lecture 28: the z-transform and its roc properties

ECE 8443 – Pattern RecognitionEE 3512 – Signals: Continuous and Discrete

• Objectives:Relationship to the Laplace TransformRelationship to the DTFTStability and the ROCROC PropertiesTransform Properties

• Resources:MIT 6.003: Lecture 22Wiki: Z-TransformCNX: Definition of the Z-TransformCNX: PropertiesRW: PropertiesMKim: Applications of the Z-Transform

LECTURE 28: THE Z-TRANSFORMAND ITS ROC PROPERTIES

Audio:URL:

EE 3512: Lecture 28, Slide 2

Definition Based on the Laplace Transform

• The z-Transform is a special case of the Laplace transform and results from applying the Laplace transform to a discrete-time signal:

• Let us consider how this transformation maps the s-plane into the z-plane:

s = j:

s = + j:

Recall, if a CT systemis stable, its poles lie in the left-half plane.

Hence, a DT system isstable if its poles areinside the unit circle.

The z-Transform behavesmuch like the Laplacetransform and can beapplied to difference equationsto produce frequency and time domain responses.

n

n

n

n

z

ts

nxt

tntst znxX(z)ettnxdtetxsX ][)(lim)()(

][0

circleunit the tomaps axis-1 jezeez tjtjts

circleunit theof inside the tomaps (LHP)0)( ttj eez


ROC and the Relationship to the DTFT

• We can derive the DTFT by setting z = rej:

• The ROC is the region for which:

Depends only on r = |z| just like the ROC in the s-plane for the Laplace transform depended only on Re{}.

If the unit circle is in the ROC, then the DTFT, X(ej), exists.

Example: (a right-sided signal)

If :

The ROC is outside a circle of radius a,and includes the unit circle, which meansits DTFT exists. Note also there is a zeroat z = 0.

nn

nn

n

nr

rnxrnxrnxX(z)

][e)][(e][ j-jez j

F

n

nrnx ][

][][ nuanx n

?1

)(1

)(][

1

1

0

1

0

az

az

azzaznuaX(z)n

n

n

nn

n

nn

azaz or,,11

11

1

az

X(z)


Stability and the ROC

• For a > 0: azaz

X(z)nuanx n

for1

1][][

1

• If the ROC is outside the unit circle, the signal is unstable.

1for1

1

][][

1

zz

X(z)

nunx • If the ROC includes the unit circle, the signal is stable.


Stability and the ROC (Cont.)

• For a < 0: azaz

X(z)nuanx n

for1

1][][

1

• If the ROC is outside the unit circle, the signal is unstable.

1for1

1

][][

1

zz

X(z)

nunx • If the ROC includes the unit circle, the signal is stable.


More on ROC

• Example:

If:

The z-Transform is the same, but the region of convergence is different.

?1

)(11

]1[)(

signal)sided(left]1[][

1

1

11

za

za

zaznuazX

nuanx

n

n

n

nn

n

azza or,,11

az

zaz

za

za

zaza

za

zaX(z)

1

1

1

11

1

1

1

11

1

1

1

1

1

11


Stability and the ROC

• For: azaz

X(z)nuanx n

for1

1]1[][

1

• If the ROC includes the unit circle, the signal is stable.

1for1

1

][][

1

zz

X(z)

nunx • If the ROC includes the unit circle, the signal is unstable.


Properties of the ROC

• The ROC is an annular ring in the z-plane centeredabout the origin (which is equivalent to a vertical strip in the s-plane).

• The ROC does not contain any poles (similar tothe Laplace transform).

• If x[n] is of finite duration, then the ROC is the entirez-plane except possibly z = 0 and/or z = :

• If x[n] is a right-sided sequence, and if |z| = r0 is in the ROC, then all finite values of z for which |z| > r0 are also in the ROC.

• If x[n] is a left-sided sequence, and if |z| = r0 is in the ROC, then all finite values of z for which |z| < r0 are also in the ROC.

)Re()()()(][]1[][

)Re()()()(0][]1[][

1)()()(1][][][

][

1

sesXTttxzROCzzXnnx

sesXTttxzROCzzXnnx

sXttxzallROCzXnnx

znxX(z)

sT

sT

n

n

:CT:DT


Properties of the ROC (Cont.)

• If x[n] is a two-sided sequence, and if |z| = r0 is in the ROC, then the ROC consists of a ring in the z-plane including |z| = r0.

• Example:

right-sided left-sided two-sided

0][ bbnx n

bzbzbbz

zX

bz

zbnub

bzbz

nub

nubnubnx

n

n

nn

1

1

1

1

1)(

1

1

1]1[

1

1][

][][][

111

11

1


Properties of the Z-Transform

• Linearity:

Proof:

• Time-shift:

Proof:

What was the analog for CT signals and the Laplace transform?

• Multiplication by n:

Proof:

][][][][ 2121 zbXzaXnbxnax

][][][][])[][(][][ 21212121 zXzXznbxznaxznbxnaxnbxnaxn

nn

n

n

n

Z

][][ 00 zXznnx n

][][][

][][][

000

0 )(00

zXzzmxzzzmx

zmxznnxnnx

n

m

mn

m

nm

m

nm

n

n

Z

dz

zdXznnx

][][

][][)(

][)(

][

1 nnxznxndz

zdXzznxn

dz

zdX

znxX(z)

n

n

n

n

n

n

Z


Summary

• Definition of the z-Transform:

• Explanation of the Region of Convergence and its relationship to the existence of the DTFT and stability.

• Properties of the z-Transform:

Linearity:

Time-shift:

Multiplication by n:

• Basic transforms (see Table 7.1) in the textbook.

n

nznxX(z) ][

][][][][ 2121 zbXzaXnbxnax

][][ 00 zXznnx n

dz

zdXznnx

][][

lecture 28: the z-transform and its roc properties

Documents