5 l lEC533: Digital Signal Processing

Lecture 5The Z-Transform

5.1 - Introduction

• The Laplace Transform (s domain) is a valuable tool for l & d l &representing, analyzing & designing continuos-time signals &

systems.• The z-transform is convenient yet invaluable tool for representing,The z transform is convenient yet invaluable tool for representing,

analyzing & designing discrete-time signals & systems.• The resulting transformation from s-domain to z-domain is called

z-transformz-transform.

• The relation between s-plane and z-plane is described below :z = esTz e

• The z-transform maps any point s = σ + jω in the s-plane to z-plane (r θ).

5.2 – The Z-Transform

For continuous-time signal,

Ti D i S‐DomainTime Domain S‐Domain

For discrete-time signal,

Time Domain Z‐DomainƵ

Ƶ-1

Causal System

where,

5.2.1 – Z-Transform Definition

• The z-transform of sequence x(n) is defined by

∞∑=∞

−∞=

−

n

nznxzX )()( Two sided z transformBilateral z transform

∑∞

)()( nX

For causal system

• The z transform reduces to the Discrete Time Fourier transform

∑==

−

0)()(

n

nznxzX One sided z transformUnilateral z transform

• The z transform reduces to the Discrete Time Fourier transform (DTFT) if r=1; z = e−jω.

DTFT

( ) ( )j j n

nX e x n eω ω

∞−

=−∞

= ∑

5.2.2 – Geometrical interpretation of z-transformz transform

• The point z = rejω is a pvector of length r from origin and an angle ω with

Im z

j

Im z

jrespect to real axis.

rω

z = rejωj

R

rω

z = rejωj

R• Unit circle : The contour |z| = 1 is a circle on the z-

l ith it di

-1 1 Re z-1 1 Re z

plane with unity radius -j-j

DTFT is to evaluate z-transform on a unit circle.

5.2.3 – Pole-zero Plot

• A graphical representation of z-transform on z-plane

Im z

j– Poles denote by “x” and– zeros denote by “o”

Re z-1 1 Re z

-j

Example

Find the z-transform of,

Solution:

It’s a geometric sequence

Recall: Sum of a Geometric Sequence

where, a: first term,  r: common ratio,n: number of terms   

5.3 – Region Of Convergence (ROC)

• ROC of X(z) is the set of all values of z for which X(z) attains a finite value.

• Give a sequence, the set of values of z for which the z-transform converges, i.e., |X(z)|<∞, is called the region of convergence.

∞∞∞<∑=∑=

−∞=

−

−∞=

−

n

n

n

n znxznxzX |||)(|)(|)(|

∑ ∞<∞

−nrnx |)(|∑ ∞<−∞=n

rnx |)(|Im

ROC is an annual ring centered on

Re

the origin.

+− << xx RzR ||r

}|{ +−ω <<== xx

j RrRrezROC

Ex. 1 Find the z-transform of the following sequence{2 3 7 4 0 0 }x = {2, -3, 7, 4, 0, 0, ……..}

4732][)( 321 zzzznxzX n ++== −−−∞

−∑

0||4732

4732][)(

23

zzzz

zzzznxzXn

>++−

=

++−==−∞=∑

origin. except the plane-complex entire theis ROC The

0||, 3

z

zz

>=

Ex. 2 Find the z-transform of δ [n]

∑∞

plane.- entire theof consisting ROCan with

1][)(

z

znzXn

n == ∑−∞=

−δ

Ex 3 Find the z-transform of δ [n -1]Ex. 3 Find the z transform of δ [n 1]

1]1[)( 1 ==−= −∞

−∑ zzznzX nδ

. 0except plane- entire theof consisting ROCan with =−∞=

zzzn

Ex. 4 Find the z-transform of δ [n +1]

∞

exceptplane-entiretheofconsistingROCanwith

]1[)(

∞=

=+= ∑∞

−∞=

−

zz

zznzXn

nδ

infinity.at pole a is therei.e.,,except plane-entiretheofconsisting ROCan with ∞=zz

Ex.5 Find the z-transform of the following right-sided sequence(causal)(causal)

