The z-transform - Continuous and Discrete Time Signals and Systems - page 589

Solution
We use the frequency-differentiation property,

d
d z

1
1 − α z

←→ −

k α k x [ k ]

z ,

− 1

which reduces to

− 1

α z

←→

k α k x [ k ]

ROC: z > α.

(1 − α z

− 1 ) 2

13.4.6 Time convolution

If x 1 [ k ] and x 2 [ k ] are two arbitrary functions with the following z-transform
pairs:

←→

x 1 [ k ] X 1 ( z ) , ROC: R 1

and

←→

x 2 [ k ] X 2 ( z ) , ROC: R 2 ,

then the convolution property states that

←→

x 1 [ k ] ∗ x 2 [ k ] X 1 ( z ) X 2 ( z ) , ROC: at least R 1 ∩ R 2 . (13.24)

The convolution property is valid for both unilateral and bilateral z-transforms.
The overall ROC of the convolved signals may be larger than the intersection
of regions R 1 and R 2 because of the possible cancelation of some poles of the
convolved sequences.

Proof
By definition, the convolution of two sequences is given by

m =−∞ x 1 [ m ] x 2 [ k − m ] .
Taking the z-transform of both sides yields

∞

x 1 [ k ] ∗ x 2 [ k ] =

m =−∞ x 1 [ m ] x 2 [ k − m ] z

∞ ∞

←→

− k .

x 1 [ k ] ∗ x 2 [ k ]

k =−∞

By interchanging the order of the two summations on the right-hand side of the
transform pair, we obtain

m =−∞ x 1 [ m ] k =−∞ x 2 [ k − m ] z

∞ ∞

←→

− k .

x 1 [ k ] ∗ x 2 [ k ]

Substituting p = k − m in the inner summation leads to

m =−∞ x 1 [ m ]

∞

p =−∞ x 2 [ p ] z

∞

←→

− ( p + m )

x 1 [ k ] ∗ x 2 [ k ]