EEL6502: HW5

Due Wednesday, April 10 in class at 3pm. Late homework will not be accepted.

1. Once again, consider the adaptive noise cancellation problem where the primary signal is zero-mean Gaussian noise with variance ( ) and the reference signal is equal to one ( ). Remember that you only need one tap weight for this problem. Suppose someone decides to implement a variable step rate algorithm where for k>0. Derive an exact expression for the variance of the weight value as a function of k. Discuss the significance of this result.
2. Consider the homogeneous problem , where (assume c>0) and where is the 3-periodic sequence Initialize the weights to zero.
   1. Show that the Sign-Sign algorithm

      

      is always divergent in the sense that ( )

   2. Find limits on for the conventional LMS algorithm to converge.
3. Show that the direct minimization of

   

   leads to

   

   where

   

   and

   

4. (Extra Credit) Prove the matrix inversion lemma. This lemma was discussed in class and is also listed in the handout from Clarkson's book.

   (turn over)

5. The rest of this assignment deals with a very simplified filter design problem. We would like to reject frequencies at f2 while preserving the signals at surrounding frequencies of f1 and f3. We will generate the data using the following code in MATLAB

   isize=300; basis=1:isize; %change isize to whatever length you need;
   d=.1*randn(1,isize);x=.1*randn(1,isize);
   f1=1/100; f2=sqrt(2)/100; f3=sqrt(3)/100;
   for f= [f1 f2 f3]  x=x+sin(2*pi*f* 1:isize); end
   for f= [f1 f3]  d=d+sin(2*pi* 1:isize); end

   Estimate the speed of convergence of the LMS algorithm as a function of assuming two taps. You may use MATLAB to verify your result.
6. Estimate the speed of convergence in the limit of a very large number (L+1) taps. (You should not have to run a program to get this value)
7. Estimate the speedup you expect from using RLS for this problem for (a) 2 taps (b) many taps. Explain.

   The remaining questions must be answered with MATLAB or similar program.

8. Choose reasonable values for and the number of taps and run the LMS algorithm you have used in previous assignments. Give analytic and measured values of convergence rate and misadjustment for the value of and the number of taps you choose you choose.
9. Implement the RLS algorithm (either the standard or the exponentially weighted version) and solve the same filter design problem given above. Compare the misadjustment and convergence rates to that of the LMS implementation.
10. Show the frequency plots for the desired and filter outputs. Plot the frequency response for the filter that you come up with (this does not mean to just plot the fft of ). Does this method succeed in blocking signals at f2 while keeping signals at f1 and f3?