How to interpret this formula from dsp? Can you describe me what this formula does and what result in the end? 
$$\delta(x)=\begin{cases}
0,  & \text{if $x \neq 0$} \\
1, & \text{if $x = 0$}  \\
\end{cases}$$
$$x(t)=\sum_{k=-\infty}^{\infty}x(t_k)\delta(t-t_k)$$
 A: $\delta(x)$ is also known as the Kronecker Delta Function. Looking at this page will help. I'm posting as an answer because it does help with achieving the desired sum. 
A: Just to keep things sorted out, let us denote the expression $x(t)$ occurring on the left by $\hat x(t)$.  Presumably $t$ is one of the $t_k$, say $t_l$; so then the formula reads
$\hat x(t_l) = \sum_{k = - \infty}^{k = \infty} x(t_k) \delta(t_l - t_k), \tag{1}$
and if the given definition of $\delta$ is applied in this equation, we see that $\delta(t_l - t_k) = 0$ unless $l = k$; then it is $1$.  So evidently 
$ \sum_{k = - \infty}^{k = \infty} x(t_k) \delta(t_l - t_k) = x(t_l), \tag{2}$
whence
$\hat x(t_l) = x(t_l); \tag{3}$
apparently the formula yields the identity map, so we may as well have written $x(t_l)$ on the left in any event; the notation $\hat x$ now looks somewhat superfluous, though I think it was a useful distinction while things were being sorted out.
Well, I hope this helps.  Cheers, 
and as always,
Fiat Lux!!!
