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sol2

# sol2 - Boston University Department of Electrical and...

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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. 1–3 textbook 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) 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 0 ( t ) and s 1 ( t ) are orthogonal complex baseband signals of bandwidth W , then s 0 ( t ) cos(2 π f 0 t ) and s 1 ( t ) cos(2 π f 0 t ) are also orthogonal signals provided f 0 is larger than W . (b) Give an example for which the statement fails when f 0 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 π f 0 t ) ←→ 1 2 ( G ( f - f 0 ) + G ( f + f 0 )) Given (i) s 0 s 1 , namely Z -∞ s 0 ( t ) s * 1 ( t ) dt = 0 (ii) s 0 ( t ) and s 1 ( t ) are bandlimited to W Hz. This means that, S 0 ( f ) = 0 , S 1 ( f ) = 0 , ∀| f | > W > 0 . 1

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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. (a) [1pt] Find the joint pmf of X , Z .

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sol2 - Boston University Department of Electrical and...

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