H(z) = (z2 + 0.5z - 0.5)/(z2 + 1.5z + 0.5)

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SUMMARY

The discussion focuses on the transfer function H(z) = (z² + 0.5z - 0.5)/(z² + 1.5z + 0.5) and its manipulation into a more manageable form. Participants confirm that the initial approach of multiplying fractions by z1 and z2 is correct, leading to the expression H(z) = 1/(1 + 0.5z⁻¹) - 1/(1 + z⁻¹ + 0.5z⁻²). The final step involves combining these fractions to derive a difference equation, which is essential for further analysis.

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  • Understanding of transfer functions in control systems
  • Familiarity with z-transform techniques
  • Knowledge of partial fraction decomposition
  • Basic algebraic manipulation of rational functions
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  • Learn about partial fraction decomposition in the context of control theory
  • Explore difference equations and their solutions
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asdf12312
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Homework Statement


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Homework Equations


H(z) = Y(z)/X(z)

The Attempt at a Solution


I realized this wasn't in partial fraction form because the 1+z-1+0.5z-2 has non-real roots. I multiplied the 1st fraction part by z1 and the 2nd fraction by z2, then I combined them into one fraction and I think I am able to get a difference equation at the end, but is the way I am doing it right? Or is there an easier way to do this problem that I am missing.
 
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asdf12312 said:
I multiplied the 1st fraction part by z1 and the 2nd fraction by z2, then I combined them into one fraction

Yes, that's how you do it.

H(z) = 1/(1+0.5z-1) - 1/(1+z-1+0.5z-2) = z/(z+0.5) - z2/(z2+z+0.5).

Combine the fractions.
 

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