How to substitute variable in a summation with both limits as infinity?

Question

The book "Signals and Systems" says by substituting $r = n - k$, we obtain the equation. $$ \sum_{k=-\infty}^{\infty} x(k)h(n-k) = \sum_{r=-\infty}^{\infty} x(n-r)h(r) $$

Please explain how to obtain the equation in details.

Answer 1

• If $r=n-k$, then $k=n-r$, giving $x(k) \mapsto x(n-r)$.
• Similarly, since $k=n-r$, $h(n-k) \mapsto h(r)$.
• Of course, this summation re-indexing is linear, so each term maps to one term only in the new sum, keeping everything valid, and ensuring the limits remain the same.

Visually, you can think of this reversing the summation (thus the $-k$), and then shifting it by $n$ units.