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...
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?
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?
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...
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...
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)...
Suppose V is a zero-mean Gaussian random variable, and define the random processes X(t) = Vt and Y(t) = V2t for −∞ < t < ∞. a)Find the crosscorrelation function for these two random processes. b)Are these random processes jointly wide-sense stationary?
Suppose V is a zero-mean Gaussian random variable, and define the random processes X(t) = Vt and Y(t) = V2t for −∞ < t < ∞. a)Find the crosscorrelation function for these two random processes. b)Are these random processes jointly 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.
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).