EEL 6545 Final, Dec. 10. 2 hours. Open book and notes. page 1
1. (6 points) If X is uniformly distributed between 0 and 1, and Y = 2X+3, draw and label the graph of the pdf for Y.
2. (6 points) If X is normal with zero mean and unit standard deviation and Y = 2X+3, write out the formula for the pdf for Y.
3. A system is described by the difference equation x(n) + 0.5x(n-2) = v(n) ,
where v(n) is zero mean white noise with unit power. Please note the "n-2". The process is run until it becomes stationary.
a. (10 points) What are the values of the autocorrelation function rX(0), rX(1), rX(2), rX(3), rX(4)?
b. (5 points) What is the standard deviation of X(n)?
c. (5 points) What is the mean of X(n)?
d. (10 points) Use these 9 autocorrelation values to estimate the power spectral density of X.
e. (10 points) Use the transfer function approach to compute the power spectral density of X.
4. Using Bartlett's method, the autocorrelation for a zero-mean stationary process is found to be
4 / [5(-2)|k/2| ] if k is even
rX(k) = {
0 if k is odd .
Write out a few terms of this to be sure you understand rX(2) = -0.4 , rX(3)=0.
a. (10 points) What is the difference equation for an ARMA(2,0) process driven by zero mean white noise with unit power that models this autocorrelation function? Please carry out the computation even if you see a short cut.
b. (5 points) Read the statement of this problem again and give your opinionwhether or not the process is ergodic; state your justification.
c. (5 points) How many values of the autocorrelation function does your model fit?
5. (14 points) The autocorrelation function of a stationary zero mean process X is the same as given in Problem 4. Assume that x(n) is measured exactly; in other words, the measurement process can be described by y(n) = x(n) + w(n), where w(n) is zero mean noise with power zero. (Usually the theory provides estimators in terms of measured values, not true values – here the measured values are the same as the true values.)
Construct the minimum mean square error predictor for x(n+1) as a linear function of x(n) and x(n-1) (which are the same as y(n) and y(n-1)).
6. I am going to ask you to rework problem one from last week, but inserting one extra stage (part (a)). Your final answer to part (b) should work out to be the same as my answer last week.
a. (7 points) Starting at n=0, a marker is placed at x=0: no uncertainty. Then a coin is flipped; if it is heads, the marker moved to x = +1; if it is tails, the marker is moved to x = -1. It is not shifted. Its position is measured with a very crude instrument, having a standard deviation of ∞. (You may use 10100 if this makes you uncomfortable.) The result of the measurement is "x(1) = 0", but remember than it has no validity. What is the minimum mean square error estimate for the position of the marker after the first flip?
b. (7 points) After a second flip of the coin, the marker is moved one unit to the right if the coin reads heads, and 1 to the left if it reads tails; but then it is shifted one-third of the way back to the origin. The position of the coin is now measured to be +0.5, with a better measuring device that is unbiased and has a standard deviation error of only 1. What is the minimum mean square error estimate for the position of the marker after the second flip?
Remember that the Kalman estimator can be constructed for any process satisfying
d(n) = a(n-1) d(n-1) + noise(n),
measured value(n) = d(n) + noise(n),
and all the noises are uncorrelated. The a's can change. You should use these equations to model the transition after the first flip, and then the transition after the second flip - the a's will be different and the noise powers will be different.