# hw3 - 6.003 Homework 3 Due at the beginning of recitation...

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6.003 Homework 3 Due at the beginning of recitation on Wednesday, September 30, 2009 . Problems 1. Complex numbers a. Evaluate the real and imaginary parts of j j . b. Evaluate the real and imaginary parts of (1 j 3) 12 . c. Express the real part of e 5 in terms of sin θ and cos θ . 2. Characterizing Systems Consider the system represented by the following difference equation: y [ n ] = 3 y [ n 1] 2 y [ n 2] + x [ n ] + x [ n 1] . a. Write a program to tabulate the first 20 values of the unit-sample response. b. Find the system functional. c. Expand the system function in the previous part using partial fractions. d. Express the unit-sample response of the system as a weighted sum of fundamental modes. e. Repeat parts a-d for the following difference equation: y [ n ] = 3 y [ n 1] 2 y [ n 2] + x [ n ] Point out important differences with the previous parts. 3. Black’s Equation Consider the general form of a feedback problem: + F G X Y Notice the minus sign on the adder: it indicates that the lower input is subtracted rather than added. a. Determine the system functional H = Y X . This result is known as Black’s equation.

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