Consider a random process of the form , X (t) b cos (2t ) Where is a constant , is a uniform random variable over 0, 2 , and is a random variable which is independent of and has a PDF, f () Find the PSD, SXX (f) in terms of f () In so doing, prove that for any S (f) which is a valid PSD function, we can always construct a random process with PSD equal to S (f)

Question: Consider a random process of the form , X (t) = b cos (2t + ) Where is a constant , is a uniform random

Consider a random process of the form ,
X (t) = b cos (2πΨt + θ)
Where is a constant θ, is a uniform random variable over [0, 2π], and Ψ is a random variable which is independent of and has a PDF, fΨ (Ψ). Find the PSD, SXX (f) in terms of fΨ (Ψ). In so doing, prove that for any S (f) which is a valid PSD function, we can always construct a random process with PSD equal to S (f).

Step by Step Solution

★★★★★

3.34 Rating (160 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Where FT refers to the Fourier Transform operator Note that Therefore Hence for any valid PSD ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

589-M-S-S-M (595).docx

120 KBs Word File

Students Have Also Explored These Related Statistics Questions!

Consider a random process Z (t) = X (t) + Y (t). (a) Find an expression for SZZ (f) in terms of SXX (f), SYY (f) and SXY (f). (b) Under what conditions does SZZ (f) = SXX (f) + SYY (f)?

Consider a random sinusoidal process of the form X (t) = bcos (2ft + ), where has an arbitrary PDF f(), . Analytically determine how the PSD of X (t) depends on f(). Give an intuitive explanation...

Consider a random variable X with density function (a) Find p = E(X) and 2 = E[(X - p)2]. (b) Demonstrate that Chebyshev's theorem holds for k = 2 and k = 3.

the density function of a Random Variable X is defined as follows. The expected value of X is: fx(x) 0.4 0.3 6. Select one: O a. 6.4 O b. none of these O c. 4.2 O d. 6.5

Consider a random process () defined by: () = 2 cos 0 where is a constant and is a uniform random variable over (0,1). (a) Sketch three typical sample functions of (). (b) Find the mean (). (c) Find...

You are given a Random Process, X(t) = sin(t + ) where is a constant and is a uniform random variable over the range 0 to 2. a) Find F(x,t) and and f(x,t). Let t=1 and plot it f(x,1). b) We select a...

2. Let X(t) and Y(t) be both zero-mean WSS random processes. Consider the random process Z(t)=x(t)+Y(t). Determine 3. (a) Auto correlation function and Power spectral density of Z(t) if X(t) and Y(t)...

Determine the hazard rate function for the random variable Exercise 8 (#1.22). Let v be a measure on a o-field F on ? and f and g be Borel functions with respect to F. Show that (i) if f fdy exists...

In creating solutions for business problems using predictive analytics, it is important to consider the following in identifying the right statistical analysis to perform: A) Whether the variables...

List four techniques for comparing the costs and benefits of a proposed system.

Problems 33-38, use Table II to find each indefinite integral. e- dx

Which type of artery has the least resistance, thickest walls, the fastest blood flow?

What does it mean for a company to manage diversity effectively? How would you know if a company is doing a good job of managing diversity?

Suppose 40 cars start at the Indianapolis 500. In how many ways can the top 3 cars nish the race?

In how many ways can the top 2 horses nish in a 10-horse race?

Four members from a 20-person committee are to be selected randomly to serve as chairperson, vice-chairperson, secretary, and treasurer. The rst person selected is the chairperson; the second, the...

Neal Enterprises has no debt. Its current total value is $77.2 million. Assume debt proceeds are used to repurchase equity. Requirement 1: Ignoring taxes, what will the companys value be if it sells...

c. At the 10% significance level, what is your conclusion about the overall significance of the model? I (Click to select) | Ho because the p-value is (Click to select) | than the significance level....

You are the CFO of a company called hamburger. Unfortunately, the firms headquarters (where all the records are kept) have recently been destroyed by fire. So, your task is to recreate the cash flow...