Confusion on nested convolutions ($x_1[n]\ast x_2[n]\ast\cdots\ast x_N[n]$)

Question

I was wondering what would be the correct way to explicitly write nested convolutions from the definition of discrete convolution.

Consider a case for the convolution of 4 functions:

$$x_1[n]\ast x_2[n]\ast x_3[n]\ast x_4[n]$$

How would I write the summation form of this convolution?

I know that I should begin from:

$$x_1[n]\ast x_2[n]\ast x_3[n]\ast x_4[n]=\sum_{k=-\infty}^{\infty}x_1[k]\sum_{l=-\infty}^{\infty}x_2[l]\sum_{p=-\infty}^{\infty}x_3[p]x_4[n-p]$$

But I am not sure how I should proceed. I suspect that I should somehow "propagate" the summation variable from the outer to the inner convolutions...

So, what would be the correct and maybe most compact way to write this? I believe that an example for a 4-function convolution should be enough for me to draw a general procedure for $N$ discrete functions functions as well as for the continuous case.

Thank you!!

Answer 1

Note that by $y[n] = x[n]\star h[n]$ we mean $$y[n] = \sum_{m=-\infty}^{\infty}x[m]h[n-m] = $$ Maybe the better notation for avoiding confusion is $(x\star h)[n]$ which emphasize on computing the convolution then evaluating at $n$. See this for better explanation.

Assuming that convolution is associative(which is true under certain conditions) we have $$(x_1[n] \star x_2[n])\star x_3[n] = x_1[n] \star (x_2[n]\star x_3[n])$$ So we can compute $x_1[n] \star x_2[n]\star x_3[n] \star x_4[n]$ by different orderings. For example first $y_1[n] = x_1[n] \star x_2[n]$ then $y_2[n] = y_1[n] \star x_3[n]$ and finally $y_3[n] = y_2[n] \star x_4[n]$.