Benchmarking Compressed Sensing, Super-Resolution, and Filter Diagonalization

Signal processing techniques have been developed that use different strategies to bypass the Nyquist sampling theorem in order to recover more information than a traditional discrete Fourier transform. Here we examine three such methods: filter diagonalization, compressed sensing, and super-resolution. We apply them to a broad range of signal forms commonly found in science and engineering in order to discover when and how each method can be used most profitably. We find that filter diagonalization provides the best results for Lorentzian signals, while compressed sensing and super-resolution perform better for arbitrary signals.

💡 Research Summary

The paper presents a systematic benchmark of three modern signal‑reconstruction techniques that aim to surpass the Shannon‑Nyquist limit when only a limited number of time‑domain samples are available. The methods compared are compressed sensing (CS), super‑resolution (SR), and filter diagonalization (FD). After a concise theoretical overview, the authors evaluate each method on a suite of test signals drawn from the Sparco toolbox together with several additional signals of interest (Gaussian pulse, a sum of random Lorentzians, and the Jacob’s Ladder signal). All signals are generated as continuous functions of time and sampled uniformly at 4096 points over a 1‑second interval (Δt = 1/4096 s, giving a Nyquist frequency of 2048 Hz).

Theoretical background

• The discrete Fourier transform (DFT) provides a direct mapping between uniformly sampled time data and frequency components but is limited by the sampling interval: the maximum recoverable frequency is 1/(2Δt) and high resolution requires long acquisition times.
• CS and SR belong to the family of L1‑norm minimization techniques. They assume the signal can be expressed sparsely in a known basis (e.g., complex exponentials, sines, damped cosines). The reconstruction problem becomes an underdetermined linear system f = Gλ, solved by minimizing ‖λ‖₁ subject to a small residual ‖f − Gλ‖₂ < η. CS samples the full time domain at random points, while SR samples a short, uniformly spaced segment; both exploit the Restricted Isometry Property to guarantee accurate recovery with far fewer measurements than the ambient dimension.
• FD takes a different route: it models the signal as the expectation value of a unitary propagator U = e^{-iΩτ}. By constructing a Krylov basis from the measured samples, the propagator and its overlap matrix are expressed entirely in terms of the data. Solving the generalized eigenvalue problem U B = λ S B yields eigenvalues λ = e^{-iωτ} and eigenvectors that directly give the frequencies ω, damping rates γ, and amplitudes λ of Lorentzian components. FD therefore assumes the signal is a sum of (possibly damped) Lorentzian peaks.

Experimental protocol
For each method the authors progressively undersample the 4096‑point data set in steps of 64 points. CS uses the same number of points but selects them randomly; SR uses equally spaced points; FD also uses equally spaced points but requires a pre‑tuned frequency grid (0–20 kHz) whose density and number of filter vectors are optimized on the full‑signal case and then held fixed. Reconstruction error is quantified by the relative 2‑norm over the full 4096‑point grid, which is equivalent to the error in the frequency domain by Parseval’s theorem.

Results

1. Lorentzian‑type signals (single or multiple damped Lorentzians): FD dramatically outperforms CS and SR. With as few as ~10 % of the total samples, FD recovers the exact peak positions, widths, and amplitudes, reflecting the perfect match between the FD model and the underlying physics. CS and SR, which rely on generic sparse bases, produce broader, less accurate spectra and suffer from over‑fitting when the basis does not capture the Lorentzian shape.
2. Non‑Lorentzian or less sparse signals (Gaussian pulse, random Lorentzian sum, Jacob’s Ladder): CS and SR achieve lower errors than FD. CS, benefitting from random sampling, distributes information uniformly and can reconstruct the signal with only ~5 % of the samples while keeping the error below 5 %. SR also performs well, delivering high frequency resolution from short, uniformly spaced windows. FD, on the other hand, is sensitive to the choice of the frequency grid and to numerical conditioning; it tends to generate spurious peaks or inflated linewidths for signals that do not conform to a pure Lorentzian model.
3. Noise robustness: L1‑based methods are inherently tolerant to moderate noise when the regularization parameter and tolerance η are properly chosen. FD includes a conditioning step (using two successive powers of the propagator) to discard non‑shared eigenvalues, which mitigates but does not eliminate sensitivity to noise and to linear dependence in the Krylov basis.

Computational considerations

• CS requires solving a large‑scale convex optimization problem (Basis Pursuit or LASSO). The computational cost scales roughly as O(N M) where N is the number of measurements and M the size of the chosen basis. Modern solvers and GPU acceleration make the approach feasible for many practical problems, albeit slower than the other two methods.
• SR solves a smaller L1 problem because the measurement matrix is a sub‑sampled DFT; it is therefore faster than CS while retaining similar accuracy for short‑window data.
• FD’s dominant cost is constructing the Krylov and overlap matrices (O(N K) operations, K being the number of filter vectors) and solving a K × K generalized eigenvalue problem. Increasing K improves resolution but can cause numerical instability; the authors mitigate this by a two‑step eigenvalue filtering procedure.

Conclusions and practical guidance

• When the underlying physics suggests that the signal consists of a sum of Lorentzian (or damped exponential) components—common in NMR, spectroscopy, scattering, and certain imaging applications—filter diagonalization is the most efficient and accurate technique.
• For signals that are sparse but not well described by Lorentzians—such as broadband pulses, arbitrary waveforms, or data where a suitable sparse basis is known—compressed sensing and super‑resolution are preferable. CS is optimal when random sampling across the full time domain is experimentally feasible; SR is advantageous when only a short, uniformly sampled segment can be acquired.
• The choice of sampling strategy (random vs. uniform) and the prior basis (Fourier, wavelet, polynomial, etc.) are decisive factors that must be aligned with the experimental constraints and the expected signal structure.

Overall, the paper delivers a clear, data‑driven comparison that equips scientists and engineers with actionable recommendations for selecting the most appropriate sub‑Nyquist reconstruction method based on signal characteristics and measurement limitations.