Department of Electrical Engineering University of Arkansas ELEG 5173L Digital Signal Processing Ch. 2 The Z-Transform Dr. Jingxian Wu [email protected] 2 OUTLINE • The Z-Transform • Properties • Inverse Z-Transform • Z-Transform of LTI system 3 Z-TRANSFORM • Bilateral Z-transform X (z) x(n)z n n • Unilateral Z-transform X (z) x(n)z n n0 • Z-transform: – Can simplify the analysis of discrete-time LTI systems – Analyze the system in z-domain instead of time domain – Doesn’t have any physical meaning (the frequency domain representation of discrete-time signal can be obtained through discrete-time Fourier transform) – Counterpart for continuous-time systems: Laplace transform. 4 Z-TRANSFORM • Example: find Z-transforms – 1. x(n) (n) n 1 – 2. x(n) u(n) 2 5 Z-TRANSFORM • Example n 1 – 3. x(n) u(n 1) 2 • Region of convergence (ROC) 6 Z-TRANSFORM: CONVERGENCE • ROC of causal signal x(n) nu(n) • ROC of anti-causal signal x(n) nu(n 1) 7 Z-TRANSFORM • Example: find the Z-transforms for the following signals n n 1 1 x(n) u(n) u(n) 2 3 8 Z-TRANSFORM • Example: find the Z-transforms for the following signals 3n , n 2 n x(n) 1 , n 2 3 9 Z-TRANSFORM: TRANSFORM TABLE 10 Z-TRANSFORM: TRANSFORM TABLE 11 OUTLINE • The Z-Transform • Properties • Inverse Z-Transform • Z-Transform of LTI system 12 PROPERTIES • Linearity – If Zx1(n) X1(z) Zx2 (n) X 2 (z) – Then Za1x1(n) a2x2(n) a1X1(z) a2 X2 (z) 13 PROPERTIES • Time Shifting – Let x ( n ) be a causal sequence with the Z-transform X (z) – Then n0 1 n0 n0 m Zx(n n0 ) z X (z) z x(m)z m0 1 n0 n0 m Zx(n n0 ) z X (z) z x(m)z mn0 14 PROPERTIES • Example – Solve the difference equation with initial condition y(1) 3 1 y(n) y(n 1) (n) 2 15 PROPERTIES • Example – Solve the difference equation with initial condition y(1) 1 y(0) 1 2 y(n 2) y(n 1) y(n) u(n) 9 16 PROPERTIES • Frequency scaling – If Zx(n) X (z) – Then Zan x(n) X (a1z) 17 PROPERTIES • Example – Find the Z-transform of x(n) an cosnu(n) 18 PROPERTIES • Differentiation with respect to z – If Zx(n) X (z) – Then d Znx(n) z X (z) dz 19 PROPERTIES • Example – Find the Z-transform of y(n) n(n 1)u(n) 20 PROPERTIES • Initial value lim X (z) x(0) z • Final value lim(1 z1)X (z) x() z1 21 PROPERTIES • Example – Find the initial value and final value of the following signal. z2 2z 3 X (z) (z 1)(z 0.5)(z2 z 1) 22 PROPERTIES • Convolution – If Zh(n) H(z) Zx(n) X (z) – Then Zx(n) h(n) X (z)H(z) • Example – Find the convolution of the following two sequences x(n) 1,2,0,1 y(n) 1,3,1 23 PROPERTIES 24 OUTLINE • The Z-Transform • Properties • Inverse Z-Transform • Z-Transform of LTI system 25 INVERSE Z-TRANSFORM • Review z az anu(n) nanu(n) z a (z a)2 • Inverse Z-Transform by partial fraction expansion – Expand X(z) in the form of z , az , etc. z a (z a)2 1 1 1 A A A X (z) X (z) 0 