# Matrix pencil method

The matrix pencil method is a numerical technique that estimates the poles and residues of a sum of damped or undamped complex exponentials from uniformly sampled data by solving a generalized eigenvalue problem. It is used widely in signal processing and electromagnetics for high-resolution parameter extraction from noisy transient records.<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> The method reformulates exponential fitting as a generalized eigenvalue problem for the nonlinear parameters (frequencies and damping) and a structured linear system for the linear coefficients (amplitudes and phases), and it is considered a stable computational approach to the exponential analysis problem.<sup>[2](https://link.springer.com/article/10.1007/s11075-023-01564-3)</sup>

| Key fact | Detail |
|---|---|
| Signal model | Sum of M damped complex sinusoids; with signed damping exponent \( \alpha_i \) (positive for growth, negative for decay), angular frequency \( \omega_i \), and sample interval \( T_s \), the method estimates poles \( z_i = e^{(\alpha_i + j \omega_i) T_s} \) (cyclic frequency \( f_i = \omega_i / (2 \pi) \)) plus amplitudes and phases<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> |
| Core computation | Generalized eigenvalues of the pencil \( Y_1 - z Y_0 \) built from shifted Hankel matrices<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup><sup> • </sup><sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC6260881/)</sup> |
| Validity condition | \( M \le L \le N - M \), where \( L \) is the pencil parameter and \( N \) the sample count<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> |
| Noise robustness | With optimal L, variance efficiency relative to the Cramér–Rao bound is 27/24 = 1.125, independent of N<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> |
| Cost | One SVD of an \( (N-L) \times L \) data matrix and M eigenvalues of an \( M \times M \) matrix<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> |
| Sample requirement | 2n samples for n exponentials in the noiseless case (dilation \( \sigma = 1 \)), with \( \lvert \operatorname{Im}(\phi_i) \rvert \Delta < \pi \)<sup>[2](https://link.springer.com/article/10.1007/s11075-023-01564-3)</sup> |
| Introducing paper | Y. Hua and T.K. Sarkar, IEEE Transactions on Acoustics, Speech, and Signal Processing, 1990<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> |

## How it works

The method assumes the measured sequence \( y_k = x_k + n_k \), where \( x_k \) is a sum of M damped complex sinusoids with amplitudes \( \lvert b_i \rvert \), phases, damping factors \( a_i \), and frequencies \( w_i \). The goal is to estimate the poles \( z_i = e^{a_i + j w_i} \).<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup>

From the samples one forms two shifted Hankel data matrices, \( Y_0 \) and \( Y_1 \), one being the other with its first or last column removed. Each signal pole \( z_i \) is a rank-reducing number of the matrix pencil \( X_1 - z X_0 \), meaning the pencil loses rank when z equals a pole, provided \( M \le L \le N - M \). The name "pencil" refers to this family of matrices parameterized by the scalar z.<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> In practice, with a rank-\(M\) truncated SVD \( Y_0 = U_M \Sigma_M V_M^H \), the poles are estimated as the eigenvalues of \( A = \Sigma_M^{-1} U_M^H Y_1 V_M \), since the truncation suppresses the noise-driven small singular values of the raw Hankel matrices.<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC6260881/)</sup>

