sol2 - Boston University Department of Electrical and...

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Unformatted text preview: Boston University Department of Electrical and Computer Engineering ENG EC515 Digital Communication ( Fall 2009 ) Solution to Problem set 2 Posted: Fri 18 Sep Suggested reading: Class-notes, handouts, and Ch. 13 textbook Notation: ( 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 R x exp(- t 2 / 2) dt , pmf is the acronym for probability mass function and cdf the acronym for cumulative distribution function. Problem 2.1 (a) Prove that if s ( t ) and s 1 ( t ) are orthogonal complex baseband signals of bandwidth W , then s ( t ) cos(2 f t ) and s 1 ( t ) cos(2 f t ) are also orthogonal signals provided f is larger than W . (b) Give an example for which the statement fails when f is smaller than W . (Hint: Use the fact that orthogonality in time domain has a corresponding expression in the frequency domain.) Solution: Note that from Fourier Transform formulas it follows that g ( t ) G ( f ) = g ( t ) cos(2 f t ) 1 2 ( G ( f- f ) + G ( f + f )) Given (i) s s 1 , namely Z - s ( t ) s * 1 ( t ) dt = (ii) s ( t ) and s 1 ( t ) are bandlimited to W Hz. This means that, S ( f ) = , S 1 ( f ) = , | f | > W > . 1 Problem 2.2 [8pts] In the following example, we shall see that pairwise independence of random variables is not the same as independence. Let X be a binary message random variable which is equally likely to be a 0 or a 1. When X is passed through a particular communication channel, it is hit by a binary noise random variable Z which is indepen- dent of X and is also equally likely to take the values 0 and 1. The output of this channel is given by the random variable Y = X Z where denotes the binary XOR operation, that is, Y = 0 whenever X = Z and 1 otherwise....
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This note was uploaded on 11/09/2010 for the course ECE 515 taught by Professor Venkateshsaligrama during the Fall '09 term at BU.

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sol2 - Boston University Department of Electrical and...

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