# Lesson 54 Autocorrelation Function

## Theory

In this lesson, we introduce a summary of a random process that is closely related to the mean and autocovariance functions. This function plays a crucial role in signal processing.

Definition 54.1 (Autocorrelation Function) The autocorrelation function $$R_X(s, t)$$ of a random process $$\{ X(t) \}$$ is a function of two times $$s$$ and $$t$$. It specifies $\begin{equation} R_X(s, t) \overset{\text{def}}{=} E[X(s)X(t)]. \tag{54.1} \end{equation}$

By the shortcut formula for covariance (29.2), $$E[X(s) X(t)] = \text{Cov}[X(s), X(t)] + E[X(s)] E[X(t)]$$, so the autocorrelation function can be related to the mean function (50.1) and autocovariance function $R_X(s, t) = C_X(s, t) + \mu_X(s) \mu_X(t).$

For a stationary process (Definition 53), the autocorrelation function only depends on the difference between the times $$\tau = s - t$$: $R_X(\tau) = C_X(\tau) + \mu_X^2$

For a discrete-time process, we notate the autocorrelation function as $R_X[m, n] \overset{\text{def}}{=} C_X[m, n] + \mu_X[m] \mu_X[n].$

Let’s calculate the autocorrelation function of some random processes.

Example 54.1 (Random Amplitude Process) Consider the random amplitude process $\begin{equation} X(t) = A\cos(2\pi f t) \tag{50.2} \end{equation}$ introduced in Example 48.1.

Its autocorrelation function is $\begin{equation} R_X(s, t) = C_X(s, t) + \mu_X(s) \mu_X(t) = (1.25 + 2.5^2) \cos(2\pi f s) \cos(2\pi f t). \tag{54.2} \end{equation}$

Example 54.2 (Poisson Process) Consider the Poisson process $$\{ N(t); t \geq 0 \}$$ of rate $$\lambda$$, defined in Example 47.1.

Its autocorrelation function is $\begin{equation} R_X(s, t) = C_X(s, t) + \mu_X(s) \mu_X(t) = \lambda\min(s, t) + \lambda^2 s t. \tag{54.3} \end{equation}$

Example 54.3 (White Noise) Consider the white noise process $$\{ Z[n] \}$$ defined in Example 47.2, which consists of i.i.d. random variables with mean $$\mu = E[Z[n]]$$ and variance $$\sigma^2 \overset{\text{def}}{=} \text{Var}[Z[n]]$$.

We showed in Example 53.3 that this process is stationary, so its autocorrelation function depends only on $$k = m - n$$: $\begin{equation} R_Z[k] = C_Z[k] + \mu_Z^2 = \sigma^2\delta[k] + \mu^2. \tag{54.4} \end{equation}$

Example 54.4 (Random Walk) Consider the random walk process $$\{ X[n]; n\geq 0 \}$$ from Example 47.3.

Its autocorrelation function is: $\begin{equation} R_X[m, n] = C_X[m, n] + \underbrace{\mu_X[m]}_{0} \mu_Y[n] = \sigma^2 \min(m, n). \tag{54.5} \end{equation}$

Example 54.5 Consider the stationary process $$\{X(t)\}$$ from Example 53.5, whose mean and autocovariance functions are \begin{align*} \mu_X &= 2 & C_X(\tau) &= 5e^{-3\tau^2}. \end{align*}

Its autocorrelation function likewise depends on $$\tau = s - t$$ only: $\begin{equation} R_X(\tau) = C_X(\tau) + \mu_X^2 = 5 e^{-3\tau^2} + 4. \tag{54.6} \end{equation}$

## Essential Practice

For these questions, you may want to refer to the mean and autocovariance functions you calculated in Lessons 50 and 52.

1. Consider a grain of pollen suspended in water, whose horizontal position can be modeled by Brownian motion $$\{B(t); t \geq 0\}$$ with parameter $$\alpha=4 \text{mm}^2/\text{s}$$, as in Example 49.1. Calculate the autocorrelation function of $$\{ B(t); t \geq 0 \}$$.

2. Radioactive particles hit a Geiger counter according to a Poisson process at a rate of $$\lambda=0.8$$ particles per second. Let $$\{ N(t); t \geq 0 \}$$ represent this Poisson process.

Define the new process $$\{ D(t); t \geq 3 \}$$ by $D(t) = N(t) - N(t - 3).$ This process represents the number of particles that hit the Geiger counter in the last 3 seconds. Calculate the autocorrelation function of $$\{ D(t); t \geq 3 \}$$.

3. Consider the moving average process $$\{ X[n] \}$$ of Example 48.2, defined by $X[n] = 0.5 Z[n] + 0.5 Z[n-1],$ where $$\{ Z[n] \}$$ is a sequence of i.i.d. standard normal random variables. Calculate the autocorrelation function of $$\{ X[n] \}$$.

4. Let $$\Theta$$ be a $$\text{Uniform}(a=-\pi, b=\pi)$$ random variable, and let $$f$$ be a constant. Define the random phase process $$\{ X(t) \}$$ by $X(t) = \cos(2\pi f t + \Theta).$ Calculate the autocorrelation function of $$\{ X(t) \}$$.

5. Let $$\{ X(t) \}$$ be a continuous-time random process with mean function $$\mu_X(t) = -1$$ and autocovariance function $$C_X(s, t) = 2e^{-|s - t|/3}$$. Calculate the autocorrelation function of $$\{ X(t) \}$$.