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Unformatted text preview: Assignment 9 1. A random process is deﬁned by Y (t) = G(t − T ) where g (t) is the rectangular pulse of following ﬁgure, and T is a uniformly distributed random variable in the interval (0,1). 1 0 1 t (a) Find the pmf of Y (t). (b) Find mY (t) and CY (t1 , t2 ). 2. The random process H (t) is deﬁned as the ”hard-limited” version of X (t): H (t) = +1 if X (t) ≥ 0 −1 if X (t) < 0. (a) Find the pdf, mean, and autocovariance of H (t) if X (t) is the sinusoid with a random amplitude presented in Example 9.2 in the text book. (b) Find the pdf, mean,and autocovariance of H (t) if X (t) is the sinusoid with random phase presented in Example 9.9 in the text book. (c) Find a general expression for the mean of H (t) in terms of the cdf of X (t) 3. Let Xn consist of an iid sequence of Poisson random variables with mean α. (a) Find the pmf of the sum process Sn . (b) Find the joint pmf of Sn and Sn+k . 1 4. Customers deposit $1 in a vending machine according to a Poisson process with rate λ. The machine issues an item with probability p. Find the pmf for the number of items dispensed in time t. 5. Packets arrive at a multiplexer at two ports according to independent Poisson processes of rates λ1 = 1 and λ2 = 2 packets/second, respectively. (a) Find the probability that a message arrives ﬁrst on line 2. (b) Find the pdf for the time until a message arrives on the either line. (c) Find the pmf for N (t), the total number of messages that arrive in an interval of length t. (d) Generalize the result of part c for the ”merging” of k independent Poisson processes of rates λ1 , . . . , λk , respectively: N (t) = N1 (t) + · · · + Nk (t). 6. Let X (t) be a zero-mean Gaussian random process with autocovariance function given by CX (t1 , t2 ) = 4e−2|t1 −t2 | . 7. Find the joint pdf of X (t) and X (t + s). 2 ...
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This note was uploaded on 01/15/2011 for the course ECE 616 taught by Professor Khkjk during the Winter '10 term at Concordia Canada.
- Winter '10