Inverse Z Transform
Table of Contents
Inverse Z Transform
Finding the time domain signal x(n) from its Z-transform X(z) is known as the inverse Z-transform.
There are several ways to carry out an inverse Z transform given a Z domain function:
- Long Division
- Direct Computation
- Partial Fraction Expansion with Table Lookup
- Direct Inversion
We will only ever use direct computation and partial fraction expansion on a regular basis.
By Long Division
Consider the function to comprehend how the inverse Z Transform can be obtained through long division.
If we perform long division
we can see that
.
So the sequence f[k] is given by
F={1, 0.5, 0.25, ……}
Upon inspection
Note: We already knew this because the form of F(z) (the exponential function) is one with which we are familiar. Although this method requires a lot of manual labor, it can be reduced to a computer-solvable algorithm.
By Direct Computation
We consider transfer functions in the Z domain and show the necessity of this method and its application. At that time, we’ll present this approach.
Inverse Z Transform by Partial Fraction Expansion
This method breaks down a complex fraction into forms found in the Z Transform table using partial fraction expansion. Here is a description of partial fractions in case you don’t know what they are.
As an example consider the function
We rewrite the fraction before expanding it for reasons that will become clear shortly by dividing the left side of the equation by “z.”
So Now we can perform a partial fraction expansion
So Our table of Z Transforms does not include these fractions. However, if we move the “z” from the left side of the equation’s denominator into the right side’s numerator, we obtain forms that are listed in the table of Z Transforms; for this reason, we first divided the equation by “z.”
So
or
F={2, 4, 5, 5.5,……}
Example
Verify the previous example by long division.
So
and the sequence f[k] is given by
f={2, 4, 5, ……}
Inverse Z Transform by Direct Inversion
For this method to work, complex plane contour integration techniques are needed. more specifically
.
So G must be in the region of convergence of the function. So This method, which is outside the purview of this document, uses residue theory and complex analysis.