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For example, how can I determine if the convolution of $x(t)$ with $y(t)$ is equal to $0$?

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  • $\begingroup$ Circular or linear convolution? And if linear, do you care about the initial/ending transients being zero or not? $\endgroup$ – hotpaw2 Apr 27 '16 at 4:38
  • $\begingroup$ linear convolution. I just need an example to understand under what circumstances I can end with 0 after convolving to signals together. The output should be zero. thanks! $\endgroup$ – Carlos M. Navarro Apr 27 '16 at 4:41
  • $\begingroup$ Consider a rectangular pulse rect$\left(\frac tT\right)$ of duration $T$ and a DC-free periodic signal of period $T$. For any delay, the convolution integral evaluates to $T$ times the average value of the periodic signal, that is, the convolution integral has value $0$. $\endgroup$ – Dilip Sarwate Apr 30 '16 at 14:07
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Time-domain convolution is frequency-domain multiplication. If at all frequencies at least one of the signals is zero-valued in frequency domain, then the convolution of the two signals will be zero-valued at all frequencies, and at all times. Except for a zero signal, no finite-length signal has a continuous run of frequency domain zeros, so your choices are limited to infinite-length signals such as periodic signals with no coinciding non-zero harmonics and ideal lowpass–highpass filter impulse response pairs.

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    $\begingroup$ For periodic signals with no coinciding non-zero harmonics, e.g, $\cos t$ and $\cos \pi t$, there is also the question of whether the convolution integral converges. $\endgroup$ – Dilip Sarwate Apr 30 '16 at 14:11

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