From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals

Conventional sub-Nyquist sampling methods for analog signals exploit prior information about the spectral support. In this paper, we consider the challenging problem of blind sub-Nyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum. Our primary design goals are efficient hardware implementation and low computational load on the supporting digital processing. We propose a system, named the modulated wideband converter, which first multiplies the analog signal by a bank of periodic waveforms. The product is then lowpass filtered and sampled uniformly at a low rate, which is orders of magnitude smaller than Nyquist. Perfect recovery from the proposed samples is achieved under certain necessary and sufficient conditions. We also develop a digital architecture, which allows either reconstruction of the analog input, or processing of any band of interest at a low rate, that is, without interpolating to the high Nyquist rate. Numerical simulations demonstrate many engineering aspects: robustness to noise and mismodeling, potential hardware simplifications, realtime performance for signals with time-varying support and stability to quantization effects. We compare our system with two previous approaches: periodic nonuniform sampling, which is bandwidth limited by existing hardware devices, and the random demodulator, which is restricted to discrete multitone signals and has a high computational load. In the broader context of Nyquist sampling, our scheme has the potential to break through the bandwidth barrier of state-of-the-art analog conversion technologies such as interleaved converters.Comment: 17 pages, 12 figures, to appear in IEEE Journal of Selected Topics in Signal Processing, the special issue on Compressed Sensin

Sub-Nyquist Sampling: Bridging Theory and Practice

Sampling theory encompasses all aspects related to the conversion of continuous-time signals to discrete streams of numbers. The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. In modern applications, an increasingly number of functions is being pushed forward to sophisticated software algorithms, leaving only those delicate finely-tuned tasks for the circuit level. In this paper, we review sampling strategies which target reduction of the ADC rate below Nyquist. Our survey covers classic works from the early 50's of the previous century through recent publications from the past several years. The prime focus is bridging theory and practice, that is to pinpoint the potential of sub-Nyquist strategies to emerge from the math to the hardware. In that spirit, we integrate contemporary theoretical viewpoints, which study signal modeling in a union of subspaces, together with a taste of practical aspects, namely how the avant-garde modalities boil down to concrete signal processing systems. Our hope is that this presentation style will attract the interest of both researchers and engineers in the hope of promoting the sub-Nyquist premise into practical applications, and encouraging further research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin

Xampling: Signal Acquisition and Processing in Union of Subspaces

We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two. Analog compression that narrows down the input bandwidth prior to sampling with commercial devices. A nonlinear algorithm then detects the input subspace prior to conventional signal processing. A representative union model of spectrally-sparse signals serves as a test-case to study these Xampling functions. We adopt three metrics for the choice of analog compression: robustness to model mismatch, required hardware accuracy and software complexities. We conduct a comprehensive comparison between two sub-Nyquist acquisition strategies for spectrally-sparse signals, the random demodulator and the modulated wideband converter (MWC), in terms of these metrics and draw operative conclusions regarding the choice of analog compression. We then address lowrate signal processing and develop an algorithm for that purpose that enables convenient signal processing at sub-Nyquist rates from samples obtained by the MWC. We conclude by showing that a variety of other sampling approaches for different union classes fit nicely into our framework.Comment: 16 pages, 9 figures, submitted to IEEE for possible publicatio

Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors

The rapid developing area of compressed sensing suggests that a sparse vector lying in an arbitrary high dimensional space can be accurately recovered from only a small set of non-adaptive linear measurements. Under appropriate conditions on the measurement matrix, the entire information about the original sparse vector is captured in the measurements, and can be recovered using efficient polynomial methods. The vector model has been extended to a finite set of sparse vectors sharing a common non-zero location set. In this paper, we treat a broader framework in which the goal is to recover a possibly infinite set of jointly sparse vectors. Extending existing recovery methods to this model is difficult due to the infinite structure of the sparse vector set. Instead, we prove that the entire infinite set of sparse vectors can recovered by solving a single, reduced-size finite-dimensional problem, corresponding to recovery of a finite set of sparse vectors. We then show that the problem can be further reduced to the basic recovery of a single sparse vector by randomly combining the measurement vectors. Our approach results in exact recovery of both countable and uncountable sets as it does not rely on discretization or heuristic techniques. To efficiently recover the single sparse vector produced by the last reduction step, we suggest an empirical boosting strategy that improves the recovery ability of any given sub-optimal method for recovering a sparse vector. Numerical experiments on random data demonstrate that when applied to infinite sets our strategy outperforms discretization techniques in terms of both run time and empirical recovery rate. In the finite model, our boosting algorithm is characterized by fast run time and superior recovery rate than known popular methods.Comment: 21 Pages, 9 figures. Submitted to the IEEE for possible publicatio

Psychological factors causing nonadherence to safety regulations in Israel’s stone and marble fabrication industry: Unveiling the source of worker noncompliance

Background: Silicosis remains a lung disease which may cause severe incapacitation and even be fatal. We examined why stone processors in Israel, though aware that regular occupational unprotected exposure to harmful silica dust might cause silicosis, choose to work without protection, in defiance of legislation and employer instructions. The study seeks to identify and map the psychological factors that non cooperative processors use, to justify ignoring safety guidelines. Understanding the inner logic behind nonadherence in a scientific and nonjudgmental way could enhance efforts to reduce unsafe behavior among stone and marble processors, including ASW (Artificial Stone Workers). Methods: This qualitative study included semi-structured in-depth interviews with 25 stone processors. The interview transcripts were processed and analyzed by the authors who identified the leading resistance themes underlying noncompliance. Results: The current study found that although interviewees made an initial declarative statement that protection from dust is important due to the perceived and acknowledged danger, as the interview progressed the interviewees displayed increasing reservations, showing that despite their recurrent declarations of understanding the danger of not using protective measures—not all of them do so in practice. Their responses show that the processors have knowledge and awareness of occupational illnesses associated with exposure to silica dust and that they had full access to the relevant protective measures. The responses also reveal the perceptions, personality traits and defense mechanisms around which processors have built a psychological narrative to justify their noncompliant behavior. We found that ASW are well aware of the risks and dangers of their occupation yet they almost completely deny personal responsibility and blame others for the consequences of their behavior (External locus of control). Their predominant emotional reaction was anger. Each worker’s response was governed by a “personal and unique narrative” that represents a defense mechanism for nonadherence to safety measures. Conclusions: Given the psychological motivators, the main conclusion of the study is that it takes more than just enhancing the awareness of workers to the importance of using protective measures to create a sustainable change in the safety climate at stone processing plants. Therefore, it is necessary that all players execute their roles in full, in order to ensure that nonadherent behavior is not only acknowledged by fabricators as endangering their health but it has also immediate implications related to their employment, freedom to operate and responsibility