Step Response from the Impulse Response

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The discussion revolves around deriving the step response from the impulse response using Laplace transforms. The main confusion arises regarding the term involving e^(at) in the book's answer, which is labeled as option A. Participants note that the time derivative of the step response corresponds to the impulse response, suggesting that integrating the impulse response should yield the step response. There is also a concern about the impulse response function h(t) = e^{-a|t|}, as it appears to violate causality by being non-zero for t < 0. The conversation highlights the importance of understanding causality in system responses.
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Homework Statement


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Homework Equations


Use Laplace then multiply in laplace domain then do laplace inverse

The Attempt at a Solution


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Book answer is A
I don't get the first term of eat in option A.[/B]
 

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The way I know how to do this is the time derivative of the step response is the impulse response, so that the integral of the impulse response is the step response. I am a little puzzled by the ## h(t)= e^{-a |t|} ## impulse response function. Normally, I think the impulse response should be zero for ## t<0 ## , or it violates causality.
 

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