# What is the covariance called when it is not divided by N?

I noticed that in signal processing they have this term called cross-covariance. The cross covariance function produces covariances of two functions with different lags. At the center of the vector there are no lags and thus it's just covariance. With one exception though, the cross covariance function is not divided by $N$. The naming is thus very misleading indeed as we are not talking about the same covariance anymore.

This kind of "covariance" rears it's head time and time again in different places, so I was just wondering what is it called?

Here are the exact formulas: \begin{align} {\rm Covariance}(X,Y) &= \frac{Σ(X_n - X_{\rm avg})(Y_n - Y_{\rm avg})}{N} \\ \text{"Cross covariance" covariance}(X,Y), \text{no lags} &= Σ(X_n - X_{\rm avg})(Y_n - Y_{\rm avg}) \end{align}

• Please deabbreviate DSP. May 11, 2015 at 21:01
• @amoeba Digital signal processing, but I use it as an abbreviation for signal processing in general.
– Dole
May 11, 2015 at 21:34

• On the Wikipedia page for cross-covariance, the formula I see is ${\rm Cov}(X_t, Y_s) = E[(X_t-\mu_t)(Y_s-\nu_s)]$. That notation is for the expected value, which does imply averaging / dividing by N. Outside of that, I don't know why people would use different terms. It would be incorrect, but in context they may want to stress the conceptual continuity. May 11, 2015 at 18:36