1. Lecture 25 Outline: Z Transforms, Discrete-time Systems Analysis
Announcements:
Last HW will be posted Friday, due Thu June 7, no late HWs
Final exam announcements on next slide
Sided signals and their ROCs
Inverting Z-transforms
LTI Systems Analysis using Z-Tranforms
Feedback in LTI Systems
Example: 2nd order system with feedback

2. Final Exam Details
Time/Location: Monday June 11, 3:30-6:30 in this room. Food afterwards.
Open book and notes – you can bring any written material you wish to the exam. Calculators not allowed. PDF browsing devices allowed.
Covers all class material; emphasis on post-MT material (lectures 14 on)
See lecture ppt slides for material in the reader that you are responsible for
Practice final will be posted by Friday, worth 25 extra credit points for “taking” it (not graded).
Can be turned in any time up until exam, Solns given when you turn in your answers
My final review on W 6/6 in class. Georgia Th 6/7 4-6pm (review + OHs)
OHs next week and before final:
Me: 6/4, Mon, 3-4; 6/6, Wed, 12-1:30 (here); 6/8, Fri, by appt. (request by Thu 5pm)
Regular Tuesday section and TA OHs next week plus Th 6/7 ~5-6pm (Georgia) Sun, 6/10, 12-2pm (John), Mon 6/11, 12-2pm (Malavika)

3. Review of Last Lecture Bilateral Z Transform Relation with DTFT:
Region of Convergence (ROC):
All z s.t. X(z) exists, depends only on r, bounded by circles
Rational Z-Transforms defined by poles and zeroes
Plots poles and zeroes as  or o on z-plane
Properties and transform pairs of z-transforms
Right, left and two-sided exponential examples
True for some rR
Defn:

4. ROC for “Sided” Signals
Right-sided: x[n]=0 for n<a for some a
ROC is outside circle associated with largest pole
Example: RH exponential: x[n]=anu[n]
Left-sided: x[n]=0 for n>a for some a
ROC is inside circle associated with smallest pole
Example: LH exponential: x[n]=-anu[-n-1]
Two-sided: neither right or left sided
Can be written as sum of RH and LH sided signals
ROC is a circular strip between two poles
Example: 2-sided exponential: x[n]=b|n|, b<1

5. Inversion of Rational Laplace Transforms, … …
Extract the Strictly Proper Part of X(s)
If M<N, is strictly proper, proceed to next step
If MN, perform long division to get , where
Invert D(s) to get time signal ( ; )
Follows from z-transform table
The second term is strictly proper
Perform a partial fraction expansion:
Invert partial fraction expansion term-by-term (check ROC)
For right-sided signals: …
Obtain coefficients via residue method

6. LTI Systems Analysis using z-Transforms
LTI Analysis using convolution property
Equivalence of Systems same as for Laplace
G[z] & H[z] in Series: T[z]=G[z]H[z]; in Parallel: T[z]=G[z]+H[z]
Causality and Stability in LTI Systems
System is causal if h[n]=0, n<0, so if h[n] right-sided; has its ROC outside circle associated with largest pole.
Causal system (BIBO) stable if bounded inputs yield bounded outputs; iff h[n] absolutely summable  H(ejW) exists H(z) is stable  H(z) strictly proper and poles inside unit circle, i.e. r=1ROC (see Reader)
LTI Systems described by difference equations
Easily implemented with delay elements: d(n-N)z-N
H[z] called the transfer
function of the system
ROCROCxROCh

7. Feedback in LTI Systems+
Equivalent System
Motivation for Feedback
Can make an unstable system stable; make a system less sensitive to disturbances; make it closer to ideal
Can have negative effect: make a stable system unstable
Transfer Function T(z) of Feedback System:
Example: Population Growth: y[n]=2y[n-1]+x[n]-r[n]
Stable for b<.5
Y(z)=G(z)E(z); E(z)=X(z)±Y(z)H(z)

8. Example: Second Order System with Feedback (ppt only if no time in lecture, pp 43-46.5 in notes)
Z-Transform
DTFT:
Exists when r<1
Lowpass filter for q=0
Bandpass filter for q=p/2
Highpass filter for q=p
Impulse Response (LPF: q=0)
Poles at re±jq
r = 0.8, q = p/2
r = 0.8, q = 0
r = 0.8, q = p/2
r = 0.8, q = 0

9. Main Points
One-Sided signals have sided ROCs outside/inside a circle. Two sided signals have circular strips as ROCs
Convolution and other properties of z-transforms allow us to study input/output relationship of LTI systems
A causal system with H(z) rational is stable if & only if all poles of H(z) lie inside unit circle circle (all poles have |z|<1)
ROC defined implicitly for causal stable LTI systems
Systems of difference equations easily characterized using z-transforms
Feedback stabilizes unstable systems; obtains desired transfer function