# hw4 - 6.003 Homework 4 The following problems are intended...

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6.003 Homework 4 The following problems are intended to help you prepare for the upcoming exam. Please do them by Wednesday, October 7, 2009 . You need not submit your answers: they will NOT be graded. Solutions will be posted. Problems 1. Avoiding excitation Consider the system described by the following diﬀerence equation: y [ n ] = x [ n ] + 5 2 y [ n 1] y [ n 2] . Find an input x [ n ] such that the output y [ n ] is proportional to ( 1 2 ) n for large values of n . Try to minimize the number of non-zero samples in x [ n ]. 2. Periodic system Consider this variant of the Fibonacci system: y [ n ] = y [ n 1] y [ n 2] + x [ n ] where x [ n ] represents the input and y [ n ] represents the output. a. Compute the unit-sample response and show that it is periodic. What is the period? b. Determine the poles of the system. c. Decompose the system functional into partial fractions, and use the result to deter- mine a closed-form expression for h [ n ], the unit-sample response. 3. Closed-loop poles

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hw4 - 6.003 Homework 4 The following problems are intended...

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