EEL6502: HW4

Due Monday March 4 in class at 3pm. Late homework will not be accepted. Note, this due-date is the day before EXAM I which is scheduled for Tuesday, March 5th in room MAT 0010, period E1-E3, 7:20pm-10:10pm. (I do not expect anyone to actually need all of this time).

1. Problem 12-1 in Widrow & Stearns
2. Problem 12-2 in Widrow & Stearns
3. A zero-mean stationary random signal is corrupted by an additive zero-mean stationary noise which is uncorrelated with . A second measurement of the form

   

   is made where is a zero-mean stationary measurement noise (uncorrelated with and )

   1. Draw a block diagram to illustrate how an LMS adaptive filter can be used to enhance the signal .
   2. Find the infinite two-sided, Wiener approximation for the filter W(z).
   3. Derive an expression for the filter output spectrum and the error spectrum
   4. If , derive an explicit form for the filter.
4. We have used two microphones to record two channels of a voice signal corrupted by low frequency noise from an electronic alarm clock. The file hw4.mat in the course directory contains two variables d (voice signal plus noise) and x (primarily noise). A few words are spoken with pauses in between. The noise is evident in the pauses between the words. Write an adaptive filter algorithm to reduce the amount of noise in the signal. You can prototype your algorithm on the first few thousand points before running on the entire signal. Draw a block diagram of your system and describe how you chose values for and the number of taps in the adaptive filter. As usual, turn in all of your MATLAB code. You only need show plots of:
   • The original signal (d) and filtered signal (e). You should see a noticeable reduction of the noise on this plot.
   • The original signal and filtered signal zoomed up so that the details of the signals can be seen.
   • The values of each of the weights (make sure your weight values have converged).
   Feel free to turn in other plots that might be interesting. If you want, you can listen to the signals if your computer supports the MATLAB ``play'' command. The signals were sampled at 11025Hz.