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?
Best Answer
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\$.