# Correlation of discrete time periodic sequences

So I was reading this textbook - Digital Signal Processing by Proakis and Manolakis. The authors define the cross-correlation operation for 2 energy signals $$x(n)$$ and $$y(n)$$ in section 2.6.1 as:

$$r_{xy}(l)=\sum_{n=-\infty}^{\infty}x(n)y(n-l)$$

In section 2.6.3, they define the cross-correlation operation of two power signals as:

$$r_{xy}(l)=\lim_{M\to\infty}\frac{1}{2M+1}\sum_{n=-M}^{M}x(n)y(n-l)$$

My question is, why do we need to have a separate definition for power signals? Are these two definitions the same? If they aren't same, what is the intuition behind defining them like that?

• Thank you @DanBoschen for answering. I understand everything that you have written, but I don't quite see how this leads to power signals having a separate definition. I mean, the equations don't have an energy term, it is just $x(n)\times y(n-l)$. Can you please elaborate? – Aditya DS May 6 '20 at 12:35
• Right, now I get it. So $x(n)y(n)\propto x^{2}(n)$ or $x(n)y(n)\propto y^{2}(n)$. And since we are summing it, it is similar to calculating the energy. And if that diverges (in case of power signal), we can't use the first equation. Did I understand properly? Thanks a lot!! – Aditya DS May 6 '20 at 13:04