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Identifiability and Joint Recovery of a Structured Signal Ensemble From Sub-Nyquist Samples
Kumari Priyanka, Satish Mulleti
TL;DR
Existing continuous-time approaches do not natively capture common-innovation structure with unknown spectral supports, despite the value of exploiting correlation to reduce aggregate sampling. The paper derives identifiability conditions and uses parametric structured dictionaries for joint recovery; experiments report a 25% aggregate rate reduction under those conditions and approximately 74% when all channels are sub-Nyquist.
Problem
Existing continuous-time frameworks do not natively capture common-innovation structure when spectral supports are unknown, limiting models for correlated analog signals.
Source: §1 · p. 1Method
The paper models signals as shared and signal-specific components, derives identifiability conditions, and jointly recovers the ensemble with parametric structured dictionaries.
Source: §1 · p. 1Results
Takeaways & Limitations
The framework supports recovery below the aggregate Nyquist rate even when the common bandwidth is unknown.
Source: §5 · p. 4The model order is treated as a fixed design parameter, and estimating unknown time shifts is left for future work.
Source: §4.1 · p. 3
Abstract
from arXiv · showhide
Efficient acquisition of correlated continuous-time signals is critical in applications such as distributed sensor networks and array processing. However, existing correlation models often fail to capture the shared structure of physical signals measured in close proximity. In this paper, we model each signal in an ensemble as the sum of a hidden common lowpass component and a signal-specific innovation highpass component with disjoint spectral supports, where the common bandwidth is unknown. We first derive theoretical identifiability conditions guaranteeing unique decomposition and reconstruction from subsampled observations. Further, we propose a practical recovery algorithm and a joint reconstruction framework that leverages parametric structured dictionaries parameterized by the unknown common bandwidth. Numerical experiments on an ensemble of four signals demonstrate exact reconstruction with a 25% aggregate sampling rate reduction under identifiability conditions, and robust recovery with up to a 74% rate reduction when all channels are sampled below the Nyquist rate, significantly reducing data acquisition and hardware overhead.
1. INTRODUCTION
The paper addresses the lack of continuous-time approaches that natively capture common-innovation structure when spectral supports are unknown. It develops identifiability guarantees and a joint recovery method for signals with a shared component and channel-specific innovations.
1. INTRODUCTION
- Existing continuous-time approaches use low-rank, graph, deconvolution, or stochastic-correlation formulations but do not natively model this unknown-support common-innovation structure.Source: §1 · p. 1
- The model represents continuous-time bandlimited signals as a shared common component plus unique innovations with disjoint spectral supports and unknown common bandwidth.Source: §1 · p. 1
- For N ≥3, sufficient conditions guarantee unique decomposition and identification from sub-Nyquist samples despite an unknown common bandwidth.Source: §1 · p. 1
- Parametric continuous-time dictionaries encode the common and innovation components, and joint least-squares optimization estimates them from finite samples.Source: §1 · p. 1
- 25% aggregate sampling-rate reduction accompanies highly accurate reconstruction when two of four signals are Nyquist-sampled and two are sub-Nyquist-sampled.The reported reconstruction error ranges from approximately −300 dB to −120 dB; when all channels are sub-Nyquist, the experiments report approximately 74% aggregate rate reduction.Source: §1 · p. 1
2. PROBLEM FORMULATION
Each continuous-time signal is modeled as a shared low-frequency component plus a signal-specific innovation occupying the remaining spectral bands. The common bandwidth and component signals are unknown, and the goal is to identify and reconstruct them from subsampled observations.
2. PROBLEM FORMULATION
- Each signal is the sum of a shared component and a signal-specific innovation, whose spectral support lies outside the common component’s band.The common component is bandlimited to [−Ω0, Ω0], with 0 < Ω0 < Ωm; the innovation support is [−Ωm, −Ω0)∪(Ω0, Ωm].Source: §2 · p. 2
- Without exploiting correlation, recovering N bandlimited signals requires an aggregate sampling rate of at least NΩm/π.Source: §2 · p. 2
- The common bandwidth Ω0, common component, and innovation components are unknown, while the maximum bandwidth and observations are known.Source: §2 · p. 2
- The goal is to establish conditions for unique identification from sub-Nyquist samples and develop a practical method to estimate Ω0 and reconstruct the ensemble.Source: §2 · p. 2
3. STRUCTURAL ASSUMPTIONS AND IDENTIFIABILITY
Under structural assumptions, ensembles of at least three signals are uniquely identifiable from measurements whose aggregate sampling rate is sub-Nyquist. The proof uses Nyquist-sampled channels to identify the common bandwidth and component, then reconstructs the remaining channels’ innovations.
