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Partial Fractions Decomposition and Inverse Z Transform of H(z) = (z^2) / (z^2 - 1)

  1. Jul 19, 2009 #1
    1. The problem statement, all variables and given/known data
    What is the inverse Z transform of
    [tex]
    H(z)=\frac{z^2}{z^{2}-1}
    [/tex]


    2. Relevant equations
    Z transform table can be found at http://en.wikipedia.org/wiki/Z-transform" [Broken]


    3. The attempt at a solution

    [tex]
    H(z)=\frac{z^2}{z^{2}-1} = 1+\frac{1}{z^{2}-1}
    [/tex]

    [tex]
    \frac{1}{z^{2}-1}=\frac{A}{z+1}+\frac{B}{z-1}
    [/tex]

    [tex]
    1=A(z-1)+B(z+1)=z(A+B)+(-A+B)
    [/tex]

    [tex]
    0=(A+B), 1=(-A+B)
    [/tex]

    [tex]
    A=-0.5, B=0.5
    [/tex]

    [tex]
    H(z)=\frac{z^2}{z^{2}-1} = 1+\frac{1}{z^{2}-1} = 1+\frac{-0.5}{z+1}+\frac{0.5}{z-1}
    [/tex]


    [tex]
    h(k)=Z^{-1}\{H(z)\} = Z^{-1}\{1\} + Z^{-1}\{\frac{-0.5}{z+1}\} + Z^{-1}\{\frac{0.5}{z-1}\} = \delta(n) + ? + ?
    [/tex]

    I don't know how to figure out what the inverse of [itex]\frac{-0.5}{z+1}[/itex] and [itex]\frac{0.5}{z-1}[/itex]. Any help?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
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