
(a) The z-domain transfer function of the discrete-time system is
H(z) ¼ H(s)j
s (2=T)((z1)=(zþ1))
¼
1
ts þ 1
s (2=T)((z1)=(zþ1))
(4:674)
¼
1
t[(2=T)((z 1)=(z þ 1))] þ 1
(4:675)
¼
T(z þ 1)
(2t þ T)z (2t T)
(4:676)
) H(z) ¼
Y(z)
U(z)
¼
T(1 þ z
1
)
(2t þ T) (2t T)z
1
(4:677)
Inverting Equation 4.677 produces the difference equation
(2t þ T)y
k
(2t T)y
k1
¼ T(u
k
þ u
k1
), k ¼ 1, 2, 3, ... (4:678)
(b) v
s
¼2p=T ¼2p=0.25 ¼8p rad=s, v
N
¼v
s
=2 ¼8p=2 ¼4p rad=s.
(c) The continuous-time output y(t) is obtained by inverse Laplace transformation of
Y(s) ¼ H(s)U(s) ¼
1
ts þ 1
v
s
2
þ v
2
(4:679)
Following partial fraction expansion and inverse Laplace transformation, the result ...