## How it is done

1. Form an \( (N-L) \times (L+1) \) Hankel matrix Y from the noisy measurements, with \( M \le L \le N - M \).<sup>[4](https://eusipco2025.org/wp-content/uploads/pdfs/0002637.pdf)</sup>
2. Choose the pencil parameter L. Values between \( N/3 \) and \( 2N/3 \) are reported as good choices;<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> a perturbation analysis gives the narrower variance-minimizing range \( N/3 \le L \le N/2 \).<sup>[5](https://arxiv.org/html/2502.17047v1)</sup> In antenna work, \( L \) between \( K/3 \) and \( K/2 \) (\( K \) = number of samples) is used to filter noise.<sup>[6](https://hal.science/hal-01407331v1/document)</sup>
3. Compute the SVD \( Y_0 = U \Sigma V^H \) and truncate small singular values to estimate the model order M and suppress noise; the poles are then the eigenvalues of \( A = \Sigma^{-1} U^H Y_1 V \) rather than of the raw pencil.<sup>[4](https://eusipco2025.org/wp-content/uploads/pdfs/0002637.pdf)</sup> SVD-based rank reduction is what makes the eigenvalue identification robust against noise.<sup>[7](https://www.osti.gov/biblio/2331204)</sup>
4. Convert eigenvalues to physical parameters: damping \( \alpha = \log(\lvert \lambda \rvert)/T_s \) and frequency \( f = \operatorname{atan2}(\operatorname{imag}(\lambda), \operatorname{real}(\lambda))/(2 \pi T_s) \).<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC6260881/)</sup>
5. Recover the coefficients \( c_j \) as a least-squares solution of the overdetermined Vandermonde system built from the estimated poles.<sup>[8](https://www-user.tu-chemnitz.de/~potts/paper/LAA_Prony.pdf)</sup>

The algorithm requires an SVD of an \( (N-L) \times L \) data matrix and M eigenvalues of an \( M \times M \) matrix, which is computationally more efficient than polynomial (SVD-Prony) methods that must find L roots of an L-degree polynomial.<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup>

## Origin

The method descends from the classical Prony technique, which approximates uniformly sampled data as a sum of complex exponentials by converting a nonlinear fit into linear equations plus polynomial root-finding; the exponential-fitting problem itself is at least two centuries old.<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC6260881/)</sup><sup> • </sup><sup>[9](https://doi.org/10.1109/74.370583)</sup> An important precursor is the SVD-Prony method of R. Kumaresan and D. Tufts, published in IEEE Transactions on [Acoustics](https://www.edgepedia.org/acoustics), Speech, and Signal Processing in 1982, which addressed estimation of exponentially damped sinusoids and pole-zero modeling in noise.<sup>[10](https://doi.org/10.1109/tassp.1982.1163974)</sup> Hua and Sarkar's earlier generalized pencil-of-function (GPOF) method extracted poles of an electromagnetic system from its transient response via a generalized eigenvalue problem, with advantages over the Prony method in both computation and noise sensitivity, approaching the [Cramér–Rao bound](https://www.edgepedia.org/cramer-rao-bound) above the SNR threshold.<sup>[11](https://doi.org/10.1109/8.18710)</sup> The matrix pencil method itself was introduced by Y. Hua and T.K. Sarkar in "Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise," IEEE Transactions on Acoustics, Speech, and Signal Processing, 1990.<sup>[12](https://doi.org/10.1109/29.56027)</sup>

## Variants

**Forward-backward pencil.** Applying the pencil to forward-backward averaged data is more robust to noise than the forward-only pencil when signal poles lie on the unit circle; in early formulations the pencil parameter was fixed to M.<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> Yanhui Liu, Qing Huo Liu, and Zaiping Nie applied the forward-backward matrix pencil method to reduce the number of elements in shaped-beam pattern synthesis, in IEEE Transactions on Antennas and Propagation in 2009.<sup>[13](https://doi.org/10.1109/tap.2009.2037709)</sup>

**Noise-mitigation variants.** Named variants addressing noise sensitivity include forward-backward MP (limited to undamped exponentials), band-pass MP (effective for damped exponentials but requiring prior knowledge and more computation), total-least-squares MP (a pre-filtering step for noise mitigation), and total forward-backward MP, which surpasses the FFT in variance beyond a specific SNR threshold at the cost of larger bias.<sup>[5](https://arxiv.org/html/2502.17047v1)</sup>

**Multidimensional and structured extensions.** Y. Hua's 1992 paper "Estimating two-dimensional frequencies by matrix enhancement and matrix pencil" (IEEE Transactions on Signal Processing) extended the approach to two-dimensional frequency estimation.<sup>[14](https://doi.org/10.1109/78.157226)</sup> A short-time matrix pencil method for extracting early- and late-time responses in transient analysis of scatterers was published by Reza Rezaiesarlak and Majid Manteghi in IEEE Transactions on Antennas and Propagation in 2015.<sup>[15](https://doi.org/10.1109/tap.2015.2477096)</sup> Multiscale generalizations introduce dilation \( \sigma \) and translation \( \tau \), extending the pencil formulation to trigonometric, hyperbolic, polynomial, Gaussian, sinc, and gamma functions.<sup>[2](https://link.springer.com/article/10.1007/s11075-023-01564-3)</sup>

