1) Consider a normal WSS process X(t) with E{X(t)3 - 0 and Xx b) E(lX(t1) -...
and is X(t) a WSS process? 6.11 Sinusoid with random phase. Consider a random process x(t)-A cos(wot + ?), where wo are nonrandom positive constants and o is a RV uniformly distributed over A and (0, ?), i.e., ? ~11(0, ?). (a) Find the mean function 2(t) of X(t).
Show that the following properties hold if X(t) is a WSS process with finite second order moments then (a) \Rxx(1)| < Rxx(0) (b) \Rxy(t) = V Rxx(0)Ryy(0) (c) Rxx(T) = Rxx(-1)
Exercise 5. Let X(t) be a WSS process with correlation function 1-Irl, if-1-1S1 0,otherwise. Rx(T) = It is known that when X (t) is input to a system with transfer function H(), the system output Y(t) has a correlation function Ry(T) sin TT = =-TT Find the transfer function H(u
Problem 3 Consider the Gaussian process, X(t), with zero mean and a utocorrela- t ) i,2 tion function Rx(t1, t2 mini 1. Find the covariance matrix of the random variables X(1) and X (2) 2. Write an expression for the joint PDF of X(1) and X(2) Problem 3 Consider the Gaussian process, X(t), with zero mean and a utocorrela- t ) i,2 tion function Rx(t1, t2 mini 1. Find the covariance matrix of the random variables X(1) and X (2)...
A. For each of the following randomn processes, state whether it is wide-sense stationary (WSS) and why in 1-3 Sentences (a) A Poisson random process N(t) with mean function mN () =M and autocovariance function CN(t,t2) = Ati. (b) A Gaussian random process W (t) with mean function mw (t) = 3t and autocovariance function Cw (l,t,) = 9e 2t2 0 and antocorrelation function (c) An exponential random process Z(t) with mean function mz(1) RZ(t1,t2) = e 42 Ll A....
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,...
Let X(t) X(t) be a Gaussian random process with μ X (t)=0 μX(t)=0 and R X ( t 1 , t 2 )=min( t 1 , t 2 ) RX(t1,t2)=min(t1,t2) . Find P(X(4)<3|X(1)=1) P(X(4)<3|X(1)=1) .
Q1) Let X(t) be a zero-mean WSS process with X(t) is input to an LTI system with Let Y(t) be the output. a) Find the mean of Y(t) b) Find the PSD of the output SY(f) c) Find RY(0) ------------------------------------------------------------------------------------------------------------------------- Q2) The random process X(t) is called a white Gaussian noise process if X(t) is a stationary Gaussian random process with zero mean, and flat power spectral density, Let X(t) be a white Gaussian noise process that is input to...
7) (20 pts) Let X(t)-At be a random process, such that A is N(0, 1). (b) Find the auto-correlation function of X, Rx(t1, t2) E[X(ti)X(t2)
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t) Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...