Page 474 - Mechanical Engineers' Handbook (Volume 2)

4 z-Transforms 465

Table 2 Table of z-Transforms a
No. F(s) ƒ(nT) F(z)
1 — 1, n 0; 0, n 0 1
2 — 1, n k;0, n k z k
1 z
3 1(nT)
s z 1
1 Tz
4 nT
s 2 (z 1) 2
2
1 1 Tz(z 1)
5 (nT) 2
s 3 2! 2(z 1) 3
2
1 1 Tz(z 4z 1)
3
6 (nT) 3
s 4 3! 6 (z 1) 4
1 ( 1) m 1 m 1 ( 1) m 1 m 1 z
7 lim e unT lim
s m a→0 (m 1)! a m 1 a→0 (m 1)! a m 1 z e aT
1 z
8 e anT
s a z e aT
1 Tze aT
9 nTe anT
)
(s a) 2 (z e aT 2
1 1 T 2 z(z e aT )
2 anT
10 (nT) e (e aT )
(s a) 3 2 2 (z e aT 3
)
1 ( 1) m 1 m 1 ( 1) m 1 m 1 z
11 (e anT )
(s a) m (m 1)! a m 1 (m 1)! a m 1 z e aT
a z(1 e aT )
12 1 e anT
s(s a) (z 1)(z e aT )
a 1 z[(aT 1 e aT )z (1 e aT aTe aT )]
13 (anT 1 e anT )
2
s (s a) a a(z 1) (z e aT )
2
b a (e aT e bT )z
14 (e anT e bnT )
(s a)(s b) (x e aT )(z e bT )
s z[z e aT (1 aT)]
15 (1 anT)e anT
)
(s a) 2 (z e aT 2
a 2 z[z(1 e aT aTe aT ) e 2aT e aT aTe aT ]
16 1 e anT (1 anT)
)
s(s a) 2 (z 1)(z e aT 2
(b a)s z[z(b a) (be aT ae bT )]
17 be bnT ae anT
(s a)(s b) (z e aT )(z e bT )
a z sin aT
18 sin anT
2
2
s a 2 z (2 cos aT)z 1
s z(z cos aT)
19 cos anT
2
2
s a 2 z (2 cos aT)z 1
s a z(z e aT cos bT)
20 e anT cos bnT
2
2
(s a) b 2 z 2e aT (cos bT)z e 2eT
b e anT sin bnT ze aT sin bT
21
(s a) b 2 z 2e aT (cos bT)z e 2aT
2
2
2
a b 2 1 e a sin bnT z(Az B)
22 anT cos bnT
2
2
2
s((s a) b ) b (z 1)(z 2e aT (cos bT)z e 2aT )
a
A 1 e aT cos bT e aT sin bT
b
a
B e 2aT e aT sin bT e aT cos bT
b
a F(s) is the Laplace transform of ƒ(t) and F(z) is the transform of ƒ(nT). Unless otherwise noted, ƒ(t) 0, t 0, and the region of convergence
of F(z) is outside a circle r z such that all poles of F(z) are inside r.