Evaluating the convolution using the convolution integral I am having trouble evaluating the convolution of two signals using the convolution integral.I want to find the convolution of two signals x and h where,
$$
   x(t) = \begin{cases}
          e^{-at} & \text{$t > 0$} \\
          0 & \text{$t < 0$ } \\
         \end{cases}
$$
$$
   h(t) = \begin{cases}
          e^{-bt} & \text{$t > 0$} \\
          0 & \text{$t < 0$ } \\
         \end{cases}
$$
using the convolution integral
$$ 
    y(t) = x(t)*h(t) = \int_{-\infty}^{\infty} h(\tau)x(t - \tau) d\tau
$$
Which will mean that:
$$
     h(\tau) = \begin{cases}
          e^{-b\tau} & \text{$\tau > 0$} \\
          0 & \text{$\tau < 0$ } \\
         \end{cases}
$$
$$
   x(t - \tau) = \begin{cases}
                  e^{-at}e^{a \tau} & \text{$t > \tau$} \\
                  0 & \text{$t < \tau$ } \\
         \end{cases}
$$
But how do I proceed from here? I don't know how to handle the $x(t - \tau)$ function which is non-zero only when $t > \tau$.
 A: Write each of the signals as
$$e^{-k \tau} \theta(\tau)$$
where $k$ is either of $a$ or $b$, and $\theta(\tau)$ is the Heaviside step function, zero when $\tau < 0$ and $1$ when $\tau > 0$.  The convolution integral may then be written as
$$\int_{-\infty}^{\infty} d\tau \, e^{-a \tau} \theta(\tau) \, e^{-b (t-\tau)} \theta(t-\tau)$$
Now, the product of the two Heavisides in the integral is zero outside the interval $[0,t]$.  Therefore, we may write the convolution integral as
$$\int_0^t d\tau \, e^{-a \tau} \, e^{-b (t-\tau)} = e^{-b t} \int_0^t d\tau \, e^{-(a-b) \tau} $$
which is
$$\frac{1}{a-b} e^{-b t} \left (1-e^{-(a-b) t} \right ) = \frac{e^{-b t}-e^{-a t}}{a-b}$$
A: Assuming that $a,b>0$, then,
$$
\widehat f_a(\omega) = \int_0^{+\infty}e^{-(a+i\omega)t}dt=\frac{1}{a+i\omega}.
$$
So
\begin{align}
\widehat{f_a\ast f_b}=&\frac{1}{(a+i\omega)(b+i\omega)}\\
f_a\ast f_b =& \frac{1}{2\pi}\int_{\mathbb R}\frac{e^{i\omega t}}{(a+i\omega)(b+i\omega)}d\omega.
\end{align}
If $t>0$, we can compute this using a half-circular contour $C_+$ in the upper-half plane: such an integral is equal to $i2\pi$ times the sum of the residues at $a$ and $b$, that lie in the upper-half plane, where the integrand has simple ploes for $a\neq b$. This gives
\begin{align}
-\frac{1}{2\pi}\oint_{C_+}\frac{e^{i\zeta t}}{(\zeta-ia)(\zeta-ib)}d\zeta=
-\frac{i2\pi}{2\pi}\left( \frac{e^{-at}}{ia-ib}+\frac{e^{-bt}}{ib-ia}\right)=\frac{e^{-bt}-e^{-at}}{a-b};
\end{align}
as the radius $R$ of the half-circle tends to infinity only the real line integral contributes:
$$
(f_a\ast f_b)(t)=\frac{e^{-bt}-e^{-at}}{a-b}.
$$
In case $a=b$, the residue of the double pole yields precisely
$
(f_a\ast f_b)(t)=te^{-at},
$
which is just the limit of the above expression as $b\to a$.
If $t\le0$, we can use a similar contour but in the lower-half plane, getting zero since there are no singularities there.
Finally
$$
(f_a\ast f_b)(t) = \frac{e^{-bt}-e^{-at}}{a-b}\theta(t).
$$
A: Use LAPLACE Transform property
$$e^{-at} u(t) \rightleftharpoons \frac{1}{s+a}$$
$$e^{-bt} u(t) \rightleftharpoons \frac{1}{s+b}$$
$$X(s) = \frac{1}{(s+a)(s+b)}$$
Now partial fraction
$$X(s) = \frac{1}{(s+a)(s+b)} = \frac{A}{s+a} + \frac{B}{s+b}$$
$$A = \frac{1}{b-a}$$
$$B =\frac{1}{a-b}$$
$$x(t) = \frac{1}{b-a} [e^{-at} - e^{-bt}] u(t)$$
