# Periodic signal filtering

If I convolve a periodic signal $x(t)$ with period = 1sec with an aperiodic signal $h(t)$ whose Fourier transform $H(f)$ is exactly equal to 1 at frequencies $f = 0,\pm 1,\pm 2, \ldots \textrm{Hz}$, but has some arbitrary and finite values at non-integers, will the resulting waveform $y(t) = x(t) \star h(t)$ be the same as $x(t)$?

I think it should, but I want to validate my reasoning.

My reasoning is that since $x(t)$ is periodic with period=1sec, its spectrum is discrete and is non-zero at integer values.

So if $H(f)$ equals 1 at integers, it should not distort $x(t)$ regardless of what its values are in between integers.

Note that I am referring to frequency in terms of Hertz and not $\omega$ (angular frequency).

Is this reasoning valid?

• Yes you are right. Eventhough it's important to remember the amount of idealization being assumed while arriving at that conclusion. Aug 2, 2016 at 17:20
• The formula for the output signal of an LTI system excited with a periodic input signal given in this answer should help you to answer your question. Aug 2, 2016 at 18:41