrete-time variable n ¼4=0.025 ¼160 and
from M-file ‘‘Chap8_Fig2_17.m,’’ x
A
(160) ¼0.05603. Setting x(t
0
) ¼x
A
(160) ¼0.5603 and solving
for t
0
.
x(t
0
) ¼
1
v
(1 cos vt
0
) ¼ x
A
(160) ¼ 0:5603 (8:89)
t
0
¼
1
2
cos
1
[1 2(0:5603)] þ
2p
v
¼ 3:9874 s(8:90)
The simulated response x