27 research outputs found
On Unlimited Sampling
Shannon's sampling theorem provides a link between the continuous and the
discrete realms stating that bandlimited signals are uniquely determined by its
values on a discrete set. This theorem is realized in practice using so called
analog--to--digital converters (ADCs). Unlike Shannon's sampling theorem, the
ADCs are limited in dynamic range. Whenever a signal exceeds some preset
threshold, the ADC saturates, resulting in aliasing due to clipping. The goal
of this paper is to analyze an alternative approach that does not suffer from
these problems. Our work is based on recent developments in ADC design, which
allow for ADCs that reset rather than to saturate, thus producing modulo
samples. An open problem that remains is: Given such modulo samples of a
bandlimited function as well as the dynamic range of the ADC, how can the
original signal be recovered and what are the sufficient conditions that
guarantee perfect recovery? In this paper, we prove such sufficiency conditions
and complement them with a stable recovery algorithm. Our results are not
limited to certain amplitude ranges, in fact even the same circuit architecture
allows for the recovery of arbitrary large amplitudes as long as some estimate
of the signal norm is available when recovering. Numerical experiments that
corroborate our theory indeed show that it is possible to perfectly recover
function that takes values that are orders of magnitude higher than the ADC's
threshold.Comment: 11 pages, 4 figures, copy of initial version to appear in Proceedings
of 12th International Conference on Sampling Theory and Applications (SampTA
One-Bit-Aided Modulo Sampling for DOA Estimation
Modulo sampling or unlimited sampling has recently drawn a great deal of
attention for cutting-edge applications, due to overcoming the barrier of
information loss through sensor saturation and clipping. This is a significant
problem, especially when the range of signal amplitudes is unknown or in the
near-far case. To overcome this fundamental bottleneck, we propose a
one-bit-aided (1bit-aided) modulo sampling scheme for direction-of-arrival
(DOA) estimation. On the one hand, one-bit quantization involving a simple
comparator offers the advantages of low-cost and low-complexity implementation.
On the other hand, one-bit quantization provides an estimate of the normalized
covariance matrix of the unquantized measurements via the arcsin law. The
estimate of the normalized covariance matrix is used to implement blind
integer-forcing (BIF) decoder to unwrap the modulo samples to construct the
covariance matrix, and subspace methods can be used to perform the DOA
estimation. Our approach named as 1bit-aided-BIF addresses the near-far problem
well and overcomes the intrinsic low dynamic range of one-bit quantization.
Numerical experiments validate the excellent performance of the proposed
algorithm compared to using a high-precision ADC directly in the given set up
HDR Imaging With One-Bit Quantization
Modulo sampling and dithered one-bit quantization frameworks have emerged as
promising solutions to overcome the limitations of traditional
analog-to-digital converters (ADCs) and sensors. Modulo sampling, with its
high-resolution approach utilizing modulo ADCs, offers an unlimited dynamic
range, while dithered one-bit quantization offers cost-efficiency and reduced
power consumption while operating at elevated sampling rates. Our goal is to
explore the synergies between these two techniques, leveraging their unique
advantages, and to apply them to non-bandlimited signals within spline spaces.
One noteworthy application of these signals lies in High Dynamic Range (HDR)
imaging. In this paper, we expand upon the Unlimited One-Bit (UNO) sampling
framework, initially conceived for bandlimited signals, to encompass
non-bandlimited signals found in the context of HDR imaging. We present a novel
algorithm rigorously examined for its ability to recover images from one-bit
modulo samples. Additionally, we introduce a sufficient condition specifically
designed for UNO sampling to perfectly recover non-bandlimited signals within
spline spaces. Our numerical results vividly demonstrate the effectiveness of
UNO sampling in the realm of HDR imaging.Comment: arXiv admin note: text overlap with arXiv:2308.0069
Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain
Recovering a sparse signal from its low-pass projections in the Fourier
domain is a problem of broad interest in science and engineering and is
commonly referred to as super-resolution. In many cases, however, Fourier
domain may not be the natural choice. For example, in holography, low-pass
projections of sparse signals are obtained in the Fresnel domain. Similarly,
time-varying system identification relies on low-pass projections on the space
of linear frequency modulated signals. In this paper, we study the recovery of
sparse signals from low-pass projections in the Special Affine Fourier
Transform domain (SAFT). The SAFT parametrically generalizes a number of well
known unitary transformations that are used in signal processing and optics. In
analogy to the Shannon's sampling framework, we specify sampling theorems for
recovery of sparse signals considering three specific cases: (1) sampling with
arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels
and, (3) recovery from Gabor transform measurements linked with the SAFT
domain. Our work offers a unifying perspective on the sparse sampling problem
which is compatible with the Fourier, Fresnel and Fractional Fourier domain
