Reconstruction of Bandlimited Functions from Unsigned Samples

We consider the recovery of real-valued bandlimited functions from the absolute values of their samples, possibly spaced nonuniformly. We show that such a reconstruction is always possible if the function is sampled at more than twice its Nyquist rate, and may not necessarily be possible if the samples are taken at less than twice the Nyquist rate. In the case of uniform samples, we also describe an FFT-based algorithm to perform the reconstruction. We prove that it converges exponentially rapidly in the number of samples used and examine its numerical behavior on some test cases

Compressive Sensing of Analog Signals Using Discrete Prolate Spheroidal Sequences

Compressive sensing (CS) has recently emerged as a framework for efficiently capturing signals that are sparse or compressible in an appropriate basis. While often motivated as an alternative to Nyquist-rate sampling, there remains a gap between the discrete, finite-dimensional CS framework and the problem of acquiring a continuous-time signal. In this paper, we attempt to bridge this gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSS's), a collection of functions that trace back to the seminal work by Slepian, Landau, and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's form a highly efficient basis for sampled bandlimited functions; by modulating and merging DPSS bases, we obtain a dictionary that offers high-quality sparse approximations for most sampled multiband signals. This multiband modulated DPSS dictionary can be readily incorporated into the CS framework. We provide theoretical guarantees and practical insight into the use of this dictionary for recovery of sampled multiband signals from compressive measurements

Arbitrary Phase Vocoders by means of Warping

The Phase Vocoder plays a central role in sound analysis and synthesis, allowing us to represent a sound signal in both time and frequency, similar to a music score – but possibly at much finer time and frequency scales – describing the evolution of sound events. According to the uncertainty principle, time and frequency are not independent variables so that any time-frequency representation is the result of a compromise between time and frequency resolutions, the product of which cannot be smaller than a given constant. Therefore, finer frequency resolution can only be achieved with coarser time resolution and, similarly, finer time resolution results in coarser frequency resolution.While most of the conventional methods for time-frequency representations are based on uniform time and uniform frequency resolutions, perception and physical characteristics of sound signals suggest the need for nonuniform analysis and synthesis. As the results of psycho-acoustic research show, human hearing is naturally organized in nonuniform frequency bands. On the physical side, the sounds of percussive instruments as well as piano in the low register, show partials whose frequencies are not uniformly spaced, as opposed to the uniformly spaced partial frequencies found in harmonic sounds. Moreover, the different characteristics of sound signals at the onset transients with respect to stationary segments suggest the need for nonuniform time resolution. In the effort to exploit the time-frequency resolution compromise at its best, a tight time-frequency suit should be tailored to snuggly fit the sound body.In this paper we overview flexible design methods for phase vocoders with nonuniform resolutions. The methods are based on remapping the time or the frequency axis, or both, by employing suitable functions acting as warping maps, which locally change the characteristics of the time-frequency plane. As a result, the sliding windows may have time dependent duration and/or frequency dependent bandwidth. As an example, in a constant Q frequency band allocation, the ratios of center band frequencies over bandwidth remains constant, so that the frequency bands become wider and wider as center frequency increases, similarly to the frequency distance of 12-tone scale notes or of octaves.While time-frequency allocation can be performed in an arbitrary way, the ability to reconstruct the original signal from Vocoder analysis data is essential in sound processing and transformation applications. Moreover, even the analysis or the production of spectrograms benefits from the perfect reconstruction property if one needs to be confident that no important information is hidden, which serves to completely describe the signal

Noise Effects on a Proposed Algorithm for Signal Reconstruction and Bandwidth Optimization

The development of wireless technology in recent years has increased the demand for channel resources within a limited spectrum. The system\u27s performance can be improved through bandwidth optimization, as the spectrum is a scarce resource. To reconstruct the signal, given incomplete knowledge about the original signal, signal reconstruction algorithms are needed. In this paper, we propose a new scheme for reducing the effect of adding additive white Gaussian noise (AWGN) using a noise reject filter (NRF) on a previously discussed algorithm for baseband signal transmission and reconstruction that can reconstruct most of the signal’s energy without any need to send most of the signal’s concentrated power like the conventional methods, thus achieving bandwidth optimization. The proposed scheme for noise reduction was tested for a pulse signal and stream of pulses with different rates (2, 4, 6, and 8 Mbps) and showed good reconstruction performance in terms of the normalized mean squared error (NMSE) and achieved an average enhancement of around 48%. The proposed schemes for signal reconstruction and noise reduction can be applied to different applications, such as ultra-wideband (UWB) communications, radio frequency identification (RFID) systems, mobile communication networks, and radar systems

Quantization and Compressive Sensing

Quantization is an essential step in digitizing signals, and, therefore, an indispensable component of any modern acquisition system. This book chapter explores the interaction of quantization and compressive sensing and examines practical quantization strategies for compressive acquisition systems. Specifically, we first provide a brief overview of quantization and examine fundamental performance bounds applicable to any quantization approach. Next, we consider several forms of scalar quantizers, namely uniform, non-uniform, and 1-bit. We provide performance bounds and fundamental analysis, as well as practical quantizer designs and reconstruction algorithms that account for quantization. Furthermore, we provide an overview of Sigma-Delta ($\Sigma\Delta$) quantization in the compressed sensing context, and also discuss implementation issues, recovery algorithms and performance bounds. As we demonstrate, proper accounting for quantization and careful quantizer design has significant impact in the performance of a compressive acquisition system.Comment: 35 pages, 20 figures, to appear in Springer book "Compressed Sensing and Its Applications", 201

