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SIGNALS AND SYSTEMS
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Because we are assuming zero initial conditions, y(- 1) = 0, and because the input consists of a linear combination of a scaled unit sample and a scaled delayed step, the solution to the difference equation is simply
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where h(n) and s(n) are the unit sample and unit step response, respectively. To find the unit sample response, we write the difference equation in the form
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The characteristic equation for this difference equation is
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and the homogeneous solution is y(n)=Avn n>O Because the input x(n) = S(n) is equal to zero for n > 0, the particular solution is zero (all that the unit sample does is set the initial condition at n = 0). Evaluating the difference equation at n = 0, we have
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Therefore, it follows that A = 1 in the homogeneous solution, and that the unit sample response is
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The step response may now be found by convolving h(n) with u(n):
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Thus, the total solution is
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We now want to find the value of d so thal after 360 equal monthly payments the mortgage is paid off. In other words. we want to find d such that
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Solving,ford, we have
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With v = % and p = 100,000, we have d = 877.57 The total payment to the bank after 30 years is C = (877.57)(360) = 315,925.20
Every second, each a particle within a reactor splits into eight P particles and each , 3 particle splits into L Y an L particle and two P particles. Schematically,
a+8P P-a+2/3
SIGNALS AND SYSTEMS
[CHAP. 1
Given that there is a single a particle in the reactor at time n = 0, find an expression for the total number of particles within the reactor at time n. Let a ( n ) and B ( n ) be the number of a particles and B particles within the reactor at time n. The behavior within the reactor may be described by the following pair of coupled difference equations:
Before we can solve these difference equations, we must uncouple them. Therefore, let us derive a single difference equation for B(n). From the first equation we see that a ( n ) = # (n - I). Substituting this relation into the second difference equation, we have
B(n+ 1)=8B(n1)+2B(n)
or, equivalently,
B ( n ) = 2B(n - 0 The characteristic equation for this difference equation is
+ 8B(n
- 2)
which gives the following homogeneous solulion
Similarly, because a ( n ) = B(n - I), the solution for a ( n ) is
With the initial conditions a ( 0 ) = I and B(0) = 0, we may solve for A1 and A2 as follows:
and the solutions for a ( n ) and B ( n ) are
a ( n ) = i(4)" B ( n ) = :(4)"
+ !(-2)"
- (-2)" 3
n 20 n 20
Because we are interested in the total number of particles within the reactor at time n , with
Supplementary Problems
Discrete-Time Signals 1.41
Find the period N of the sequence
CHAP. 11
SIGNALS AND SYSTEMS
The input to a linear shift-invariant system is periodic with period N . (a) Show that the output of the system is also periodic with period N. (b) If the system is linear but shift-varying, is the output guaranteed to be periodic (c) If the system is nonlinear but shift-invariant, is the output guaranteed to be periodic
1.43 1.44
If x(n) = 0 for n < 0, and the odd part is x,,(n) = n(0.5)1n1, x(n) given that x(0) = 1. find Find the conjugate symmetric part of the sequence
1.45 1.46
If x(n) is odd, what is y(n) = x2(n)
If x(n) = 0 for n < 0, Pe is the power in the even part of x(n), and Po is the power in the odd part, which of the following statements are true (a) Pc Po (b) Po 2 Pe (c) Pe = Po
(d) None of the above are true.
Express the sequence x(n) = as a sum of scaled and shifted unit steps. (-1 (0
-2 5 11 5 2 else
Synthesize the triangular pulse
as a sum of scaled and shifted pulses,
Discrete-Time Systems
Listed below are several systems that relate the input x(n) to the output y(n). For each, determine whether the system is linear or nonlinear, shift-invariant or shift-varying, stable or unstable, causal or noncausal, and invertible or noninvertihle.
(e) y(n) = median(x(n - l), x(n), x(n
+ I))
SIGNALS AND SYSTEMS
[CHAP. 1
Given below are the unit sample responses of several linear shift-invariant systems. For each system, determine the conditions on the parameter a in order for the system to be stable. (a) h ( n ) = a n u ( - n )
(h) h ( n ) = a " [ u ( n )- u(n - 100))
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