1 2 (z a )(z a ) 1 2 z z(z a1)(z a2 ) z z a1 z a2 X (z) X (z) X (z) A z A (z a ) A (z a ) 0 z 1 1 z 2 2 z z0 za1 za2 1 z z X (z) A0 A1 A2 z(z a1)(z a2 ) z a1 z a2 26 INVERSE Z-TRANSFORM • Example – Find the inverse Z-Transform of 1 X (z) 1 1 1 z z z 2 2 4 27 INVERSE Z-TRANSFORM • Solve the difference equation 3 1 y(n) y(n 1) y(n 2) (n) 4 8 28 INVERSE Z-TRANSFORM • Example – Find the convolution anu(n) bnu(n) 29 INVERSE Z-TRANSFORM • Example – Find the following convolutions anu(n) (n 1) anu(n) (n 1) 30 OUTLINE • The Z-Transform • Properties • Inverse Z-Transform • Z-Transform of LTI system 31 LTI SYSTEM • Transfer function of discrete-time LTI system – ad y(n) x(n) h(n) Y(z) X (z)H(z) Y (z) H(z) X (z) • Transfer function of discrete-time LTI system N M ak y(n k) bk x(n k) k0 k0 – Z-transform on both sides: N M k k ak z Y (z) bk z X (z) k0 k0 M k bk z k0 H (z) N k ak z k0 32 LTI SYSTEM • Example – Let the step response of a LTI system be as follows. Find the transfer function n n 6 1 1 2 1 y(n) u(n) u(n) u(n) 5 2 2 15 4 33 LTI SYSTEM • Zeros and poles (z z )(z z )(z z ) H(z) M M 1 1 (z pN )(z pN 1)(z p1) – Zeros: z1, z2 ,, zM – Poles: p1, p2 ,, pN • Stability – A discrete-time LTI system is stable if all the poles are inside the unit circle. – A discrete-time LTI system is unstable if at least one pole is on or outside the unit circle. – Review: a continuous-time LTI system is stable if all the poles are on the left half plane. 34 LTI SYSTEM • Example – Consider a LTI system described by the difference equation. Find the transfer function and the zeros and poles. Is the system stable? 1 y(n) 2y(n 1) 2y(n 2) x(n) x(n 1) 2 35 LTI SYSTEM • Example – Find the transfer function of the system shown in the following diagram. If k = 1, is the system stable? 0.8z H (z) (z 0.5)(z 0.8) 36 LTI SYSTEM • Matlab 2 z2 – Example H(z) 1 3z1 2z2 % numerator coefficients b = [2, 0, 1]; % denominator coefficients a = [1, 3, 2]; [r, p, k] = residuez(b, a) r = [4.5, -3], p =[-2, -1], k = 0.5 r1 r2 4.5 3 H(z) 1 1 k 1 1 0.5 1 p1z 1 p2z 1 (2)z 1 (1)z 37 LTI SYSTEM • Matlab 2 z2 – Example (Cont’d) H(z) 1 3z1 2z2 % numerator coeffcients b = [2, 0, 1]; % denominator coefficients a = [1, 3, 2]; % partial fraction expansion [r, p, k] = residuez(b, a) % find the zeros z = roots(b); % plot the poles and zeros zplane(b, a); % find the output to the system with an input x x = [2, 1, -2, 3]; y = filter(b, a, x); 38 LTI SYSTEM • Matlab 2 z2 – Example (multiple poles) H(z) 1 2 n 1 4z az 4z na u(n) 2 % numerator coeffcients (z a) b = [2, 0, 1]; % denominator coefficients a = [1, 4, 4]; % partial fraction expansion [r, p, k] = residuez(b, a) r = [-0.5, 2.25], p =[-2, -2], k = 0 r1 r2 0.5 2.25 H(z) 1 1 2 1 1 2 1 p1z (1 p2z ) 1 (2)z [1 (2)z ] h(n) 0.5(2)n u(n) 2.25(n 1)(2)n1u(n 1).