EECE 301 Note Set 33a PFE for Z Transform

EECE 301 Note Set 33a PFE for Z Transform - 1 Partial...

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1 Partial Fraction Expansion via MATLAB The “residue” function of MATLAB can be used to compute the partial fraction expansion (PFE) of a ratio of two polynomials. This can be used or Laplace transforms or Z transforms, although we will illustrate it with Z transforms here. The “residue” command gives three pieces of information: ± the residues are given in output vector r, ± the poles are given in output vector p, ± the so-called direct terms are given in output vector k. When the order of the numerator polynomial is less than the order of the denominator polynomial there will be no direct terms. When the order of the numerator equals the order of the denominator there will be one direct term;. When the order of the numerator is one greater than the order of the denominator there will be two direct terms. When the order of the numerator is two greater than the order of the denominator there will be three direct terms; etc. That is, when the order of the numerator is p greater than the order of the denominator (with p 0) there will be p+1 direct terms . The “residue” command requires two input vectors: ± one holding the coefficients of the numerator and ± one holding the coefficients of the numerator. The right-most element in these vectors corresponds to the z 0 coefficient, the next element to the left is the z 1 coefficient, etc., until you reach the highest power; if a power is not present it has a zero coefficient.
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2 It is easiest to explain how to use “residue” by giving examples. Note : We will be expanding “ H ( z ) divided by z” , like is done when trying to find the inverse Z transform – the division by z is a trick used to ensure that after PFE we get a form for which it is usually easy to
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EECE 301 Note Set 33a PFE for Z Transform - 1 Partial...

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