Two important attributes of analog to digital converters (ADCs) are its sampling rate and dynamic range. The sampling rate should be greater than or equal to the Nyquist rate for bandlimited signals with bounded energy. It is also desired that the signals' dynamic range should be within that of the ADC's; otherwise, the signal will be clipped. A modulo operator has been recently suggested prior to sampling to restrict the dynamic range. Due to the nonlinearity of the modulo operation, the samples are distorted. Existing recovery algorithms to recover the signal from its modulo samples operate at a high sampling rate and are not robust in the presence of noise. In this paper, we propose a robust algorithm to recover the signal from the modulo samples which operates at lower sampling rate compared to existing techniques. We also show that our method has lower error compared to existing approaches for a given sampling rate, noise level, and dynamic range of the ADC. Our results lead to less constrained hardware design to address dynamic range issues while operating at the lowest rate possible.
Reconstruction of Piecewise-Constant Sparse Signals for Modulo Sampling
Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科）；Institute of Engineering, Tokyo University of Agriculture and Technology（东京农工大学工程学院）
The latest theoretical advances in the field of unlimited sampling framework (USF) show the potential to avoid clipping problems of analog-to-digital converters (ADC). To date, most of the related works have focused on real-valued modulo samples, but little has been reported about the impact of quantization. In this paper, we study more practical USF system where modulo samples are quantized to a finite number of bits. In particular, we present a new requirement about the lower bound of sampling rate to ensure exact recovery of original signals from quantized modulo samples. The minimum sampli…
We consider the problem of recovering a continuous-time bandlimited signal from the discrete-time signal obtained from sampling it every $T_s$ seconds and reducing the result modulo $Δ$, for some $Δ>0$. For $Δ=\infty$ the celebrated Shannon-Nyquist sampling theorem guarantees that perfect recovery is possible provided that the sampling rate $1/T_s$ exceeds the so-called Nyquist rate. Recent work by Bhandari et al. has shown that for any $Δ>0$ perfect reconstruction is still possible if the sampling rate exceeds the Nyquist rate by a factor of $πe$. In this letter we improve upon this result an…
An alternative to conventional uniform sampling is that of time encoding, which converts continuous-time signals into streams of trigger times. This gives rise to Event-Driven Sampling (EDS) models. The data-driven nature of EDS acquisition is advantageous in terms of power consumption and time resolution and is inspired by the information representation in biological nervous systems. If an analog signal is outside a predefined dynamic range, then EDS generates a low density of trigger times, which in turn leads to recovery distortion due to aliasing. In this paper, inspired by the Unlimited S…
The Unlimited Sensing Framework (USF) is a digital acquisition protocol that allows for sampling and reconstruction of high dynamic range signals. By acquiring modulo samples, the USF circumvents the clipping or saturation problem that is a fundamental bottleneck in conventional analog-to-digital converters (ADCs). In the context of the USF, several works have focused on bandlimited function classes and recently, a hardware validation of the modulo sampling approach has been presented. In a different direction, in this paper we focus on non-bandlimited function classes and consider the well-kn…
Following the Unlimited Sampling strategy to alleviate the omnipresent dynamic range barrier, we study the problem of recovering a bandlimited signal from point-wise modulo samples, aiming to connect theoretical guarantees with hardware implementation considerations. Our starting point is a class of non-idealities that we observe in prototyping an unlimited sampling based analog-to-digital converter. To address these non-idealities, we provide a new Fourier domain recovery algorithm. Our approach is validated both in theory and via extensive experiments on our prototype analog-to-digital conve…
We consider the problem of recovering $n$ i.i.d samples from a zero mean multivariate Gaussian distribution with an unknown covariance matrix, from their modulo wrapped measurements, i.e., measurement where each coordinate is reduced modulo $Δ$, for some $Δ>0$. For this setup, which is motivated by quantization and analog-to-digital conversion, we develop a low-complexity iterative decoding algorithm. We show that if a benchmark informed decoder that knows the covariance matrix can recover each sample with small error probability, and $n$ is large enough, the performance of the proposed blind …
