16 research outputs found
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
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
A Modulo-Based Architecture for Analog-to-Digital Conversion
Systems that capture and process analog signals must first acquire them
through an analog-to-digital converter. While subsequent digital processing can
remove statistical correlations present in the acquired data, the dynamic range
of the converter is typically scaled to match that of the input analog signal.
The present paper develops an approach for analog-to-digital conversion that
aims at minimizing the number of bits per sample at the output of the
converter. This is attained by reducing the dynamic range of the analog signal
by performing a modulo operation on its amplitude, and then quantizing the
result. While the converter itself is universal and agnostic of the statistics
of the signal, the decoder operation on the output of the quantizer can exploit
the statistical structure in order to unwrap the modulo folding. The
performance of this method is shown to approach information theoretical limits,
as captured by the rate-distortion function, in various settings. An
architecture for modulo analog-to-digital conversion via ring oscillators is
suggested, and its merits are numerically demonstrated
Denoising modulo samples: k-NN regression and tightness of SDP relaxation
Many modern applications involve the acquisition of noisy modulo samples of a
function , with the goal being to recover estimates of the original samples
of . For a Lipschitz function , suppose we are
given the samples where
denotes noise. Assuming are zero-mean i.i.d Gaussian's, and
's form a uniform grid, we derive a two-stage algorithm that recovers
estimates of the samples with a uniform error rate holding with high probability. The first stage
involves embedding the points on the unit complex circle, and obtaining
denoised estimates of via a NN (nearest neighbor) estimator.
The second stage involves a sequential unwrapping procedure which unwraps the
denoised mod estimates from the first stage.
Recently, Cucuringu and Tyagi proposed an alternative way of denoising modulo
data which works with their representation on the unit complex circle. They
formulated a smoothness regularized least squares problem on the product
manifold of unit circles, where the smoothness is measured with respect to the
Laplacian of a proximity graph involving the 's. This is a nonconvex
quadratically constrained quadratic program (QCQP) hence they proposed solving
its semidefinite program (SDP) based relaxation. We derive sufficient
conditions under which the SDP is a tight relaxation of the QCQP. Hence under
these conditions, the global solution of QCQP can be obtained in polynomial
time.Comment: 34 pages, 6 figure
Error analysis for denoising smooth modulo signals on a graph
In many applications, we are given access to noisy modulo samples of a smooth
function with the goal being to robustly unwrap the samples, i.e., to estimate
the original samples of the function. In a recent work, Cucuringu and Tyagi
proposed denoising the modulo samples by first representing them on the unit
complex circle and then solving a smoothness regularized least squares problem
-- the smoothness measured w.r.t the Laplacian of a suitable proximity graph
-- on the product manifold of unit circles. This problem is a quadratically
constrained quadratic program (QCQP) which is nonconvex, hence they proposed
solving its sphere-relaxation leading to a trust region subproblem (TRS). In
terms of theoretical guarantees, error bounds were derived for (TRS).
These bounds are however weak in general and do not really demonstrate the
denoising performed by (TRS).
In this work, we analyse the (TRS) as well as an unconstrained relaxation of
(QCQP). For both these estimators we provide a refined analysis in the setting
of Gaussian noise and derive noise regimes where they provably denoise the
modulo observations w.r.t the norm. The analysis is performed in a
general setting where is any connected graph.Comment: 36 pages, 2 figures. Added Section 5 (Simulations) and made minor
changes as per reviewers comment