Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform...
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Transcript of Signals and Systems Fall 2003 Lecture #22 2 December 2003 1.Properties of the ROC of the z-Transform...
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
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
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
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.
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)
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
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
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
Example #1
0 ,|| bbnx n
1 nubnubnx nn
b
zzb
nub
bzbz
nub
n
n
1 ,
1
11
,1
1
From:
11
1
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
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
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)(
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
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
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
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
)(
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
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
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
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