
(c) The continuous-time response y(t) to the input u(t) ¼2 sin 3t, t 0 is obtained from the transfer
function of the continuous-time system
H(s) ¼
Y(s)
U(s)
¼
1
ts þ 1
¼
1
0:25s þ 1
(4:666)
Y(s) ¼ H(s)U(s) ¼
1
0:25s þ 1
2
3
s
2
þ 9
(4:667)
¼
24
(s þ 4)(s
2
þ 9)
(4:668)
Inverting Y(s) by partial fractions leads to
y(t) ¼
24
25
e
t=0:25
þ
4
3
sin 3t cos 3t
(4:669)
The transient and steady-state components of y(t) are
y
tr
¼
24
25
e
t=0:25
, y
ss
¼
24
25
4
3
sin 3t cos 3t
(4:670)
(d) The magnitude and phase of the discrete-time frequency response function at v ¼3 rad=s and
T ¼0.025 s are obtained from Equations 4.664 and 4.665, respectively.
jH(e
j3(0:025)
)j¼
0:1
[1:81 1:8 cos 3(0:025)] ...