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 \(\{ X(t) \}\).