COL+UNIV-2012+SPRING-RECTANGULAR+PULSE+TRAIN-1+JAN+2012

# COL+UNIV-2012+SPRING-RECTANGULAR+PULSE+TRAIN-1+JAN+2012 - 1...

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Random Signals and Noise ELEN E4815 Columbia University Spring Semester 2012 27 November 2011 Professor I. Kalet The Autocorrelation Function of the Binary Rectangular Random Pulse Train 1

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THE RANDOM RECTANGULAR PULSE TRAIN t 0 T sec A -A x(t)= Σ a n p(t-nT+t 0 ) - All the a n ’s are independent variables which may 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 2
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 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 a m p(t-mT+t 0 )} x ( τ )= Σ Σ E {a n a m }E{p(t+

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