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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 constantcoeffi 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
(P6.21)
(a) Assume that a solution to eq. (P6.21) 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. (P6.21). P6.3 Consider the LCCDE dy(t) + 1 y(t) = x(t), 2 dt x(t) = e t u(t) (P6.31)
(a) Determine the family of signals y(t) that satisfies the associated homogeneous equation. (b) Assume that for t > 0, one solution of eq. (P6.31), 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. (P6.31), show that y 1(t) = [2e t/2  2e']u(t)
is one solution for all t.
P6i
Signals and Systems
P62
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 timeinvariant.
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 P63
P6.6
Consider the following differential equation governing an LTI system. dx(t) dytt) dt + ay(t) = b di + cx(t) dt dt
(P6.61)
(a) Draw the direct form I realization of eq. (P6.61). (b) Draw the direct form II realization of eq. (P6.61).
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 P64
P6.9 Consider the following difference equation:
y[n]  ly[n

1] = x[n]
(P6.91) (P6.92)
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. (P6.91) 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...
P6.1 Suppose that y 1(t) and y 2(t) both satisfy the homogeneous linear constantcoeffi 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
(P6.21)
(a) Assume that a solution to eq. (P6.21) 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. (P6.21). P6.3 Consider the LCCDE dy(t) + 1 y(t) = x(t), 2 dt x(t) = e t u(t) (P6.31)
(a) Determine the family of signals y(t) that satisfies the associated homogeneous equation. (b) Assume that for t > 0, one solution of eq. (P6.31), 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. (P6.31), show that y 1(t) = [2e t/2  2e']u(t)
is one solution for all t.
P6i
Signals and Systems
P62
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 timeinvariant.
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 P63
P6.6
Consider the following differential equation governing an LTI system. dx(t) dytt) dt + ay(t) = b di + cx(t) dt dt
(P6.61)
(a) Draw the direct form I realization of eq. (P6.61). (b) Draw the direct form II realization of eq. (P6.61).
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 P64
P6.9 Consider the following difference equation:
y[n]  ly[n

1] = x[n]
(P6.91) (P6.92)
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. (P6.91) 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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