pset01-1

# pset01-1 - GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of...

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GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 6254 Spring 2010 Problem Set #1 Assigned: January 18, 2010 Due Date: January 26, 2010 Reading Assignment : Chapter 4, Sections 4.1 - 4.4.1 (pp. 129-154). Quote of the Week The purpose of models is not to fit the data but to sharpen the questions. Samuel Karlin, 11th R.A. Fisher Memorial Lecture, 1983. PROBLEM 1.1: In Pad´ e’s method, we model a signal x ( n ) as the unit sample response of a filter that has a system function of the form H ( z ) = B q ( z ) A p ( z ) and the filter coefficients a p ( k ) and b q ( k ) are found so that x ( n ) = h ( n ) for n = 0 , 1 , . . . , p + q . In this problem, we consider some generalizations of Pad´ e’s method. (a) One way to generalize the approach of Pad´ e is to consider matching x ( n ) to the unit sample response h ( n ) over other intervals. For example, instead of forcing x ( n ) = h ( n ) for n [0 , p + q ], suppose we would like to find values for a p ( k ) and b k ( q ) so that x ( n ) = h ( n ) for n = n 1 , n 1 + 1 , . . . , n 1 + p + q , i.e. over an interval of length p + q + 1 that begins at n = n 1 rather than n = 0. Is this possible? If so, derive the equations that must be solved and work an example by defining a signal and then finding the model. If it is not possible, explain why

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