561_hmk3 - ECE 561 Digital Communications Systems Homework...

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ECE 561 – Digital Communications Systems Homework #3 1. A random variable, X, is defined by the probability density function: f X ( x ) = 2 e 2 x , x 0 a. What is the mean of X? b. What is the cumulative distribution function F X (x)? c. What is the probability that X is between 0 and 1? d. What is the probability that X is between 1 and 2? 2. The probability density function of a Gaussian random variable, X, is given by: p X ( x ) = 1 3 2 π e x 4 ( ) 2 18 a. What is the mean of X? b. What is the variance of X? c. P(X 2) d. P(X > 7) e. P( -2 X 10) 3. Sketch the impulse response of the matched filter for each of the signals shown below. s 1 (t) t T 1 s 2 (t) t T 2 T/2 -1 s 3 (t) t T 2 T/2 a. b. c.
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4. For the orthogonal signal set: a. If s 1 is applied to the correlator shown below, sketch the outputs r 1 and r 2 for 0 t T. b. Repeat part a for the input s 2 . c.
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561_hmk3 - ECE 561 Digital Communications Systems Homework...

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