32  Power Spectral Density

• 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 other aspects of the circuit (e.g., the resistance $R$) are held constant.
• The (instantaneous) power dissipated by the circuit is then $$\text{Power}(t)=I^2(t)R=\frac{V^2(t)}{R}$$ In other words, the power is proportional to the square of the signal.
• We’ll assume a nominal resistance of $R=1$ so we can consider voltage and current interchangeably.
• Throughout, let $X(t)$ denote a wide sense stationary (WSS) stochastic process with autcorrelation function $R_X(\tau )=E[X(t)X(t+\tau )]$.
  • $X(t)$ represents a random voltage or current signal
  • $X^2(t)$ represents instantaneous power of the signal at time $t$
• The expected power (a.k.a. (ensemble) average power) of a WSS stochastic process $X(t)$, denoted $P_X$, is $$P_X=\text{E}[X^2(t)]=R_X(0)$$ The expected power is constant in time because the process is WSS
• The power spectral density (psd) (a.k.a. power spectrum) of a WSS stochastic process $X(t)$, denoted $S_X(f)$, indicates how the power $P_X$ is distributed across the frequency domain.
  • $f$ denotes frequency, measured in Hertz (1/s)
• The autocorrelation function $R_X(\tau )$ is a function of time $\tau$, while the power spectral density $S_X(f)$ is a function of frequency $f$.
• Fourier transforms provide the bridge between the time and frequency domains. To compute the psd, we use
• Wiener-Khinchin Theorem. If $X(t)$ is a WSS stochastic process, then its power spectral density and its autocorrelation function are a Fourier transform pair $$\begin{array}{rl}{S}_X(f)& =\mathcal{F}\{R_X(\tau )\}\\ S_X(f)& \stackrel{\mathcal{F}}{⟷}{R}_X(\tau )\end{array}$$
• Recall: the Fourier transform of a function $g(t)$ of time results in a function $G(f)$ of frequency, given by $$G(f)=\mathcal{F}\{g(t)\}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(t)e^{-j2\pi ft}dt$$ where $j=\sqrt{-1}$

Example 32.1 Consider the random process $X(t)={10}^4\cos (2000\pi t+\mathrm{\Theta })$, where $\mathrm{\Theta }$ is a random variable with a Uniform distribution on the interval $(0,2\pi )$. This process is WSS with mean 0 and autocorrelation function $$R_X(\tau )=\frac{{10}^8}{2}\cos (2000\pi \tau )$$

1. Find the expected power in $X(t)$.
   
   
   
   
   
   

2. How is this power “distributed” in the frequency domain?
   
   
   
   
   

3. Find and graph the power spectral density of $X(t)$.
   
   
   
   
   

• Let $S_X(f)$ be the power spectral density of a WSS process $X(t)$. Then
• $S_X(f)\ge 0$ for all frequencies $f$
• $S_X(f)=S_X(-f)$; that is, $S_X(f)$ is an even function
• The (ensemble) average power in $X(t)$ is $$P_X={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{S}_X(f)df$$
• The (ensemble) average power of $X(t)$ in any frequency band $[f_1,f_2]$, for $0<f_1<f_2$, is given by $$P_X[f_1,f_2]=2{\int }_{f_1}^{f_2}{S}_X(f)df$$

Example 32.2 Suppose $X(t)$ is a WSS process with autocorrelation function $$R_X(\tau )=30+100\exp (-40,000{\tau }^2)$$

1. What does the autocorrelation function tell you about the behavior of the process $X$?
   
   
   
   
   

2. Determine the average power in $X(t)$.
   
   
   
   
   

3. Determine and graph the power spectral density of $X(t)$.
   
   
   
   
   

4. Determine the average power in $X(t)$ in the frequency band [5 Hz, 10 Hz].
   
   
   
   
   

5. Determine the average power in X(t) below 20 Hz.
   
   
   
   
   

Example 32.3 Suppose $X_1(t)$ and $X_2(t)$ are independent, zero-mean, WSS stochastic processes with respective autocorrelation functions $R_1(\tau )=20\text{tri}(\tau /5)$ and $R_2(\tau )=6\cos (70\pi \tau )$. Consider the process $X(t)=X_1(t)+X_2(t)+4$, which is WSS with autocorrelation function $$R_X(\tau )=20\text{tri}(\tau /5)+6\cos (70\pi \tau )+16$$

1. What do the autocorrelation functions $R_1$, $R_2$, and $R_X$ tell you about the processes $X_1$, $X_2$, and $X$?
   
   
   
   
   

2. Determine the expected power in $X_1$, $X_2$, and $X$.
   
   
   
   
   

3. Determine and graph the power spectral density of $X(t)$.
   
   
   
   
   
   

4. Determine the expected power in $X(t)$ below 25 Hz.
   
   
   
   
   

• $X(t)$ is a (pure) white noise process if its power spectral density is constant across all frequencies $$S_X(f)=c\text{ for all }f\in (-\mathrm{\infty },\mathrm{\infty })$$
• $X(t)$ is called a band-limited white noise process if its power spectral density is constant across a frequency band $(-B,B)$ $$S_X(f)=\{\begin{array}{ll}c,& -B<f<B\\ 0,& \text{otherwise}\end{array}$$

Example 32.4 Let $X(t)$ be a pure white noise process.

1. Find the autocorrelation function.
   
   
   
   
   

2. Can such a process physically exist? Why or why not?
   
   
   
   
   

Example 32.5 Let $X(t)$ be a band-limited white noise process.

1. Find the expected power in this process.
   
   
   
   
   

2. Find the autocorrelation function of this process.