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...

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...

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