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
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 nth coefficient is x[n]. This can more easily be seen by comparing the Laurent series to the Z-transform formula:
