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Partial Fraction decomposition

  1. Nov 18, 2013 #1

    I am currently in a signals and systems course where we're looking at taking the Fourier Transform with [itex]Y(\omega) = H(\omega)X(\omega)[/itex].

    If [itex]Y(\omega) = \frac{\omega + 2}{(\omega + 1 )^{2} (\omega + 3)} = \frac {A_{11}}{(\omega +1)} + \frac {A_{12}}{(\omega + 1)^{2}} + \frac {A_{21}}{(\omega + 3)} [/itex]

    Oppenheim, author of Signals and Systems, states that you can use the following equation to solve for the value of A.

    [itex] A_{ik} = \frac {1}{(\sigma_{i} - k)!} \left[\frac {d^{\sigma_{i} - k}} {dv^{\sigma_{i} - k}} \left[(v - \rho_{i})^{\sigma_{i}}G(v)\right]\right] [/itex]

    i and k are the subscripts that are associated with the values of the numerator values. This example is from Oppenheim's signal and systems book. This is NOT a homework problem.

    Could someone help me understand the second equation which should be used to determine the numerator values? Once I understand how to use this equation I can them extrapolate it to my homework problems.

    In the book they have the following to solve for [itex] A_{11} [/itex]

    [itex] A_{11} = \frac {1}{(2-1)!} \left[\frac {d} {dv} \left[(v + 1)^{2}G(v)\right]\right] [/itex]

  2. jcsd
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