EEL6502: HW1

Due Monday January 22 in class at 3pm. Late homework will not be accepted. Anytime you use Matlab (or similar progam) you should hand in all of the files you used to generate your answers.

1. The sequences and are related by the following difference equation:

   

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

2. Do problems 6, 7 and 8 in Chapter 2 in Widrow & Stearns. Note that the solutions for these problems are given at the end of the chapter.
3. Consider the second-order autoregressive (AR) process described by

   

   where and are real-valued constant coefficients, and 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:

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   1. Compute analytically the variance of the AR process for each case.
   2. Use Matlab and measure the variance of the noise process in each case.
   3. Plot samples of the corresponding time variation of the AR process.
   4. Use Matlab to plot the autocorrelation function r(l) for each case.
4. Download the file hw1_dat from the course ftp directory. (You will be given information in class about how to download this file) In this file two variables are defined: inx and des for the input and desired signal respectively. Computer the optimal Wiener filter weights assuming an order of 5.
5. Write a paragraph that discusses exactly what is accomplished by the Wiener filter in the last question-you may want to substantiate your ideas with frequency plots.