
An approximate expression for H(e
jvT
) is obtained using the asymptotic approximation for H(e
jvT
)
in Equation 8.99.
H(e
jvT
)
K 1 þ jv 1 þ e
I
( jvT)
k
1 l 1=jv 1 þ e
I
( jvT)
k
, vT 1 (8:104)
K
jv 1 þ e
I
( jvT)
k
l
, vT 1 (8:105)
Using trapezoidal integration, k ¼ 2, e
I
¼1=12 from Table 8.4,
H(e
jvT
)
K
jv[1 þ (1=12)(vT)
2
] l
, vT 1 (8:106)
The exact expression for H(e
jvT
) is obtained from
H(z) ¼ H(s)
s 1=H
I
(z)
¼
K
s l
s
2
T
z1
zþ1
(8:107)
) H(e
jvT
) ¼
K
(2=T)( (z 1)=(z þ 1)) l
z e
jvT
(8:108)
¼
KT(e
jvT
þ 1)
(2 lT)e
jvT
(2 þ lT)
(8:109)
The use of trapezoidal integration for digital simulation of linear continuous-time systems is referred
to