Page 388 - Mechanical Engineers' Handbook (Volume 2)

8 Model Classifications 379

The z-transforms for many sampled functions can be expressed in closed form. A listing
of the transforms of several commonly encountered functions is given in Table 12. Properties
of the z-transform are listed in Table 13.
Pulse Transfer Functions
The transfer function concept developed for continuous systems has a direct analog for
sampled-data systems. For a continuous system with sampled output u(t) and sampled input
y(t), the pulse or discrete transfer function G(z) is defined as the ratio of the z-transformed
output Y(z) to the z-transformed input U(z), assuming zero initial conditions. In general, the
pulse transfer function has the form

Y(z) b bz 1 bz 2 bz m
m
0
2
1
G(z)
U(z) 1 az 1 az 1 az n
1
n
2
Zero-Order Hold
The zero-order data hold is the most common mathematical approach to D/A conversion,
that is, to creating a piecewise continuous approximation u(t) of the form
Table 12 z-Transform Pairs
X(s) x(t)or x(k) X(z)
1 1 (t) 1
2 e kTs (t kT) z k
1 z
3 1(t)
s z 1
1 Tz
4 t
s 2 (z 1) 2
1 z
5 e at
s a z e aT
a (1 e aT )z
6 1 e at
s(s a) (z 1)(z e aT )
z sin T
7 sin t
2
s 2 z 2z cos T 1
2
8 s z(z cos T)
cos t
2
2
s 2 z 2z cos T 1
1 Tze aT
9 te at
)
(s a) 2 (z e aT 2
ze aT sin T
10 e at sin t
2
2
(s a) 2 z 2ze aT cos T e 2aT
s a z ze aT cos T
2
11 e at cos t
(s a) 2 z 2ze aT cos T e 2aT
2
2
2
2 Tz(z 1)
12 t 2
s 3 (z 1) 3
z
13 a
z a
z
14 a cos k
k
z a