{ "id": "ep2ghvcfz5", "slug": "matrix-pencil-method", "title": "Matrix pencil method", "updated": "2026-10-25", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis" }, { "id": "physical.math.analysis.numerical_analysis_and_computation", "label": "Numerical analysis and computation", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis.numerical_analysis_and_computation" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.analysis.numerical_analysis_and_computation", "label": "Non-geographic · 1946 to 2000: Numerical analysis and computation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis.numerical_analysis_and_computation", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis" }, { "id": "geo.nongeo.t1946.physical.math.analysis.numerical_analysis_and_computation", "label": "Numerical analysis and computation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis.numerical_analysis_and_computation" } ] } ], "excerpt": "The matrix pencil method is a numerical technique that estimates the poles and residues of damped complex exponentials from uniformly sampled data, widely used in signal processing and electromagnetics.", "snippet": "The matrix pencil method is a numerical technique that estimates the poles and residues of damped complex exponentials from uniformly sampled data, widely used in signal processing and electromagnetics.", "node": "physical.math.analysis.numerical_analysis_and_computation", "markdown": "# Matrix pencil method\n\nThe 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.[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) 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.[2](https://link.springer.com/article/10.1007/s11075-023-01564-3)\n\n| Key fact | Detail |\n|---|---|\n| 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[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) |\n| Core computation | Generalized eigenvalues of the pencil \\( Y_1 - z Y_0 \\) built from shifted Hankel matrices[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) • [3](https://pmc.ncbi.nlm.nih.gov/articles/PMC6260881/) |\n| Validity condition | \\( M \\le L \\le N - M \\), where \\( L \\) is the pencil parameter and \\( N \\) the sample count[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) |\n| Noise robustness | With optimal L, variance efficiency relative to the Cramér–Rao bound is 27/24 = 1.125, independent of N[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) |\n| Cost | One SVD of an \\( (N-L) \\times L \\) data matrix and M eigenvalues of an \\( M \\times M \\) matrix[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) |\n| Sample requirement | 2n samples for n exponentials in the noiseless case (dilation \\( \\sigma = 1 \\)), with \\( \\lvert \\operatorname{Im}(\\phi_i) \\rvert \\Delta < \\pi \\)[2](https://link.springer.com/article/10.1007/s11075-023-01564-3) |\n| Introducing paper | Y. Hua and T.K. Sarkar, IEEE Transactions on Acoustics, Speech, and Signal Processing, 1990[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) |\n\n## How it works\n\nThe 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} \\).[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)\n\nFrom 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.[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) 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.[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC6260881/)\n\n## How it is done\n\n1. Form an \\( (N-L) \\times (L+1) \\) Hankel matrix Y from the noisy measurements, with \\( M \\le L \\le N - M \\).[4](https://eusipco2025.org/wp-content/uploads/pdfs/0002637.pdf)\n2. Choose the pencil parameter L. Values between \\( N/3 \\) and \\( 2N/3 \\) are reported as good choices;[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) a perturbation analysis gives the narrower variance-minimizing range \\( N/3 \\le L \\le N/2 \\).[5](https://arxiv.org/html/2502.17047v1) In antenna work, \\( L \\) between \\( K/3 \\) and \\( K/2 \\) (\\( K \\) = number of samples) is used to filter noise.[6](https://hal.science/hal-01407331v1/document)\n3. 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.[4](https://eusipco2025.org/wp-content/uploads/pdfs/0002637.pdf) SVD-based rank reduction is what makes the eigenvalue identification robust against noise.[7](https://www.osti.gov/biblio/2331204)\n4. 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) \\).[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC6260881/)\n5. Recover the coefficients \\( c_j \\) as a least-squares solution of the overdetermined Vandermonde system built from the estimated poles.[8](https://www-user.tu-chemnitz.de/~potts/paper/LAA_Prony.pdf)\n\nThe 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.[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf)\n\n## Origin\n\nThe 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.[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC6260881/) • [9](https://doi.org/10.1109/74.370583) 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.[10](https://doi.org/10.1109/tassp.1982.1163974) 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.[11](https://doi.org/10.1109/8.18710) 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.[12](https://doi.org/10.1109/29.56027)\n\n## Variants\n\n**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.