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Assignment5 - ECE 2200 / Spring 2009: Assignment Five 1 ECE...

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Unformatted text preview: ECE 2200 / Spring 2009: Assignment Five 1 ECE 2200 / Spring 2009 SAMPLED-DATA SYSTEMS ASSIGNMENT FIVE 1 1. Often it is desirable to add a component to an existing system in order to make it produce a specific behavior. In order to properly design this component knowledge of the transfer function of the already existing components is necessary. Unfortu- nately, in many practical circumstances foreknowledge of a key component’s transfer function parameters is incomplete or unavailable. Consider the system with input x and output y described (adequately) by the difference equation model y [ k ] = b x [ k ] + b 1 x [ k- 1] + b 2 x [ k- 2] + b 3 x [ k- 3] with the values of the b i unknown, but records of the input and output from sample 0 (at time zero) through sample N (at time NT s with T s the sample period) are available { x [0] ,x [1] ,x [2] ,...,x [ N ] } { y [0] ,y [1] ,y [2] ,...,y [ N ] } . The first output for which sufficient input samples are available is y [3], i.e. y [3] = b x [3] + b 1 x [2] + b 2 x [1] + b 3 x [0] Similarly, y [4] = b x [4] + b 1 x [3] + b 2 x [2] + b 3 x [1] and y [5] = b x [5] + b 1 x [4] + b 2 x [3] + b 3 x [2] Collecting the parameters and outputs into column vectors B = [ b ,b 1 ,b 2 ,b 3 ] T and Y = [ y [3] ,y [4] ,y [5] ,y [6] ,...,y [ N ]] T results in Y = Φ B where Φ = x [3] x [2] x [1] x [0] x [4] x [3] x [2] x [1] x [5] x [4] x [3] x [2] x [6] x [5] x [4] x [3] . . . . . . . . . . . . x [ N ] x [ N- 1] x [ N- 2] x [ N- 3] 1 Updated April 9, 2009. ECE 2200 / Spring 2009: Assignment Five 2 In practice, the actual system typically has some other contributors to its output that are not captured by the finite-dimensional parametric model. For example, the output may be measured in a noisy environment or the system actually has more...
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This note was uploaded on 03/02/2011 for the course ECE 2200 taught by Professor Johnson during the Spring '05 term at Cornell.

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Assignment5 - ECE 2200 / Spring 2009: Assignment Five 1 ECE...

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