## Applications

**Electromagnetic scattering and SEM poles.** The singularity expansion method models the late-time response of an object as a sum of decaying exponentials characterized by poles and residues.<sup>[6](https://hal.science/hal-01407331v1/document)</sup> The matrix pencil method estimates these natural resonances from transient responses recorded along multiple look directions, producing a single pole estimate without averaging waveforms, using a total-least-squares SVD-based formulation described as most robust in the presence of random noise.<sup>[16](https://www.math.ucdavis.edu/~saito/data/sonar/SarkarMultiLook.pdf)</sup>

**Power systems and other fields.** Prony analysis, matrix pencil, and the [eigensystem realization algorithm](https://www.edgepedia.org/eigensystem-realization-algorithm) are the three major methods for identifying power-system eigenvalues from measurement data. [Matrix pencil](https://www.edgepedia.org/matrix-pencil) was applied to power systems in 2005 to estimate dominant oscillation modes, and has been shown more capable of mode extraction than Prony analysis for noisy power-system signals.<sup>[7](https://www.osti.gov/biblio/2331204)</sup> The method has also been applied to high-resolution de-embedding of microwave and antenna measurements, direction-of-arrival estimation, [deep-level transient spectroscopy](https://www.edgepedia.org/deep-level-transient-spectroscopy), medical percussion signal analysis, and anechoic chamber evaluation.<sup>[17](https://onlinelibrary.wiley.com/doi/10.1002/9781119076230.ch2)</sup>

## Limitations and alternatives

**Noise thresholds and extraneous poles.** Additive noise makes the Hankel submatrices full rank, producing \( L - M \) extraneous eigenvalues that contain only noise-related information, so the model order must be determined before parameter estimation.<sup>[5](https://arxiv.org/html/2502.17047v1)</sup> For a single damped sinusoid, the Kumaresan–Tufts method gives better z-plane localization of the mode, working for SNR above 5 dB, while the matrix pencil method works for SNR above 10 dB; the matrix pencil method exhibits lower variance but greater bias in the damping factor, and the damping estimate is biased for both methods.<sup>[18](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2004/defevent/papers/cr1145.pdf)</sup> The SNR threshold above which theoretical variance expressions hold is reported differently: first-order perturbation results are described as good approximations at roughly 30 dB and above,<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> while a later analysis finds the mean-square-error expression valid beyond about 10 dB for a damping factor of −0.1, with the threshold depending on damping.<sup>[19](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2009/contents/papers/1569187944.pdf)</sup>

**Parameter choice.** The method is least sensitive to noise when \( L \) lies between \( N/3 \) and \( 2N/3 \) and most sensitive when \( L \) equals \( M \) or \( N - M \);<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> the optimal L and the number of equations used for amplitude estimation depend strongly on the damping factor, with a heuristic choice of \( L \in [N/3, N/2] \) keeping variances near minimum.<sup>[19](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2009/contents/papers/1569187944.pdf)</sup>

**Comparison with other methods.** The matrix pencil approach has lower variance of the parameter estimates than polynomial-type methods such as Prony's, and is computationally more efficient.<sup>[9](https://doi.org/10.1109/74.370583)</sup> Unlike the polynomial method, the pencil method does not require all poles to be inside or outside the unit circle to separate extraneous poles.<sup>[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)</sup> [Polynomial](https://www.edgepedia.org/polynomial) methods have difficulty obtaining roots when the number of poles M exceeds 50, whereas the pencil method finds poles directly as eigenvalues of a single matrix in one step.<sup>[20](https://scholarsmine.mst.edu/cgi/viewcontent.cgi?article=5114&context=masters_theses)</sup>

