The Inverse Z Transform

Given a Z domain function, there are several ways to perform an inverse Z Transform:

• Long Division
• Direct Computation
• Partial Fraction Expansion with Table Lookup
• Direct Inversion

The only two of these that we will regularly use are direct computation and partial fraction expansion.

Inverse Z Transform by Long Division

To understand how an inverse Z Transform can be obtained by long division, consider the function

If we perform long division

we can see that

.

So the sequence f[k] is given by

Upon inspection

Note: We already knew this because the form of F(z) is one that we have worked with previously (i.e., the exponential function).

This technique is laborious to do by hand, but can be reduced to an algorithm that can be easily solved by computer.

Inverse Z Transform by Direct Computation

The need for this technique, as well as its implementation, will be made clear when we consider transfer functions in the Z domain.  We will present this method at that time.

Inverse Z Transform by Partial Fraction Expansion

This technique uses Partial Fraction Expansion to split up a complicated fraction into forms that are in the Z Transform table.  If you are unfamiliar with partial fractions, here is an explanation.

As an example consider the function

For reasons that will become obvious soon, we rewrite the fraction before expanding it by dividing the left side of the equation by "z."

Now we can perform a partial fraction expansion

These fractions are not in our table of Z Transforms.  However if we bring the "z" from the denominator of the left side of the equation into the numerator of the right side, we get forms that are in the table of Z Transforms; this is why we performed the first step of dividing the equation by "z."

So

or

Example

Verify the previous example by long division.

So

and the sequence f[k] is given by

Inverse Z Transform by Direct Inversion

This method requires the techniques of contour integration over a complex plane.  In particular

.

 The contour, G, must be in the functions region of convergence.  This technique makes use of Residue Theory and Complex Analysis and is beyond the scope of this document. 

Laplace transform

The Laplace transform is a powerful tool formulated to solve a wide variety ofinitial-value problems. The strategy is to transform the difficult differential equations into simple algebra problems where solutions can be easily obtained. One then applies the Inverse Laplace transform to retrieve the solutions of the original problems. This can be illustrated as follows: Initial-Value Problems ODE's or PDE's Algebra Problems Difficult Very Easy Solutions of Initial-Value Problems    Solutions of Algebra Problems

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Definition of the Laplace Transform

For a function  defined on , its Laplace transform is denoted as  obtained by the following integral: where  is real and  is called the Laplace Transform Operator. Conditions for the Existence  of a Laplace Transform of f(t) = F(s) 1)  is piecewise continuous on . 2)  is of exponential order as . That is, there exist real constants , , and  such that for all . Note that conditions 1 and 2 are sufficient, but not necessary, for  to exist.

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Definition of the Inverse Laplace Transform

If the Laplace transform of  is , then the we say that the Inverse Laplace Transform of  is . Or, where  is called the Inverse Laplace Transform Operator. Conditions for the Existence of an Inverse Laplace Transform of F(s) = f(t) 1) . 2)  is finite.