Find discrete convolution of $f$.

Question

I am trying to understand convolution and I am stuck in this particular problem in which I have to find the discrete convolution of $f[x] = [1,2,1,0,1,0-1,-2]$ with the signal $g[x] = [1, 1, -1,-1]$. I have no idea on how to use the following formula $$ f * g[m] = \sum_{m=-\infty}^{\infty} f[n]g[m-n] $$ in order to find the discrete convolution of $f$. Can anyone help me?

Answer 1

Considering the sequence $f(x)$ exists from $0$ to $7$, and $g(x)$ exists from $0$ to $3$,

$f*g(n) = \sum_{m=-\infty}^{\infty} f(m)g(n-m) = \sum_{m=-\infty}^{\infty} g(m)f(n-m)$.

$f*g(n)= \sum_{m=0}^{3} g(m)f(n-m)$.

$f*g(0)= \sum_{m=0}^{3} g(m)f(0-m) = g(0)f(0) = 1$

$f*g(1)= \sum_{m=0}^{3} g(m)f(1-m) = g(0)f(1)+g(1)f(0) = 3$, and so on.

This is if you go by formula. If the 2 sequences are less in length, then there is a matrix method to solve easily.