P9.3 A random process X(t) has the following member functions: x1 (t) -2 cos(t), x2(t)2 sin(t),...
2. Consider the random process x(t) defined by x(t) a cos(wt 6), where w and 0 are constants, and a is a random variable uniformly distributed in the range (-A, A). a. Sketch the ensemble (sample functions) representing x(t). (2.5 points). b. Find the mean and variance of the random variable a. (5 points). c. Find the mean of x(t), m(t) E((t)). (5 points). d. Find the autocorrelation of x(t), Ra (t1, t2) E(x (t)x2 )). (5 points). Is the...
2. Consider the random process x(t) defined by x(t) a cos(wt + 6).where w and a are constants, and 0 is a random variable uniformly distributed in the range (-T, ) Sketch the ensemble (sample functions) representing x(t). (2.5 points). a. b. Find the mean and variance of the random variable 0. (2.5 points). Find the mean of x(t), m (t) E(x(t)). (2.5 points). c. d. Find the autocorrelation of x(t), R (t,, t) = E(x, (t)x2 (t)). (5 points)....
Problem 1 A Poisson process is a continuous-time discrete-valued random process, X(t), that counts the number of events (think of incoming phone calls, customers entering a bank, car accidents, etc.) that occur up to and including time t where the occurrence times of these events satisfy the following three conditions Events only occur after time 0, i.e., X(t)0 for t0 If N (1, 2], where 0< t t2, denotes the number of events that occur in the time interval (t1,...
t + τ Proof From Definition 10.17, RİT (r) yields Rn(t) = Elx()r(t + τ)]. Making the substitution u Since X(0) and Y(O) are jointly wide sense stationary, Ryr(u, -t for random sequences Rx-r). The proof is similar i: 10.11X(t) is a wide sense stationary stochastic process with autocorrelation function Rx(r). (2) Express the autocorrelation function of Y(C) in terms of Rx(r) Is r) wide sense (2) Express the cross-correlation function of x(t) and Y (t) in terms of Rx(t)...
Please answer all the questions thank you ne 10. 2019 4. A random process Z(t) is given by, Z(t) = Kt, where K is a random variable The probability dessity function for K is given below. Use this information to answer the questions below (20 points k <-1 0 fK(k)=-k-1sks k> 1 0 (a) Find the mean function for Z(t). (b) Find the autocovariance function for Z(e). (c) Is this process wide sense stationary (WSS)? Explain your answer in 2-3...
ne 10. 2019 4. A random process Z(t) is given by, Z(t) = Kt, where K is a random variable The probability dessity function for K is given below. Use this information to answer the questions below (20 points k <-1 0 fK(k)=-k-1sks k> 1 0 (a) Find the mean function for Z(t). (b) Find the autocovariance function for Z(e). (c) Is this process wide sense stationary (WSS)? Explain your answer in 2-3 sentences. ne 10. 2019 4. A random...
(13 points) The random process X(t) consists of the following two sample functions which are equally likely: x(t,sı)=e?, x(t,52)=-e Determine the mean and autocorrelation function of X(t), and also determine whether X(t) is wide sense stationary. (Note: no credit will be awarded for correct guesses without justification).
Q.6 Determine the autocorrelation function and power spectral density of the random process olt)= m(t) cos(21f t+), where m(t) is wide sense stationary random process, and is uniformly distributed over (0,2%) and independent of m(t).
7. Let X(t) be a wide-sense stationary random process with the autocorrelation function : Rxx(t)=et where a> 0 is a constant. Assume that X(t) amplitude modulates a carrier cos(2ttfot+0), Y(t) = X(t) cos(21tfot+0) where 0 is random variable on (-10,1t) and is statistically independent of X(t). a. Determine the autocorrelation function Ryy(t) of Y(t), and also give a sketch of it. (5 points). b. Is y(t) wide-sense stationary as well ? (5 points).
7) (20 pts) Let X(t) = At be a random process, such that A is N(0, 1). , ??(t)-EX(t)]. (a) Find mean of the random process X(t) (b) Find the auto-correlation function of X, Rx(t1,t2) - E[X (ti)X (t2)