Difference Equations and Stability Linear Systems and Signals Lecture 10 Spring 2008

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 In general, from Lathi (3.49), 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] Lathi (3.49) is similar to Lathi (2.23): slide 4-8 Lathi (3.49) 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 Unique solution  C 1 = 1, C 2 = 4 h[n] = [(-0.2) n + 4 (0.8) n ] u[n]

Example: Solution Zero-state response solution (Lathi, Ex. 3.14) 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 (slide 9-9) Continuous Time Discrete TimeCase non-repeated roots repeated roots

Stability for an LTID System 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 (see Figs and 3.24 in Lathi) 1    Unstable Stable Marginally Stable Lathi, Fig Re  Im 

Stability in Both Domains Discrete-Time Systems Marginally stable: non-repeated characteristic roots on the unit circle (discrete-time systems) or imaginary axis (continuous- time systems) 1 Unstable Stable Marginally Stable Re  Im  UnstableStable Marginally Stable Re Im Continuous-Time Systems