I have a continuous-time signal $x(t)$ that has the following properties:
- It is periodic, with a periodicity of 1 (sec).
- It has no discontinuities or infinite values at any time.
I want to come up with an LPF to pass $x(t)$ through, but with the following requirements:
- The filter is ideal (see point #3), continuous-time, with an impulse response $h(t)$ (and frequency response $H(f)$).
- The filter need not be a simple looking one like a brick-wall LPF. But $h(t)$ should be describable in closed form (piece-wise definition ok too). I am not looking for anything that involves stuff like dirac-delta function or infinite sums or something that can only be defined as an integral, etc.
- Any cutoff frequency is ok, but should be a finite number. Beyond that cutoff frequency, $H(f)$ should be 0 (not slowly reaching 0 asymptotically).
- The main requirement is that the output $y(t) = x(t)\star h(t)$, should have the same values as $x(t)$ at times $0, 1, 2, 3, \ldots$ i.e. $y(t) = x(t)$ for all integers. At other times, there is no restriction on how $y(t)$ should look like, except that it should be finite at all times.
- The filter need not be physically realizable. Just a mathematical definition is good enough for my purpose.
Of course, since I have given very few specifics, I am not expecting an exact $h(t)$ or $H(f)$ as an answer, but looking for what generic shape such a filter will have.