EEL6502: HW#2

EEL 6502  -  Spring 1998

Due Monday, February 9 at 5pm. As usual, hand in your homework in two distinct parts. Part A answers all of the questions and contains the numerical results and plots from Matlab with the appropriate descriptions. Part B contains the Matlab code use to generate results in part A.

A1

Problem 3.16 Clarkson

A2

Problem 3.22 Clarkson

A3

You are given the inputs and outputs of an unknown system. You know that it is FIR but you don't know the order. The x and d files can be found at
http://www.cnel.ufl.edu/analog/courses/EEL6502/hw2/x.asc and
http://www.cnel.ufl.edu/analog/courses/EEL6502/hw2/d.asc Use the Wiener filter formulation we discussed in class to determine the coefficients of the unknown FIR filter. What are the coefficients? (This problem is interesting because you don't know order of the system so some sort of trial and error is expected.)

A4

What type of filter is the unknown FIR filter (e.g. high-pass, band-stop, etc.)? Show a plot of some kind in matlab that justifies your answer.

A second-order autoregressive (AR) process x(n) is described

where and are real-valued constant coefficients, and w(n) is a white noise process with zero mean and unit variance. We would like to build a single-step predicter with a Wiener filter. Express each of the following in terms of the AR parameters and .

B1

The simplest prediction value is to guess the mean value at each iteration, i.e. y(n)=0. What is the value of the MSE for this predictor?

B2

Another simple estimator is to guess the previous value i.e. y(n)=x(n). What is the value of the MSE for this predictor?

B3

Derive the optimal Wiener Filter weights f(0) and f(1). Show all of your work. What is the MSE for the Wiener Filter? Can the MSE computed above be reduced by adding more taps? Explain.

B4

Problem 3.20

Do the following parts using Matlab (or similar program).

C1

Let and for the second order AR sequence. Implement the three predictors discussed in B1, B2, and B3 in Matlab. What are the three prediction errors ( ) and how do they compare to your estimates in B?

C2

Download the data file in http://www.cnel.ufl.edu/analog/courses/EEL6502/hw2/sunspots.asc This file contains the number of sunspots observed for each month between 1947 and 1991. Run your wiener filter for single-step prediction for a length L=10. What is the prediction error over the whole sequence. (you have to discard the first L of the data samples when you compute the error)

C3

How does your prediction error change with the size of the Wiener filter? Does your Wiener filter always perform better than the estimators described in B1 and B2?

C4

Is the sunspot sequence an ARMA series?