Signals and Systems Spring 2013 Room 324, Geology Palace, 13756569051, zhukaiguang@jlu.edu.cn
Chapter 10 The Z-Transform 1) Z-Transform 2) Properties of the ROC of the z-transform 3) Inverse z-transform 4) Examples 5) Properties of the z-transform 6) System Functions of DT LTI Systems a) Causality b) Stability
The z-transform Motivation: Analogous to Laplace Transform in CT We now do not restrict ourselves just to z = e jω The (Bilateral) z-transform
The ROC and the Relation Between zt and DTFT, r = z depends only on r = z, just like the ROC in s-plane only depends on Re(s) Unit circle (r = 1) in the ROC DTFT X(e jω ) exists
Example #1 This form for PFE and inverse z- transform = 1 1 az 1 = z z a That is, ROC z > a, outside a circle This form to find pole and zero locations
Example #2: Same X(z) as in Ex #1, but different ROC.
Rational z-transforms x[n] = linear combination of exponentials for n > 0 and for n < 0 Polynomials in z characterized (except for a gain) by its poles and zeros
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(e jω ) exists Rational transforms correspond to signals that are linear combinations of DT exponentials
Some Intuition on the Relation between zt and LT The (Bilateral) z-transform Can think of z-transform as DT version of Laplace transform with
More intuition on zt-lt, s-plane - z-plane relationship LHP in s-plane, Re(s) < 0 z = e st < 1, inside the z = 1 circle. Special case, Re(s) = - z = 0. RHP in s-plane, Re(s) > 0 z = e st > 1, outside the z = 1 circle. Special case, Re(s) = + z =. A vertical line in s-plane, Re(s) = constant e st = constant, a circle in z-plane.
Properties of the ROCs of z-transforms (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) (2) The ROC does not contain any poles (same as in LT).
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
ROC Properties Continued (4) If x[n] is a right-sided sequence, and if z = r o is in the ROC, then all finite values of z for which z > r o are also in the ROC.
Side by Side (5) If x[n] is a left-sided sequence, and if z = r o is in the ROC, then all finite values of z for which 0 < z < r o are also in the ROC. (6) If x[n] is two-sided, and if z = r o is in the ROC, then the ROC consists of a ring in the z-plane including the circle z = r o. What types of signals do the following ROC correspond to? right-sided left-sided two-sided
Example #1
Example #1 continued Clearly, ROC does not exist if b > 1 No z-transform for b n.
for fixed r: Inverse z-transforms
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.
ROC III: ROC II: ROC I:
Inversion by Identifying Coefficients in the Power Series Example #3: 3-1 2 0 for all other n s A finite-duration DT sequence
Example #4: (a) (b)
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 n o > 0 ROC z 0 (maybe) n o < 0 ROC z (maybe) (2) z-domain Differentiation same ROC Derivation:
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
CAUSALITY (1) h[n] right-sided ROC is the exterior of a circle possibly including z = : A DT LTI system with system function H(z) is causal the ROC of H(z) is the exterior of a circle including z =
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
Stability LTI System Stable ROC of H(z) includes the unit circle z = 1 Frequency Response H(e jω ) (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
Chapter 10 The Z-Transform 1) Geometric Evaluation of z-transforms and DT Frequency Responses 2) First-and Second-Order Systems 3) System Function Algebra and Block Diagrams 4) Unilateral z-transforms
Geometric Evaluation of a Rational z-transform Example #1: Example #2: Example #3: All same as in s-plane
Geometric Evaluation of DT Frequency Responses First-Order System one real pole
Second-Order System Two poles that are a complex conjugate pair (z 1 = re jθ =z 2* ) Clearly, H peaks near ω = ±θ
Demo: DT pole-zero diagrams, frequency response, vector diagrams, and impulse- & step-responses
DT LTI Systems Described by LCCDEs Use the time-shift property Rational ROC: Depends on Boundary Conditions, left-, right-, or two-sided. For Causal Systems ROC is outside the outermost pole
Feedback System (causal systems) System Function Algebra and Block Diagrams negative feedback configuration Example #1: z -1 D Delay
Example #2: Cascade of two systems
Unilateral z-transform Note: (1) If x[n] = 0 for n < 0, then (2) UZT of x[n] = BZT of x[n]u[n] ROC always outside a circle and includes z = (3) For causal LTI systems,
Properties of Unilateral z-transform Many properties are analogous to properties of the BZT e.g. Convolution property (for x 1 [n<0] = x 2 [n<0] = 0) But there are important differences. For example, time-shift Derivation: Initial condition
Use of UZTs in Solving Difference Equations with Initial Conditions UZT of Difference Equation ZIR Output purely due to the initial conditions, ZSR Output purely due to the input.
Example (continued) β = 0 System is initially at rest: ZSR α = 0 Get response to initial conditions ZIR