A transient electromagnetic signal can be represented as:^[3]

${\displaystyle y(t)=x(t)+n(t)\approx \sum _{i=1}^{M}R_{i}e^{s_{i}t}+n(t);0\leq t\leq T,}$ 

where

${\displaystyle y(t)}$  is the observed time-domain signal,

${\displaystyle n(t)}$  is the signal noise,

${\displaystyle x(t)}$  is the actual signal,

${\displaystyle R_{i}}$  are the residues (${\displaystyle R_{i}}$ ),

${\displaystyle s_{i}}$  are the poles of the system, defined as ${\displaystyle s_{i}=-\alpha _{i}+j\omega _{i}}$ ,

${\displaystyle z_{i}=e^{(-\alpha _{i}+j\omega _{i})T_{s}}}$  by the identities of Z-transform,

${\displaystyle \alpha _{i}}$  are the damping factors and

${\displaystyle \omega _{i}}$  are the angular frequencies.

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

${\displaystyle y[kT_{s}]=x[kT_{s}]+n[kT_{s}]\approx \sum _{i=1}^{M}R_{i}z_{i}^{k}+n[kT_{s}];k=0,...,N-1;i=1,2,...,M}$ ,

Generalized pencil-of-function estimates the optimal ${\displaystyle M}$  and ${\displaystyle z_{i}}$ 's.^[4]

For the noiseless case, two ${\displaystyle (N-L)\times L}$  matrices, ${\displaystyle Y_{1}}$  and ${\displaystyle Y_{2}}$ , are produced:^[3]

${\displaystyle [Y_{1}]={\begin{bmatrix}x(0)&x(1)&\cdots &x(L-1)\\x(1)&x(2)&\cdots &x(L)\\\vdots &\vdots &\ddots &\vdots \\x(N-L-1)&x(N-L)&\cdots &x(N-2)\end{bmatrix}}_{(N-L)\times L};}$  ${\displaystyle [Y_{2}]={\begin{bmatrix}x(1)&x(2)&\cdots &x(L)\\x(2)&x(3)&\cdots &x(L+1)\\\vdots &\vdots &\ddots &\vdots \\x(N-L)&x(N-L+1)&\cdots &x(N-1)\end{bmatrix}}_{(N-L)\times L}}$ 

where ${\displaystyle L}$  is defined as the pencil parameter. ${\displaystyle Y_{1}}$  and ${\displaystyle Y_{2}}$  can be decomposed into the following matrices:^[3]

${\displaystyle [Y_{1}]=[Z_{1}][B][Z_{2}]}$ 

${\displaystyle [Y_{2}]=[Z_{1}][B][Z_{0}][Z_{2}]}$ 

where

${\displaystyle [Z_{1}]={\begin{bmatrix}1&1&\cdots &1\\z_{1}&z_{2}&\cdots &z_{M}\\\vdots &\vdots &\ddots &\vdots \\z_{1}^{(N-L-1)}&z_{2}^{(N-L-1)}&\cdots &z_{M}^{(N-L-1)}\end{bmatrix}}_{(N-L)\times M};}$  ${\displaystyle [Z_{2}]={\begin{bmatrix}1&z_{1}&\cdots &z_{1}^{L-1}\\1&z_{2}&\cdots &z_{2}^{L-1}\\\vdots &\vdots &\ddots &\vdots \\1&z_{M}&\cdots &z_{M}^{L-1}\end{bmatrix}}_{M\times L}}$ 

${\textstyle [Z_{0}]}$  and ${\textstyle [B]}$  are ${\textstyle M\times M}$  diagonal matrices with sequentially-placed ${\textstyle z_{i}}$  and ${\textstyle R_{i}}$  values, respectively.^[3]

If ${\textstyle M\leq L\leq N-M}$ , the generalized eigenvalues of the matrix pencil

