Page 472 - Mechanical Engineers' Handbook (Volume 2)

4 z-Transforms 463

Z{aƒ(k)} aF(z) (35)
(k) and g (k) are z-transformable and and are scalars, then ƒ(k) ƒ (k)
P2. If ƒ 1 1 1
g (k) has the z-transform
1
F(z) F (z) G (z) (36)
1
1
where F (z) and G (z) are the z-transforms of ƒ (k) and g (k), respectively.
1
1
1
1
P3. If
F(z) Z{ƒ(k)}
then
Z{a ƒ(k)} F
z
k
a (37)
T1. Shifting Theorem. If ƒ(t) 0 for t 0 and ƒ(t) has the z-transform F(z), then
n
Z{ƒ(t nT)} zF(z)
and
Z{ƒ(t nT)} z F(z) ƒ(kT)z
n 1
k
n
k 0 (38)
T2. Complex Translation Theorem. If
F(z) Z{ƒ(t)}

then
at
Z{e at ƒ(t)} F(ze ) (39)
T3. Initial-Value Theorem. If F(z) Z{ƒ(t)} and if lim z→ F(z) exists, then the initial value
ƒ(0) of ƒ(t)or ƒ(k) is given by

ƒ(0) lim F(z) (40)
z→
T4. Final-Value Theorem. Suppose that ƒ(k), where ƒ(k) 0 for k 0, has the z-transform
F(z) and that all the poles of F(z) lie inside the unit circle, with the possible exception
of a simple pole at z 1. Then the final value of ƒ(k) is given by
lim ƒ(k) lim [(1 z )F(z)] (41)
1
k→ z→1

4.5 Pulse Transfer Function
Consider the LTI discrete-time system characterized by the following linear difference equa-
tion:

y(k) ay(k 1) ay(k n) bu(k) bu(k 1) bu(k m) (42)
1
n
0
1
m
where u(k) and y(k) are the system’s input and output, respectively, at the kth sampling or
at the real time kT; T is the sampling period. To convert the difference equation (42) to an
algebraic equation, take the z-transform of both sides of Eq. (42) by definition: