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Question

# Consider a LTI system whose input is x(t) and output is y(t). If y(t) and x(t) are related to each other by relation y(t)=∫t−10−∞e−(t−5−τ)x(τ)d(τ) then impulse response of system will be

A
e(t5)u(t10)
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B
e(t5)u(t5)
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C
e(t5)u(t10)
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D
e(t5)u(t5)
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Solution

## The correct option is C e−(t−5)u(t−10) Since y(t)=x(t)∗h(t) y(t)=∫∞−∞h(t−τ)x(τ)d(τ) Given that y(t)=∫t−10−∞e−(t−5−τ)x(τ)dτ Comparing we get h(t−τ)=e−(t−5−τ) for τ<t−10 So h (−τ ) = e(5+τ) for (τ<−10) Thus h (τ ) = e(5−τ), for (τ>10) ⇒ h(t)=e−(t−5)u(t−10)

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