# Lesson 55 Power of a Stationary Process

## Motivation

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.

## Theory

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