# Inverse Z Transform

Partial Fraction Expansion 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: