31610 use of z transform

8/14/2019 31610 Use of z Transform

1/18

Applications of the Z-transform

March 19, 2006


2/18

Power series method. Partial fraction expansion method.

Residue method.

1. Inverse Z-transform


3/18

H(z) =

n=

h(n)zn System Function

ROC at least the intersection of the ROCs of H(z)

and X(z). Can be larger if there are pole/zero

cancellation. e.g.

H(z) =1

z a, z > a

x(z) = z a, z =

Y(z) = 1, ROC : all z

Y(z) = H(z)X(z),

h(n)x(n) y(n) = x(n)*h(n)

2. Convolution Property and System Functions


4/18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A discrete-time linear time-invariant system function H(z) is causal when,

and only when the ROC of H(z) is the exterior of a circle and includes

z =

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

When h(n) right-sided, then ROC is the exterior of a circle:

H(z) =

n=N1

h(n)zn.

If N1 < 0, then h(N1)zN1 at z = . ROC outside a circle, butdoes not include .

Causal N1 0

3. Causality


5/18

Delay

Multiplication

Addition

Branch

x(n)

x(n)

x(n)

x(n)

x(n1)

x(n)y(n)

x(n)+y(n)

x(n)

T

y(n)

y(n)

x(n)

4. Structure of a Digital System

4.1. Symbols for Digital Operations


6/18

z1

1n+1

n

x(n) y(n)

y(n) =n

n + 1y(n 1) +

1

n + 1x(n)

We want a system that calculates:

y(n) =1

n + 1

nk=0

x(k)

4.2. Example - Cumulative Averaging System


7/18

5. DT LTI systems described by LCCDEs

Nk=0

aky(n k) =

Mk=0

bkx(n k)

Using the time-shift property:

N

k=0

akzkY(z) =

M

k=0

bkzkX(z)

Y(z) = H(z)X(z)

H(z) =Mk=0 bkzk

Nk=0 akz

k

ROC: Depends on boundary conditions, left-, right-, or two-sided. For

causal systems - ROC is outside the outermost pole.


8/18

6. Rational Z-Transform

X(z

) =

N(z)

D(z) =

b0 + b1z1 + + bMz

M

a0 + a1z1 + + aNzN

X(z) =b0

a0zNM

(z z1)(z z2) (z zM)

(z p1)(z p2) (z pN)

The transform has M finite zeros at z = z1, . . . zM, N infinite poles at

z = p1 . . . zN and N-M zeros (if N > M) or poles (if N < M) at the

origin z = 0.

A DT LTI system is causal if the ROC is the exterior of a circle outsideoutermost pole include . Thus

N M, for a causal system


9/18

1

0

1

1

0

1

0

0.5

1

1.5

2

2.5

3

3.5

4

Re

Im

|X(z)|

7. Poles - Zero Description of Discrete-Time Systems


10/18

0 1

zplane x(n)

0 1

zplane x(n)

0 1

zplane x(n)

0 1

zplane x(n)

0 1

zplane x(n)

0 1

zplane x(n)

7.1. Time-domain behavior - Single real-pole causal signal


11/18

0 1

zplane x(n)

0 1

zplane x(n)

0 1

zplane x(n)

7.2. Time-domain behavior - Complex-conjugate poles


12/18


13/18

Geometric Evaluation of a Rational z-Transform

Example #1:

Example #3:

Example #2:

All same as

in s-plane


14/18

Geometric Evaluation of DT Frequency Responses

First-Order System one real pole


15/18

Second-Order System

Two poles that are a complex conjugate pair (z1= rej=z2

*)

Clearly, |H| peaks near =


16/18

Definition:

X(z) =

n=0

x(n)zn

Characteristics:

1. No information about x(n) for n < 0.

2. Unique only for causal signals.3. Identical to the two-sided z-transform of the signal x(n)u(n).

9. The one-sided z-transform

9.1. Definition and characteristics


17/18

All properties are like for the two-sided Z-transform except for:

Shifting property:

x(n k)z

zk

X(z) +

k

n=1x(n)zn

, k > 0

x(n + k)z

zk

X(z) k1n=0

x(n)zn

, k > 0

Final value theorem

limn

x(n) = limn1

(z 1)X(z)

9.2. Properties


18/18

Example taken from Digital Signal Processing, Principles, Algorithms and Applications by Proakisand Manolakis

Example

The well known Fibonacci sequence of integer numbers is obtained by com-

puting each term as a sum of the two previous ones. The first few terms of

the sequence are:

1, 1, 2, 3, 5, 8, . . .

Determine a closed-form expression for the nth term of the Fibonacci se-

quence.

10. Solution of Difference Equations

31610 use of z transform

Documents