How to find the inverse system of a given one

Question

what is the inverse formula of y[n]=x[n]*x[n+1] ?

And how can I find the inverse formula/system of a given one in general? I'm having some troubles with this when it comes to some formulas.

Answer 1

If you know $y[n]$, in order to find $x$ you also need a value, e.g. $x[0]$. Then apply $$x[m]=\frac {y[m-1]}{x[m-1]}$$ for $m=1,2, \dots$ to get $x[1], x[2], x[3], \dots$

Answer 2

What does it mean to find an inverse? Here we have a function, $y$, which maps from $n \rightarrow x[n]x[n+1]$. We would like to find $y^{-1}$, such that $y^{-1}[y[n]] = n$.

However, without knowing the sequence of $x$ values, there's no telling that this inverse exists. Suppose $x[i] = 0$ for all $i$. then $y[i] = 0$ for all $i$.

We would say that $y$ is not one-to-one. Meaning multiple inputs go to the same output. This means $y$ is not invertible. If we found an inverse, there would be no way to tell whether $0 = y[1]$ or $0 = y[2]$ or $0 = y[14]$, and so on. This is a more general form of the vertical line test from high-school.

In general, you are looking for a function which maps back from the outputs of your function to the inputs. Here, this is not possible for all sequences $x[1], x[2], ...$.

Hope that helps!