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.

[1] 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.

[1][2] The method has a plethora of applications in electrical engineering, particularly related to problems in computational electromagnetics, microwave engineering and antenna theory.

[1] A transient electromagnetic signal can be represented as:[3] where The same sequence, sampled by a period of

, can be written as the following: Generalized pencil-of-function estimates the optimal

is defined as the pencil parameter.

diagonal matrices with sequentially-placed

, the generalized eigenvalues of the matrix pencil yield the poles of the system, which are

denotes the Moore–Penrose inverse, also known as the pseudo-inverse.

Singular value decomposition can be employed to compute the pseudo-inverse.

If noise is present in the system,

are combined in a general data matrix,

For efficient filtering, L is chosen between

A singular value decomposition on

are unitary matrices with respective eigenvectors

is a diagonal matrix with singular values of

denotes the conjugate transpose.

Singular values after

, which are below the filtering threshold, are set to zero; for an arbitrary singular value

, the threshold is denoted by the following formula:[1]

and p are the maximum singular value and significant decimal digits, respectively.

For a data with significant digits accurate up to p, singular values below

are obtained through removing the last and first row and column of the filtered matrix

matrices are obtained as:[4] Prefiltering can be used to combat noise and enhance signal-to-noise ratio (SNR).

[1] Band-pass matrix pencil (BPMP) method is a modification of the GPOF method via FIR or IIR band-pass filters.

[1][5] GPOF can handle up to 25 dB SNR.

For GPOF, as well as for BPMP, variance of the estimates approximately reaches Cramér–Rao bound.

[3][5][4] Residues of the complex poles are obtained through the least squares problem:[1] The method is generally used for the closed-form evaluation of Sommerfeld integrals in discrete complex image method for method of moments applications, where the spectral Green's function is approximated as a sum of complex exponentials.

[1][6] Additionally, the method is used in antenna analysis, S-parameter-estimation in microwave integrated circuits, wave propagation analysis, moving target indication, radar signal processing,[1][7][8] and series acceleration in electromagnetic problems.