Frequency-Guided Sampling

• Frequency-guided sampling is a set of methodologies that allocate measurements based on a priori or data-driven frequency-domain characteristics to enhance signal reconstruction.
• Techniques such as energy-equipartition, quasi-random low-discrepancy sequences, and adaptive data-driven strategies optimize sampling for improved PSNR and convergence rates.
• Applications span compressive sensing, sub-Nyquist hardware, random field simulation, graph signal processing, and inverse imaging, demonstrating significant gains in efficiency.

Frequency-guided sampling refers to a collection of methodologies for selecting sampling locations, frequencies, or measurement designs based explicitly on a priori or data-driven knowledge of frequency-domain structure. Motivations range from optimizing compressive sensing with known spectral density, to designing sub-Nyquist hardware systems, statistically optimal simulation of random fields, and adaptively guiding sensing protocols in both signal processing and distributed sensor scenarios. The unifying principle is that, in contrast to uniform, random, or generic measurement selection, frequency-guided approaches allocate measurements preferentially in regions of spectral space with higher signal energy, information content, or statistical importance, often resulting in higher reconstruction fidelity, improved convergence rates, or resource-efficient acquisition.

1. Energy-Density-Guided and Equipartition Strategies

A canonical instantiation of frequency-guided sampling is the energy-equipartition scheme for compressive sensing with known spectral energy densities. For a signal with one-sided energy spectral density $S(f)$ supported on $[f_L, f_H]$, the cumulative energy function is

$E(f) = \int_{f_L}^f S(\nu)\,d\nu$

with total energy $E_{\rm tot} = E(f_H)$. Equipartition selects $M$ sample frequencies $\{f_k\}$ such that

$$E(f_k) = \left(k - \frac12\right)\frac{E_{\rm tot}}{M}, \quad 1 \leq k \leq M$$

i.e., each frequency interval corresponds to an equal share of the total spectral energy. This mapping is computed via inverse-cumulative lookup, often by numerical inversion of $E(\cdot)$ on a discretized frequency grid.

In compressive sensing, measurements $y = \Phi x + \eta$ are taken at the selected frequencies, with $\Phi$ the measurement matrix constructed according to the sampling pattern. Standard $\ell_1$ minimization is then used for reconstruction. Empirical analysis on monocycles and multicycle signals demonstrates that energy-equipartition sampling attains substantially higher PSNR than both uniform and random sampling for the same $M$—differences exceeding 30 dB over uniform and 20 dB over random in specific regimes (0904.1910):

Sampling scheme	PSNR (dB)
Frequency-equipartition (uniform)	−12.9
Random	−6.2
Energy-equipartition (EES)	+19.6

The observed gains arise because matching sampling density to regions of high $S(f)$ reduces aliasing and spectral “holes” critical for faithful CS reconstruction. Importantly, although the mutual coherence bound in generic CS is unchanged, the practical constant is reduced, enabling recovery with fewer samples (0904.1910).

2. Quasi-Random and Discrepancy-Minimizing Frequency Filling

For the generation of random fields with prescribed spectral statistics, particularly in simulation of spatial optical turbulence, frequency-guided sampling is realized via quasi-random (low-discrepancy) sequences such as Sobol’, Halton, or the R₂ sequence. These sequences sample the frequency domain deterministically and quasi-uniformly, with coverage error (star discrepancy) decaying as $O((\log M)^d/M)$ in $d$ dimensions—substantially outperforming pseudo-random $O(M^{-1/2})$ filling (Berdja et al., 2024).

The sampling steps:

1. Generate low-discrepancy sequence points $s_i$ in $[0,1]^d$.
2. Map to physical frequencies $f_i$ in the desired band.
3. Amplitude-weight by the target spectrum $W_\chi(f_i)$ and assign random phase.
4. Synthesize realizations via

$$\chi(r_j) = \sum_{i=1}^M A_i \cos(2\pi f_i \cdot r_j + \phi_i)$$

Practically, this eliminates low-frequency undersampling, high-frequency aliasing, and irregular coverage inherent in grid-based FFT methods. For Monte Carlo simulation of random fields, ensemble-averaged statistics such as covariance or structure functions converge as $O((\log M)^2/M)$, outperforming both pseudo-random and grid-based (“static”) FFT approaches (Berdja et al., 2024).

3. Frequency-Guided Sampling in Adaptive and Data-Driven Frameworks

Learning-based frequency-guided sampling methods optimize sampling locations via data-driven objectives. For instance, in finite-rate-of-innovation (FRI) signals, a greedy forward or backward search is employed to choose $K$ frequency bins, each time training a sparse-recovery network (e.g., LISTA) conditioned on the partially observed sample pattern. At each step, the candidate frequency leading to the lowest empirical reconstruction error on the training set is selected (Mulleti et al., 2021):

• Initialize sampling mask with no frequencies.
• At each iteration, for each candidate, retrain or update the reconstruction network parameters, evaluate reconstruction error, and select the frequency yielding minimum error.
• Continue until $K$ frequencies are chosen.

