• 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.