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

  • $\begingroup$ 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

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.

  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.