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5. Arandom prices of X(t) is known to be wide-sense stationary with E[X (t)] 11. Give one or more...
Let X(t) be a wide-sense stationary random process with the autocorrelation function :
Let X(t) be a wide-sense stationary random process with the autocorrelation function : Rxx(τ)=e-a|τ| where a> 0 is a constant. Assume that X(t) amplitude modulates a carrier cos(2πf0t+θ), Y(t) = X(t) cos(2πf0t+θ) where θ is random variable on (-π,π) and is statistically independent of X(t). a. Determine the autocorrelation function Ryy(τ) of Y(t), and also give a sketch of it. b. Is y(t) wide-sense stationary as well?
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...
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)...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay a) (4 points) Find the power spectral density of the output random process y(t), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c) (4 points) Find the output autocorrelation function Фуу(r) d) (1 points) What is the total power in the output process...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay a) (4 points) Find the power spectral density of the output random process y(t), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c) (4 points) Find the output autocorrelation function Фуу(r) d) (1 points) What is the total power in the output process...
The purpose of this assignment is to practice concepts related to the wide-sense stationary processes, filtering,...
The purpose of this assignment is to practice concepts related to the wide-sense stationary processes, filtering, auto-correlation, and power spectral density I. (20 points) Let X(1) denote a wide sense stationary process with μ,-0 and autocorrelation Rdr). Let y(1) = 2 + XUt). What is R)(tz)? Is Y(t) wide sense stationary?
A(t) is a wide-sense stationary random process and is a random variable distributed uniformly over [0,...
A(t) is a wide-sense stationary random process and is a random variable distributed uniformly over [0, 211]. Furthermore, is independent of A(t). Three random processes X(t), Y(t), and Z(t) are given by X(t) = A(t) cos(20ft + 0) Y(t) = A(t) cos(507t + 0) z(t) = X(t) + y(t) a. Show that X(t) and Y(t) are stationary in the wide sense. b. Show that Z(t) is not stationary in the wide sense.
2. (30 points) Let X(t) be a wide-sense stationary (WSS) random signal with power spectral density...
2. (30 points) Let X(t) be a wide-sense stationary (WSS) random signal with power spectral density S(f) = 1011(f/200), and let y(t) be a random process defined by Y(t) = 10 cos(2000nt + 1) where is a uniformly distributed random variable in the interval [ 027]. Assume that X(t) and Y(t) are independent. (a) Derive the mean and autocorrelation function of Y(t). Is Y(t) a WSS process? Why? (b) Define a random signal Z(t) = X(t)Y(t). Determine and sketch the...
5. Let X(t) be a random process which consist of the summation of two sinusoidal components...
5. Let X(t) be a random process which consist of the summation of two sinusoidal components as t(t) = A cos(wt) + B sin(wt), where A and B are independent zero mean random variables. (a) (5 points) Find the mean function, pat). (b) (5 points) Find the autocorrelation function Ratta). (e) (5 points) Under what conditions is i(t) wide sense stationary (WSS)?! The questions form the textbook : 1.4, 2.1, 2.4, 2.6 Some trigonometric formulas: cos(A + B) = cos...
P9.3 A random process X(t) has the following member functions: x1 (t) -2 cos(t), x2(t)2 sin(t),...
P9.3 A random process X(t) has the following member functions: x1 (t) -2 cos(t), x2(t)2 sin(t), x3(t)- 2 (cos(t) +sin(t)),x4t)cost) - sin(t), xst)sin(t) - cos(t).Each member function occurs with equal probability. (a) Find the mean function, Hx (t). (b) Find the autocorrelation function, Rx(t1,t2) (c) Is this process WSS? Is it stationary in the strict sense?
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
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)...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay a) (4 points) Find the power spectral density of the output random process y(t), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c) (4 points) Find the output autocorrelation function Фуу(r) d) (1 points) What is the total power in the output process...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay a) (4 points) Find the power spectral density of the output random process y(t), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c) (4 points) Find the output autocorrelation function Фуу(r) d) (1 points) What is the total power in the output process...
The purpose of this assignment is to practice concepts related to the wide-sense stationary processes, filtering, auto-correlation, and power spectral density I. (20 points) Let X(1) denote a wide sense stationary process with μ,-0 and autocorrelation Rdr). Let y(1) = 2 + XUt). What is R)(tz)? Is Y(t) wide sense stationary?
A(t) is a wide-sense stationary random process and is a random variable distributed uniformly over [0, 211]. Furthermore, is independent of A(t). Three random processes X(t), Y(t), and Z(t) are given by X(t) = A(t) cos(20ft + 0) Y(t) = A(t) cos(507t + 0) z(t) = X(t) + y(t) a. Show that X(t) and Y(t) are stationary in the wide sense. b. Show that Z(t) is not stationary in the wide sense.
2. (30 points) Let X(t) be a wide-sense stationary (WSS) random signal with power spectral density S(f) = 1011(f/200), and let y(t) be a random process defined by Y(t) = 10 cos(2000nt + 1) where is a uniformly distributed random variable in the interval [ 027]. Assume that X(t) and Y(t) are independent. (a) Derive the mean and autocorrelation function of Y(t). Is Y(t) a WSS process? Why? (b) Define a random signal Z(t) = X(t)Y(t). Determine and sketch the...
5. Let X(t) be a random process which consist of the summation of two sinusoidal components as t(t) = A cos(wt) + B sin(wt), where A and B are independent zero mean random variables. (a) (5 points) Find the mean function, pat). (b) (5 points) Find the autocorrelation function Ratta). (e) (5 points) Under what conditions is i(t) wide sense stationary (WSS)?! The questions form the textbook : 1.4, 2.1, 2.4, 2.6 Some trigonometric formulas: cos(A + B) = cos...
P9.3 A random process X(t) has the following member functions: x1 (t) -2 cos(t), x2(t)2 sin(t), x3(t)- 2 (cos(t) +sin(t)),x4t)cost) - sin(t), xst)sin(t) - cos(t).Each member function occurs with equal probability. (a) Find the mean function, Hx (t). (b) Find the autocorrelation function, Rx(t1,t2) (c) Is this process WSS? Is it stationary in the strict sense?
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