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Unformatted text preview: EE 482 PROBLEM SET 5 DUE: 18 April 2008 Reading assignment: Chapter 10, sections 10.7 and 10.8; Chapter 9 Problem 21: (20 points) Suppose that we suspect a linear relationship between the sequences { x ( k ) } and { y ( k ) } of the form y ( k ) = m x ( k ) + b, where m and b are unknown constants. Using N measurement points, { x ( i ) ,y ( i ) } for i = 1 N , we wish to calculate the best estimate of the constants m and b in the sense of minimizing the cost function J ( m,b ) = N k =1 e ( k ) 2 , where e ( k ) = y ( k ) y ( k ) and y ( k ) = m x ( k ) + b is the estimate of y ( k ) for a given choice of m and b . If the value of J is minimized for m = m and b = b , then m and b are called the leastsquare estimates of m and b , respectively. The necessary conditions for m and b to minimize J ( m,b ) are J ( m,b ) m m = m = 0 J ( m,b ) b b = b = 0 . 1. (10 points) Solve for m and b in terms of the measurement points and place your answer in the form = A 1 b where = m b A = a 2 2 matrix depending only on the values of x ( k ) b = a 2 1 column vector depending on both x ( k ) and y ( k ) . 2. (10 points) Find the leastsquares estimate of b and m for the N = 20 data points ( x m ( k ) ,y m ( k )) stored in the MatLab file ps5 p21 data.mat that can be downloaded from the EE 482 web page. Write a MatLab mfile that: Loads the measurement data from the file ps5 p21 data.mat. Calculates the leastsquares estimates of b and m using the equation derived in part 1. In a single figure showing y versus x , plots the measurement data ( x m ( k ) ,y m ( k )) as series of open circles and the estimated line y ( k ) = m x m ( k ) + b. Include a copy of the mfile and figure along with your solutions. Make sure that your name appears at the top of the mfile. Problem 22: (20 points) Consider an n th order linear SISO system described by the transfer function G ( z ) = Y ( z ) U ( z ) = b m z m + b m 1 z m 1 + + b z n + a n 1 z n 1 + + a ( m n ) . The corresponding difference equation representation is y ( k ) = a n 1 y ( k 1)  a y ( k n ) + b m u ( k + m n ) + + b u ( k n ) . (1) Suppose we wish to estimate the parameters a ,a 1 , a n 1 and b ,b 1 , ,b m given the sequences u ( k ) and y ( k ) for k = 1 ,...,N . As discussed in lecture, y ( k ) can be expressed as y ( k ) = T ( k ) where the regressor vector ( k ) is a n + m + 1 column vector ( k ) =...
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 Spring '08
 SCHIANO

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