3. STRUCTURAL ASSUMPTIONS AND IDENTIFIABILITY
- The common component’s Fourier transform must not vanish almost everywhere on any nonempty open interval within its bandwidth.Source: §3 · p. 2
- At least two channels are Nyquist-sampled, and their innovation spectra must differ almost everywhere on some portion of every open interval in the innovation band.Source: §3 · p. 2
- Each sub-Nyquist channel’s sampling rate must satisfy the stated interval constraint for some integer q ≥2 to prevent innovation aliasing.Source: §3 · p. 2
- For N ≥3 and assumptions (A1)–(A3), the theorem guarantees unique identification from measurements with a sub-Nyquist aggregate sampling rate.Source: §3 · p. 2
- For N ≥ 3 and |JN| > 1, the proof reconstructs fn for n∈JN from samples via Shannon-Nyquist, determines Ω0, and obtains innovation samples for n∉JN, which under Assumption (A3) uniquely determine gn(t).The common component is obtained from any reconstructed channel n∈JN by retaining its spectrum over [−Ω0, Ω0].Source: §3 · p. 2
- A joint reconstruction framework estimates Ω0 from observed subsamples while simultaneously recovering the signal ensemble.Source: §3 · p. 2
4. DICTIONARY-BASED RECONSTRUCTION METHOD
The method represents each signal with shared and channel-specific continuous-time dictionaries, then jointly estimates the common bandwidth and coefficients from finite samples. Experiments report low reconstruction error under reduced aggregate sampling, with recovery degrading near extreme common-to-total bandwidth ratios.
4.1. Finite-Dimensional Reconstruction Model
- Each signal is approximated by shifted common and innovation kernels, with fixed design parameter L; unknown time-shift estimation is left for future work.The common bandwidth is estimated along with the common and innovation coefficients, while the maximum bandwidth is assumed known.Source: §4.1 · p. 3
4.2. Dictionaries and Unknown Coefficients
- The structured dictionaries have continuous-time parametric form and non-sparse coefficients, unlike the cited conventional discrete dictionary representations.Their kernels represent the common component and each signal's innovation, with dictionary entries depending on the unknown common bandwidth.Source: §4.2 · p. 3
- The method estimates the common bandwidth by minimizing a least-squares residual over candidate bandwidths, then recovers coefficients using the corresponding dictionary.The coefficient solution is unique when the joint dictionary matrix has full column rank (N + 1)L.Source: §4.2 · p. 3
4.3. Numerical Experiments
- A 25% aggregate sampling rate reduction yielded average NMSE −302 dB under identifiability conditions; all-sub-Nyquist sampling achieved ≈74% reduction and average NMSE −217 dB.The representative reconstructions report NMSEs of −304 dB and −221 dB, respectively.Source: Figure 1 · p. 4, Figure 2 · p. 4, §4.3 · p. 4
- Across ensemble sizes and bandwidth ratios, NMSEs remain below −200 dB except near ratios 0 and 1, where ill-conditioning and rank deficiency prevent unique coefficient recovery.At the low-ratio limit, common atoms become hard to distinguish over a finite observation window; at the high-ratio limit, innovation atoms become nearly linearly dependent.Source: §4.3 · p. 4
5. CONCLUSION
The paper establishes identifiability conditions and a joint reconstruction algorithm for continuous-time signals with shared lowpass structure and signal-specific innovations. Experiments report exact recovery with a 25% aggregate sampling rate reduction and robust recovery with up to a 74% rate reduction.
5. CONCLUSION
- 25% aggregate sampling rate reduction yielded exact recovery under sufficient identifiability conditions, while robust recovery reached up to 74% when all channels were sampled sub-Nyquist.The framework also recovers signals below the aggregate Nyquist rate when the common bandwidth is unknown.Source: §5 · p. 4
- The paper established theoretical identifiability conditions and developed a joint reconstruction algorithm for continuous-time bandlimited signals under a common-innovation model.The model captures shared lowpass structure and disjoint channel-specific innovation components, with the common bandwidth unknown.Source: §5 · p. 4
- A joint optimization approach using parametric structured dictionaries parameterized by the unknown common bandwidth translates the guarantees into simultaneous recovery.Source: §5 · p. 4
- Future work will explore unknown time shifts in dictionary atoms, dictionary mismatch, and real-time hardware.Source: §5 · p. 4