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ps10-PulseMod

# ps10-PulseMod - 6.450 Principles of Digital Communication...

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6.450 Principles of Digital Communication Wednesday, November 7, 2002 MIT, Fall 2002 Handout #35 Due: Wednesday, November 14, 2001 Problem Set 10 Problem 10.1 The purpose of this problem is to show that if { N j ; j Z } is a sequence of Gaussian rv’s N (0 , σ 2 ), then the sampling theorem expansion N ( t ) = j N j sinc( t - jT ) is a stationary random process. (a) Suppose that an L 2 function x ( t ) is baseband limited to 1 / (2 T ) for some T > 0 and is the inverse Fourier transform of ˆ x ( f ). Show that for any fixed τ , x ( t - τ ) can be expressed as x ( t - τ ) = X j Z x ( jT - τ ) sinc t - jT T (b) Show that sinc t - τ T = X j Z sinc τ - jT T sinc t - jT T (c) Consider the stochastic process N ( t ) = X j Z N j sinc t - jT T where { N j ; j Z } is a set of iid N (0 , σ 2 ) rv’s. Find K N ( t, τ ) = E [ N ( t ) N ( τ )]. (d) Show that { N ( t ) } is a stationary Gaussian random process and find the covariance function ˜ K N ( t - τ ) = K N ( t, τ ). Note that the sample functions of { N ( t ) } are not L 2 , although N ( t 1 ) , N ( t 2 ) , . . . , N ( t k ) are jointly Gaussian.

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ps10-PulseMod - 6.450 Principles of Digital Communication...

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