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If h(n) is the impulse response of an LTI system T and h1(n) is the impulse response of the inverse system T-1, then which of the following is true?
- a)[h(n)*h1(n)].x(n)=x(n)
- b)[h(n).h1(n)].x(n)=x(n)
- c)[h(n)*h1(n)]*x(n)=x(n)
- d)[h(n).h1(n)]*x(n)=x(n)
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If h(n) is the impulse response of an LTI system T and h1(n) is the im...
Explanation: . If h(n) is the impulse response of an LTI system T and h1(n) is the impulse response of the inverse system T-1, then we know that h(n)*h1(n)=δ(n)=> [h(n)*h1(n)]*x(n)=x(n).
Most Upvoted Answer
If h(n) is the impulse response of an LTI system T and h1(n) is the im...
Explanation:
To understand why option C is the correct answer, let's break down the given information and the properties of LTI systems.
Impulse Response of an LTI System:
The impulse response of an LTI (Linear Time-Invariant) system, denoted as h(n), represents the output of the system when an impulse signal is applied as input. It characterizes the behavior of the system and contains all the information needed to describe the system's response to any input signal.
Inverse System:
The inverse system of an LTI system T is denoted as T-1. When the output of the inverse system T-1 is passed through the original system T, the result should be an impulse response. In other words, the impulse response of T convolved with the impulse response of T-1 should give an impulse response.
Convolution:
Given two sequences x(n) and y(n), their convolution is denoted by the symbol "*", and it is defined as the sum of the products of corresponding elements:
(x * y)(n) = Σ[x(k) * y(n-k)]
Explanation of the Correct Answer (Option C):
According to the given information, h(n) is the impulse response of the LTI system T, and h1(n) is the impulse response of the inverse system T-1.
The expression [h(n) * h1(n)] * x(n) represents the convolution of the sequences h(n) * h1(n) with the input sequence x(n).
Since the impulse response of the inverse system T-1 convolved with the impulse response of the original system T should result in an impulse response, the convolution [h(n) * h1(n)] represents the impulse response of the combined system T * T-1.
Now, if we convolve this impulse response with the input sequence x(n), we obtain the output sequence of the combined system T * T-1 when x(n) is applied as input.
If the output sequence of the combined system T * T-1 matches the input sequence x(n), it implies that the combined system is the identity system, which means it has no effect on the input signal.
Therefore, the correct answer is option C: [h(n) * h1(n)] * x(n) = x(n).
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If h(n) is the impulse response of an LTI system T and h1(n) is the impulse response of the inverse system T-1, then which of the following is true?a)[h(n)*h1(n)].x(n)=x(n)b)[h(n).h1(n)].x(n)=x(n)c)[h(n)*h1(n)]*x(n)=x(n)d)[h(n).h1(n)]*x(n)=x(n)Correct answer is option 'C'. Can you explain this answer?
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