I. (5 points) Let {X, } be a stationary series with mean μ and autocovariance function...
Let { be a zero-mean stationary process and let a and b be constants. (a) (5 points) If Xi a+bt+St+Yi, where St is a seasonal component with period 12, show that ▽12V is stationary and express its autocovariance function in terms of that of { (b) (5 points) If X1-(a + bt)Sİ + Y. where Sı is a seasonal component with period 12, show that Vi2 is stationary and express its autocovariance function in terms of that of {
1. Let {Xt} be a stationary process with mean μt = E(Xt) = 0 and autocovariance function γX(k) = E(XtXt−k) - μ2 = E(XtXt+k) - μ2. De ne Yt = 5 + 2t + Xt. (a) Find E(Yt), the mean function for Yt. (b) Find γY (k), the autocovariance function for Yt in terms of γX (k). (c) Is Yt stationary? Explain. (d) De ne a new process Wt as Wt = Yt − Yt−1. Find E(Wt) and γW (k)....
Yt = 5 − 2t + Xt, where {Xt} is stationary with mean 0 and autocovariance function γk. Now, let Wt = Yt − Yt−1. (a) Find the mean function for {Wt}. (b) Find the autocovariance function for {Wt}. (c) Is {Wt} stationary? Why or why not?
Exercise 2.31 Superposition [] Given two independent weakly stationary time series Xt and Yi) with autocovariance functions x(h) and y (h), show that Zt- Xt +Yt is also weakly stationary, with autocovariance function given by yz(h)-x(h)y(h).
Suppose Zt = 2 + Xt -2Xt-1+Xt-2, where {Xt} is zero-mean stationary series with autocovariance function. Calculate the autocovariance of Zt
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...
Let wt for t = . . .,-2,-1, 0, 1, 2, . . . be an independent and identically distributed process with wt ~ M0, σ2). and consider the time series Determine the mean and the autocovariance function of xt and state whether it is stationary
9] 2. Let(X")}, (X12Ί, . . . , { with spectral densities f:i) (w) for-1, 2, . . . , n. Let Xbe n uncorrelated stationary processes Ut i-1 for some constants αι, . . . Prove that Ut is a stationary process. Find the autocovariance function of the process {Ut} in terms of the autocovariance function for each {X)')}, when i - 1,2,... ,k. rove UW) - where fr (w) is the spectral density for the process {Σ¡ι αίΧΙ)}...
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...
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?