Generalized pencil-of-function method

Generalized pencil-of-function method (GPOF), also known as matrix pencil method, is a signal processing technique for estimating a signal or extracting information with complex exponentials. Being similar to Prony and original pencil-of-function methods, it is generally preferred to those for its robustness and computational efficiency.

The method was originally developed by Yingbo Hua and Tapan Sarkar for estimating the behaviour of electromagnetic systems by its transient response, building on Sarkar's past work on the original pencil-of-function method. The method has a plethora of applications in electrical engineering, particularly related to problems in computational electromagnetics, microwave engineering and antenna theory.

A transient electromagnetic signal can be represented as:

where

The same sequence, sampled by a period of ${\displaystyle T_{s}}$, can be written as the following:

Generalized pencil-of-function estimates the optimal ${\displaystyle M}$ and ${\displaystyle z_{i}}$'s.

For the noiseless case, two ${\displaystyle (N-L)\times L}$ matrices, ${\displaystyle Y_{1}}$ and ${\displaystyle Y_{2}}$, are produced:

where ${\displaystyle L}$ is defined as the pencil parameter. ${\displaystyle Y_{1}}$ and ${\displaystyle Y_{2}}$ can be decomposed into the following matrices: