# Linear convolution

We compute the linear convolution of two causal signals with finite support.

Convolution is defined by $(x \star y)[n] = \sum_{k\in\mathbf Z} x[k] y[n-k]$

By dragging the coefficients of $$x$$ and $$y$$, check that

• the response of a filter with impulse response $$y$$ to an impulse $$x = \delta$$ is $$z = y$$,
• the response of a filter with impulse response $$y$$ an impulse at index $$k$$ is the input $$x$$ shifted by $$k$$ samples,
• the convolution of two signals can be obtained by decomposing a signal as a sum of impulses, and adding the responses to these impulses.