Rational z-Transforms and Its Inverse Reference Rational z-Transforms and Its Inverse Reference: Professor Deepa Kundur Sections 3.3 and 3.4 of University of Toronto John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th edition, 2007. Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 1 / 26 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 2 / 26 Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Why Rational? I X (z) is a rational function iff it can be represented as the ratio of two polynomials in z −1 (or z): Rational z-Transforms −1 −2 −M b0 + b1z + b2z + ··· + bM z X (z) = −1 −2 −N a0 + a1z + a2z + ··· + aN z I For LTI systems that are represented by LCCDEs, the z-Transform of the unit sample response h(n), denoted H(z) = Zfh(n)g, is rational Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 3 / 26 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 4 / 26 Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Poles and Zeros Poles and Zeros of the Rational z-Transform Let a0; b0 6= 0: −1 −2 −M B(z) b0 + b1z + b2z + ··· + bM z X (z) = = −1 −2 −N A(z) a0 + a1z + a2z + ··· + aN z I zeros of X (z): values of z for which X (z) =0 −M M M−1 b0z z + (b1=b0)z + ··· + bM =b0 = −N N N−1 a0z z + (a1=a0)z + ··· + aN =a0 I poles of X (z): values of z for which X (z) = 1 b (z − z )(z − z ) ··· (z − z ) = 0 z −M+N 1 2 M a0 (z − p1)(z − p2) ··· (z − pN ) QM (z − z ) = Gz N−M k=1 k QN k=1(z − pk ) Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 5 / 26 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 6 / 26 Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Poles and Zeros of the Rational z-Transform Poles and Zeros of the Rational z-Transform QM (z − zk ) X (z) = Gz N−M k=1 where G ≡ b0 QN a0 Example: k=1(z − pk ) 2z 2 − 2z + 1 X (z) = z I X (z) has M finite zeros at z = z1; z2;:::; zM 16z 3 + 6z + 5 1 1 1 1 I X (z) has N finite poles at z = p1; p2;:::; pN (z − ( 2 + j 2 ))(z − ( 2 − j 2 )) = (z − 0) 1 3 1 3 1 (z − ( 4 + j 4 ))(z − ( 4 − j 4 ))(z − (− 2 )) I For N − M 6= 0 I if N − M > 0, there are jN − Mj zeros at origin, z = 0 1 3 1 I if N − M < 0, there are jN − Mj poles at origin, z = 0 poles: z = 4 ± j 4 ; − 2 zeros: z = 0; 1 ± j 1 Total number of zeros= Total number of poles 2 2 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 7 / 26 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 8 / 26 Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Pole-Zero Plot Pole-Zero Plot and Conjugate Pairs Example: poles: z = 1 ± j 3 ; − 1 , zeros: z = 0; 1 ± j 1 4 4 2 2 2 I For real time-domain signals, the coefficients of X (z) are necessarily real unit I complex poles and zeros must occur in conjugate pairs circle I note: real poles and zeros do not have to be paired up unit 0.5 circle 0.5 -0.5 0.5 2z2 − 2z +1 X (z) = z =) 16z3 +6 z +5 -0.5 0.5 -0.5 -0.5 POLE POLE ZERO ZERO Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 9 / 26 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 10 / 26 Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Pole-Zero Plot and the ROC Pole-Zero Plot and the ROC I Recall, for causal signals, the ROC will be the outer region of a I Therefore, for a causal signal the ROC is the smallest disk (origin-centered) circle encompassing all the poles. 0.5 I ROC cannot necessarily include poles ( X (pk ) = 1) * -0.5 0.5 -0.5 ROC Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 11 / 26 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 12 / 26 Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Causality and Stability Pole-Zero Plot, Causality and Stability I Recall, I For stable systems, the ROC will include the unit circle. LTI system is () P1 jh(n)j < 1 stable n=−∞ I Moreover, 1 1 X −n X −n jH(z)j = j h(n)z j ≤ jh(n)z j I For stability of a n=−∞ n=−∞ 1 causal system, the X = jh(n)j for jzj = 1 poles must lie inside n=−∞ the unit circle. I It can be shown: LTI system is () P1 jh(n)j < 1 () ROC of H(z) contains stable n=−∞ unit circle Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 13 / 26 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 14 / 26 Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms The System Function The System Function of LCCDEs N M Z X X h(n) ! H(z) y(n) = − ak y(n − k) + bk x(n − k) time-domain Z! z-domain k=1 k=0 N M impulse response Z! system function X X Zfy(n)g = Zf− ak y(n − k) + bk x(n − k)g k=1 0.5 k=0 Z N M y(n) = x(n) ∗ h(n) ! Y (z) = X (z) · H(z) X X Zfy(n)g = − ak Zfy(n − k)g + bk Zfx(n − k)g -0.5 0.5 k=1 k=0 N M Therefore, X −k -0.5 X −k Y (z) Y (z) = − ak z Y (z)+ bk z X (z) H(z) = X (z) ROCk=1 k=0 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 15 / 26 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 16 / 26 Rational z-Transforms and Its Inverse 3.3 Rational z-Transforms Rational z-Transforms and Its Inverse 3.4 Inversion of the z-Transform The System Function of LCCDEs N M X −k X −k Y (z) + ak z Y (z) = bk z X (z) k=1 k=0 " N # M Inversion of the z-Transform X −k X −k Y (z) 1 + ak z = X (z) bk z k=1 k=0 PM −k Y (z) k=0 bk z H(z) = = h i X (z) PN −k 1 + k=1 ak z LCCDE ! Rational System Function Many signals of practical interest have a rational z-Transform. Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 17 / 26 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 18 / 26 Rational z-Transforms and Its Inverse 3.4 Inversion of the z-Transform Rational z-Transforms and Its Inverse 3.4 Inversion of the z-Transform Inversion of the z-Transform Expansion into Power Series Three popular methods: Example: 1. Contour integration: I 1 n−1 x(n) = X (z)z dz −1 2πj C X (z) = log(1 + az ); jzj > jaj 1 1 X (−1)n+1anz −n X (−1)n+1an = = z −n −1 n n 2. Expansion into a power series in z or z : n=1 n=1 1 X −k X (z) = x(k)z By inspection: n+1 n k=−∞ (−1) a n ≥ 1 x(n) = n and obtaining x(k) for all k by inspection. 0 n ≤ 0 3. Partial-fraction expansion and table look-up. Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 19 / 26 Professor Deepa Kundur (University of Toronto) Rational z-Transforms and Its Inverse 20 / 26 Rational z-Transforms and Its Inverse 3.4 Inversion of the z-Transform Rational z-Transforms and Its Inverse 3.4 Inversion of the z-Transform Partial-Fraction Expansion Partial-Fraction Expansion 1. Find the distinct poles of X (z): p1; p2;:::; pK and their Example: Find x(n) given poles of X (z) at p1 = −2 and a double corresponding multiplicities m1; m2;:::; mK . pole at p2 = p3 = 1; specifically, 1 2. The partial-fraction expansion is of the form: X (z) = (1 + 2z −1)(1 − z −1)2 K A A A X (z) z 2 −R X 1k 2k mk = z X (z) = + 2 + ··· + m 2 z − pk (z − pk ) (z − pk ) k z (z + 2)(z − 1) k=1 z 2 A A A = 1 + 2 + 3 where pk is an mk th order pole (i.e., has multiplicity mk ) and R (z + 2)(z − 1)2 z + 2 z − 1 (z − 1)2 is selected to make z −R X (z) a strictly proper rational function.