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.