# Why is autocorrelation used without normalization in signal processing field?

According to the wikipedia(Link), autocorrelation has two definition.

In statistics, the definition of the autocorrelation between times $s$ and $t$ is like the following: $$\displaystyle R(s,t) = \frac{\mathbb{E}[(X_t-\mu_t)(X_s-\mu_s]}{\sigma_t\sigma_s}$$

However, in signal processing, the above definition is often used without the normalizaiton.

According to digital communication written by Bernard Sklar, $$R_x(\tau)=\int_{-\infty}^{\infty}x(t)x(t+\tau)du$$

$$R_X(\tau)=\mathbb{E}\{X(t)X(t+\tau)\}$$

Where \begin{array}{a} \tau & \mbox{ is the difference between } t \mbox{ and }s.\\ x(t) & \mbox{ is real-valued energy signal.}\\ X(t) & \mbox{ is a random process. }\end{array}

• sometimes all we care about are relative values of the autocorrelation. normalizing it simply means that it's relative to $R_x(0)$. – robert bristow-johnson Jan 22 '16 at 5:51

Autocorrelation (sometimes also termed aucovariance) is often used in the context of stationnary time signals. As the $\sigma_t$ and $\sigma_s$ are "equal". Since the autocorrelation is symmetric and peaks at $\tau=0$, $R_x(\tau)$ (as noted by @robert bristow-johnson) is often regarded as the natural reference. In a lot of applications, one mostly look at its scale-variant properties: as a measure on how signal chunks look similar, or modeling its decay, or by fitting a linear model through it.