Differential Equation

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6 Systems Represented by Differential and Difference Equations Recommended Problems
P6.1 Suppose that y 1(t) and y 2(t) both satisfy the homogeneous linear constant-coeffi­ cient differential equation (LCCDE) dy(t) + ay(t) = 0

dt

Show that y 3 (t) = ayi(t) + 3y2 (t), where a and # are any two constants, is also a solution to the homogeneous LCCDE. P6.2 In this problem, we consider the homogeneous LCCDE d 2yt + 3 dy(t) + 2y(t) = 0 dt 2 dt

(P6.2-1)

(a) Assume that a solution to eq. (P6.2-1) is of the form y(t) = es'. Find the qua­ dratic equation that s must satisfy, and solve for the possible values of s. (b) Find an expression for the family of signals y(t) that will satisfy eq. (P6.2-1). P6.3 Consider the LCCDE dy(t) + 1 y(t) = x(t), 2 dt x(t) = e- t u(t) (P6.3-1)

(a) Determine the family of signals y(t) that satisfies the associated homogeneous equation. (b) Assume that for t > 0, one solution of eq. (P6.3-1), with x(t) as specified, is of the form y 1(t) = Ae-, t > 0

Determine the value of A. (c) By substituting into eq. (P6.3-1), show that y 1(t) = [2e -t/2 - 2e-']u(t)

is one solution for all t.

P6-i

Signals and Systems
P6-2

P6.4

Consider the block diagram relating the two signals x[n] and y[n] given in Figure P6.4.

x[n]

+

1 y[n]

1 2

Figure P6.4

Assume that the system described in Figure P6.4 is causal and is initially at rest. (a) Determine the difference equation relating y[n] and x[n]. (b) Without doing any calculations, determine the value of y[ -5] when x[n] = u[n]. (c) Assume that a solution to the difference equation in part (a) is given by y[n] = Kanu[n]

when x[n] = b[n]. Find the appropriate value of K and a, and verify that y[n] satisfies the difference equation. (d) Verify your answer to part (c) by directly calculating y[O], y[l], and y[2]. P6.5

Figure P6.5 presents the direct form II realization of a difference equation. Assume that the resulting system is linear and time-invariant.

x[n]

O +

r0n] D

y[n]

+1
3

-2
Figure P6.5

(a) Find the direct form I realization of the difference equation. (b) Find the difference equation described by the direct form I realization.

(c) Consider the intermediate signal r[n] in Figure P6.5. (i) Find the relation between r[n] and y[n]. (ii) Find the relation between r[n] and x[n]. (iii) Using your answers to parts (i) and (ii), verify that the relation between y[n] and x[n] in the direct form II realization is the same as your answer to part (b).

Systems Represented by Differential and Difference Equations / Problems P6-3

P6.6

Consider the following differential equation governing an LTI system. dx(t) dytt) dt + ay(t) = b di + cx(t) dt dt

(P6.6-1)

(a) Draw the direct form I realization of eq. (P6.6-1). (b) Draw the direct form II realization of eq. (P6.6-1).

Optional Problems
P6.7

Consider the block diagram in Figure P6.7. The system is causal and is initially at rest.

r [n]
x [n] + D y [n]

-4
Figure P6.7

(a) Find the difference equation relating x[n] and y[n].
(b) For x[n] = [n], find r[n] for all n. (c) Find the system impulse response.

P6.8

Consider the system shown in Figure P6.8. Find the differential equation relating x(t) and y(t).

x(t)

+

a

r(t)

+

y t

a
Figure P6.8

b

Signals and Systems P6-4

P6.9 Consider the following difference equation:
y[n] - ly[n
-

1] = x[n]

(P6.9-1) (P6.9-2)

with
x[n] = K(cos gon)u[n]

Assume that the solution y[n] consists of the sum of a particular solution y,[n] to eq. (P6.9-1) for n 0 and a homogeneous solution yjn] satisfying the equation Yh[flI
-

12Yhn -

1]

=0.

(a) If we assume that Yh[n] = Az", what value must be chosen for zo? (b) If we assume that for n 0, y,[n] = B cos(Qon + 0),

what are the values of B and 0? [Hint: It is convenient to view x[n] = Re{Kej"onu[n]} and y[n] = Re{Ye"onu[n]}, where Y is a complex number to be...
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