The Inverse -Transform By Contour Integration

The Cauchy Residue Theorem:

Let  be a function of the complex variable  and  be a closed path in the -plane. If the derivative  exists on and inside the contour  and if  has no poles at , then



More generally, if the -order derivative of  exists and  has no poles at , then


the values on the right hand side are called the residues of the pole at  (what is left of  after you remove the pole).

For , we can show:


where,

This can be applied to the inverse -transform:



Example: Find the inverse -transform of



Using the contour integral,



1. For ,  has only zeros and hence no poles inside . The only pole occurs at . Hence,



2. If ,  has an nth order pole at . For , we have



you can show that .
The Inverse Z-Transform By Power Series Expansion

Given a  -transform, expand  into a power series of the form:



By uniqueness of the -transform, . When  is rational, the process can be performed by long division.

Example:






Note the implication of this: a pole can be approximated by a collection of zeros. If , the power series can be truncated after a few terms.
Example:










Consider more complicated cases:



The Inverse Z-Transform By Partial Fraction Expansion

In this approach, we factor  into a weighted sum of simpler polynomials:



 can be easily found using the principle of linearity and superposition:



Let us illustrate with a simple example:


therefore,


which implies that


Note the difference in the signal flow graphs:




Direct Form Realization:	2 multiplies/accumulates
	2 registers for delays
Parralel Form Realization:	4 multiplies, 3 additions
	2 registers for delays

2 multiplies/accumulates
2 registers for delays
Repeated roots are a little more complicated:


therefore,


which implies that






what about  ???
Derivation of z-Transform of a Sinewave


Note:


Therefore, if

and,


Thus, for the damped cosine, , the transform is: