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Let P be linearity, Q be time-invariance, R be the causality and S be the with memory. A continuous-time system has the input-output relationship: y(t) = t x(t) where y(t) is the output and x(t) is the input. The system has the properties.
- a)P, R, S but not Q
- b)P, Q, R but not S
- c)P, R but not Q, S
- d)R, S, Q but not P
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let P be linearity, Q be time-invariance, R be the causality and S be ...
Answer:
To determine the properties of the given continuous-time system, let's analyze each property individually:
P (Linearity):
A system is linear if it satisfies the properties of additivity and homogeneity.
Additivity: If the input to the system is a sum of two signals, the output should be the sum of the individual outputs for each signal. Let's verify this property for the given system.
Let x1(t) and x2(t) be two input signals, and y1(t) and y2(t) be the corresponding outputs. According to the given system, y1(t) = tx1(t) and y2(t) = tx2(t).
Now, let's consider the input x(t) = x1(t) + x2(t). The output y(t) for this input can be calculated as y(t) = tx(t) = t(x1(t) + x2(t)).
Expanding this, we get y(t) = tx1(t) + tx2(t), which is equal to y1(t) + y2(t). Hence, the system satisfies the additivity property and is linear.
R (Causality):
A system is causal if the output at any time depends only on the present and past values of the input.
In the given system, the output y(t) is directly proportional to the input x(t) multiplied by t. Since the output at any time t depends on the input x(t) at that particular time and all the previous values of x(t), the system is causal.
Q (Time-Invariance):
A system is time-invariant if a time shift in the input signal results in the same time shift in the output signal.
Let's consider a time shift of Δt in the input signal x(t), denoted as x(t - Δt). The corresponding output for this shifted input can be calculated as y(t - Δt) = t(x(t - Δt)).
However, if we substitute t = t - Δt in the original expression for y(t), we get y(t - Δt) = (t - Δt)x(t - Δt).
Comparing the two expressions for y(t - Δt), we can observe that they are not equal. Hence, the given system is not time-invariant.
S (With Memory):
A system is said to have memory if the output at any time depends on past and/or future values of the input.
In the given system, the output y(t) is directly proportional to the input x(t) multiplied by t. Since the output depends on the present value of t, the system does not have memory.
Based on the above analysis, the given continuous-time system has the properties P (Linearity), R (Causality), and S (Without Memory), but does not have the property Q (Time-Invariance). Hence, the correct answer is option 'c' - P, R but not Q, S.
To determine the properties of the given continuous-time system, let's analyze each property individually:
P (Linearity):
A system is linear if it satisfies the properties of additivity and homogeneity.
Additivity: If the input to the system is a sum of two signals, the output should be the sum of the individual outputs for each signal. Let's verify this property for the given system.
Let x1(t) and x2(t) be two input signals, and y1(t) and y2(t) be the corresponding outputs. According to the given system, y1(t) = tx1(t) and y2(t) = tx2(t).
Now, let's consider the input x(t) = x1(t) + x2(t). The output y(t) for this input can be calculated as y(t) = tx(t) = t(x1(t) + x2(t)).
Expanding this, we get y(t) = tx1(t) + tx2(t), which is equal to y1(t) + y2(t). Hence, the system satisfies the additivity property and is linear.
R (Causality):
A system is causal if the output at any time depends only on the present and past values of the input.
In the given system, the output y(t) is directly proportional to the input x(t) multiplied by t. Since the output at any time t depends on the input x(t) at that particular time and all the previous values of x(t), the system is causal.
Q (Time-Invariance):
A system is time-invariant if a time shift in the input signal results in the same time shift in the output signal.
Let's consider a time shift of Δt in the input signal x(t), denoted as x(t - Δt). The corresponding output for this shifted input can be calculated as y(t - Δt) = t(x(t - Δt)).
However, if we substitute t = t - Δt in the original expression for y(t), we get y(t - Δt) = (t - Δt)x(t - Δt).
Comparing the two expressions for y(t - Δt), we can observe that they are not equal. Hence, the given system is not time-invariant.
S (With Memory):
A system is said to have memory if the output at any time depends on past and/or future values of the input.
In the given system, the output y(t) is directly proportional to the input x(t) multiplied by t. Since the output depends on the present value of t, the system does not have memory.
Based on the above analysis, the given continuous-time system has the properties P (Linearity), R (Causality), and S (Without Memory), but does not have the property Q (Time-Invariance). Hence, the correct answer is option 'c' - P, R but not Q, S.
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Let P be linearity, Q be time-invariance, R be the causality and S be the with memory. A continuous-time system has the input-output relationship: y(t) = t x(t) where y(t) is the output and x(t) is the input. The system has the properties.a)P, R, S but not Qb)P, Q, R but not Sc)P, R but not Q, Sd)R, S, Q but not PCorrect answer is option 'C'. Can you explain this answer?
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Let P be linearity, Q be time-invariance, R be the causality and S be the with memory. A continuous-time system has the input-output relationship: y(t) = t x(t) where y(t) is the output and x(t) is the input. The system has the properties.a)P, R, S but not Qb)P, Q, R but not Sc)P, R but not Q, Sd)R, S, Q but not PCorrect answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Let P be linearity, Q be time-invariance, R be the causality and S be the with memory. A continuous-time system has the input-output relationship: y(t) = t x(t) where y(t) is the output and x(t) is the input. The system has the properties.a)P, R, S but not Qb)P, Q, R but not Sc)P, R but not Q, Sd)R, S, Q but not PCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P be linearity, Q be time-invariance, R be the causality and S be the with memory. A continuous-time system has the input-output relationship: y(t) = t x(t) where y(t) is the output and x(t) is the input. The system has the properties.a)P, R, S but not Qb)P, Q, R but not Sc)P, R but not Q, Sd)R, S, Q but not PCorrect answer is option 'C'. Can you explain this answer?.
Let P be linearity, Q be time-invariance, R be the causality and S be the with memory. A continuous-time system has the input-output relationship: y(t) = t x(t) where y(t) is the output and x(t) is the input. The system has the properties.a)P, R, S but not Qb)P, Q, R but not Sc)P, R but not Q, Sd)R, S, Q but not PCorrect answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Let P be linearity, Q be time-invariance, R be the causality and S be the with memory. A continuous-time system has the input-output relationship: y(t) = t x(t) where y(t) is the output and x(t) is the input. The system has the properties.a)P, R, S but not Qb)P, Q, R but not Sc)P, R but not Q, Sd)R, S, Q but not PCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P be linearity, Q be time-invariance, R be the causality and S be the with memory. A continuous-time system has the input-output relationship: y(t) = t x(t) where y(t) is the output and x(t) is the input. The system has the properties.a)P, R, S but not Qb)P, Q, R but not Sc)P, R but not Q, Sd)R, S, Q but not PCorrect answer is option 'C'. Can you explain this answer?.
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