Transfer function to impulse response

  • Thread starter yoran
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  • #1
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Homework Statement


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.

Homework 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].

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.
 

Answers and Replies

  • #2
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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:
  • #3
118
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Ok thank you now I think I can solve it.
 

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