Special Topics in Modern Telecommunication Systems


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Θέμα Ανοιξης 2002: Θεωρία ΠιΘανοτήτων και Στοχαστικών Διαδικασιών

Θέμα Project Ανοιξης 2002: Γραμμική εκτίμηση ελάχιστου μέσου τετραγωνικού σφάλματος.

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1). Consider two random vectors X (m x 1) and Y (n x 1). Suppose that you observe X and you wish to estimate Y on the basis of X. For simplicity, you wish to work with a linear estimate of the form AX+B, where A is n x m and B is n x 1. Derive (from scratch) expressions for the optimal A, B, in the sense of minimizing the mean squared error E[||Y - (AX+B)||^2]. (||Χ||^2 = Χ'*Χ = sum of squares of the elements of X). This is similar to the scalar case that we worked out in class, only the algebra is a bit more complicated. Note that I would like to see your complete derivation of the formulas for A, B, including spelling out any assumptions that you may possibly need in your derivation.

2). As a special case, consider a scalar Y (i.e., n=1). Applications of the above result can be found in numerous practical problems, from prediction (think of X as holding past samples of a certain time series and Y being the next sample which you have not yet seen; or even a sample "further down the road"), to channel equalization (think of X as holding observations from the channel output and Y being a particular channel input sample that you are interested in), and so on. Note that X, Y can be drawn from the same time series or from different (but correlated!) time series. Browse the web and select a time series (or pair thereof) that is of interest to you. As a starting point, take a look at http://www.mhhe.com/engcs/electrical/viniotis/projdata.mhtml

3). Decide what is X, Y for your particular set of data (clearly, there are many meaningful choices, depending on what it is that you want to do with your data). Split your data in half. Use the first part of the data to estimate all parameters necessary to calculate A, B. Explain how you will go about estimating these parameters and why.

4). Run your optimal estimator on the rest of the data. Do you obtain reasonable results? How can you judge this?

5). Recall that X is an m x 1 vector. Does the choice of m affect your results in step 4 above? Why? Is there a trade-off here, or should you make m as large as possible?

6). Suppose now that you wish to implement your solution in practice. Assess the computational complexity of your "batch" solution above. Then do a literature search to see if there exist any clever ways to reduce this computational burden and make it suitable for on-line implementation. Report your findings along with suitable pointers to the literature. Be brief and to-the-point.

7). Submit a complete report containing your derivations, code listings, any associated plots that you deem worthwhile including, plus references.

Hint: To get started (and also for help with Matlab), look at the material (problem statement, discussion, data, and Matlab code) under Recitations 7 and 8 at the URL http://www.ece.umn.edu/users/nikos/public_html/EE3025HOME/EE3025.html

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