The Autocorrelation Function of the Binary Rectangular Random Pulse Train

# The Autocorrelation Function of the Binary Rectangular Random Pulse Train

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RANDOM SIGNALS AND NOISE ELEN E4815y Columbia University Spring Semester 2008 The Autocorrelation Function of the Binary Rectangular Random Pulse Train 13 February 2008 I. Kalet t 0 T sec A -A 1

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THE RANDOM RECTANGULAR PULSE TRAIN x(t)= Σ a n p(t-nT+t 0 ) - All the a n ’s are independent variable which take on the values of ± 1 with equal probability (In fact they can take on one of M possible values and the the same proof would hold) The time offset, t 0 , is an independent variable with a uniform probability density function, f(t 0 ),= 1/T for –T/2 t 0 T/2. f(t 0 ) 1/T -T/2 0 T/2 t 0 The pulse, p(t), is a rectangular pulse of height, A, and width equal to T seconds. (However, except for the final result we never make use of the fact that it is a rectangular pulse. In fact, it can be any pulse, even lasting for more than T seconds. p(t) A 0 T t 2
The autocorrelation function, x ( τ ) is given by the equation x ( τ )=E{x(t+ τ ) x(t)} x ( τ )=E{ Σ a n p(t+ τ -nT+t 0 ) Σ

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## This note was uploaded on 01/22/2011 for the course ELWN E4815Y taught by Professor Yossi during the Spring '10 term at Punjab Engineering College.

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The Autocorrelation Function of the Binary Rectangular Random Pulse Train

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