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Consider a continuous-time system with input x(t) and output y(t) given by y(t) = x(t) cos(t)
This system is
This system is
- a)linear and time-invariant
- b)non-linear and time-invariant
- c)linear and time-varying
- d)non-linear and time-varying
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Consider a continuous-time system with input x(t) and output y(t) give...
given y(t) = x(t) cos(t)
It satisfies both additivity and Homogentity principles, so it is linear
If the input is delayed by t0
It satisfies both additivity and Homogentity principles, so it is linear
If the input is delayed by t0
Most Upvoted Answer
Consider a continuous-time system with input x(t) and output y(t) give...
Linear and Time-Varying System
To determine the linearity and time-invariance of the given system, let's analyze its properties one by one.
Linearity:
A system is considered linear if it satisfies the properties of superposition and homogeneity.
Superposition:
In a linear system, if x1(t) produces y1(t) and x2(t) produces y2(t), then any linear combination of these inputs, ax1(t) + bx2(t), should produce a corresponding linear combination of the outputs, ay1(t) + by2(t).
Let's test this property for the given system:
y1(t) = x1(t) * cos(t)
y2(t) = x2(t) * cos(t)
Now, let's consider the linear combination of the inputs:
y(t) = a * (x1(t) * cos(t)) + b * (x2(t) * cos(t))
Expanding the equation:
y(t) = a * x1(t) * cos(t) + b * x2(t) * cos(t)
Comparing this with the output produced by the linear combination of individual inputs:
ay1(t) + by2(t) = a * (x1(t) * cos(t)) + b * (x2(t) * cos(t))
Since both equations are the same, the system satisfies the superposition property and is linear in nature.
Time-Varying:
A system is considered time-invariant if its behavior remains the same over time. In other words, a time shift in the input results in the same time shift in the output.
To check the time-invariance property, let's consider a time-shifted input:
y(t - t0) = x(t - t0) * cos(t - t0)
Comparing this with the original output equation:
y(t) = x(t) * cos(t)
We can see that the time shift in the input also affects the output. Therefore, the system does not exhibit time-invariance.
Conclusion:
Based on the analysis, the given system is linear due to satisfying the superposition property. However, it is time-varying because a time shift in the input results in a corresponding time shift in the output. Therefore, the correct answer is option 'C' - linear and time-varying.
To determine the linearity and time-invariance of the given system, let's analyze its properties one by one.
Linearity:
A system is considered linear if it satisfies the properties of superposition and homogeneity.
Superposition:
In a linear system, if x1(t) produces y1(t) and x2(t) produces y2(t), then any linear combination of these inputs, ax1(t) + bx2(t), should produce a corresponding linear combination of the outputs, ay1(t) + by2(t).
Let's test this property for the given system:
y1(t) = x1(t) * cos(t)
y2(t) = x2(t) * cos(t)
Now, let's consider the linear combination of the inputs:
y(t) = a * (x1(t) * cos(t)) + b * (x2(t) * cos(t))
Expanding the equation:
y(t) = a * x1(t) * cos(t) + b * x2(t) * cos(t)
Comparing this with the output produced by the linear combination of individual inputs:
ay1(t) + by2(t) = a * (x1(t) * cos(t)) + b * (x2(t) * cos(t))
Since both equations are the same, the system satisfies the superposition property and is linear in nature.
Time-Varying:
A system is considered time-invariant if its behavior remains the same over time. In other words, a time shift in the input results in the same time shift in the output.
To check the time-invariance property, let's consider a time-shifted input:
y(t - t0) = x(t - t0) * cos(t - t0)
Comparing this with the original output equation:
y(t) = x(t) * cos(t)
We can see that the time shift in the input also affects the output. Therefore, the system does not exhibit time-invariance.
Conclusion:
Based on the analysis, the given system is linear due to satisfying the superposition property. However, it is time-varying because a time shift in the input results in a corresponding time shift in the output. Therefore, the correct answer is option 'C' - linear and time-varying.
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Consider a continuous-time system with input x(t) and output y(t) given by y(t) = x(t) cos(t)This system isa)linear and time-invariantb)non-linear and time-invariantc)linear and time-varyingd)non-linear and time-varyingCorrect answer is option 'C'. Can you explain this answer?
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