Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 28

for > 1. The inverse z -transform is defined using Cauchy's theorem (deriva-

tion details in Akay, 1994) as

| z |

1
2 πj

X ( z ) z n− 1 d z x ( n )= (2.42)

Γ

with respect to the z -transform (Equation 2.39). The contour of integration
Γ is counterclockwise and encloses the singularities of X assuming that the
integral exists. In general it is easier to recover the digital signal x ( n )ifthe
transform X has the rational function form:

φ ( z )

X ( z ) z n− 1 =

(2.43)

( z − p 1 ) m 1 ( z − p 2 ) m 2 ··· ( z − p k ) m k

where p i are the poles of the system. Then solving Equation 2.42 we get using
the residue theorem

k

X ( z ) z n− 1 x ( n )=

z = p i {

res }

i =1

k

d m i − 1

d m i − 1 ( z

p i ) m i X ( z ) z n− 1

1
( m i −

= lim

z = p i

− (2.44)

1)!

i =1

The z -transform has several properties, which we list here. Let x 1 ( n ) and
x 2 ( n ) be two arbitrary digital sequences, then:

a. Linearity .For a, b ∈ C , the following holds:

Z ( ax 1 ( n )+ bx 2 ( n )) = aZ ( x 1 ( n )) + bZ ( x 2 ( n )) (2.45)

and similarly for its inverse

Z − 1 ( ax 1 ( n )+ bx 2 ( n )) = aZ − 1 ( x 1 ( n )) + bZ − 1 ( x 2 ( n ))

(2.46)

b. Shift or Delay . The z -transform of a delayed causal sequence x ( n − d )is

∞

d ) z −n

Z [ x ( n − d )] = x ( n − (2.47)

n =0

where d> 0 is the sequence delay or shift. Let n = n − d and we can

substitute into Equation 2.47 to get

∞

d ) z − ( n + d ) Z [ x ( n − d )] = x ( n −

n =0

∞

= z −d x ( n ) z −n = z −d X ( z ) (2.48)

n =0