Filter implementation and alternatives I'm learning of digital filters and found a question asking why a filter with a impulse response 

$$h[n]= \begin{cases} 
      \frac{1}{n^2+3n+2} &amp; n\geq 0 \\
     0 &amp; n&lt;0 \\
   \end{cases}
$$

cannot be implemented. My only guess is because of the infinite amount of computation required, but the problem in the book states it is not only beacuse of this. Also, which would be good alternative filters with a finite complexity such that thay would have a close frequency response to this filter?

Cheers
Only rational transfer functions can be implemented by a combination of adders, multipliers, and delay elements. The transfer function corresponding to this impulse response (i.e., its $\mathcal{Z}$-transform) is not rational (see
link).

Note that filters with infinitely long impulse responses can be implemented by recursion, so the infinite length by itself is no problem, as long as the transfer function is rational. A way to approximate this impulse response by a realizable filter is to truncate the impulse response by multiplying it with some (right-sided) window function, resulting in a finite impulse response (FIR) filter, which can always be implemented.