Lesson 55 Power of a Stationary Process


This lesson is the first in a series of lessons about the processing of random signals. The two most common kinds of random signals that are studied in electrical engineering are voltage signals \(\{ V(t) \}\) and current signals \(I(t)\). These signals are often modulated while the other aspects of the circuit (e.g., the resistance) are held constant. The (instantaneous) power dissipated by the circuit is then \[ \text{Power}(t) = I(t)^2 R = \frac{V(t)^2}{R}. \] In other words, the power is proportional to the square of the signal.


For this reason, the (instantaneous) power of a general signal \(\{ X(t) \}\) is defined as \[ \text{Power}_X(t) = X(t)^2. \] When the signal is random, it makes sense to report its expected value, rather than its

Definition 55.1 (Expected Power) The expected power of a random process \(\{ X(t) \}\) is defined as \[ E[X(t)^2]. \]

Notice that the expected power is related to the autocorrelation function (54.1) by \[ E[X(t)^2] = R_X(t, t). \]

For a stationary process, the autocorrelation function only depends on the difference between the times, \(R_X(\tau)\), so the expected power of a stationary process is \[ E[X(t)^2] = R_X(0). \]

Since most noise signals are stationary, we will only calculate expected power for stationary signals.

Example 55.1 (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 \(\sigma^2 \overset{\text{def}}{=} \text{Var}[Z[n]]\).

This process is stationary, with autocorrelation function \[ R_Z[k] = \sigma^2\delta[k] + \mu^2, \] so its expected power is \[ R_Z[0] = \sigma^2\delta[0] + \mu^2 = \sigma^2 + \mu^2. \]
Example 55.2 Consider the process \(\{X(t)\}\) from Example 53.5. This process is stationary, with autocorrelation function \[ R_X(\tau) = 5 e^{-3\tau^2} + 4, \] so its expected power is \[ R_X(0) = 5 e^{-3 \cdot 0^2} + 4 = 9. \]

Essential Practice

For these questions, you may want to refer to the autocorrelation function you calculated in Lesson 54.

  1. 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 expected power in \(\{ D(t); t \geq 3 \}\).

  2. 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 expected power in \(\{ X[n] \}\).

  3. 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 expected power in \(\{ X(t) \}\).

  4. 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 expected power of \(\{ X(t) \}\).