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I am trying to do convolution of a function \$ x(t)\$ with \$e^{-t}\delta(t)\$

Here are the steps I followed:

\$ x(t)e^{-t}\delta(t) = \int x(\tau)e^{t+\tau}\delta(t-\tau)d\tau =e^t\int x(\tau)e^{\tau}\delta(t-\tau)d\tau=e^t x(t)e(t)=e^{2t}x(t)\$

But if I put back \$x(t)=\delta(t)\$, we wont get impulse response.

Can some one tell me where I made a mistake in convolution?

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2 Answers 2

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There is a slight mistake: it's not \$e^{t+\tau}\$ in the convolution, but \$e^{-t+\tau}\$. You replace the original \$t\$ with \$t-\tau\$, and \$-(t-\tau)=-t+\tau\$.

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  • \$\begingroup\$ i have done it in that way first but i was getting \$ x(t) \$ answer \$\endgroup\$
    – Aditya
    Sep 5, 2014 at 17:23
  • \$\begingroup\$ \$x(t)*e^{-t}\delta(t)=e^{-t} \int e^{\tau} x(\tau) \delta(t-\tau) d\tau = e^{-t} e^{t} x(t)=x(t)\$ \$\endgroup\$
    – Aditya
    Sep 5, 2014 at 17:26
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\$ e^{-t}\delta \left ( t \right ) \$ is simply \$\delta \left ( t \right )\$. That should make it easier, no? FWIW, the solutions to many "problems" that courses present you with on convolution involve either simplifying the functions you'll be convolving by simple reduction (like this one), or simplifying by playing games with the limits of integration.

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