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Problem 3 Consider the Gaussian process, X(t), with zero mean and a utocorrela- t ) i,2 tion function Rx(t1, t2 mini 1....
Autocorrelation of an X(t) random process is Rxx (t1, t2) = 4e-t-t2 This a Gaussian process...
Autocorrelation of an X(t) random process is Rxx (t1, t2) = 4e-t-t2 This a Gaussian process with mean zero. a) [6p] Is this process wide sense stationary? Briefly explain. b) [9p] Calculate the probability P (X(2)> 1) using the Table at the cover. c) [10p] Calculate approximately the probability P(X(2) > X(4) + 1). Some useful relations 1. Var(X(t)) = E({€)) - (E(X(t))) 2. R(X(t)X(t) = ELX(t-)X(02)]| 3. Var(X(c) +X)) = Var( (t) ) + Var (X (t2) - 2Cov(X...
Let X(t) and Y(t) be independent, wide-sense stationary random process with zero means and the same covariance function...
Let X(t) and Y(t) be independent, wide-sense stationary random process with zero means and the same covariance function Cx(t) Let Z(t) be defined by Z(t) = X(t)coswt + Y(t)sinwt Find the joint pdf of X(t1) and X(t2) in part b
324. Consider the random process X(t) = A + Bt2 for - <t < oo, where...
324. Consider the random process X(t) = A + Bt2 for - <t < oo, where A and B are two statistically independent Gaussian random variables, each with zero mean and variance o?. a) Plot two sample functions of X(t). b) Find E{X(0)} c) Find the autocorrelation function Rx(t,t +T). d) Find the pdf of the random variable Y = X(1). e) Is X(t) a Gaussian process? Prove your result.
Let X(t) X(t) be a Gaussian random process with μ X (t)=0 μX(t)=0 and R X...
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) .
3.34. Let fXc(t)) and (X,(t)J denote two statistically independent zero n stationary Gaussian random processes with...
3.34. Let fXc(t)) and (X,(t)J denote two statistically independent zero n stationary Gaussian random processes with common power spec- tral density given by SX (f) = SX (f) = 112B(f) watt/Hz. Define x(t) = Xe(t) cos(2tht)--Xs(t) sin(2tht) where fo 》 (a) Is X(t) a Gaussian process? (b) Find the mean E(X (t), autocorrelation function Rx (t,t + T), and power spectral density Sx(f) of the process X(t) (c) Find the pdf of X(O) (d) The process X(t) is passed through...
Problem #1 below. 2. Assume that the random variables X and Y of Prob. 1, are...
Problem #1 below.
2. Assume that the random variables X and Y of Prob. 1, are jointly Gaussian, both are zero mean, both have the same variance o2, and additionally are statistically independent. Use this information to obtain the joint pdf fzv(z,w) of Prob. 1. Verify that this joint pdf is alial 1. Let X and Y be two random variables with known joint PDF fx(x,y). Define two new random variables through the transformations Determine the joint pdf fzw(z, w)...
7) (20 pts) Let X(t) = At be a random process, such that A is N(0,...
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)
Problem 1 A Poisson process is a continuous-time discrete-valued random process, X(t), that counts the number of events...
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,...
3.34. Let (X.(t) and (x.(e)) denote two statistically independent zero mean stationary Gaussian r...
3.34. Let (X.(t) and (x.(e)) denote two statistically independent zero mean stationary Gaussian random processes with common power spec- tral density given by Ste (f) = S, (f) = 112B(f) Watt/Ha Define X (t) X( t) cos(2 fo t) - Xs (t) sin(2r fot) ) - Xs(t) sin(2T fot where fo》 B (c) Find the pdf of X(0). (d) The process X(t) is passed through an ideal bandpass filter with transfer function given by otherwise. Let Y(t) denote the output...
PROBLEM 2.15] Consider the linear mapping T : R4 → R3, defined as T1 T2 |...
PROBLEM 2.15] Consider the linear mapping T : R4 → R3, defined as T1 T2 | = 5x1 + 2x2 + 7x3 + 24 42:1 + 322 + 713-214 ) T4 (i) Write the corresponding matrix [T]. (i) Find a basis of Range(T).1 (ii Find a basis of Null(T).[1) (iv) Find the rank of T in 3 different ways.[1 (v) Show that T satisfies the rank theorem. 1
Autocorrelation of an X(t) random process is Rxx (t1, t2) = 4e-t-t2 This a Gaussian process with mean zero. a) [6p] Is this process wide sense stationary? Briefly explain. b) [9p] Calculate the probability P (X(2)> 1) using the Table at the cover. c) [10p] Calculate approximately the probability P(X(2) > X(4) + 1). Some useful relations 1. Var(X(t)) = E({€)) - (E(X(t))) 2. R(X(t)X(t) = ELX(t-)X(02)]| 3. Var(X(c) +X)) = Var( (t) ) + Var (X (t2) - 2Cov(X...
324. Consider the random process X(t) = A + Bt2 for - <t < oo, where A and B are two statistically independent Gaussian random variables, each with zero mean and variance o?. a) Plot two sample functions of X(t). b) Find E{X(0)} c) Find the autocorrelation function Rx(t,t +T). d) Find the pdf of the random variable Y = X(1). e) Is X(t) a Gaussian process? Prove your result.
3.34. Let fXc(t)) and (X,(t)J denote two statistically independent zero n stationary Gaussian random processes with common power spec- tral density given by SX (f) = SX (f) = 112B(f) watt/Hz. Define x(t) = Xe(t) cos(2tht)--Xs(t) sin(2tht) where fo 》 (a) Is X(t) a Gaussian process? (b) Find the mean E(X (t), autocorrelation function Rx (t,t + T), and power spectral density Sx(f) of the process X(t) (c) Find the pdf of X(O) (d) The process X(t) is passed through...
Problem #1 below.
2. Assume that the random variables X and Y of Prob. 1, are jointly Gaussian, both are zero mean, both have the same variance o2, and additionally are statistically independent. Use this information to obtain the joint pdf fzv(z,w) of Prob. 1. Verify that this joint pdf is alial 1. Let X and Y be two random variables with known joint PDF fx(x,y). Define two new random variables through the transformations Determine the joint pdf fzw(z, w)...
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)
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,...
3.34. Let (X.(t) and (x.(e)) denote two statistically independent zero mean stationary Gaussian random processes with common power spec- tral density given by Ste (f) = S, (f) = 112B(f) Watt/Ha Define X (t) X( t) cos(2 fo t) - Xs (t) sin(2r fot) ) - Xs(t) sin(2T fot where fo》 B (c) Find the pdf of X(0). (d) The process X(t) is passed through an ideal bandpass filter with transfer function given by otherwise. Let Y(t) denote the output...
PROBLEM 2.15] Consider the linear mapping T : R4 → R3, defined as T1 T2 | = 5x1 + 2x2 + 7x3 + 24 42:1 + 322 + 713-214 ) T4 (i) Write the corresponding matrix [T]. (i) Find a basis of Range(T).1 (ii Find a basis of Null(T).[1) (iv) Find the rank of T in 3 different ways.[1 (v) Show that T satisfies the rank theorem. 1
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