EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Difference Equations and Stability

Example: Second-Order Equation y[n+2] y[n+1] y[n] = 5 x[n+2] with y[-1] = 0 and y[-2] = 6.25 and x[n] = 4 -n u[n] Zero-input response Characteristic polynomial   = (  + 0.2) (  - 0.8) Characteristic equation (  + 0.2) (  - 0.8) = 0 Characteristic roots  1 = -0.2 and  2 = 0.8 Solution y 0 [n] = C 1 (-0.2) n + C 2 (0.8) n Zero-state response

Example: Impulse Response h[n+2] h[n+1] h[n] = 5  [n+2] with h[-1] = h[-2] = 0 because of causality General form of impulse response h[n] = (b N /a N )  [n] + y 0 [n] u[n] Since a N = and b N = 0, h[n] = y 0 [n] u[n] = [C 1 (-0.2) n + C 2 (0.8) n ] u[n] Discrete-time version of slides 5-9 and 5-10 balances impulsive events at origin

Example: Impulse Response Need two values of h[n] to solve for C 1 and C 2 h[0] h[-1] h[-2] = 5  [0]  h[0] = 5 h[1] h[0] h[-1] = 5  [1]  h[1] = 3 Solving for C 1 and C 2 h[0] = C 1 + C 2 = 5 h[1] = -0.2 C C 2 = 3  C 1 = 1, C 2 = 4 h[n] = [(-0.2) n + 4 (0.8) n ] u[n] Useful result:

Example: Solution Zero-state response solution y s [n] = h[n] * x[n] = {[(-0.2) n + 4(0.8) n ] u[n]} * (4 -n u[n]) y s [n] = [-1.26 (4) -n (-0.2) n (0.8) n ] u[n] Total response: y[n] = y 0 [n] + y s [n] y[n] = [C 1 (-0.2) n + C 2 (0.8) n ] + [-1.26 (4) -n (-0.2) n (0.8) n ] u[n] With y[-1] = 0 and y[-2] = 6.25 y[-1] = C 1 (-5) + C 2 (1.25) = 0 y[-2] = C 1 (25) + C 2 (25/16) = 6.25 Solution: C 1 = 0.2, C 2 = 0.8

Repeated Roots For r repeated roots of Q(  ) = 0 y 0 [n] = (C 1 + C 2 n + … + C r n r-1 )  n Similar to continuous-time case Continuous Time Discrete TimeCase non-repeated roots repeated roots

Stability of Zero-Input Response Asymptotically stable if and only if all characteristic roots are inside unit circle. Unstable if and only if one or both of these conditions exist: At least one root outside unit circle Repeated roots on unit circle Marginally stable if and only if no roots are outside unit circle and no repeated roots are on unit circle 1    Unstable Asymptotically Stable Marginally Stable Re  Im  Discrete-Time Systems

Stability of Zero-Input Response Discrete-Time Systems Marginally stable: non-repeated characteristic roots on the unit circle (discrete-time systems) or imaginary axis (continuous- time systems) 1 Unstable Asymptotically Stable Marginally Stable Re  Im  UnstableAsymptotically Stable Marginally Stable Re Im Continuous-Time Systems