based results. In deriving our results, we introduce the SAFT series (analogous
to the Fourier series) and the short time SAFT, and study convolution theorems
that establish a convolution--multiplication property in the SAFT domain.Comment: 42 pages, 3 figures, manuscript under revie
Joint transmit and receive beamforming design in full-duplex integrated sensing and communications
Integrated sensing and communication (ISAC) has been envisioned as a solution to realize the sensing capability required for emerging applications in wireless networks. For a mono-static ISAC transceiver, as signal transmission durations are typically much longer than the radar echo round-trip times, the radar returns are drowned by the strong residual self interference (SI) from the transmitter, despite adopting sufficient SI cancellation techniques before digital domain - a phenomenon termed the echo-miss problem. A promising approach to tackle this problem involves the ISAC transceiver to be full-duplex (FD), and in this paper we jointly design the transmit and receive beamformers at the transceiver, transmit precoder at the uplink user, and receive combiner at the downlink user to simultaneously (a) maximize the uplink and downlink communication rate, (b) maximize the transmit and receive radar beampattern power at the target, and (c) suppress the residual SI. To solve this optimization problem, we proposed a penalty-based iterative algorithm. Numerical results illustrate that the proposed design can effectively achieve up to 60 dB digital-domain SI cancellation, a higher average sum-rate, and more accurate radar parameter estimation compared with previous ISAC FD studies
Fourier-Domain Inversion for the Modulo Radon Transform
Inspired by the multiple-exposure fusion approach in computational
photography, recently, several practitioners have explored the idea of high
dynamic range (HDR) X-ray imaging and tomography. While establishing promising
results, these approaches inherit the limitations of multiple-exposure fusion
strategy. To overcome these disadvantages, the modulo Radon transform (MRT) has
been proposed. The MRT is based on a co-design of hardware and algorithms. In
the hardware step, Radon transform projections are folded using modulo
non-linearities. Thereon, recovery is performed by algorithmically inverting
the folding, thus enabling a single-shot, HDR approach to tomography. The first
steps in this topic established rigorous mathematical treatment to the problem
of reconstruction from folded projections. This paper takes a step forward by
proposing a new, Fourier domain recovery algorithm that is backed by
mathematical guarantees. The advantages include recovery at lower sampling
rates while being agnostic to modulo threshold, lower computational complexity
and empirical robustness to system noise. Beyond numerical simulations, we use
prototype modulo ADC based hardware experiments to validate our claims. In
particular, we report image recovery based on hardware measurements up to 10
times larger than the sensor's dynamic range while benefiting with lower
quantization noise (12 dB).Comment: 12 pages, submitted for possible publicatio
Time Encoding via Unlimited Sampling: Theory, Algorithms and Hardware Validation
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 Sensing
Framework (USF), we propose a new EDS architecture that incorporates a modulo
nonlinearity prior to acquisition that we refer to as the modulo EDS or MEDS.
In MEDS, the modulo nonlinearity folds high dynamic range inputs into low
dynamic range amplitudes, thus avoiding recovery distortion. In particular, we
consider the asynchronous sigma-delta modulator (ASDM), previously used for low
power analog-to-digital conversion. This novel MEDS based acquisition is
enabled by a recent generalization of the modulo nonlinearity called
modulo-hysteresis. We design a mathematically guaranteed recovery algorithm for
bandlimited inputs based on a sampling rate criterion and provide
reconstruction error bounds. We go beyond numerical experiments and also
provide a first hardware validation of our approach, thus bridging the gap
between theory and practice, while corroborating the conceptual underpinnings
of our work.Comment: 27 pgs, 11 figures, IEEE Trans. Sig. Proc., accepted with minor
revision
Modulation For Modulo: A Sampling-Efficient High-Dynamic Range ADC
In high-dynamic range (HDR) analog-to-digital converters (ADCs), having many
quantization bits minimizes quantization errors but results in high bit rates,
limiting their application scope. A strategy combining modulo-folding with a
low-DR ADC can create an efficient HDR-ADC with fewer bits. However, this
typically demands oversampling, increasing the overall bit rate. An alternative
method using phase modulation (PM) achieves HDR-ADC functionality by modulating
the phase of a carrier signal with the analog input. This allows a low-DR ADC
with fewer bits. We've derived identifiability results enabling reconstruction
of the original signal from PM samples acquired at the Nyquist rate, adaptable
to various signals and non-uniform sampling. Using discrete phase demodulation
algorithms for practical implementation, our PM-based approach doesn't require
oversampling in noise-free conditions, contrasting with modulo-based ADCs. With
noise, our PM-based HDR method demonstrates efficiency with lower
reconstruction errors and reduced sampling rates. Our hardware prototype
illustrates reconstructing signals ten times greater than the ADC's DR from
Nyquist rate samples, potentially replacing high-bit rate HDR-ADCs while
meeting existing bit rate needs.Comment: 12 Pages, 13 Figure