Shannon's sampling theorem is one of the cornerstone topics that is well understood and explored, both mathematically and algorithmically. That said, practical realization of this theorem still suffers from a severe bottleneck due to the fundamental assumption that the samples can span an arbitrary range of amplitudes. In practice, the theorem is realized using so-called analog-to-digital converters (ADCs) which clip or saturate whenever the signal amplitude exceeds the maximum recordable ADC voltage thus leading to a significant information loss. In this paper, we develop an alternative parad…
The Unlimited Sensing Framework (USF) was recently introduced to overcome the sensor saturation bottleneck in conventional digital acquisition systems. At its core, the USF allows for high-dynamic-range (HDR) signal reconstruction by converting a continuous-time signal into folded, low-dynamic-range (LDR), modulo samples. HDR reconstruction is then carried out by algorithmic unfolding of the folded samples. In hardware, however, implementing an ideal modulo folding requires careful calibration, analog design and high precision. At the interface of theory and practice, this paper explores a com…
Samsung Research Institute（三星研究院）；Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
Simultaneous orthogonal matching pursuit (SOMP) and block OMP (BOMP) are two widely used techniques for sparse support recovery in multiple measurement vector (MMV) and block sparse (BS) models respectively. For optimal performance, both SOMP and BOMP require \textit{a priori} knowledge of signal sparsity or noise variance. However, sparsity and noise variance are unavailable in most practical applications. This letter presents a novel technique called generalized residual ratio thresholding (GRRT) for operating SOMP and BOMP without the \textit{a priori} knowledge of signal sparsity and noise…
University of Bremen（不来梅大学）；University of Hamburg（汉堡大学）；Imperial College London（伦敦帝国理工学院）；Technical University of Munich（慕尼黑工业大学）
Recently, experiments have been reported where researchers were able to perform high dynamic range (HDR) tomography in a heuristic fashion, by fusing multiple tomographic projections. This approach to HDR tomography has been inspired by HDR photography and inherits the same disadvantages. Taking a computational imaging approach to the HDR tomography problem, we here suggest a new model based on the Modulo Radon Transform (MRT), which we rigorously introduce and analyze. By harnessing a joint design between hardware and algorithms, we present a single-shot HDR tomography approach, which to our …
Self-reset analog-to-digital converters (ADCs) are used to sample high dynamic range signals resulting in modulo-operation based folded signal samples. We consider the case where each vertex of a graph (e.g., sensors in a network) is equipped with a self-reset ADC and senses a time series. Graph sampling allows the graph time series to be represented by the signals at a subset of sampled vertices and time instances. We investigate the problem of recovering bandlimited continuous-time graph signals from folded signal samples. We derive sufficient conditions to achieve successful recovery of the…
We present DeepFPC, a novel deep neural network designed by unfolding the iterations of the fixed-point continuation algorithm with one-sided l1-norm (FPC-l1), which has been proposed for solving the 1-bit compressed sensing problem. The network architecture resembles that of deep residual learning and incorporates prior knowledge about the signal structure (i.e., sparsity), thereby offering interpretability by design. Once DeepFPC is properly trained, a sparse signal can be recovered fast and accurately from quantized measurements. The proposed model is evaluated in the task of direction-of-a…
In this paper we introduce a new sampling and reconstruction approach for multi-dimensional analog signals. Building on top of the Unlimited Sensing Framework (USF), we present a new folded sampling operator called the multi-dimensional modulo-hysteresis that is also backwards compatible with the existing one-dimensional modulo operator. Unlike previous approaches, the proposed model is specifically tailored to multi-dimensional signals. In particular, the model uses certain redundancy in dimensions 2 and above, which is exploited for input recovery with robustness. We prove that the new opera…
