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# fin - ECE 562 Fall 2011 December 1 2011 Final(Max score =...

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ECE 562 Fall 2011 December 1, 2011 Final (Max score = 130 pts) Assigned: December 1, 2011 Due: December 9, 2011 at 1:00 PM in room 106 CSL Note: This is an open-book, take-home exam. You can use your class notes, the text book, any of the references that I’ve put on reserve in the library, and the HW sets and solutions. The contents of the exam are confidential until 1 PM on December 9, 2011. Until that time, you must not show this examination to, or discuss its contents with, any person other than Prof. Veeravalli. This also applies to your solutions to the exam of course. You may use Matlab or similar software packages to do your numerical computations. 1. (15 pts) Signal Space and Performance in AWGN. Consider the signal set shown below: E s 0 2 t 1 0 . 5 1 . 5 0 2 t 1 0 . 5 1 . 5 0 2 t 1 0 . 5 1 . 5 s 2 ( t ) s 3 ( t ) s 4 ( t ) 0 s 1 ( t ) 2 t 1 0 . 5 1 . 5 E s E s E s (a) Find an orthonormal basis for this signal set and express the signals in terms of this basis. (b) Find the measure of goodness ζ for the signal constellation. (c) Suppose this signal set is used for communication over an AWGN channel with noise PSD N 0 . Assuming equally likely symbols, find the IUB and NNA approx- imations for the average probability of symbol error P e in terms of γ s = E s /N 0 . (d) Now find an exact expression for P e in terms of γ s . c V.V. Veeravalli, 2011 1

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2. (25 pts) V.29 Modem. The V.29 standard for 9.6 Kbit/s modems uses the following 16-ary signal constellation: -1 1 3 5 -1 -5 -3 1 3 5 -5 -3 Assume equally likely symbols. (a) Carefully draw the MPE decision regions for this constellation (b) Find the IUB for P e as a function of γ s . (c) Find the NNA for P e as a function of γ s . (d) Find the NNA to P e for a (square) 16-QAM constellation as a function of γ s . (e) Comparing (c) and (d), which constellation is more power efficient at high SNR? (f) Determine whether you can label the signal points using three bits so that nearest neighbors differ by at most one bit (Gray coding). If so, find such a labeling. If not, state why not and find a labeling that minimizes the maximum number of bit transitions between neighbors.
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