Signals and Systems - Electrical Engineering

9.4 One-Sided Z-Transform 541

Table 9.1 One-Sided Z-Transforms

Function of Time Function ofz, ROC

1. δ[n] 1, wholez-plane


2. u[n]
   1
   1 −z−^1
   , |z|> 1


3. nu[n]
   z−^1
   ( 1 −z−^1 )^2
   , |z|> 1


4. n^2 u[n]
   z−^1 ( 1 +z−^1 )
   ( 1 −z−^1 )^3
   , |z|> 1


5. αnu[n],|α|< 1
   1
   1 −αz−^1
   , |z|>|α|


6. nαnu[n],|α|< 1
   αz−^1
   ( 1 −αz−^1 )^2
   , |z|>|α|


7. cos(ω 0 n)u[n]^1 −cos(ω^0 )z

− 1

1 −2 cos(ω 0 )z−^1 +z−^2

, |z|> 1

8. sin(ω 0 n)u[n]
   sin(ω 0 )z−^1
   1 −2 cos(ω 0 )z−^1 +z−^2
   , |z|> 1


9. αncos(ω 0 n)u[n],|α|< 1
   1 −αcos(ω 0 )z−^1
   1 − 2 αcos(ω 0 )z−^1 +z−^2
   , |z|> 1


10. αnsin(ω 0 n)u[n],|α|< 1
    αsin(ω 0 )z−^1
    1 − 2 αcos(ω 0 )z−^1 +z−^2
    , |z|>|α|

Table 9.2Basic Properties of One-Sided Z-Transform

Causal signals and constants αx[n],βy[n] αX(z),βY(z)

Linearity αx[n]+βy[n] αX(z)+βY(z)

Convolution sum (x∗y)[n]=

∑

kx[n]y[n−k] X(z)Y(z)

Time shifting—causal x[n−N]Ninteger z−NX(z)

Time shifting—noncausal x[n−N] z−NX(z)+x[−1]z−N+^1

x[n]noncausal,Ninteger +x[−2]z−N+^2 +···+x[−N]

Time reversal x[−n] X(z−^1 )

Multiplication byn n x[n] −z

dX(z)

dz

Multiplication byn^2 n^2 x[n] z^2

d^2 X(z)

dz^2

+z

dX(z)

dz

Finite difference x[n]−x[n−1] ( 1 −z−^1 )X(z)−x[−1]

Accumulation

∑n

k= 0 x[k]

X(z)

1 −z−^1

Initial value x[0] zlim→∞X(z)

Final value nlim→∞x[n] lim

z→ 1

(z− 1 )X(z)