Convolution of $e^{-|t|}$ and a Band-pass

Question

Can someone explain how you would solve this convolution:

$$\sigma(t)=\text{ Heaviside}-\text{Function}$$ $$f(t)=e^{-|t|}$$ $$g(t)=\sigma(t+T)-\sigma(t-T)$$

$$(f*g)(t)=\int_{-\infty}^{\infty}e^{-|\tau|}(\sigma(t-\tau+T)-\sigma(t-\tau-T)) d\tau$$

Is this method correct:

1) defining exp-function Piece-wise:

$$ (f*g)(t)=\int_{-\infty}^{0}e^{\tau}(\sigma(t-\tau+T)-\sigma(t-\tau-T))d\tau+ \int_{0}^{\infty}e^{-\tau}(\sigma(t-\tau+T)-\sigma(t-\tau-T)) d\tau $$

what would be the next steps?

thanks

Answer 1

$$\begin{align}(f*g)(t)&=\int_{-\infty}^{\infty}e^{-|\tau|}(\sigma(t-\tau+T)-\sigma(t-\tau-T)) d\tau\\ &=\int_{t-\tau-T}^{t-\tau+T}e^{-|\tau|}d\tau \end{align}$$

Can you do the rest?