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 \[\begin{align*} S_{X}(f) & \quad = \,\, \mathcal{F}\{R_{X}(\tau)\}\\ S_X(f) & \stackrel{\mathcal{F}}{\longleftrightarrow} R_X(\tau) \end{align*}\]
• Recall: the Fourier transform of a function \(g(t)\) of time results in a function \(G(f)\) of frequency, given by \[ G(f)= \mathcal{F}\{g(t)\} = \int_{-\infty}^\infty g(t) e^{-j2\pi f t}dt \] where \(j=\sqrt{-1}\)

Example 32.1 Consider the random process \(X(t) = 10^4\cos(2000\pi t + \Theta)\), where \(\Theta\) 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 \[ P_X = \int_{-\infty}^\infty S_{X}(f)\, df \]
• 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 \[ S_{X}(f) = c \text{ for all } f \in(-\infty,\infty) \]
• \(X(t)\) is called a band-limited white noise process if its power spectral density is constant across a frequency band \((-B, B)\) \[ S_{X}(f) = \begin{cases} c, & -B < f< B\\ 0, & \text{otherwise} \end{cases} \]

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.