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Unformatted text preview: COMER ECE 600 Homework 11 1. Show that if is a random variable with ( ) = E[ei ] and (1) = (2) = 0, then the process X(t) = cos( t+ ) is wide-sense stationary. 2. Consider a zero-mean strict-sense stationary random process with R X (τ ) = 50 cos(20πτ ) + 18 cos(30πτ )
as input to the following system: X( t ) Gain A 2 Second Delay + + Y( t ) a. Find the variance of X(t). b. What value of A will minimize the mean square value of Y(t)? What is the mean square value of Y(t) for this value of A? 3. White noise X(t) with spectral density 1 V2/Hz is input to a linear time-invariant system with impulse response h(t) = u(t) – u(t-1). If the output of the system is Y(t), then a. Determine RXY(t1,t2). b. Determine RYY(t1,t2). 4. Let X1, …, Xn be jointly Gaussian random variables with the same mean covariance function and with σ 2,
Cov(X i , X j ) = ρσ 2 , 0, where | | < 1. i= j | i − j |= 1 otherwise, a. Find the mean and variance of S n = X1 + + Xn . b. Find the characteristic function of Sn. c. Find the mean and variance of Sn – Sm for n > m. 2. Suppose that a secretary receives phone calls that arrive according to a Poisson process with a rate of 10 calls per hour. What is the probability that no calls go unanswered if the secretary is away from the office for the first and last 15 minutes of an hour? 3. Let N(t) be a Poisson process. Find the covariance of N(t1) and N(t2), where t1 < t2. ...
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This note was uploaded on 01/27/2010 for the course ECE 600 taught by Professor Staff during the Fall '08 term at Purdue University-West Lafayette.
- Fall '08