${\displaystyle [Y_{2}]-\lambda [Y_{1}]=[Z_{1}][B]([Z_{0}]-\lambda [I])[Z_{2}]}$ 

yield the poles of the system, which are ${\displaystyle \lambda =z_{i}}$ . Then, the generalized eigenvectors ${\displaystyle p_{i}}$  can be obtained by the following identities:^[3]

${\displaystyle [Y_{1}]^{+}[Y_{1}]p_{i}=p_{i};}$     ${\displaystyle i=1,...,M}$ 

${\displaystyle [Y_{1}]^{+}[Y_{2}]p_{i}=z_{i}p_{i};}$     ${\displaystyle i=1,...,M}$ 

where the ${\displaystyle ^{+}}$  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, ${\textstyle [Y_{1}]}$  and ${\textstyle [Y_{2}]}$  are combined in a general data matrix, ${\textstyle [Y]}$ :^[3]

${\displaystyle [Y]={\begin{bmatrix}y(0)&y(1)&\cdots &y(L)\\y(1)&y(2)&\cdots &y(L+1)\\\vdots &\vdots &\ddots &\vdots \\y(N-L-1)&y(N-L)&\cdots &y(N-1)\end{bmatrix}}_{(N-L)\times (L+1)}}$ 

where ${\displaystyle y}$  is the noisy data. For efficient filtering, L is chosen between ${\textstyle {\frac {N}{3}}}$  and ${\textstyle {\frac {N}{2}}}$ . A singular value decomposition on ${\textstyle [Y]}$  yields:

${\displaystyle [Y]=[U][\Sigma ][V]^{H}}$ 

In this decomposition, ${\textstyle [U]}$  and ${\textstyle [V]}$  are unitary matrices with respective eigenvectors ${\textstyle [Y][Y]^{H}}$  and ${\textstyle [Y]^{H}[Y]}$  and ${\textstyle [\Sigma ]}$  is a diagonal matrix with singular values of ${\textstyle [Y]}$ . Superscript ${\textstyle H}$  denotes the conjugate transpose.^[3][4]

Then the parameter ${\textstyle M}$  is chosen for filtering. Singular values after ${\textstyle M}$ , which are below the filtering threshold, are set to zero; for an arbitrary singular value ${\textstyle \sigma _{c}}$ , the threshold is denoted by the following formula:^[1]

${\displaystyle {\frac {\sigma _{c}}{\sigma _{max}}}=10^{-p}}$ ,

${\textstyle \sigma _{max}}$  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 ${\textstyle 10^{-p}}$  are considered noise.^[4]

${\textstyle [V_{1}']}$  and ${\textstyle [V_{2}']}$  are obtained through removing the last and first row and column of the filtered matrix ${\textstyle [V']}$ , respectively; ${\textstyle M}$  columns of ${\textstyle [\Sigma ]}$  represent ${\textstyle [\Sigma ']}$ . Filtered ${\textstyle [Y_{1}]}$  and ${\textstyle [Y_{2}]}$  matrices are obtained as:^[4]

${\displaystyle [Y_{1}]=[U][\Sigma '][V_{1}']^{H}}$ 

${\displaystyle [Y_{2}]=[U][\Sigma '][V_{2}']^{H}}$ 

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]

${\displaystyle {\begin{bmatrix}y(0)\\y(1)\\\vdots \\y(N-1)\end{bmatrix}}={\begin{bmatrix}1&1&\cdots &1\\z_{1}&z_{2}&\cdots &z_{M}\\\vdots &\vdots &\ddots &\vdots \\z_{1}^{N-1}&z_{2}^{N-1}&\cdots &z_{M}^{N-1}\end{bmatrix}}{\begin{bmatrix}R_{1}\\R_{2}\\\vdots \\R_{M}\end{bmatrix}}}$ 

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.^[9]

• Estimation of signal parameters via rotational invariance techniques
• Generalized eigenvalue problem
• Matrix pencil
• MUSIC (algorithm)
• Prony's method