Discrete-Time Signals and Systems The z-transform and Its Application Dr. Deepa Kundur University of Toronto Reference: Sections 3. - 3.4 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th edition, 2007. Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 2 / 36 The Direct z-transform Region of Convergence Direct z-transform: Notation: X (z) x(n)z n n X (z) {x(n)} the region of convergence (ROC) of X (z) is the set of all values of z for which X (z) attains a finite value The z-transform is, therefore, uniquely characterized by:. expression for X (z) 2. ROC of X (z) x(n) X (z) Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 3 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 4 / 36
Power Series Convergence Power Series Convergence For a power series, For a power series, f (z) a n (z c) n a 0 + a (z c) + a 2 (z c) 2 + n0 f (z) n0 a n (z c) n a 0 + a (z c) + a 2 (z c) 2 + there exists a number 0 r such that the series convergences for z c < r, and diverges for z c > r may or may not converge for values on z c r. there exists a number 0 r such that the series convergences for z c >r, and diverges for z c <r may or may not converge for values on z c r. Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 5 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 6 / 36 Region of Convergence Region of Convergence: r > r 2 Consider X (z) n n x(n)z n x(n)z n + x( n )z n + n 0 }{{} ROC: z < r x(n)z n n0 x(n)z n n0 }{{} ROC: z > r 2 x(0) }{{} ROC: all z Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 7 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 8 / 36
Region of Convergence: r < r 2 ROC Families: Finite Duration Signals Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 9 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 0 / 36 ROC Families: Infinite Duration Signals z-transform Properties Property Time Domain z-domain ROC Notation: x(n) X (z) ROC: r 2 < z < r x (n) X (z) ROC x 2 (n) X (z) ROC 2 Linearity: a x (n) + a 2 x 2 (n) a X (z) + a 2 X 2 (z) At least ROC ROC 2 Time shifting: x(n k) z k X (z) ROC, except z 0 (if k > 0) and z (if k < 0) z-scaling: a n x(n) X (a z) a r 2 < z < a r Time reversal x( n) X (z ) r < z < r 2 Conjugation: x (n) X (z ) ROC dx (z) z-differentiation: n x(n) z r d < z < r Convolution: x (n) x 2 (n) X (z)x 2 (z) At least ROC ROC 2 among others... Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 2 / 36
Convolution Property x(n) x (n) x 2 (n) X (z) X (z) X 2 (z) Convolution using the z-transform Basic Steps:. Compute z-transform of each of the signals to convolve (time domain z-domain): X (z) {x (n)} X 2 (z) {x 2 (n)} 2. Multiply the two z-transforms (in z-domain): X (z) X (z)x 2 (z) 3. Find the inverse z-transformof the product (z-domain time domain): x(n) {X (z)} Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 3 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 4 / 36 Common Transform Pairs Common Transform Pairs Signal, x(n) z-transform, X (z) ROC δ(n) All z 2 u(n) z z > 3 a n u(n) az 4 na n u(n) az ( az ) 2 5 a n u( n ) az 6 na n az u( n ) 7 cos(ω 0 n)u(n) 8 sin(ω 0 n)u(n) 9 a n cos(ω 0 n)u(n) 0 a n sin(ω 0 n)u(n) ( az ) 2 z cos ω 0 z < a z < a 2z cos ω 0 +z 2 z > z sin ω 0 2z cos ω 0 +z 2 z > az cos ω 0 2az cos ω 0 +a 2 z 2 az sin ω 0 2az cos ω 0 +a 2 z 2 Signal, x(n) z-transform, X (z) ROC δ(n) All z 2 u(n) z z > 3 a n u(n) az 4 na n u(n) az ( az ) 2 5 a n u( n ) az 6 na n az u( n ) 7 cos(ω 0 n)u(n) 8 sin(ω 0 n)u(n) 9 a n cos(ω 0 n)u(n) 0 a n sin(ω 0 n)u(n) ( az ) 2 z cos ω 0 z < a z < a 2z cos ω 0 +z 2 z > z sin ω 0 2z cos ω 0 +z 2 z > az cos ω 0 2az cos ω 0 +a 2 z 2 az sin ω 0 2az cos ω 0 +a 2 z 2 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 5 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 6 / 36
Why Rational? Poles and eros X (z) is a rational function iff it can be represented as the ratio of two polynomials in z (or z): X (z) b 0 + b z + b 2 z 2 + + b M z M a 0 + a z + a 2 z 2 + + a N z N zeros of X (z): values of z for which X (z) 0 poles of X (z): values of z for which X (z) For LTI systems that are represented by LCCDEs, the z-transform of the unit sample response h(n), denoted H(z) {h(n)}, is rational Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 7 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 8 / 36 Poles and eros of the Rational z-transform Poles and eros of the Rational z-transform Let a 0, b 0 0: X (z) B(z) A(z) b 0 + b z + b 2 z 2 + + b M z M a 0 + a z + a 2 z 2 + + a N z ( ) N b0 z M z M + (b /b 0 )z M + + b M /b 0 a 0 z N z N + (a /a 0 )z N + + a N /a 0 b 0 z M+N (z z )(z ) (z z M ) a 0 (z p )(z p 2 ) (z p N ) Gz N M M k (z z k) N k (z p k) X (z) Gz N M M k (z z k) N k (z p k) where G b 0 a 0 Note: finite does not include zero or. X (z) has M finite zeros at z z,,..., z M X (z) has N finite poles at z p, p 2,..., p N For N M 0 if N M > 0, there are N M zero at origin, z 0 if N M < 0, there are N M poles at origin, z 0 Total number of zeros Total number of poles Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 9 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 20 / 36
