PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

476 Practical MATLAB® Applications for Engineers

f. Differentiation in frequency

nf n z

d

dz

()↔ {()}Fz

g. Frequency shifting (modulation theorem)

f(n)[e−jwo n] ↔ F(ejwoz)

h. Convolution in time

f 1 (n) ⊗ f 2 (n) ↔ F 1 (z) F 2 (z)

i. Convolution in frequency

f(n) f 2 (n) ↔ ___^1
2 πj

. [F 1 (z) ⊗ F 2 (z)]

R.5.52 Some useful observations about the ZT are summarized as follows:

a. Let f(n) ↔ F(z), then

f(n − 1) ↔ z−^1 F(z)

f(n − 2) ↔ z−^2 F(z)

f(n − b) ↔ z−bF(z) for any integer b

f(n + a) ↔ zaF(z) for any integer a

Note that delaying a sequence f(n) by b time units in the time domain is equiva-

lent to multiplying its transform F(z) by z−b.

b. Since Z[δ(n)] = 1, then by defi nition of the ZT

Zn nzn

n

[( )]


( )

 



∑

and recall that

()n

n

n



10

00







then Z[δ(n)] = 1 , since z^0 = 1.

c. Recall (from Chapter 1 of this book) that any arbitrary sequence f(n) can be

expressed as

fn fk n k

k

()
()( )





∑ 

and since the transform of an impulse is known, as well as a shifted impulse,

then ZT of any arbitrary sequence f(n) can be easily evaluated.