Unlimited Sensing Framework (USF)

• The Unlimited Sensing Framework (USF) is a signal acquisition paradigm that uses a modulo operation to fold high-dynamic-range signals into a bounded range before digitization.
• USF eliminates clipping and saturation by mapping signals into a fixed interval, enabling accurate recovery of both weak and strong signal components.
• USF integrates advanced recovery algorithms and predictable Cramér–Rao Bound scaling to achieve simultaneous high dynamic range and resolution in spectral estimation.

The Unlimited Sensing Framework (USF) is a signal acquisition and reconstruction paradigm that redefines analog-to-digital conversion by introducing analog-domain modulo (folding) nonlinearities prior to quantization. By mapping high-dynamic-range (HDR) signals into a bounded range before digitization, USF fundamentally eliminates the clipping, saturation, and dynamic range–resolution trade-offs inherent to conventional digitization. At its theoretical and practical core, USF replaces one-dimensional linearity with a nonlinear modulo mapping—preserving amplitude information implicitly and enabling recovery algorithms to reconstruct or estimate signals far beyond a conventional ADC's nominal input range. USF is foundational for contemporary advances in spectral estimation, sparse recovery, and hardware design, facilitating simultaneously high dynamic range and high digital resolution.

1. Foundational Principles and Modulo Nonlinearity

In USF, the analog front-end applies a modulo operation to the input signal before sampling. For a folding threshold λ > 0, the mapping is defined by: $$\mathcal{M}_\lambda(g) = 2\lambda \left( \left\{ \frac{g + \lambda}{2\lambda} \right\} - \frac{1}{2} \right),$$ where $\{ \cdot \}$ denotes the fractional part. This operation folds any input amplitude into the fixed interval [–λ, λ]. The practical implications are twofold: (1) the system acquires signals of arbitrary amplitude without hardware saturation, and (2) the digital bit budget may be allocated to achieve high resolution rather than to prevent clipping.

This approach stands in contrast to standard ADCs, which are fundamentally limited by the simultaneous requirements for wide dynamic range and fine quantization; exceeding the threshold λ leads to irreversible information loss by clipping. By folding the signal instead, USF precludes this loss—though at the expense of introducing a structured ambiguity (the "residue" or folding error), which must be algorithmically resolved.

2. Unlimited Sensing in Spectral Estimation

USF enables a new framework for spectral estimation—USF-SpecEst—by facilitating the unbiased recovery of frequency and amplitude parameters of sparse or multi-sinusoidal signals, even under extreme amplitude disparities. The core insight is that unlimited sampling does not merely prevent clipping; it changes the statistical structure of the estimation problem. The analog signal

$g(t) = \sum_{k=1}^K a_k \sin(\omega_k t + \phi_k)$

is mapped by the modulo operator prior to digitization: $y[n] = \mathcal{M}_\lambda[g(nT)] + w[n]$ with additive noise $w[n]$. Whereas standard spectral estimation is forced to trade resolution for dynamic range, USF decouples these constraints, and the folding residue—though nonlinear—can be treated as a random impulsive process that is independent of the desired parameters when oversampling is sufficient.

3. Theoretical Performance: Cramér–Rao Bounds

The principal theoretical result in USF-based spectral estimation is the derivation of Cramér–Rao Bounds (CRBs) for parameter estimation from modulo-folded, noisy samples. The authors show that, asymptotically (number of samples $N \rightarrow \infty$), and under suitable oversampling, the CRBs for estimating amplitude, frequency, and phase parameters are scaled versions of the conventional Gaussian CRBs. Specifically, for a single sinusoid,

$$\lim_{N \rightarrow \infty} \begin{bmatrix} \operatorname{CRB}(a_1) \ \operatorname{CRB}(\omega_1 T) \ \operatorname{CRB}(\phi_1) \end{bmatrix} = \frac{2\gamma}{N \operatorname{PSNR}} \begin{bmatrix} a_1^2 \ (\sqrt{12}/N)^2 \ 2^2 \end{bmatrix}, \qquad \gamma = (1 - \cos(\omega_1 T))^{-1}$$

with $\operatorname{PSNR} = a_1^2/\sigma^2$ (signal-to-noise ratio). Thus,

$$\lim_{N \rightarrow \infty} \operatorname{CRB}_{\text{USF}} = \gamma \cdot \lim_{N \rightarrow \infty} \operatorname{CRB}_{\text{conventional}}$$

This scaling factor $\gamma$ depends on the sampling interval and frequency content but does not otherwise penalize asymptotic estimation performance, highlighting that modulo folding does not fundamentally degrade the achievable accuracy for spectral parameters (Guo et al., 6 May 2025).

