EEL6502: HW#1

EEL 6502  -  Spring 1998

Due Friday January 23 at 3pm. Late homework will not be accepted. Anytime you use Matlab (or similar program) you should hand in all of the files you used to generate your answers.

1. Evaluate the autocorrelation function of

   

   where w(n) is a zero-mean white noise sequence with variance and where the 's are constants.

2. The sequences y(n) and x(n) are related by the following difference equation:

   

   where a is a constant. Evaluate the autocorrelation function of y(n) in terms of the autocorrelation of x(n). (Call the autocorrelation functions and ).

3. A zero mean, white noise source w(n) with variance is input to a system h(n) with

   

   Find the correlation function of the output of the system.

4. Consider the second-order autoregressive (AR) process x(n) described by

   

   where and are real-valued constant coefficients, and w(n) is a white noise process with zero mean. The noise process has a variance of one. In this problem the AR coefficients are varied in such a way that the characteristic equation of the process has complex conjugate roots. In particular, we wish to investigate the use of three sets of AR coefficients:

   AR1

   

   AR2

   

   AR3

   

   Compute (analytically) the variance of each AR process x(n).

   (continued on back)

5. Answer the following questions using Matlab.
   1. Numerically generate a sample of each AR process in problem 4 above. (Don't try to show too many points on your plots or else all details will be lost).
   2. Roughly how many samples need to be thrown away before you consider the processes to be stationary? Explain your definition.
   3. Numerically calculate the variance of each noise process. Compare your results to the analytical calculations above.
   4. Use Matlab to plot the power spectrum for each AR process. There are several ways of creating these plots, describe your method.

Final notes:

• Your homework should be in two distinct parts. Part 1 should show the answers, plots, hand calculations etc. that you need to answer the questions. Make sure the the axes on all plots are properly labeled. Part 2 should contain all of the Matlab code that you have written to generate the answers in the first part.
• During the homework assignments, some amount of communication between students is fine and even encouraged. However, the final solutions and all the code that you turn in should be your own.
• You do not necessarily need to use Matlab, you can use any computer language you like as long as you can perform the equivalent computations.
• Good luck!