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hw5 - Boston University Department of Electrical and...

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Boston University Department of Electrical and Computer Engineering ENG EC515 Digital Communication ( Fall 2009 ) Problem set 5 Assigned: Mon 26 Oct Due: Wed 4 Nov Suggested reading: Class-notes, handouts, and Ch. 1–4 textbook O ffi ce hours: M, W 4:00–6:00pm PHO438 Notation: δ ( t ) , t R denotes the continuous-time Dirac-delta (impulse) function, P ( E ) denotes probabil- ity of event E , E ( · ) denotes the mathematical expectation operator, Q ( x ) denotes the Q-function defined as Q ( x ) = 1 2 π R x exp( - t 2 / 2)d t , pmf is the acronym for probability mass function, cdf the acronym for cumu- lative distribution function, WSS the acronym for Wide-Sense Stationary, and psd the acronym for power spectral density. Problem 5.1 (Irrelevant data, dimensionality reduction) One of three ( K = 3) messages is transmitted over a vector (real, zero-mean) AWGN channel with noise variance σ 2 . The messages need not be equally likely. Specifically, M ∈ M = { 1 , 2 , 3 } and M p M where p M need not be uniform. The channel codewords (or symbols) are given by C
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