Poles and eros of the Rational z-transform Example: X (z) 2z + 6z 3 + 6z + 5 (z ( 2 (z 0) + j ))(z ( j )) 2 2 2 (z ( + j 3))(z ( j 3))(z ( )) 4 4 4 4 2 Pole-ero Plot Example: poles: z 4 ± j 3 4, 2, zeros: z 0, 2 ± j 2 unit circle poles: z 4 ± j 3 4, 2 zeros: z 0, 2 ± j 2 POLE ERO Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 2 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 22 / 36 Pole-ero Plot Graphical interpretation of characteristics of X (z) on the complex plane ROC cannot include poles; assuming causality... Pole-ero Plot For real time-domain signals, the coefficients of X (z) are necessarily real complex poles and zeros must occur in conjugate pairs note: real poles and zeros do not have to be paired up X (z) z2 2z + 6z 3 + 6z + 5 ROC ROC Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 23 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 24 / 36
Pole-ero Plot Insights For causal systems, the ROC will be the outer region of the smallest (origin-centered) circle encompassing all the poles. For stable systems, the ROC will include the unit circle. ROC Causal? Yes. Stable? Yes. For stability of a causal system, the poles will lie inside the unit circle. The System Function Therefore, h(n) time-domain impulse response y(n) x(n) h(n) H(z) H(z) Y (z) X (z) z-domain system function Y (z) X (z) H(z) Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 25 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 26 / 36 The System Function of LCCDEs The System Function of LCCDEs y(n) a k y(n k) + b k x(n k) k {y(n)} { {y(n)} Y (z) k0 a k y(n k) + k a k {y(n k)} + k a k z k Y (z) + k k0 b k x(n k)} k0 b k {x(n k)} k0 b k z k X (z) Y (z) + Y (z) [ a k z k Y (z) k + ] a k z k k H(z) Y (z) X (z) b k z k X (z) k0 X (z) b k z k k0 M [ k0 b kz k + ] N k a kz k LCCDE Rational System Function Many signals of practical interest have a rational z-transform. Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 27 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 28 / 36
Inversion of the z-transform Expansion into Power Series Three popular methods: Contour integration: x(n) X (z)z n dz 2πj C Expansion into a power series in z or z : X (z) x(k)z k k and obtaining x(k) for all k by inspection Partial-fraction expansion and table lookup Example: By inspection: X (z) log( + az ), ( ) n+ a n z n ( ) n+ a n n n n x(n) n { ( ) n+ a n n n 0 n 0 z n Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 29 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 30 / 36. Find the distinct poles of X (z): p, p 2,..., p K and their corresponding multiplicities m, m 2,..., m K. 2. The partial-fraction expansion is of the form: X (z) K k ( Ak + A ) 2k z p k (z p k ) + + A mk 2 (z p k ) m k where p k is an m k th order pole (i.e., has multiplicity m k ). 3. Use an appropriate approach to compute {A ik } Example: Find x(n) given poles of X (z) at p 2 and a double pole at p 2 p 3 ; specifically, X (z) X (z) z ( + 2z )( z ) 2 (z + 2)(z ) 2 (z + 2)(z ) 2 A z + 2 + A 2 z + A 3 (z ) 2 Note: we need a strictly proper rational function. DO NOT FORGET TO MULTIPLY BY z IN THE END. Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 3 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 32 / 36
(z + 2) A (z + 2) + A 2(z + 2) (z + 2)(z ) 2 z + 2 z A (z ) 2 + A 2(z + 2) z A 4 9 + A 3(z + 2) (z ) 2 + A 3(z + 2) (z ) 2 z 2 (z ) 2 A (z ) 2 (z + 2)(z ) 2 z + 2 (z + 2) A (z ) 2 z + 2 A 3 3 + A 2(z ) 2 z + A 3(z ) 2 (z ) 2 + A 2 (z ) + A 3 z Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 33 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 34 / 36 (z ) 2 A (z ) 2 (z + 2)(z ) 2 z + 2 d dz (z + 2) [ ] (z + 2) + A 2(z ) 2 z + A 3(z ) 2 (z ) 2 A (z ) 2 + A 2 (z ) + A 3 z + 2 d [ ] A (z ) 2 + A 2 (z ) + A 3 dz z + 2 z A 2 5 9 Therefore, assuming causality, and using the following pairs: X (z) 4 9 a n u(n) na n u(n) + 2z + 5 9 az az ( az ) 2 z + 3 x(n) 4 9 ( 2)n u(n) + 5 9 u(n) + 3 nu(n) [ ( 2) n+2 + 5 9 9 + n ] u(n) 3 z ( z ) 2 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 35 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 36 / 36