Empirically, such joint sampling/reconstruction optimization significantly reduces reconstruction NMSE and improves support hit-rates relative to random or heuristic, non-adaptive selection—even without knowledge of the pulse spectrum $h$ (Mulleti et al., 2021).

Similarly, in adaptive graph signal sampling, selection of sensor subsets is guided by spectral properties of the underlying graph Laplacian or covariance structure: for each partition, sensors are selected to maximize the smallest eigenvalue of a sample-set adaptive GFT, thereby minimizing the worst-case bandlimited model mismatch error (Pakiyarajah et al., 2024).

4. Sub-Nyquist Sampling, Hardware Architectures, and Aliasing Control

Frequency-guided sampling also manifests as sub-Nyquist measurement protocols leveraging hardware properties to target or disambiguate specific frequency bands. In two prominent cases:

• Photonic frequency-oriented sub-sampling: An ultrashort pulse train is dispersed and modulated by the RF signal, mapped in time via a chromatic dispersive element. Coherent optical gating with a delayed local oscillator enables selection of a narrow “slice” of the RF spectrum. The center frequency is tuned by optical delay, and pre-filter bandwidth can be made less than the A/D sampling rate. Coherent $I/Q$ demodulation further enables imaging rejection exceeding 26 dB (Hao et al., 2017).
• Joint Sub-Nyquist Frequency Estimation: For frequency estimation, two channels are sampled at rates $F_{\text{Nyquist}}/u_1$ and $F_{\text{Nyquist}}/u_2$ with $u_1, u_2$ coprime; subspace methods (e.g., ESPRIT, MUSIC) resolve frequency candidates in each channel, and their intersection via the Chinese Remainder Theorem or pseudo-spectrum screening eliminates alias ambiguity. Reliable unaliased estimation is achievable at sampling rates far below Nyquist (Huang et al., 2015).

5. Frequency-Guided Sampling in Modern Generative and Inverse Models

Frequency guidance can be incorporated into modern generative frameworks, e.g., diffusion models, for inverse imaging problems. In Frequency-Guided Posterior Sampling (FGPS) with diffusion-based priors for image restoration, a time-varying low-pass filter is applied in Fourier space to the data at each reverse sampling step. The filter’s frequency cutoff is scheduled (linear or exponential) from low to high frequencies in line with the underlying data’s power-law spectrum. This “curriculum” aligns the restoration sequence from global structure to fine details, minimizing approximation errors arising from likelihood-score mismatches. FGPS achieves state-of-the-art FID, LPIPS, PSNR, and SSIM on a variety of inverse tasks, outperforming conventional diffusion-based approaches and problem-specific networks with negligible extra computational cost (Thaker et al., 2024).

In iterative spectral acquisition such as protein NMR, uncertainty quantification by a diffusion-inpainting network is used to adaptively decide which frequency–time rows to acquire next, focusing measurements where uncertainty is highest. Empirically, this protocol halves the hallucination rate of spurious peaks and reduces acquisition time by 20–40% compared to Poisson-gap sampling (Goffinet et al., 6 Feb 2025).

6. Frequency-Guided Sampling in Stream Analytics

In large-scale stream analytics for frequency-based statistics (e.g., distinct counts, frequency-capped sums), the $\ell$-capped sample framework produces compact sketches whose sampling probabilities are directly tied to key frequencies. Each key $x$ is scored and the $k$ lowest-seed keys are retained, guaranteeing unbiased estimates for any monotone nondecreasing frequency function $f(w_x)$. Multi-objective samples allow for simultaneous accurate recovery of statistics for multiple frequency caps, with coefficient-of-variation guarantees scaling as $O((qk)^{-1/2})$ when the cap and guiding parameter align (Cohen, 2015).

7. Extensions and Generalizations

Frequency-guided sampling extends to scenarios such as:

• Warping-based sampling: Transforming non-band-limited signals via nonlinear frequency warping to become band-limited, then sampling and reconstructing in a frequency-adapted orthonormal basis, yielding superior convergence for fractal or heavy-tail spectra (Lafon et al., 2017).
• Graph signal sampling: Adapting GFT and inner product to the selected sampling set, localizing model-mismatch error to high-frequency components and optimizing the sample set for spectral efficacy (Pakiyarajah et al., 2024).
• Statistical simulation: Deterministic low-discrepancy frequency selection in random field simulation accelerates convergence of ensemble statistics compared to random or grid-based selection, with precise control over spectral coverage (Berdja et al., 2024).

The consistent outcome is that by aligning measurement, simulation, or reconstruction resources to the frequency structure of the underlying signal or statistical process, frequency-guided sampling realizes substantial gains in rates, accuracy, or efficiency across diverse domains.