Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform...

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Signals and Systems Fall 2003 Lecture #22 2 December 2003 operties of the ROC of the z-Transform verse z-Transform amples operties of the z-Transform stem Functions of DT LTI Systems Causality Stability

Transcript of Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform...

Page 1: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Signals and SystemsFall 2003

Lecture #222 December 2003

1. Properties of the ROC of the z-Transform2. Inverse z-Transform3. Examples4. Properties of the z-Transform5. System Functions of DT LTI Systems

a. Causalityb. Stability

Page 2: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

The z-Transform

-depends only on r = |z|, just like the ROC in s-plane only depends on Re(s)

• Last time:

•Unit circle (r = 1) in the ROC ⇒DTFT X(ejω) exists

•Rational transforms correspond to signals that are linear combinations of DT exponentials

nxZznxzXnxn

n

)(

n

nj rnxrez ||at which ROC

Page 3: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Some Intuition on the Relation between zT and LT

The (Bilateral) z-Transform

Can think of z-transform as DTversion of Laplace transform with

)}({)()()( txdtetxsXtx st L

TenTx nsT

n nxT

)()(lim

0

n

nsT

TenxT )(lim

0

}{)( nxzznxzXnxn

n

sTez

Let t=nT

Page 4: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

More intuition on zT-LT, s-plane - z-plane relationshipzesT

)( plane-sin axis jsj plan-zin circleunit a1 Tjez

• LHP in s-plane, Re(s) < 0 |⇒ z| = | esT| < 1, inside the |z| = 1 circle. Special case, Re(s) = -∞ ⇔|z| = 0.

• RHP in s-plane, Re(s) > 0 |⇒ z| = | esT| > 1, outside the |z| = 1 circle. Special case, Re(s) = +∞ ⇔|z| = ∞.•A vertical line in s-plane, Re(s) = constant⇔| esT| = constant, a

circle in z-plane.

Page 5: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Properties of the ROCs of z-Transforms

(2) The ROC does notcontain any poles (same as in LT).

(1) The ROC of X(z) consists of a ring in the z-plane centered about the origin (equivalent to a vertical strip in the s-plane)

Page 6: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

More ROC Properties

(3) If x[n] is of finite duration, then the ROC is the entire z-plane, except possibly at z = 0 and/or z = ∞.

Why?

Examples:

CT counterpart

2

1

)(N

Nn

nznxzX

stzn all ROC 1)( all ROC 1

seeTtzzn sT )( 0 ROC 1 1

seeTtzzn sT )( ROC 1

Page 7: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

ROC Properties Continued

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

1

1

0

1

nfaster tha converges

Nn

n

n

Nn

rnx

rnx

Page 8: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

What types of signals do the following ROC correspond to?

right-sided left-sided two-sided

(5) If x[n] is a left-sided sequence, and if |z| = ro is in the ROC, then all finite values of z for which 0 < |z| < ro are also in the ROC.

(6) If x[n] is two-sided, and if |z| = ro is in the ROC, then the ROC consists of a ring in the z-plane including the circle |z| = ro.

Side by Side

Page 9: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Example #1

0 ,|| bbnx n

1 nubnubnx nn

b

zzb

nub

bzbz

nub

n

n

1 ,

1

11

,1

1

From:

11

1

Page 10: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Example #1 continued

Clearly, ROC does not exist if b > 1 ⇒ No z-transform for b|n|.

bzb

zbbzzX

1 ,

1

1

1

1)(

111

Page 11: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Inverse z-Transforms

for fixed r:

ROC },][{)()( jnj rezrnxreXzX

dereXreXrnx njjjn

2

1 )(2

1)}({F

dnz

erreXnx njnj

2)(

2

1

dzzj

ddjredzrez jj 11

dzzzXj

nx n 1)(2

1

Page 12: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Example #2

Partial Fraction Expansion Algebra: A = 1, B = 2

Note, particular to z-transforms:

1) When finding poles and zeros, express X(z) as a function of z.

2) When doing inverse z-transform using PFE, express X(z) as a function of z-1.

1111

12

3

11

4

11

3

11

4

11

6

53

3

1

4

16

53

)(

z

B

z

A

zz

z

zz

zzzX

nxnxnx

zzzX

21

11

][

3

11

2

4

11

1)(

Page 13: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

ROC III:

ROC II:

ROC I:

nunx

nunx

z

n

n

3

12

4

1

singnal sided-right- 3

1

2

1

13

12

4

1

singnal sided-two- 3

1

4

1

2

1

nunx

nunx

z

n

n

13

12

4

1

singnal sided-left- 4

1

2

1

nunx

nunx

z

n

n

Page 14: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

4

1

3

23 )( 43

nx

x

x

x

z-zzzX

Inversion by Identifying Coefficients in the Power Series

Example #3:

nznx oft coefficien -

n

nznxzX )(

3

-1

2

0 for all other n’s— A finite-duration DT sequence

Page 15: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Example #4:

(a)

(b)

zzaz

azazaz

zX

,i.e.,1for convergent

)(11

1)(

1

2111

nuanx n

azeiza

zazaa

zazaza

zaza

azzX

.,.,1for convergent

))(1(

1

1

1

1 )(

1

33221

2111

11

1

1 nuanx n

Page 16: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Properties of z-Transforms

(1) Time Shifting

The rationality of X(z) unchanged, different from LT. ROC unchanged except for the possible addition or deletion of the origin or infinity

no> 0 ⇒ ROC z ≠ 0 (maybe)no< 0 ⇒ ROC z ≠ ∞ (maybe)

(2) z-Domain Differentiation same ROC

Derivation:

),(00 zXznnx n

dz

zdXznnx

)(

n

n

n

n

znnxdz

zdX

znxzX

1)(

)(

n

nznnxdx

zdXz

)(

Page 17: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Convolution Property and System Functions

Y(z) = H(z)X(z) , ROC at least the intersection of the ROCs of H(z) and X(z),

can be bigger if there is pole/zero cancellation. e.g.

H(z) + ROC tells us everything about system

nx nh nhnxny

zzY

zazzX

azaz

zH

all ROC 1)(

,)(

,1

)(

Function System The )(

n

nznhzH

Page 18: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

CAUSALITY

(1) h[n] right-sided⇒ROC is the exterior of a circle possiblyincluding z = ∞:

A DT LTI system with system function H(z) is causal ⇔ the ROC ofH(z) is the exterior of a circle including z = ∞

1

)(Nn

nznhzH

. include doesbut , circle a outside ROC

at ][ rerm then the,0 If 111

not

zzNhN N

0Causal 1 N

No zm terms with m>0 =>z=∞ ROC

Page 19: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Causality for Systems with Rational System Functions

A DT LTI system with rational system function H(z) is causal⇔ (a) the ROC is the exterior of a circle outside the outermost pole;

and (b) if we write H(z) as a ratio of polynomials

then

NM

azazaza

bzbzbzbzH

NN

NN

MM

MM

if ,at poles No

)(01

11

011

1

)(

)()(

zD

zNzH

)(degree)(degree zDzN

Page 20: Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.

Stability

• LTI System Stable⇔ ROC of H(z) includes

the unit circle |z| = 1⇒ Frequency Response H(ejω) (DTFT of h[n]) exists.

• A causal LTI system with rational system function is stable ⇔ all poles are inside the unit circle, i.e. have magnitudes < 1

nnh