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y =Hf y[k] + 2y[k - 1] = I [k - 1] 1/ z. 6 16 9 T ime-Domain A nalysis o f D iscrete-Time S ystems
with the input f (k] = e-ku[k] a nd the auxiliary condition y [-1] = o. Hint: You will
have to determine the auxiliary condition y[O] using the iterative method. 9 .5-a ( a) Using the classical method, solve
y[k + 2] + 3y[k + 1] + 2y[k] = f [k + 2] + 3 f[k + 1] + 3f(k] with the input f [k] = (3)k a nd the auxiliary conditions y[O] = 1, y = 3.
( b) Repeat ( a) if the auxiliary conditions are y [-1] = y [-2] = 1. Hint: Using t he
iterative method, determine y[O] and y.
9 .5-4 Using t he classical method, solve
y[k] + 2y[k - 1] + y[k - 2] = 2 f[k]- f[k - 1] with the input f [k] = 3 - ku[k] a nd the auxiliary conditions y[O]
9 .5-5 = 2 a nd y = -¥. Using t he classical method, solve
(E2 - E + 0.16)y[k] = E f[k] with the input f [k] = (0.2)ku[k] and the auxiliary conditions y[O]
The input is a n atural mode of the system.
9 .5-6 = 1, y = 2. + 0.16)y[k] = E f(k] with the input f (k] = c os(¥ + t )u[k] and the initial conditions y [-1]
Hint: Find y[O] and y iteratively.
9 .6-1 Hint: Using t he classical method, solve
(E2 - E Fourier Analysis o f
D iscrete-Time Signals = y [-2] = o. Each of the following equations specifies an LTID system. Determine whether these
systems are asymptotically stable, unstable, or marginally stable. + 2] + 0.6y[k + 1 ]- 0.16y[k] = f [k + 1 ]- 2f[k]
( b) (E2 + 1)(E2 + E + 1)y[k] = E f[k]
( c) (E - 1)2(E + 4)y[k] = (E + 2 )f[k]
( d) y[k] + 2y[k - 1] + 0.96y[k - 2] = 2 f[k - 1] + 3 f[k ( e) (E2 - 1)(E2 + 1)y[k] = f [k]
( a) y[k 3] 9 .6-2 In Sec. 9.6 we showed t hat for BIBO stability in an LTID system, it is sufficient for
its impulse response h[k] to satisfy Eq. (9.75). Show t hat this is also a necessary
condition for the system to be BIBO-stable. In other words, show t hat if Eq. (9.75)
is not satisfied, there exists a bounded input t hat produces unbounded output.
Hint: Assume t hat a system exists for which h[k] violates Eq. (9.75), yet its o utput
is bounded for every bounded input. Establish contradiction in this statement by
considering an input f [k] defined by f [k! - m] = 1 when h[m] > 0 and f [k! - m] =
- 1 when h[m] < 0, where k! is some fixed integer. 9 .6-3 Show t hat a marginally stable system is BIBO-unstable. Verify your result by considering a system with characteristic roots on t he unit circle a nd show t hat for t he input
of the form of the natural mode (which is bounded), t he response is unbounded. I n C hapters 3 , 4, a nd 6, we s tudied t he w ays o f r epresenting a c ontinuous-time
s ignal a s a s um o f s inusoids o r e xponentials. I n t his c hapter we s hall d iscuss s imilar
d evelopment for d iscrete-time s ignals. O ur a pproach is p arallel t o t hat u sed for
c ontinuous-time s ignals. We first r epresent a p eriodic f [k] a s a F ourier s eries formed
b y a d iscrete-time e xponential ( or sinusoid) a nd i ts h armonics. L ater we e xtend t his
r epresentation t o a n a peri...
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