Prove that the absolute value of autocorrelation function is bound by 1 for a second-order stationary process.
Prove that the absolute value of autocorrelation function is bound by 1 for a second-order stationary...
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?
5. A stationary random process V (t) having an autocorrelation function Sin(101) Rv.v. (1) - is applied to the network shown below T 692 4 MF 1 mH a) Find Sv.v,(w). b) Find |H(w)|? c) Find Sv.v.(w).
2. (20 points) Use first principles to find the autocorrelation function for the stationary process defined by Y, – ste, - 3. (20 points) Identify the following as specific ARIMA models. That is, what are p, d, and 4 and what are the values of the parameters (the 4's and o's)? (a) Y = Y.-0.257,-2 +e, -0.le, -1. (b) Y - 27,,-Y2+, (c) , -0.5, -0,5% 2+e, -- 0.5e,.. +0.250,-2. [Solution]
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...
Use your knowledge of the relationship between spectral density and autocorrelation function in order to answer the following questions. Show your work for full credit. Determine the spectral density of a process with autocorrelation function Rx(t) = 3e-2t a) Determine the spectral density of a process with autocorrelation function Rx(t)-2 sinc(0.51) b) c) Determine the autocorrelation function of a process with spectral density Sx (f) 2 sinc2(f/2) 12 Determine the autocorrelation function ofa process with spectral density Sx(a)-A+ d) Use...
Suppose that a stationary time series, {Y], has an autocorrelation function of the form ρ,-φκ for k > 0, where φ is a constant in the range (-1,+1) (a) Show that Var(Y)--LT n(1-ф) (Hint: Use Equation (3.2.3) on page 28, the finite geometric sum and the related sum (b) If n is large, argue that Var()- (C) Plot ( 1 + φ)/(1-0) for φ over the range-l to +1. Interpret the plot in terms 1n of the precision in estimating...
1. For each function in question 1 of section 6.1 exercises, now use second- order conditions to determine whether each stationary value you found is a maximum, minimum, or point of inflection. y function in the neighborhood of the s (a) y = x3 – 3x2 + 1 (b) y = x4 - 4x3 + 16x - 2 (C) y = 3x3 – 3x - 2 (d) y = 3x4 - 10x3 + 6x2 +1 (e) y = 2x/(x +1)...
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).
Problem 2 Prove the following bound known as the Chemoff bound: Let X be a random variable with moment generating function X (s) defined for s > 0, Then for any a and any s > 0, Hint: To prove the bound apply Markov's inequality with X replaced by e) Apply the се Chemoff bound in case X is a standard normal random variable and a > 0. Find the value of s >0 that gives the sharpest bound, i.e,...
Prove analytically that control loop of an ideal second order process (K/(As2+Bs+1)) controlled by a proportional controller is always stable. Prove analytically that a typical first order process (Ke-Ls/(Ts+1)) with delay in the same control loop may not be stable. Useful information: A, B, T, L always have positive values. e-a = (1-a/2)/(1+a/2)