1. Lecture 26 Outline: Z Transforms Announcements: Reading: “8: z-transforms” pp. 1-50 (no inverse or unilateral z transforms) HW 9 posted, due 6/3 midnight (no late HWs); section Mon 5/30 7-8pm Final exam announcements on next slide OceanOne Robot Tour will be after class today Bilateral Z Transform and its ROC Rational Z Transforms Z-Transform Properties and Pairs ROC for Sided Signals Inversion of Z-Tranforms Systems Analysis using Z-Tranforms

2. Final Exam Details Time/Location: Tuesday June 7, 8:30-11:30am in this room. Open book and notes – you can bring any written material you wish to the exam. Calculators and electronic devices not allowed. Covers all class material; emphasis on post-MT material (lectures 15 on) See lecture ppt slides for material in the reader that you are responsible for Practice final will be posted by next Thursday, worth 25 extra credit points for “taking” it (not graded). Can be turned in any time up until you take the exam Solutions given when you turn in your answers In addition to practice MT, we will also provide additional practice problems/solns Instead of final review, we will provide extra OHs for me and the TAs to go over course material, practice MT, and practice problems My extra OHs: Wed 6/1 2-3:30, Thurs 6/2: 1:30-3, Fri 6/3: 9:30-11, Mon 6/6: 11-12:30 Alon: Friday 6/3 5-6 PM, Monday 6/6 5-6 PM; Mainak: Thursday 6/2 3-4 PM, Monday 6/6 3-4 PM; Jeremy: Thursday 6/2 1-2 PM, Monday 6/6 7-8 PM

3. Review of Last Lecture Causality, Stability, & Feedback in LTI Systems Causality and Stability in LTI Systems System is causal if h(t)=0, t<0, so is h(t) right-sided; has its ROC to the right of the right-most pole. System is BIBO stable if bounded inputs yield bounded outputs; true iff h(t) absolutely integrable  H(j  ) exists LTI Systems described by differential equations Example: 2 nd order Lowpass Systems Feedback in LTI Systems   ±

4. Bilateral Z-Transform Discrete Time; Analysis very similar to Laplace Generalizes Discrete-Time Fourier Transform Always exists within a Region of Convergence Used to study systems/signals w/out DTFTs Relation with DTFT: Region of Convergence (ROC): All values of z such that X(z) exists Depends only on r (vs.  in Laplace) l Circles instead of planes Example: Defn: Smallest r : X(z) exists Circles in z-plane Plotted on z-plane Z True for some r  R

5. Rational Z Transforms Numerator and Denominator are polynomials Can factor as product of monomials  s are zeros (where X(z)=0),  s are poles (where X(z)=  ) ROC cannot include any poles If X(z) real, a’s and b’s are real  all zeros of X(z) are real or occur in complex-conjugate pairs. Same for the poles. b m /a n, zeros, poles, and ROC fully specify X(z) Example: 2-sided exponential

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

9. 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 For right-sided signals:, Obtain coefficients via residue method ppt only

10. LTI Systems Analysis using z-Transforms LTI Analysis using convolution property Equivalence of Systems same as for Laplace Systems G[z] and H[z] in Parallel: T[z]=G[z]H[z] Systems G[z] and H[z] in Series: T[z]=G[z]+H[z] Causality and Stability in LTI Systems System is causal if h[n]=0, n<0, so is h[n] right-sided; has its ROC outside circle associated with largest pole. System BIBO stable if bounded inputs yield bounded outputs; true iff h[n] absolutely summable  H(e j  ) exists  poles inside unit circle LTI Systems described by differential equations H[ z ] called the transfer function of the system ROC ROC  ROC x  ROC h

11. Feedback in LTI Systems Motivation for Feedback Can make an unstable system stable Can make transfer function closer to desired (ideal) one Can make system less sensitive to disturbances Can have negative effect: make a stable system unstable Transfer Function T(z) of Feedback System: Equivalent System Y(z)=G(z)E(z); E(z)=X(z)±Y(z)H(z)   ±

12. Main Points Z-transform generalizes DTFT, similar analysis as Laplace. Includes the ROC which is defined by circles in z-plane Expressing the z-transform in rational form allows inverse via partial fraction expantion Also allows z-transform characterization via pole-zero plot 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