4. Methodology and Algorithmic Innovations

USF-based spectral estimation relies on both theoretical analysis and tailored algorithms to handle measurements processed by the modulo nonlinearity. The main methodologies include:

• Residue Decomposition and Modeling: The continuous signal is decomposed as $g = \mathcal{M}_{\lambda}(g) + \varepsilon$, where the residue $\varepsilon$ (corresponding to foldings) constitutes a sparse, impulsive error. For sufficiently small sampling intervals (oversampling), the number of foldings per sample becomes negligible.
• Noise Modeling and Fisher Information: The residue is modeled as impulsive noise, typically approximated by a Bernoulli process. When combined with Gaussian measurement noise, the overall hybrid process is shown to converge to Gaussian behavior as the fraction of folding events tends to zero.
• Algorithm Adaptation: Established spectral estimation techniques, such as the matrix pencil method and variants of Prony's method, are demonstrated to remain effective when directly applied to modulo samples. Under oversampling, with an informed model of the residue, the performance of these algorithms approaches the scaled CRB.

5. Numerical Validation

The theoretical results are substantiated with comprehensive numerical experiments:

• Single Sinusoid Tests: For ω₁T = 1.05, a₁ = 1, and a folding threshold close to the amplitude (λ ≈ a₁), simulations with N=100 display three regimes for frequency estimation accuracy as a function of PSNR. At high SNR, recovery variance is residue dominated; at moderate SNR, recovery aligns with the CRB; at very low SNR, high-variance outlier events predominate.
• Multiple Sinusoids: With, e.g., ω₁T = 0.63 and ω₂T = 1.00, the empirical CRBs scale as γₖ = (1 – cos(ωₖT))⁻¹ for each frequency component, matching theoretical predictions.

Performance closely tracks the derived bounds except in regions with a high density of folding (e.g., high signal amplitude or low λ), where deviations reflect physical limits on the number of measurable folding events and increased impulsive interference.

6. Practical Implications and Impact

USF-based spectral estimation delivers critical advantages:

• Simultaneous High Dynamic Range and Resolution: With up to a 60-fold dynamic range increase and improved quantization noise properties demonstrated in hardware (Zhu et al., 26 Oct 2024), USF enables the recovery of weak and strong spectral components without the traditional limitations of clipping or sub-optimal quantization.
• Predictable Performance Benchmarks: The prevalence of a simple scaling relation for the CRB in modulo-processed signals provides an explicit and reliable prediction of estimation accuracy, facilitating robust system design and performance benchmarking.
• Seamless Algorithmic Integration: The effective Gaussian nature of the impulse-plus-noise model ensures that conventional spectral estimation algorithms, with minor adaptation, remain applicable.

These features position USF as an enabling technology for advanced applications—especially where extreme amplitude disparities and super-resolution requirements are coupled, such as radar, tomography, full-duplex communications, and time-of-flight imaging.

7. Limitations and Future Directions

While the scaling law for the CRB holds asymptotically and under sufficient oversampling, a few caveats merit attention:

• In practical finite-sample settings, especially with high densities of folding or non-Gaussian residual events, performance may saturate above the theoretical CRB.
• Recovery performance can deteriorate if the impulse residue is not adequately modeled or if foldings cluster (leading to statistical dependence).
• Extension of the theory to non-sinusoidal or structured signals, multi-dimensional scenarios, and under-sampled regimes remains an active area, as does the design of hybrid hardware–algorithm systems optimizing the scaling factor γ and managing hardware-induced nonidealities.

Continued progress is anticipated in the development of robust recovery algorithms for more general classes of signals and in the integration of USF architectures in low-power, high-performance digitization systems.