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Spring 2008
Linear Systems and Signals
Lecture 10
Difference Equations and Stability
Example: Second-Order Equation
• y[n+2] - 0.6 y[n+1] - 0.16 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 g2 - 0.6 g - 0.16 = (g + 0.2) (g - 0.8)
Characteristic equation
(g + 0.2) (g - 0.8) = 0
Characteristic roots
g1 = -0.2 and g2 = 0.8
Solution
y0[n] = C1 (-0.2)n + C2 (0.8)n
• Zero-state response
n  
y s n  h
xn
Impulse
Response
Input
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Example: Impulse Response
• h[n+2] - 0.6 h[n+1] - 0.16 h[n] = 5 d[n+2]
with h[-1] = h[-2] = 0 because of causality
• In general, from Lathi (3.49),
h[n] = (bN/aN) d[n] + y0[n] u[n]
• Since aN = -0.16 and bN = 0,
h[n] = y0[n] u[n] = [C1 (-0.2)n + C2 (0.8)n] u[n]
• Lathi (3.49) is similar to Lathi (2.23): slide 4-8
y[n  1]  m[n  1] u[n  1]
 m[n  1] u[n]  d [n]
Lathi (3.49) balances
 m[n  1] u[n]  m[n  1] d [n]
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impulsive events at origin
y[n]  m[n] u[n]
Example: Impulse Response
• Need two values of h[n] to solve for C1 and C2
h[0] - 0.6 h[-1] - 0.16 h[-2] = 5 d[0]  h[0] = 5
h[1] - 0.6 h[0] - 0.16 h[-1] = 5 d[1]  h[1] = 3
• Solving for C1 and C2
h[0] = C1 + C2 = 5
h[1] = -0.2 C1 + 0.8 C2 = 3
Unique solution  C1 = 1, C2 = 4
• h[n] = [(-0.2)n + 4 (0.8)n] u[n]
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Example: Solution
• Zero-state response solution (Lathi, Ex. 3.14)
ys[n] = h[n] * x[n] = {[(-0.2)n + 4(0.8)n] u[n]} * (4-n u[n])
ys[n] = [-1.26 (4)-n + 0.444 (-0.2)n + 5.81 (0.8)n] u[n]
• Total response: y[n] = y0[n] + ys[n]
y[n] = [C1(-0.2)n + C2(0.8)n] +
[-1.26 (4)-n + 0.444 (-0.2)n + 5.81 (0.8)n] u[n]
• With y[-1] = 0 and y[-2] = 6.25
y[-1] = C1 (-5) + C2(1.25) = 0
y[-2] = C1(25) + C2(25/16) = 6.25
Solution: C1 = 0.2, C2 = 0.8
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Repeated Roots
• For r repeated roots of Q(g) = 0
y0[n] = (C1 + C2 n + … + Cr nr-1) gn
• Similar to continuous-time case (slide 9-9)
Continuous
Time
Discrete
Time
e t u (t )
g nu[n]
t m e t u (t )
n g u[n]
m
n
Case
non-repeated
roots
repeated
roots
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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
Im g
Marginally Stable
Unstable
g
|g|
b
Re g
-1
1
Stable
Lathi, Fig. 3.22
• Marginally stable if and only if no roots are
outside unit circle and no repeated roots are on
unit circle (see Figs. 3.22 and 3.24 in Lathi)
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Stability in Both Domains
Im 
Im g
Marginally
Stable
Marginally
Stable
Unstable
-1
1
Re 
Re g
Stable
Stable
Discrete-Time Systems
Unstable
Continuous-Time Systems
Marginally stable: non-repeated characteristic roots on the unit
circle (discrete-time systems) or imaginary axis (continuoustime systems)
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