[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) 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.[13](https://doi.org/10.1109/tap.2009.2037709)\n\n**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.[5](https://arxiv.org/html/2502.17047v1)\n\n**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.[14](https://doi.org/10.1109/78.157226) 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.[15](https://doi.org/10.1109/tap.2015.2477096) Multiscale generalizations introduce dilation \\( \\sigma \\) and translation \\( \\tau \\), extending the pencil formulation to trigonometric, hyperbolic, polynomial, Gaussian, sinc, and gamma functions.[2](https://link.springer.com/article/10.1007/s11075-023-01564-3)\n\n## Applications\n\n**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.[6](https://hal.science/hal-01407331v1/document) 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.[16](https://www.math.ucdavis.edu/~saito/data/sonar/SarkarMultiLook.pdf)\n\n**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.[7](https://www.osti.gov/biblio/2331204) 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.[17](https://onlinelibrary.wiley.com/doi/10.1002/9781119076230.ch2)\n\n## Limitations and alternatives\n\n**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.[5](https://arxiv.org/html/2502.17047v1) 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.[18](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2004/defevent/papers/cr1145.pdf) 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,[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) 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.[19](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2009/contents/papers/1569187944.pdf)\n\n**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 \\);[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) 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.[19](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2009/contents/papers/1569187944.pdf)\n\n**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.[9](https://doi.org/10.1109/74.370583) Unlike the polynomial method, the pencil method does not require all poles to be inside or outside the unit circle to separate extraneous poles.[1](https://intra.ece.ucr.edu/~yhua/MPM.pdf) [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.[20](https://scholarsmine.mst.edu/cgi/viewcontent.cgi?article=5114&context=masters_theses)\n\n## References\n\n1. [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)\n2. [Multiscale matrix pencils for separable reconstruction problems (Cuyt & Lee, Numerical Algorithms, 2023)](https://link.springer.com/article/10.1007/s11075-023-01564-3)\n3. [Coding Prony's method in MATLAB and applying it to biomedical signal filtering (PeerJ / PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC6260881/)\n4. [SAMP++: Robust Structure-Aware Matrix Pencil (EUSIPCO 2025)](https://eusipco2025.org/wp-content/uploads/pdfs/0002637.pdf)\n5. [Structure-Aware Matrix Pencil Method (arXiv, 2025)](https://arxiv.org/html/2502.17047v1)\n6. [Comparison of TLS-Prony and TLS-Matrix Pencil methods applied on antenna backscattered responses (DGA / HAL)](https://hal.science/hal-01407331v1/document)\n7. [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)\n8. [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)\n9. [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)\n10. [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)\n11. [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)\n12. [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)\n13. [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)\n14. [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)\n15. [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)\n16. [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)\n17. [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)\n18. [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)\n19. [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)\n20. [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)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Numerical analysis and computation*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 25, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://intra.ece.ucr.edu/~yhua/MPM.pdf", "https://www.math.ucdavis.edu/~saito/data/sonar/SarkarMultiLook.pdf" ], "url": "https://www.edgepedia.org/matrix-pencil-method", "markdown_url": "https://www.edgepedia.org/matrix-pencil-method.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Matrix pencil method\", Edgepedia (EdgeChat), https://www.edgepedia.org/matrix-pencil-method. Edgepedia Community License 1.0.", "credit_md": "\"[Matrix pencil method](https://www.edgepedia.org/matrix-pencil-method)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/matrix-pencil-method](https://www.edgepedia.org/matrix-pencil-method). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Matrix pencil method\", Edgepedia (EdgeChat), https://www.edgepedia.org/matrix-pencil-method. Edgepedia Community License 1.0.", "speakable": "The matrix pencil method is a numerical technique that estimates the poles and residues of damped complex exponentials from uniformly sampled data, widely used in signal processing and electromagnetics." }