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Homework Help: Transfer function to impulse response

  1. Mar 24, 2008 #1
    1. The problem statement, all variables and given/known data
    Given the transfer function of a linear, time-invariant system
    [tex]H(z)=\frac{z^2+5z}{z^2+5}[/tex]
    compute the impulse response.

    2. Relevant equations
    We are supposed to compute the inverse z-transform with partial fraction decomposition but the problem here is the irreducible quadratic function [tex]z^2+5[/tex].

    3. The attempt at a solution
    In our table of inverse z-transforms they are only functions of the the type
    [tex]\frac{z^{m+1}}{(z-a)^{m+1}}[/tex]
    I tried this.
    [tex]H(z)=\frac{z^2+5z}{z^2+5}=\frac{z^2}{z^2+5}+5\frac{z}{z^2+5}[/tex]
    I can compute the inverse z-transform of [tex]\frac{z^2}{z^2+5}[/tex] just fine, but how do I compute the inverse z-transform of [tex]\frac{z}{z^2+5}[/tex]

    Thanks.
     
  2. jcsd
  3. Mar 24, 2008 #2
    It's not irreducible. [itex]z^2+5=(z-\sqrt{5}i)(z+\sqrt{5}i)[/itex]. Complex numbers are an important part of z transforms.
     
    Last edited: Mar 24, 2008
  4. Mar 24, 2008 #3
    Ok thank you now I think I can solve it.
     
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