Inverse Z Transform

  

Introduction 

Power Series Method 

Partial Fraction Expansion Method 

Residue Method 

 

 Introduction

The Inverse Z Transform allows us to recover the discrete time sequence from the Z function:

 

For the calculations the above integration formula is rarely used. 

If X(z)  is of the form:

 

the Inverse Z Transform is obtained directly from the coefficients. For example:

 

 

In most cases X(z) is in fractional form:

 

For these cases, three methods can be used to calculate the Inverse Z Transform: Power Series, Partial Fractions Expansion, Residues.

 

Power Series Method

Applying long division to the z fractions we obtain power series. The Inverse Z Transform, which is the discrete time sequence, is the coefficients of the power series:

 

 

 

Example 1:

 

Applying long division we have:

 

So the Inverse Z Transform is:

 

 

Computer Form for Power Series Method

Reforming the long division we obtain the following recursive forms, which are appropriate for computer programming:

 

 

 

 

Generally the formula is:

 

A pseudocode for this method could be the following:

x(0)=a(0)/b(0)

for(n=1; n<=total_returns_we_want; n=n+1)

{

summation=0

s=j

for(k=1; k<=s; k=k+1)

{

summation=summation+x(n-k)*b(k)

}

x(n)=(a(n)-summation)/b(0)

}

 

Example 2:

 

Applying long division we have:

 

So the Inverse Z Transform is:

 

Also, using the recursive method we have:

 

 

 

 

Partial Fraction Expansion Method

In the Partial Fraction Expansion Method the Z function is expanded into a sum of simple fractions. The Inverse Z Transform of each partial fraction is then obtained using the transformations tables. (For more about the Partial Fraction Expansion have a look at the appropriate menu selection Mathematics/Algebra/Partial Fractions). This method leads to a closed form solution.

 

It is necessary that the order of the numerator N is lower than the order of the denominator M. If that is not the case, then a long division must be done so that N<=M.

The first step is to find the roots of the denominator and to factorize it. We have two cases: single roots and repeated roots.

Single Roots Case:

For the single roots case we have the following procedure:

 

 

To calculate the coefficients we use the formula (Heaviside Cover Up Method):

 

Then:

 

 

Multiple Roots Case:

For the multiple roots case we have the following procedure (for example double root):

 

 

To calculate the coefficients we use the formulas (Heaviside Cover Up Method):

 

*For the repeated roots coefficients, mind the sequence of the coefficients at the partial fractions.

Then:

 

 

 

 

Example 1:

 

 

 

 

 

Calculation of the partial fraction coefficients:

 

 

So:

 

 

 

 

 

Example 2:

 

 

 

 

 

Calculation of the partial fraction coefficients:

 

 

 

So:

 

 

 

 

 

Residue Method

In the Residue method the Inverse Z Transform is obtained by calculating the contour integral:

 

Using Cauchy’s residue theorem, this contour integration can be calculated as:

 

These residues are given by the formula:

 

For simple poles the above residue formula becomes:

 

For double poles the residue formula becomes:

 

 

Example:

 

 

Calculating the Residues:

 

 

So: