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Unformatted text preview: ECE 1520 Data Communications Fall 2011 Problem Set 2 Due in class, Wednesday, Oct. 5th, 2011. 1. I had wanted to show this in class, but then didn’t want to take the time: Consider a lowpass Gaussian WSS random process, X ( t ) with bandwidth W and power spectral density S x ( f ) = braceleftbigg 1  f  ≤ W  f  > W Now, define a discrete random process X [ n ] as a sampled version of X ( t ), i.e., X [ n ] = X ( nT s ) where T s is the sampling period. (a) Is the discrete process WSS, i.e., is E [ X [ n ] X * [ n + m ]] = R x [ m ] (independent of the value of n )? Prove your answer. If ‘yes’, find the relationship between R x [ m ] and the original autocorrelation function R x ( τ ). (b) Show that if T s just meets Nyquist criterion, i.e., T s = 1 / 2 W , the samples are i.i.d. Gaussian random variables. 2. Q6.3 from the text: The alphabet of a DMS, X , is { a 1 ,a 2 ,...,a n } with correspond ing probabilities { p 1 ,p 2 ,... ,p n } . Show that H ( X ) is maximized when p...
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 Spring '09
 Probability, Signal Processing, Probability theory, Stationary process, Gaussian random variables, lowpass Gaussian WSS

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