## References

1. [Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise (Hua & Sarkar, IEEE Trans. Acoust., Speech, Signal Process., 38(5):814-824, 1990)](https://intra.ece.ucr.edu/~yhua/MPM.pdf)
2. [Multiscale matrix pencils for separable reconstruction problems (Cuyt & Lee, Numerical Algorithms, 2023)](https://link.springer.com/article/10.1007/s11075-023-01564-3)
3. [Coding Prony's method in MATLAB and applying it to biomedical signal filtering (PeerJ / PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC6260881/)
4. [SAMP++: Robust Structure-Aware Matrix Pencil (EUSIPCO 2025)](https://eusipco2025.org/wp-content/uploads/pdfs/0002637.pdf)
5. [Structure-Aware Matrix Pencil Method (arXiv, 2025)](https://arxiv.org/html/2502.17047v1)
6. [Comparison of TLS-Prony and TLS-Matrix Pencil methods applied on antenna backscattered responses (DGA / HAL)](https://hal.science/hal-01407331v1/document)
7. [A tutorial on data-driven eigenvalue identification: Prony analysis, matrix pencil, and eigensystem realization algorithm (Almunif, Fan & Miao, Int. Trans. Electr. Energy Syst., 2020)](https://www.osti.gov/biblio/2331204)
8. [Parameter estimation for nonincreasing exponential sums by Prony-like methods (Potts & Tasche, Linear Algebra Appl.)](https://www-user.tu-chemnitz.de/~potts/paper/LAA_Prony.pdf)
9. [Using the matrix pencil method to estimate the parameters of a sum of complex exponentials (Sarkar & Pereira, IEEE Antennas and Propagation Magazine, 1995), aggregator copy](https://doi.org/10.1109/74.370583)
10. [R. Kumaresan, D. Tufts (1982). Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise. IEEE Transactions on Acoustics Speech and Signal Processing.](https://doi.org/10.1109/tassp.1982.1163974)
11. [Generalized pencil-of-function method for extracting poles of an EM system from its transient response (Hua & Sarkar, IEEE Trans. Antennas and Propagation, 1989), aggregator record](https://doi.org/10.1109/8.18710)
12. [Y. Hua, T.K. Sarkar (1990). Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise. IEEE Transactions on Acoustics Speech and Signal Processing.](https://doi.org/10.1109/29.56027)
13. [Yanhui Liu, Qing Huo Liu, Zaiping Nie (2009). Reducing the Number of Elements in the Synthesis of Shaped-Beam Patterns by the Forward-Backward Matrix Pencil Method. IEEE Transactions on Antennas and Propagation.](https://doi.org/10.1109/tap.2009.2037709)
14. [Y. Hua (1992). Estimating two-dimensional frequencies by matrix enhancement and matrix pencil. IEEE Transactions on Signal Processing.](https://doi.org/10.1109/78.157226)
15. [Reza Rezaiesarlak, Majid Manteghi (2015). Accurate Extraction of Early-/Late-Time Responses Using Short-Time Matrix Pencil Method for Transient Analysis of Scatterers. IEEE Transactions on Antennas and Propagation.](https://doi.org/10.1109/tap.2015.2477096)
16. [Application of the matrix pencil method for estimating the SEM (singularity expansion method) poles (IEEE Trans. Antennas and Propagation)](https://www.math.ucdavis.edu/~saito/data/sonar/SarkarMultiLook.pdf)
17. [Modern Characterization of Electromagnetic Systems and Its Associated Metrology, Chapter 2: Matrix Pencil Method (Sarkar et al., 2021)](https://onlinelibrary.wiley.com/doi/10.1002/9781119076230.ch2)
18. [Statistical analysis of the Kumaresan-Tufts and Matrix Pencil methods in estimating a damped sinusoid (EUSIPCO 2004)](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2004/defevent/papers/cr1145.pdf)
19. [First-order analysis of the mode and amplitude estimates of a damped sinusoid using Matrix Pencil (EUSIPCO 2009)](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2009/contents/papers/1569187944.pdf)
20. [Comparison of Matrix Pencil and Prony analysis for estimating electromechanical modes in noisy signals for power system applications (Missouri S&T master's thesis)](https://scholarsmine.mst.edu/cgi/viewcontent.cgi?article=5114&context=masters_theses)

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