1.34. Consider the system shown in Fig. 1-35. Determine whether it is (a) memoryless, (b) causal, (c) linear, (d) time-invariant, or (e) stable. (a) From Fig. 1-35 we have y(t) = T{x(t)} = x(1) cos wet Since the value of the output y(t) depends on only the present values of the input x(t), the system is memoryless. (b) Since the output y(t) does not depend on the future values of the input x(t), the system is causal. (c) Let x(t)=a,x(t) + a2x(t). Then y(t) = T{x(t)} = [a₁x₁(1) + α₂x2(t)] cos wet =α₁₁(1) cos wc + a2x2(1) cos cl = a₁y(t)+ay2(1) Thus, the superposition property (1.68) is satisfied and the system is linear. Multiplier y(t) = x(t) cos w Χ Ncos w Fig. 1-35
1.34. Consider the system shown in Fig. 1-35. Determine whether it is (a) memoryless, (b) causal, (c) linear, (d) time-invariant, or (e) stable. (a) From Fig. 1-35 we have y(t) = T{x(t)} = x(1) cos wet Since the value of the output y(t) depends on only the present values of the input x(t), the system is memoryless. (b) Since the output y(t) does not depend on the future values of the input x(t), the system is causal. (c) Let x(t)=a,x(t) + a2x(t). Then y(t) = T{x(t)} = [a₁x₁(1) + α₂x2(t)] cos wet =α₁₁(1) cos wc + a2x2(1) cos cl = a₁y(t)+ay2(1) Thus, the superposition property (1.68) is satisfied and the system is linear. Multiplier y(t) = x(t) cos w Χ Ncos w Fig. 1-35
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![1.34. Consider the system shown in Fig. 1-35. Determine whether it is (a) memoryless, (b)
causal, (c) linear, (d) time-invariant, or (e) stable.
(a) From Fig. 1-35 we have
y(t) = T{x(t)} = x(1) cos wet
Since the value of the output y(t) depends on only the present values of the input x(t),
the system is memoryless.
(b) Since the output y(t) does not depend on the future values of the input x(t), the system
is causal.
(c) Let x(t)=a,x(t) + a2x(t). Then
y(t) = T{x(t)} = [a₁x₁(1) + α₂x2(t)] cos wet
=α₁₁(1) cos wc + a2x2(1) cos cl
=
a₁y(t)+ay2(1)
Thus, the superposition property (1.68) is satisfied and the system is linear.
Multiplier
y(t) = x(t) cos w
Χ
Ncos w
Fig. 1-35](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47baedac-bd97-4f55-8ddc-daad1377c846%2F9e3875c0-51c0-48c5-af70-f3b6d980f4b7%2Ft9xhwtr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.34. Consider the system shown in Fig. 1-35. Determine whether it is (a) memoryless, (b)
causal, (c) linear, (d) time-invariant, or (e) stable.
(a) From Fig. 1-35 we have
y(t) = T{x(t)} = x(1) cos wet
Since the value of the output y(t) depends on only the present values of the input x(t),
the system is memoryless.
(b) Since the output y(t) does not depend on the future values of the input x(t), the system
is causal.
(c) Let x(t)=a,x(t) + a2x(t). Then
y(t) = T{x(t)} = [a₁x₁(1) + α₂x2(t)] cos wet
=α₁₁(1) cos wc + a2x2(1) cos cl
=
a₁y(t)+ay2(1)
Thus, the superposition property (1.68) is satisfied and the system is linear.
Multiplier
y(t) = x(t) cos w
Χ
Ncos w
Fig. 1-35
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