communications and Signal Processing have all seen explosive development that would not have been possible without the theory of stochastic processes

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The applications of Stochastic Processes’ Theory have grown extensively and can now be found in Mathematics, Physics, Engineering, Biology and Finance to name but a few. In Electrical Engineering, the disciplines of Control, Communications and Signal Processing have all seen explosive development that would not have been possible without the theory of stochastic processes. Instead of considering an application in a particular electrical engineering discipline, the objective here is to focus on the essential fundamental relations between the temporal response aspect and the probabilistic/ensemble structural aspect essential to stochastic processes. On the one hand, a stochastic process is a signal with probabilistic variability (“randomness”) while, on the other hand, it is a random variable with temporal variability (“evolution”). These two perspectives will be studied here by relating and contrasting three experimental random processes provided as Matlab MEX files (black_boxes.zip). Instead of drilling the theory, the approach here is to perform numerical experimentation on these three stochastic processes to develop a solid understanding of the fundamentals involved. A thorough understanding of the principles of stochastic processes is the goal with the software routines used, seen as a means to an end. 2. Assignment Suppose three real-valued discrete-time stochastic processes, represented as 𝑋(𝑡, 𝑠), 𝑌(𝑡, 𝑠) and 𝑍(𝑡, 𝑠) are given. Here, 𝑡 represents the time variable of discrete/integer instants 0, 1, 2, 3, … and 𝑠 represents the ensemble variable of real values uniformly distributed over the range [−1, 1]. These processes are provided in the form of individual MEX files, each taking as arguments 𝑡 and 𝑠 and returning the respective stochastic process’ output value. You are required to identify their stochastic dynamical similarities and differences based on: 1. Realisations and related aspects: a. Realisations of the stochastic process, i.e., for different 𝑠-values (fixed) observe different time responses. b. Time-autocorrelation1 functions for different realisations. 2. Ensemble aspects: a. Random variables manifesting at different time instants along the time history of the process. b. First-order distributions associated with different time instants. c. Second-order distributions associated with different time-instant pairs. d. Correlation between pairs of random variables at the pair taken at different times while keeping the time difference fixed. Repeat this for different choices of the time separation/difference. Then calculate the ensemble-autocorrelation1 function. e. Degree of stationarity. 3. Frequency spectral aspects: a. Non-statistical power spectral density of different realisations. b. Statistical power spectral density of the stochastic process. 4. Any other aspect that might help to develop insight. Although beyond the scope of this course, one could consider aspects such as sensitivity on initial conditions, Lyapunov exponents etc. Bonus Task Recognise one of the above stochastic processes and provide formal calculations of its corresponding first-order probability density, statistical-autocorrelation and power spectral density. 3. Submission You are required to submit a short report – 2 to 4 pages – detailing the approach you have taken to solve the problem and the results that demonstrate your final conclusions. Your report MUST include: • All relevant calculations and plots supporting the similarities and differences you discovered. • An introduction, conclusion and reference section (amongst others). An abstract is not necessary for such a short report.

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