Polimi CS - Notes

2 - Stochastic Processes

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Lecture 01 - February 18: Stochastic Processes and White Noise

1. Stochastic Processes

Definition [Stochastic Process]

A stochastic process (SP) is an infinite sequence of random variables, all defined on the same probabilistic space:

where:

  • is the time index

  • represents the outcome of a random experiment

  • Each fixed gives one realization (a deterministic signal)

  • Each fixed gives a random variable with some probability distribution 1

1.1 Key Concepts

Extension of Random Variables: The definition extends the notion of “random variable” to signals. Setting fixes the whole sequence, giving a deterministic signal 2.

Important Note

Even if , , are different signals, if they are realizations of the same stochastic process , they are said to be stochastically equivalent (generated by the same SP, but not identical!)3.

1.2 Wide-Sense Characterization

To fully describe a SP, we would need to know how the probability distribution of evolves over time. This is TOO DIFFICULT FOR MODELING PURPOSES4.

From now on, we’ll work with the so-called WIDE-SENSE CHARACTERIZATION of SP (mean and variance only). If the process is Gaussian, with wide-sense characterization we could work only with mean and variance , with some approximation to simplify the computations (simpler than working with the full pdf)5.

Definition [Mean Value of a SP]

The mean value of a stochastic process is:

This represents the average behavior of the SP (mean signal over , expectation)6.

Definition [Covariance Function of a SP]

The covariance function of a stochastic process is:

Special case: If :

Note

The covariance function is a function of two time instances. By their distance , we can see the dependence:

This could be useful to make predictions on the future! 7

2. Stationary Stochastic Processes

In mechatronics, we mainly work with stationary stochastic processes (SSP)8.

Definition [Stationary Stochastic Process]

A stationary stochastic process (SSP) is a SP with:

  1. (constant mean)

The second condition means that doesn’t depend on the specific times , but only on their difference 9. This means:

even if and 10.

2.1 Properties of the Covariance Function

For a stationary process with covariance function :

  1. Positivity:

  2. Non-increasing from zero:

    Moving to the future, the correlation decreases as increases11.

  3. Even function:

2.2 Observations

  1. For a given SSP , we will write (for ) and (or ) to indicate its mean and covariance function.

  2. Two SSPs and are wide-sense equivalent if:

  3. The covariance function is different from the correlation function if 12.

3. Examples

Example [Constant Realization Process - Stationary]

Consider where (i.e., is a Gaussian variable with mean 1 and variance 3)13.

Question: Is the process stationary?

Answer: Yes.

Verification:

Fixing becomes a number, so all the realizations are constant.

  1. Constant realization: Each outcome gives a horizontal line14.

  2. Distribution: The values are distributed as by definition15.

Mean: The mean is a constant:

Covariance:

Note: They are the same because is Gaussian.

Conclusion: Because both and do not depend on (they are constant), the process is SSP16.

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Example [Linear Process - Non-Stationary]

Consider with 18.

Question: Is the process stationary?

Answer: No.

Verification:

Mean:

Covariance:

In particular, the variance at time is:

Actually, more generally:

Conclusion: Because the variance is not constant (it grows with time), this process is not stationary19.

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4. White Noise

Definition [White Noise]

A SSP is called white noise (WN) with mean and variance if the following holds:

a) Mean value:

b) Variance:

c) Covariance:

We will write to define a white noise signal21.

4.1 Properties of White Noise

  • White noise is an unpredictable process (no correlation with the future)22.

  • Zero covariance for means that knowing gives no information about for any 23.

  • A constant signal (like the one represented in the figure) is possible but very unlikely. This is the so-called constant realization and is one of the possible (but very unlikely) realizations of the WN24.

  • We don’t need to know the full pdf to characterize white noise - the wide-sense characterization is sufficient25.

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5. Summary

Key Takeaways

  • A stochastic process extends the concept of random variables to signals over time27.

  • Wide-sense characterization uses only mean and covariance, simplifying analysis28.

  • Stationary processes have constant mean and covariance that depends only on time difference .

  • The covariance function is maximum at , non-increasing, and even30.

  • White noise is a stationary process with zero correlation at non-zero time lags.

Posso fare altro per aiutarti con questi appunti sui processi stocastici?

\begin{document}
\begin{tikzpicture}
    \node [draw, rectangle, minimum width=2cm, minimum height=1.5cm] (system) {$\begin{array}{c} x_1(t) \\ x_2(t) \\ \vdots \\ x_n(t) \end{array}$};  
    \node [left=1.5cm] (input) {};  
    \node [right=1.5cm] (output) {};  
      
    \draw[->] (input) -- node[above]{$u(t)$} node[below,font=\small]{input} (system);  
    \draw[->] (system) -- node[above]{$y(t)$} node[below,font=\small]{output} (output);  
    \node[above=0.1cm] {state $x(t)$};  
\end{tikzpicture}
\end{document}

Graph View