Sampling and aliasing of stochastic processes

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Problem 4:10 from Lindgren G., Rootzén H., Sandsten M. "Stationary Stochastic Processes for Scientists and Engineers" [LRS14]More problems in Chapter 4 "Spectral representations"

(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}

(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 from Lindgren G., Rootzén H., Sandsten M. "Stationary Stochastic Processes for Scientists and Engineers" [LRS14]More problems in Chapter 4 "Spectral representations"

(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.

(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?

Problem 6:6 from Lindgren G., Rootzén H., Sandsten M. "Stationary Stochastic Processes for Scientists and Engineers" [LRS14]More problems in Chapter 6 "Linear filters -- general theory"

(a)(b)

A time continuous random process, $\left\{ X(t), t \in \mathbb {R}\right\}$ , consists of a sum of three sinusoids:

X(t) = \sum _{k=1}^{3} A_{k} \cos (2\pi f_{k} t + \phi _{k}), \quad -\infty < t < \infty ,

where $\mathbb {E}\left[A_{1}^{2}\right] = 1$ , $\mathbb {E}\left[A_{2}^{2}\right] = 4$ , $\mathbb {E}\left[A_{3}^{2}\right] = 1$ , and the phases $\phi_{1}$ , $\phi_{2}$ , and $\phi_{3}$ are independent and uniformly distributed random variables in the interval $(-\pi , \pi ]$ . The frequencies are $f_{1} = 4.5$ , $f_{2} = 7$ , and $f_{3} = 11$ kHz. The time continuous process is sampled. By mistake the sampling frequency is chosen $1/d = f_{s} = 5$ kHz.

(a)

Which frequencies are present in the sampled process?

(b)

The time discrete process is filtered with a filter with impulse response $h(n) = \frac{1}{2}\operatorname {sinc}(n/2)$ . Which frequencies are present in the output signal after filtering?

References

[LRS14] Lindgren G., Rootzén H., Sandsten M. Stationary Stochastic Processes for Scientists and Engineers. CRC Press (Taylor & Francis Group), 2014.