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A system with an input x(t) and output y(t) is described by the relation: y(t) = tx(t). This system is
- a)Linear and time-invariant,
- b)Linear and time varying.
- c)None-iinear and time invariant.
- d)Non-linear and time varying.
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
A system with an input x(t) and output y(t) is described by the relati...
Explanation:
Linear systems satisfy the properties of superposition and homogeneity, whereas time-invariant systems have the same response for a given input, regardless of when the input is applied.
Let's analyze the given system:
y(t) = tx(t)
1. Superposition:
The system is not additive since it doesn't satisfy the superposition property. If we apply two inputs, say x1(t) and x2(t), then the output will be:
y1(t) = tx1(t) and y2(t) = tx2(t)
y1(t) + y2(t) = tx1(t) + tx2(t)
y1(t) + y2(t) ≠ t(x1(t) + x2(t))
Thus, the system is not linear.
2. Homogeneity:
The system satisfies the homogeneity property since applying a scalar value to the input results in a scaled output.
y(at) = atx(t) = a(tx(t))
Thus, the system is homogeneous.
3. Time-invariance:
The system is clearly time-varying because the output depends on the value of time t, which means the system's response changes with time.
Therefore, the system is linear and time-varying, which matches with option B.
Conclusion:
The system described by y(t) = tx(t) is linear and time-varying.
Linear systems satisfy the properties of superposition and homogeneity, whereas time-invariant systems have the same response for a given input, regardless of when the input is applied.
Let's analyze the given system:
y(t) = tx(t)
1. Superposition:
The system is not additive since it doesn't satisfy the superposition property. If we apply two inputs, say x1(t) and x2(t), then the output will be:
y1(t) = tx1(t) and y2(t) = tx2(t)
y1(t) + y2(t) = tx1(t) + tx2(t)
y1(t) + y2(t) ≠ t(x1(t) + x2(t))
Thus, the system is not linear.
2. Homogeneity:
The system satisfies the homogeneity property since applying a scalar value to the input results in a scaled output.
y(at) = atx(t) = a(tx(t))
Thus, the system is homogeneous.
3. Time-invariance:
The system is clearly time-varying because the output depends on the value of time t, which means the system's response changes with time.
Therefore, the system is linear and time-varying, which matches with option B.
Conclusion:
The system described by y(t) = tx(t) is linear and time-varying.
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Community Answer
A system with an input x(t) and output y(t) is described by the relati...
y(t) = tx(t)
y1(t) = t.x1(t) = r1(t)
y2(t) = tx2(t) = r2(t)
y1(t) + y2(t) = t(x1(t) + x2(t))
= r1(t) + r2(t) ∴ linear
y(t) = t.x(t)
y( t - to) = (t - to) x ( t - to)
and for delayed input signal,
y(t) = t x (t - to)
y(t) ≠ y( t-to)
∴ Time varying signal
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A system with an input x(t) and output y(t) is described by the relation: y(t) = tx(t). This system isa)Linear and time-invariant,b)Linear and time varying.c)None-iinear and time invariant.d)Non-linear and time varying.Correct answer is option 'B'. Can you explain this answer?
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