Filtering of complex exponential functions In my problem the signals $x(t), h(t), w(t), y(t)$ and $y_w(t)$  are defined as follows: 

\begin{align}
x(t)&amp;=\sum_{i=1}^M a_i\delta(t-\tau_i) \\
h(t)&amp;=e^t\exp\left(-e^t\right) \\
w(t)&amp;=\begin{cases} 1, &amp; \text{when}\quad |t|\le\frac{\tau_0}{2}\\
0, \quad &amp;\text{elsewhere} \end{cases}\\
y(t)&amp;=x(t)*h(t) \\
y_w(t)&amp;=y(t)w(t)
\end{align}

Can any one recommend me a way how to get the $x(t)$ for only measuring $y_w(t)$ and knowing the $\tau_0$ by using the FFT.

Hints:
\begin{align}
X(\omega)&amp;=\sum_{i=1}^M a_ie^{j\tau_i\omega}\\
H(\omega)&amp;=\Gamma\left(1-j\omega\right)\\
W(\omega)&amp;=\tau_0\textrm{sinc}\left(\frac{\tau_0\omega}{2\pi}\right)
\end{align}
HINT:

If the given $X(\omega)$ is the Fourier transform of $x(t)$, what does $x(t)$ look like? It will turn out that removing exponential terms with certain values $\tau_i$ is equivalent to suppressing certain time intervals in the time domain.