# 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)&=\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}

• 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. – MBaz Jun 8 '15 at 13:45
• Thank you very much for your answers. I would try to show the complete problem in a second question. – AhmadR Jun 8 '15 at 13:50
• Are your $\tau_i$s integers or real-valued? – Oliver Jun 9 '15 at 6:37

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.

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.

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 $t$).