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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)&=\sum_{i=1}^M a_i\delta(t-\tau_i) \\ h(t)&=e^t\exp\left(-e^t\right) \\ w(t)&=\begin{cases} 1, & \text{when}\quad |t|\le\frac{\tau_0}{2}\\ 0, \quad &\text{elsewhere} \end{cases}\\ y(t)&=x(t)*h(t) \\ y_w(t)&=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)&=\sum_{i=1}^M a_ie^{j\tau_i\omega}\\ H(\omega)&=\Gamma\left(1-j\omega\right)\\ W(\omega)&=\tau_0\textrm{sinc}\left(\frac{\tau_0\omega}{2\pi}\right) \end{align}

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    $\begingroup$ This looks like a homework question. Can you describe what you have tried so far? Can you tell us what are you having trouble with, specifically? As a hint, what you're looking for is called a filter. $\endgroup$ – MBaz Jun 8 '15 at 13:45
  • $\begingroup$ Thank you very much for your answers. I would try to show the complete problem in a second question. $\endgroup$ – AhmadR Jun 8 '15 at 13:50
  • $\begingroup$ Are your $\tau_i$s integers or real-valued? $\endgroup$ – Oliver Jun 9 '15 at 6:37
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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.

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Your signal $x(t)$ is of the form $$x(t)=\sum_{i=1}^{M}a_{i}\delta(t+\tau_{i}).$$ Assuming $\tau_{i}>0$, then this is a sequence of Dirac impulses which are not zero only for some negative time indexes. From there, I am no expert, but I think you may be able to write the z-transfer function of the system and obtain the filter which will output your desired signal.

About negative time:

Inside a signal processing system, negative time just means "in advance with respect to a given time reference". That is, the system output depends on the current input and the future input. Of course, physically you cannot travel in the future to go get your future input, you actually need to wait for it to happen and become as well the current input. This waiting incurs a delay on the output of the system. So your system is not real-time.

Let us assume that $\tau_{i}<\tau_{j}$ for all $i<j$. If your function $x(t)$ was the impulse response of a system, then, to create the output at time $t$, you need to look at all your input from time $t$ to time $t+\tau_{M}$ then do the required processing to get your output (which actually will be produced at $t+\tau_{M}$, but corresponds to input time $t$).

Hope it helps.

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