Math Problems Vault

Problem 4:1

Suppose that the stationary process $\left\{ X(t), t \in \mathbb {R}\right\}$ has the covariance function $r_X(\tau ) = e^{-\alpha \left|\tau \right|} \cos 2\pi f_0 \tau$ , $\tau \in \mathbb {R}$ . Find its spectral density.

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Problem 4:2

(a)(b)

Determine the corresponding spectral densities to the following covariance functions of stationary processes in continuous time:

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(a)

$r_X(\tau ) = e^{-\alpha \tau^2}$ ;

(b)

$r_X(\tau ) = \dfrac {1}{1+\tau^2}$ .

Problem 4:3

A stationary process has the covariance function $r(\tau ) = \alpha_0^2 + \dfrac {\alpha_1^2}{2} \cos 2\pi f_0 \tau$ in continuous time. Find the spectral density.

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Problem 4:4

Find the covariance functions of processes with spectral densities

R_1(f) = \frac{A}{1+(2\pi f)^2}, \quad R_2(f) = \frac{A}{4+(2\pi f)^2}, \quad R_3(f) = \frac{A}{\left(1+(2\pi f)^2\right)\left(4+(2\pi f)^2\right)}.

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Problem 4:5

Figure 4.9 shows four spectral densities and four covariance functions. Match up the corresponding spectral densities and covariance functions.

#| echo: false
#| fig-cap: "Figure 4.9 Spectral densities and covariance functions to Exercise 4:5."
exec(open("figs/figs_recreated/4/001.tex.py").read())

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Problem 4:6

(a)(b)

Figure 4.10 shows realizations of two stationary processes, $U(t)$ and $V(t)$ , with the following spectral densities:

R_U(f) = \frac{200}{100^2+(2\pi f)^2}, \quad f \in \mathbb {R},

R_V(f) = \begin{cases} 1, & 2 < \left|f\right| < 2.5, \\ 0, & \text{otherwise}. \end{cases}

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(a)

Find $\operatorname {Cov}\left[ U(t), U(t+\tau ) \right]$ and $\operatorname {Cov}\left[ V(t), V(t+\tau ) \right]$ .

(b)

Which realization belongs to which process?

#| echo: false
#| fig-cap: "Figure 4.10 Realizations to Exercise 4:6."
exec(open("figs/figs_recreated/4/002.tex.py").read())

Problem 4:7

Figure 4.11 shows three realizations, three covariance functions, and three spectral densities of three different stationary processes. Determine which figures that are realizations, covariance functions, and spectral densities. Also state how they are connected.

#| echo: false
#| fig-cap: "Figure 4.11 Realizations, covariances, and spectra to Exercise 4:7."
exec(open("figs/figs_recreated/4/003.tex.py").read())

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Problem 4:8

Find the values for the constant $B$ such that

R(f) = \frac{1}{1+f^2} + \frac{B}{4+f^2}

is a spectral density in continuous time.

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Problem 4:9

(a)(b)(c)

Figure 4.12 shows realizations, covariance functions, and spectral densities of two processes. One of the processes has continuous time and one has discrete time; both realizations have been smoothed to a continuous curve.

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(a)

For each plot, A--F, determine whether it depicts a realization, a covariance function, or a spectral density. Justify your answer!

(b)

Which realization belongs to which covariance function and which spectral density?

(c)

Suppose now that we would like to estimate the expected mean value of the processes with the average of only two samples at a time distance $h$ . That is, we estimate $\mathbb {E}\left[X(t)\right] = m_X$ with $\widehat{m}_X = \frac{1}{2}\left(X(t)+X(t+h)\right)$ . Determine, for each process, an optimal value of $h$ , i.e., the $h$ giving the minimum variance of the estimator.

#| echo: false
#| fig-cap: "Figure 4.12 Realizations, covariances, and spectra of processes in Exercise 4:9."
exec(open("figs/figs_recreated/4/004.tex.py").read())

Problem 4:10

(a)(b)(c)

Suppose that $\left\{ X(t), t \in \mathbb {R}\right\}$ , $t$ in seconds, is a process of interest with the following spectral density:

R_X(f) = \begin{cases} \cos ^2 \dfrac {\pi f}{2}, & \text{for } -1 \leq f \leq 1, \\ 0, & \text{otherwise}. \end{cases}

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(a)

Draw a graph of the spectral density of the process.

(b)

In order to study the process, we need to sample it. How often should it be sampled if we want to avoid aliasing?

(c)

Compute and plot the spectral density of the sampled process, with (1) sampling once every second, (2) with five times per second?

Problem 4:11

(a)(b)

The stationary process $\left\{ X(t), t \in \mathbb {R}\right\}$ has a spectral density which we assume is constant $R_X(f) = C$ in the interval $10 \leq \left|f\right| \leq 20$ Hz and zero for all other values.

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(a)

Which is the smallest sampling frequency that can be chosen if one wants to avoid aliasing?

(b)

Sometimes it is easy to forget that in many measurements there might be a strong disturbance at the frequency $50(60)$ Hz from the electricity current. Suppose that this disturbance process can be described by

N(t) = \cos (2\pi 50 t + \phi ), \quad -\infty < t < \infty ,

where the random variable $\phi$ is uniformly distributed in the interval $0 \leq \phi < 2\pi$ . Our measurement signal consists of $Y(t) = X(t) + N(t)$ and we choose our sampling frequency to $1/d = 30$ Hz. What frequency interval is covered by the process $\left\{ X(t)\right\}$ and the disturbance $\left\{ N(t)\right\}$ , respectively, after the sampling?