5D seismic data completion and denoising using a novel class of tensor decompositions

We have developed a novel strategy for simultaneous interpolation and denoising of prestack seismic data. Most seismic surveys fail to cover all possible source-receiver combinations, leading to missing data especially in the midpoint-offset domain. This undersampling can complicate certain data processing steps such as amplitude-variation-with-offset analysis and migration. Data interpolation can mitigate the impact of missing traces. We considered the prestack data as a 5D multidimensional array or otherwise referred to as a 5D tensor. Using synthetic data sets, we first found that prestack data can be well approximated by a low-rank tensor under a recently proposed framework for tensor singular value decomposition (tSVD). Under this low-rank assumption, we proposed a complexity-penalized algorithm for the recovery of missing traces and data denoising. In this algorithm, the complexity regularization was controlled by tuning a single regularization parameter using a statistical test. We tested the performance of the proposed algorithm on synthetic and real data to show that missing data can be reliably recovered under heavy downsampling. In addition, we demonstrated that compressibility, i.e., approximation of the data by a low-rank tensor, of seismic data under tSVD depended on the velocity model complexity and shot and receiver spacing. We further found that compressibility correlated with the recovery of missing data because high compressibility implied good recovery and vice versa.National Science Foundation (U.S.). Graduate Research Fellowship (Grant DGE-0806676)National Science Foundation (U.S.). Division of Computing and Communication Foundations (Grant NSF-1319653

Nonuniform sampling problems in computed tomography

Two problems involving high-resolution reconstruction from nonuniformly sampled data in x-ray computed tomography are addressed. A technique based on the theorem for sampling on unions of shifted lattices is introduced which exploits the symmetry property in two-dimensional fan beam computed tomography and permits the reconstruction of images with twice the resolution as the standard reconstruction by increasing only the number of views per rotation. An estimate is given for the aliasing error committed in the case of non-bandlimited data. Numerical results are presented which demonstrate the improvement in the quality of images from real and simulated data. A mathematical framework is presented for analyzing the longitudinal interpolation problem in three-dimensional multislice helical computed tomography. The problem is viewed as a collection of one-dimensional nonuniform but periodic sampling problems An accurate interpolation formula based on the periodic sampling theorem is introduced. A measure of suitability of a sampling scheme is presented and candidates for so-called preferred helical pitch are identified. Numerical results from simulated data are presented which confirm the theoretical predictions

A Compressive Phase-Locked Loop

We develop a new method for tracking narrowband signals acquired through compressive sensing, called the compressive sensing phase-locked loop (CS-PLL). The CS-PLL enables one to track oscillating signals in very large bandwidths using a small number of measurements. Not only does the CS-PLL potentially operate below the Nyquist rate, it can extract phase and frequency information without the computational complexity normally associated with compressive sensing signal re-construction. The CS-PLL has a wide variety of applications, including but not limited to communications, phase tracking, robust control, sensing, and FM demodulation. In particular we emphasize the advantages of using this system in wideband surveillence systems. Our design modifies classical PLL designs to operate with CS-based sampling systems. Performance results are shown for PLLs operating on both real and complex data. In addition to explaining general performance tradeoffs, implementations using several different CS sampling systems are explored

Model-based reconstruction of magnetic resonance spectroscopic imaging

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 77-80).Magnetic resonance imaging (MRI) is a medical imaging technique that is used to obtain images of soft tissue throughout the body. Since its development in the 1970s, MRI has gained tremendous importance in clinical practice because it can produce high quality images of diagnostic value in an ever expanding range of applications from neuroimaging to body imaging to cancer. By far the dominant signal source in MRI is hydrogen nuclei in water. The presence of water at high concentration (-50M) in body tissue, combined with signal contrast modulation induced by the local environment of water molecules, accounts for the success of MRI as a medical imaging modality. As opposed to conventional MRI, which derives its signal from the water component, magnetic resonance spectroscopy (MRS) acquires the magnetic resonance signal from other chemical components, most frequently various metabolites in the brain, but also signals from tumors in breast and prostate. The spectroscopic signal arises from low concentration (-1 - 10mM) compounds, but in spite of the challenges posed by the resulting low signal-to-noise ratio (SNR), the development of MRS is motivated by the desire to directly observe signal sources other than water. The combination of MRS with spatial encoding is called magnetic resonance spectroscopic imaging (MRSI). MRSI captures not only the relative intensities of metabolite signals at each voxel, but also their spatial distributions. While MRSI has been proven to be clinically useful, it suffers from fundamental tradeoffs due to the inherently low SNR, such as long acquisition time and low spatial resolution. In this thesis, techniques that combine benefits from both model-based reconstruction methods and regularized reconstructions with prior knowledge are proposed and demonstrated for MRSI. These methods address constraints on acquisition time in MRSI by undersampling data during acquisition in combination with improved image reconstruction methods.by Itthi Chatnuntawech.S.M