Question:
The output of a system \( y(t) \) is related to its input \( x(t) \) according to the relation \( y(t) = x(t) \sin(2\pi t) \). This system is _________
The output of a system \( y(t) \) is related to its input \( x(t) \) according to the relation \( y(t) = x(t) \sin(2\pi t) \). This system is _________
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A system is time-variant if the output changes when the input is shifted in time. The system described here has a time-varying coefficient \( \sin(2\pi t) \), making it time-variant.
Updated On: Nov 25, 2025
- Linear and time-variant
- Non-linear and time-invariant
- Linear and time-invariant
- Non-linear and time-variant
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The Correct Option is A
Solution and Explanation
Step 1: Analyzing the system.
For linearity, the system must satisfy the properties of additivity and homogeneity. Since the output is the product of the input \( x(t) \) and a function of time \( \sin(2\pi t) \), the system is linear. For time-invariance, the system must produce the same output if the input is shifted in time. In this case, a time shift in \( x(t) \) will result in a corresponding time shift in the output, meaning the system is time-variant because the multiplying factor \( \sin(2\pi t) \) depends explicitly on time. Step 2: Analyzing the options.
- (A) Correct, the system is linear because the output is a product of the input and a time-varying function, and it is time-variant because the system's behavior depends explicitly on time.
- (B) Incorrect, the system is linear, not non-linear.
- (C) Incorrect, the system is time-variant, not time-invariant.
- (D) Incorrect, the system is linear and time-variant, not non-linear. Step 3: Conclusion.
Thus, the correct answer is (A) Linear and time-variant.
For linearity, the system must satisfy the properties of additivity and homogeneity. Since the output is the product of the input \( x(t) \) and a function of time \( \sin(2\pi t) \), the system is linear. For time-invariance, the system must produce the same output if the input is shifted in time. In this case, a time shift in \( x(t) \) will result in a corresponding time shift in the output, meaning the system is time-variant because the multiplying factor \( \sin(2\pi t) \) depends explicitly on time. Step 2: Analyzing the options.
- (A) Correct, the system is linear because the output is a product of the input and a time-varying function, and it is time-variant because the system's behavior depends explicitly on time.
- (B) Incorrect, the system is linear, not non-linear.
- (C) Incorrect, the system is time-variant, not time-invariant.
- (D) Incorrect, the system is linear and time-variant, not non-linear. Step 3: Conclusion.
Thus, the correct answer is (A) Linear and time-variant.
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