The Laurent Series in DSP - Rhea

The Laurent Series in DSP

Erik Jensen

Introduction:

The Laurent series is a way to descrive any analytic function that has its domain on the complex plane. Much like the Taylor Series it is a sum of a variable to a power multiplied by a corresponding coefficient. However, the Laurent series also has the ability to describe functions with poles by containing negative powers of the complex variable (represented by z) as well. The Laurent series is the link in DSP between the Discrete Fourier Transform (DFT) and the Z-Transform.

___________________________________________________________________________________________________________________________________________________ The Taylor Series

The Taylor series is a good way to begin discussion of the Laurent Series. The Taylor Series is of the form:

the coefficients for a particular function f(x) centered at c∈R can be found with:

a_n = f⁽ⁿ⁾(c)/n!

where f⁽ⁿ⁾ is the n^th derivative of f(x)

___________________________________________________________________________________________________________________________________________________ Background

There are a few terms that have to be defined to discuss the Laurent series. The first is residue, the The second

___________________________________________________________________________________________________________________________________________________ Applications in DSP

As was mentioned above, the Laurent series is actually the link between the DFT and the Z-transform. The Z-Transform is the Laurent series where if x[n] is the DFT of some signal, the n^th coefficient is x[n]. This can more easily be seen by comparing the Laurent series to the Z-transform formula:

The x[n] is in the place where a_n would be in the sum, and otherwise the formulas are the same. Having a good understanding of the Laurent series can then help with computing more difficult Z-transforms, either forward or backward. The concepts involved in finding a Laurent series are also directly applicable to finding things like the region of convergence of some signal's Z-Transform. Finally, understanding what the Z-tranform is allows you to use facts that apply to the Laurent Series to do things like use the pole-residue theorem or Cauchy's Integral formula to analyze the values of the DFT based on its Z-transform.