Complex Partial Fraction Expansion

In summary, the conversation discusses the process of expanding a function with partial fractions. The student initially follows their normal procedure but realizes they need to approach it differently in order to perform an inverse-z transform. They then encounter a discrepancy in their answer and question why they need to add a certain term. After further thought and calculations, the student figures out the mistake and corrects it.
  • #1
Number2Pencil
208
1

Homework Statement


I'm trying to expand the following with partial fractions:

[tex]H(z) = \frac{z^2 + 1.5932z + 1}{z^2 + 0.9214z + 0.5857}[/tex]


Homework Equations





The Attempt at a Solution



Just going through my normal procedure, I end up with the following:

[tex]H(z) = \frac{0.3359-j0.0857518}{z-(-0.4607+j0.6111)}+\frac{0.3359+j0.0857518}{z-(-0.4607-j0.6111)}+1[/tex]

but it turns out, to facilitate performing an inverse-z transform (which is a different thread), we need to perform the partial fraction a little different:

[tex]\frac{H(z)}{z} = \frac{z+1.5932+z^{-1}}{z^2+0.9214z+0.5857}[/tex]

I follow my normal procedures again:

[tex]\frac{H(z)}{z} = \frac{-0.354-j0.283}{z-(-0.4607+j0.6111)}+\frac{-0.354+j0.283}{z-(-0.4607-j0.6111)}[/tex]

Now this is apparently wrong, and according to my professor, I forgot to add on a 1.7074 to this answer.

My question is: Why do I need to add on the 1.7074? I thought if the degree of the denominator is higher than the numerator, there is no extra term added? And for that matter, how would I actually go about calculating this 1.7074 term?
 
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  • #2
Looks like I posted too soon...because I just figured it out.

Turns out, having a negative power in the partial fraction was a bad idea, so I multiply the top and bottom by z, giving me one extra expansion term. Sure enough, it ends up being

[tex]\frac{1.707}{z}[/tex]

And when I finally multiply H(z) by z to get the final output, the z on the bottom goes away.

Guess I just needed to look at it in fantastic LaTex
 

1. What is complex partial fraction expansion?

Complex partial fraction expansion is a mathematical method used to decompose a rational function into a sum of simpler fractions, where the denominator of each fraction is a power of a linear polynomial. This method is commonly used in integration problems in calculus and in solving linear differential equations.

2. When is complex partial fraction expansion used?

Complex partial fraction expansion is used when solving integration problems involving rational functions, as well as in solving linear differential equations with non-constant coefficients.

3. What is the difference between partial fraction expansion and complex partial fraction expansion?

The difference between partial fraction expansion and complex partial fraction expansion is that the former deals with rational functions with real coefficients, while the latter deals with rational functions with complex coefficients.

4. How do you perform complex partial fraction expansion?

To perform complex partial fraction expansion, the rational function must first be written in the form of a sum of simpler fractions, with the denominator being a power of a linear polynomial. Then, the coefficients of each term in the expansion can be found by equating the coefficients of the corresponding powers of the variable in the original function and in the expanded form.

5. What are some applications of complex partial fraction expansion?

Complex partial fraction expansion has various applications in mathematics and engineering, including in solving differential equations, signal processing, and control systems. It is also used in finding the inverse Laplace transform of a function.

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