SFEM: Strategy Frequency Estimation Method

• SFEM is a high-resolution technique that models signals as sums of damped complex exponentials using low-rank Hankel matrices.
• It reformulates frequency estimation as a rank-constrained optimization problem solved via ADMM, achieving near Cramér–Rao bound accuracy.
• SFEM is applied in signal processing, spectroscopy, radar, and quantum metrology, demonstrating robustness with noisy or incomplete data.

The Strategy Frequency Estimation Method (SFEM) is a high-resolution framework for estimating the frequencies, and when present, damping rates of complex exponential components in signals, with rigorous connections to algebraic structure, modern optimization, and practical robustness. SFEM encompasses a class of methods designed to efficiently extract frequency parameters from observed data, often under constraints of incomplete, noisy, or high-dimensional measurements. Its development and application span signal processing, uncertainty quantification, structural reliability analysis, and quantum metrology.

1. Mathematical Foundations and Classical Formulation

SFEM originates in the parametric modeling of signals as linear combinations of damped complex exponentials. For a continuous signal:

$$f_c(t) = \sum_{p=0}^{P-1} c_p\, e^{\zeta_p t} + e(t),$$

where $c_p$ are complex amplitudes, $\zeta_p$ represent complex frequencies encoding both frequency and exponential damping, and $e(t)$ is noise.

Upon discretization on equally spaced samples ($j=0,\dots,2N$):

$f(j) = f_c(t_0 + j t_s),$

the primary objective is the accurate estimation of the unknown frequency parameters $\zeta_p$. Unlike standard approaches that directly minimize residuals over $\{c_p, \zeta_p\}$, SFEM expresses the problem in terms of a generating vector $g$ and a structured Hankel matrix:

$[H(g)](j,k) = g(j+k),\quad 0 \leq j, k \leq N.$

Kronecker’s theorem guarantees that if $f$ is a sum of $P$ exponentials, $H(f)$ is exactly of rank $P$. The best $\ell_2$-approximation by a sum of exponentials is thus posed as:

$$\min_{g}\ \tfrac{1}{2}\|f-g\|_2^2 \ \text{subject to}\ \mathrm{rank}(H(g)) = P.$$

This reformulation converts difficult parameter estimation into low-rank matrix approximation, harnessing the algebraic structure of Hankel matrices (1306.2907).

2. Nonconvex Optimization via ADMM

To solve the SFEM-constrained minimization, the Alternating Direction Method of Multipliers (ADMM) is adopted. The optimization is alternately performed over the Hankel matrix variable $A = H(g)$ and the residual $r = f-g$, with an indicator function enforcing the rank constraint:

$$\min_{A,r}\ R(A) + \tfrac{1}{2}\|r\|_2^2,\ \text{subject to}\ A(j,k) + r(j+k) = f(j+k).$$

Each ADMM iteration comprises:

• Projection to Rank-$P$: $A$ is updated by projecting a matrix $B$ onto the set of rank-$P$ matrices via singular value decomposition (SVD), retaining the $P$ largest singular values (Eckart–Young theorem).
• Residual Update: $r$ is updated in closed form, exploiting the anti-diagonal Hankel structure.
• Dual Variable Update: The Lagrange multipliers are incremented according to the latest constraint violations.

Despite nonconvexity, numerical studies confirm reliable convergence to accurate estimates even under significant noise, with the method attaining the Cramér–Rao lower bound (CRB) for frequency and damping estimation error under broad conditions (1306.2907).

3. Frequency Parameter Extraction

After ADMM convergence, frequency extraction utilizes the spectral properties of the solution Hankel matrix. Being complex symmetric, $A$ admits a con-eigenvalue decomposition:

$$A = \sum_{p=0}^N s_p\, u_p u_p^\mathrm{T},\quad \quad A\,\overline{u_p} = s_p\, u_p,$$

where $s_p$ and $u_p$ are con-eigenvalues/vectors. The dominant $P$ con-eigenvectors $u_p$ have the Vandermonde form $[e^{\zeta_p 0},\ldots, e^{\zeta_p N}]^\mathrm{T}$ (up to mixing). By forming matrices from shifted rows, the product $Ū^\dagger \underline{U}$ has eigenvalues $e^{\zeta_p}$, so the desired frequencies $\zeta_p$ are computed using the principal logarithm.

This invariant-based procedure links Hankel rank structure and frequency estimation, generalizing classical Prony and matrix pencil methods, but in a robust, optimization-based setting (1306.2907).

4. Robustness, Accuracy and Extensions

Extensive simulation demonstrates that SFEM achieves:

• CRB-level accuracy: Standard deviations for frequency and damping estimates closely approach the Cramér–Rao bounds at moderate and low SNRs.
• Resolution of closely spaced components: Outperforms engineered subspace approaches like ESPRIT in signals with close, unequally weighted components.
• Tolerance of missing data: The use of weighted residuals and the low-rank Hankel framework naturally incorporate cases of arbitrary missing or censored samples, consistently yielding estimates at the CRB.

These properties are substantiated by numerical experiments on signals with both uniform and randomly missing samples, where SFEM maintains high accuracy and robustness, in marked contrast to most classical subspace methods (1306.2907).

5. Broader Applications and Adaptations

SFEM's flexibility facilitates application to various fields:

• Spectroscopy and Material Characterization: Nuclear quadrupole resonance spectroscopy, where signatures are sums of damped exponentials.
• Radar and Communications: High-resolution estimation in Doppler, spectral, and direction-of-arrival problems.
• Incomplete and Noisy Data: Any spectral estimation with missing samples or in the presence of non-white noise.

The methodology is directly extensible to stochastic differential equations by leveraging sample-based separated representations, efficiently addressing the curse of dimensionality in high-dimensional settings (1905.05802, 2008.07911). In stochastic finite element analyses, SFEM allows the computation of frequency of failure events and robust uncertainty quantification, with sample-based approaches effectively circumventing computational bottlenecks prevalent in alternatives.

6. Comparative and Contemporary Perspectives

Recent innovations have extended SFEM ideas to quantum sensing, privacy-preserving data science, and nonlinear probe design. For instance, adaptive quantum strategies demonstrate that entanglement-based frequency estimation's theoretical advantages are contingent on identifiability constraints and require adaptive update of measurement parameters to realize optimal scaling of precision (2405.06548). In another direction, nonlinearly engineered quantum probes use SFEM principles with resource accounting via the Wigner–Yanase skew information, enhancing sensitivity by scrambling frequency information across the probe’s Hilbert space (2503.01959).

Elsewhere, empirical Bayes and sample-based algorithms have been fused with SFEM paradigms to achieve efficient, consistent, and uncertainty-quantified inference in frequency estimation, often leveraging the algebraic structure (e.g., Hankel, Vandermonde, prolate spheroidal wave functions) to design scalable and reliable estimators (1910.09475).

7. Concluding Synthesis

SFEM represents a modern synthesis of structured algebraic modeling, convex and nonconvex optimization, and spectral estimation, unified in a manner that maximizes both statistical efficiency and computational tractability. Its reformulation of the frequency node estimation problem—eschewing direct nonlinear least squares in favor of low-rank matrix methods solvable by ADMM—endows the approach with robustness against noise and missing data, while remaining easy to implement. The resultant dominance in CRB-level accuracy, flexibility to missing/incomplete data, and adaptability to emerging domains explain the enduring and expanding influence of SFEM in both classical and quantum signal processing.