][][ nuanx n=

nnn )(][)( 1∑∑∞∞ n

nn

nn azznuazX )(][)(0

1∑=∑==

−

−∞=

−

Thi f fi d iThis form to find inverseZT using PFE

Ex.6 Find the z-transform of the following left-sided sequence

Ex. 7 Find the z-transform of

Rewriting x[n] as a sum of left sided and right sided sequencesRewriting x[n] as a sum of left-sided and right-sided sequencesand finding the corresponding z-transforms,

where

Notice from the ROC that the z-transformdoesn’t exist for b > 1

5.3.1 – Characteristic Families of Signals with Their Corresponding ROC

5.3.2 – Properties of ROC

• A ring or disk in the z-plane centered at the origin.g p g• The Fourier Transform of x(n) is converge absolutely iff the

ROC includes the unit circle.• The ROC cannot include any poles• The ROC cannot include any poles• Finite Duration Sequences: The ROC is the entire z-plane

except possibly z=0 or z=∞.• Right sided sequences (causal seq.): The ROC extends

outward from the outermost finite pole in X(z) to z=∞.• Left sided sequences: The ROC extends inward from theLeft sided sequences: The ROC extends inward from the

innermost nonzero pole in X(z) to z=0.• Two-sided sequence: The ROC is a  ring bounded by two 

circles passing through two pole with no poles inside the ringcircles passing through two pole with no poles inside the ring

5.4 - Properties of z-Transform)()(][][ zYbzXanybnxa :Linearity (1) +←→+

⎞⎛ z⎟⎠⎞

⎜⎝⎛←→

azXnxan ][ :Property scale- Z(4)

)1(][)( l

)()(][][)6( zXzHnxnh:nConvolutio ←→∗

)1(][)5(z

Xnx :Reversal Time ←→−

)()(][][)6( zXzHnxnh :nConvolutio ←→∗Transfer Function

5.5 - Rational z-Transform

For most practical signals, the z-transform can be expressedf l las a ratio of two polynomials

)())(()()( 21 zzzzzzzN ML −−−

where)())(()())((

)()()(

21

21

pzpzpzzzzzzzG

zDzNzX

N

M

L −−−==

roots thei.e.,, of theare ,,, gain,scalar isG

21 X(z)zeroeszzz ML

polynomialrdenominatotheofroots thei.e.,, of theare ,,, and

polynomialnumerator theof

21 X(z)polesppp NL

.polynomialr denominato theof

5.6 - Commonly used z-Transform pairs

Sequence z‐Transform ROC

δ[n] 1 All values of zδ[n] 1 All values of z

u[n] |z| > 111

1−− z

αnu[n] |z| > |α|

nαnu[n] |z| > |α|

111

−− zα

21

1−zαnαnu[n] |z| > |α|

(n+1) αnu[n] |z| > |α|

21)1( −− zα

21)1(1

−− zα

(rn cos ωon) u[n] |z| > |r|2210

10

)cos2(1)cos(1

−−

−

+−−

zrzrzr

ωω

(rn sin ωon) [n] |z| > |r|221

0

10

)cos2(1)sin(1

−−

−

+−−

zrzrzr

ωω

5.7 - Z-Transform & pole-zero distribution & Stability considerationsy

Thus, unstable

z

Mapping between S-plane & Z-plane is done as follows: L.H.S.R.H.S.stable

1) Mapping of Poles on the jω‐axis of the s‐domain to the z‐domain ω /41) Mapping of Poles on the jω axis of the s domain to the z domain

Maps to a unit circle & represents Marginally stable terms 1

ωs/4

ω /2 ω=0ω=ωs

ωs/2

3ωs/4

5.7 - Z-Transform & pole-zero distribution & Stability considerations – cont.

2) Mapping of Poles in the L.H.S. of the s‐plane to the z‐plane

y

Maps to inside the unit circle & represents stable terms & the system is stable.

3) Mapping of Poles in the R.H.S. of the s‐plane to the z‐plane

Outside the unit circle & represents unstable terms.

Discrete Systems Stability Testing StepsDiscrete Systems Stability Testing Steps

1) Find the pole positions of the z-transform.2) If any pole is on or outside the unit circle. (Unless coincides with zero on the unit

circle) The system is unstable.

5.7.1 - Pole Location and Time-domain Behavior of Causal SignalsBehavior of Causal Signals

5.7.2 - Stable and Causal SystemsC l S t ROC t d t d f th t t l

ImCausal Systems : ROC extends outward from the outermost pole.

ReR

Stable Systems : ROC includes the unit circle. Im

Re

1A stable system requires that its Fourier transform is uniformly convergent.