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Consider the following mean-square differential equation,

driven by a WSS random process  with psd

The differential equation is subject to the initial condition , where the random variable  has zero-mean, variance 5, and is orthogonal to the input random process .

(a) As a preliminary step, express the deterministic solution to the above differential equation, now regarded as an ordinary differential equation with deterministic input and initial condition , not a random variable. Write your solution as the sum of a zero-input part and a zero-state part.

(b) Now returning to the m.s. differential equation, write the solution random process  as a mean-square convolution integral of the input process  over the time interval  plus a zero-input term due to the random initial condition . Justify the mean-square existence of the terms in your solution.

(c) Write….

If a stationary random process is periodic, then we can represent it by a Fourier series with orthogonal coefficients. This is not true in general when the random process, though stationary, is not periodic. Thus, point out the fallacy in the following proposition, which purports to show that the Fourier series coefficients are always orthogonal: First take a segment of length T from a stationary random process  Repeat the corresponding segment of the correlation function periodically. This then corresponds to a periodic random process. If we expand this process in a Fourier series, its coefficients will be orthogonal. Furthermore, the periodic process and the original process will agree over the original time interval.

Certain continuous time communications channel can be modeled as signal plus an independent additive noise

(a) The receiver must process  for the purpose of determining the value of the message random variable M. Argue that a receiver which computes and bases its decision exclusively on, for 

(b) What are the issues in determining which value of  to use? Is there a best value?

Let  be a random process with constant mean  and covariance function,

(a) Show that an m.s. derivative process  exists here.

(b) Find the covariance function of 

In our derivation of the Kalman filter in Section 11.2, we assumed that the Gauss– Markov signal model (Equation 11.2-6) was zero-mean. Here we modify the Kalman filter to permit the general case of nonzero mean for . Let the Gauss–Markov signal model be

Equation 11.2-6

(c) Extend the Kalman filtering Equation 11.2-16 to the nonzero mean case by using the result of (b).

Equation 11.2-16

(d) How do the gain and error-covariance equations change?

1. What are the three major categories of resources that need to be managed during implementation?

2. What are three strategies from the modified model of Parkinson and Associates (1982) for implementing health promotion programs?

3. What are some techniques planners can use to enhance the first day of implementation? What does it mean to kick off a program?

4. What is meant by the term informed consent?

5. What can program planners do to ensure the health and safety of program participants?