A random process is defined as Y(t) X (t) X(t T), where X (t) is a wide sense stationary random process with auto correlation function R X (T) and power spectral density S x (f) (a) Show that R Y () 2R X () R X ( T) R X ( T) (b) Show that S Y (f) 4S X (f) cos 2 (fT) (c) If X (t) has auto correlation function R X () 5() where () is the unit area triangular function, and T 0 5, find and sketch the power spectral density of Y(t) as defined in the problem statement

Question: A random process is defined as Y(t) = X (t) + X(t - T), where X (t) is a wide-sense stationary random process with auto

A random process is defined as Y(t) = X (t) + X(t - T), where X (t) is a wide-sense stationary random process with auto correlation function R_X(T) and power spectral density S_x (f).

(a) Show that R_Y(τ) = 2R_X(τ) + R_X(τ + T) + R_X(τ - T).

(b) Show that S_Y(f) = 4S_X (f) cos²(πfT).

(c) If X (t) has auto correlation function R_X(τ) = 5Λ(τ) where Λ(τ) is the unit-area triangular function, and T = 0.5, find and sketch the power spectral density of Y(t) as defined in the problem statement.

Step by Step Solution

★★★★★

3.43 Rating (150 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a The auto correlation function is b Application of the time delay t... 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

Students Have Also Explored These Related Principles Communications Systems Questions!

Repeat the previous computer exercise with 20 values of uniformly distributed in the range - /4

A random process is composed of sample functions that are square waves, each with constant amplitude T 0 , and random delay Ï as sketched in Figure 7.15. The pdf of Ï is Figure 7.15 (a)...

A random process is composed of sample functions of the form where n(t) is a wide-sense stationary random process with the auto correlation function R n (Ï), and n k = n(kT s ). (a) If T s is...

Q. 13 State any two properties of auto correlation function, Q. 14 Find 4 The autocorrelation function of a stationary process is Rxx (t)= 25+ 1+672 the mean and variance of the process. Q. 15 Define...

We developed the theory of real-valued random processes that vary randomly with time t (i.e., that are defined on a 1-dimensional space in which t is a coordinate). Here we generalize a few elements...

help me plz 9.14 A complex continuous random process is defined by X(t) = Ae][wot+) +W(t) where A and Q are real parameters, wo is a known radian frequency, and W(t) is a stationary zero-mean white...

1 2.3 Definition of a Discrete Probability Function Definition: Let S be a discrete sample space from some experiment. A function P, defined on all events in S, is said to be a probability function...

A random process X (t) is defined by X (t) = A cos (2? f c t), where A is a Gaussian-distributed random variable of zero means variance ?2A. This random process is applied to an ideal integrator,...

The square wave x (t) of Figure of constant amplitude A, period T0, and delay td represents the sample function of a random process X (t). They delay is random, described by the probability density...

Consider a random process X(t) defined by X(t)= sin(2m Wt) in which the frequency W is a random variable with the probability density function 0wB otherwise Determine whether or not this process is...

Brainstorming at United Central Bank (Part I): As part of the 1989 strategic policy plan for United Central Bank, the president, Joseph P. Keith, decided to embark on weekly brainstorming meetings in...

Assume you invest $5,000 today in an investment that promises to return $9,000 in exactly 10 years. a. Use the present value technique to estimate the yield on this investment. b. If a minimum return...

Whenever the client imposes restrictions on the scope of the audit, the auditor should be concernod that manapement may be lrymg lo prevent discovery of misstatements. In such cases, the audifor will...

A profit for one person does not necessarily imply a loss for someone else. Do you agree or disagree? Explain your answer with an example.

The scalar product of vectors A and B is +48.0 m 2 . Vector A has magnitude 9.00 m and direction 28.0 o west of south. If vector B has direction 39.0 o south of east, what is the magnitude of B?

The Mars Polar Lander spacecraft was launched on January 3, 1999. On December 3, 1999, the day Mars Polar Lander touched down on the Martian surface, the positions of the earth and Mars were given by...

All the stars of the Big Dipper (part of the constellation Ursa Major) may appear to be the same distance from the earth, but in fact they are very far from each other. Figure P1.101 shows the...

Consider the following two mutually exclusive projects: The required return on these investments is 13 percent. a. What is the payback period for each project? Note: Do not round intermediate...

Q1: A company process two types of products, a first-class product, and a second-class product, and the process goes through three stages, which are inspection preparation, and packaging. The...

You want to buy $20,000 worth of shares in Tootsie Roll Industries Inc. on margin, but you only have $10,000 of your own money to invest. The remaining $10,000 is borrowed by issuing T - Bills;...