EEL6502: HW6

Due Wednesday, April 24 in class at 3pm. Late homework will not be accepted-this is the last day of class.

Exam 2 will be held during final exams week. Thursday, May 2 10am-noon in Larsen 330 (our usual classroom)

1. Consider the system identification problem with an unknown plant having the following rational system function:

   

   The adaptive filter that is used to model H(z) has two free parameters, a and b, as follows:

   

   The input to both systems is unit variance white noise and the goal is to find the values of a and b that minimize the mean-square error ( ).

   1. (optional) The mean-square error is a bimodal function for the given parameters. Show how to derive the numerical values for a and b for the global minimum and for the local minimum. (Answer: global minimum at (a,b)=(-.311,0.906) and local minimum at (0.114,-0.519)).
   2. Write down the equations for the simplified IIR LMS adaptive filter.
   3. Write down the equations for the filtered signal approach LMS adaptive filter.
   4. A further simplification that has been proposed by Feintuch is to ignore the feedback terms in the equations for the gradient estimates. This simplification results in:

      

      

      Although more efficient than the filtered signal approach, this algorithm may converge to a false minimum, even when the simplified IIR LMS algorithm and the filtered signal approach converge to a minimum. Write down the adaptive filtering equation for W(z) using Feintuch's algorithm.

   5. In order for the filter coefficients and in W(z) to converge in the mean using Feintuch's algorithm, it is necessary that

      

      and

      

      Find the values of and at the global minimum. What does this imply about Feintuch's algorithm?

   6. Find the stationary point of the Feintuch adaptive filter, i.e., the value or values of a and b for which and .
2. The hyperstable Adaptive Recursive Filter (HARF) is an IIR adaptive filtering algorithm with known convergence properties. Due to its computational complexity, a simplified version of the HARF, known as the SHARF, is often used. Although the convergence properties of HARF are not preserved in the SHARF algorithm, both algorithms are similar when the filter is adapted slowly (small ). The coefficient update equations for the SHARF algorithm are:

   

   

   where

   

   is the error signal that has been filtered with an FIR filter C(z). Suppose that the coefficients of a second-order adaptive filter

   

   are updated using the SHARF algorithm with

   

   1. Write down the SHARF adaptive filter equations for H(z).
   2. If the SHARF algorithm converges in the mean, then converges to zero, where

      

      What does this imply about the relationship between the filter coefficients and and ?

   3. What does the SHARF algorithm correspond to when c(z)=1?
3. Show that the for the gamma filter can be computed by changing at each iteration by an amount equal to given by:

   

   Refer to the paper by Principe et al. for the full description and notation for the gamma filter.