Signals and Systems - Electrical Engineering

562 C H A P T E R 9: The Z-Transform

a very important role in making this determination. Once this is done, the inverse is found by looking

for the causal and the anti-causal partial fraction expansion components in a Z-transforms table. The

coefficients of the partial fraction expansion are calculated like those in the case of causal signals.

nExample 9.21

Consider finding the inverse Z-transform of

X(z)=

2 z−^1

( 1 −z−^1 )( 1 − 2 z−^1 )^2

1 <|z|< 2

which corresponds to a noncausal signal.

Solution

The functionX(z)has two zeros atz=0, a pole atz=1, and a double pole atz=2. For the region

of convergence to be a torus of internal radius 1 and outer radius 2, we need to associate with the

pole atz=1 the region of convergence

R 1 :|z|> 1

corresponding to a causal signal, and with the pole atz=2, we associate a region of convergence

R 2 :|z|< 2

associated with an anti-causal signal. Thus, we have

1 <|z|< 2 =R 1 ∩R 2

The partial fraction expansion is then done so that

X(z)=

A

︸^1 −︷︷z−^1 ︸

R 1 :|z|> 1

+

[

B

1 − 2 z−^1

+

Cz−^1

( 1 − 2 z−^1 )^2

]

︸ ︷︷ ︸

R 2 :|z|< 2

That is, the first term hasR 1 as the region of convergence and the terms in the square brackets

haveR 2 as their region of convergence. The inverse of the first term will be a causal signal, and the

inverse of the other two terms will be an anti-causal signal.

The coefficients are found like in the case of causal signals. In this case, we have that

A=X(z)( 1 −z−^1 )|z− (^1) = 1 = 2
C=X(z)
( 1 − 2 z−^1 )^2
z−^1
|z− (^1) =0.5= 4
To calculateBwe computeX(z)and its expansion for a value ofz−^1 6=1 or 0.5. For instance,
z−^1 =0 gives
X( 0 )=A+B= 0