Nonuniform decimation and reconstruction of generalized-bandlimited MD signals

It is well known that the data rate of bandlimited signals can be reduced without loss of information. One scheme which achieves this goal is the so-called nonuniform decimation. Recent results show that a bandlimited signal can be reconstructed from a nonuniformly decimated version. Theoretical results and efficient reconstruction methods have been addressed for one-dimensional signals. For the multidimensional case, some partial results are known. In this paper, we will discuss in detail the theory and implementation of the reconstruction of generalized-bandlimited multidimensional signals from their nonuniformly decimated versions

Multirate digital filters, filter banks, polyphase networks, and applications: a tutorial

Multirate digital filters and filter banks find application in communications, speech processing, image compression, antenna systems, analog voice privacy systems, and in the digital audio industry. During the last several years there has been substantial progress in multirate system research. This includes design of decimation and interpolation filters, analysis/synthesis filter banks (also called quadrature mirror filters, or QMFJ, and the development of new sampling theorems. First, the basic concepts and building blocks in multirate digital signal processing (DSPJ, including the digital polyphase representation, are reviewed. Next, recent progress as reported by several authors in this area is discussed. Several applications are described, including the following: subband coding of waveforms, voice privacy systems, integral and fractional sampling rate conversion (such as in digital audio), digital crossover networks, and multirate coding of narrow-band filter coefficients. The M-band QMF bank is discussed in considerable detail, including an analysis of various errors and imperfections. Recent techniques for perfect signal reconstruction in such systems are reviewed. The connection between QMF banks and other related topics, such as block digital filtering and periodically time-varying systems, based on a pseudo-circulant matrix framework, is covered. Unconventional applications of the polyphase concept are discussed

Generalized Sampling: Stability and Performance Analysis

Generalized sampling provides a general mechanism for recovering an unknown input function $f(x) \in H$ from the samples of the responses of m linear shift-invariant systems sampled at 1 ⁄ mth the reconstruction rate. The system can be designed to perform a projection of f(x) onto the reconstruction subspace $V(\varphi) = span \{\varphi(x - k)\} _{ k \in Z }$ ; for example, the family of bandlimited signals with $\varphi(x) = sinc(x)$. This implies that the reconstruction will be perfect when the input signal is included in V(φ): the traditional framework of Papoulis' generalized sampling theory. Otherwise, one recovers a signal approximation $f ^{ ~ } (x) \in V(\varphi)$ that is consistent with f(x) in the sense that it produces the same measurements. To characterize the stability of the algorithm, we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a Riesz basis of V(φ), and we use the corresponding Riesz bounds to define the condition number of the system. We then use these results to analyze the stability of various instances of interlaced and derivative sampling. Next, we consider the issue of performance, which becomes pertinent once we have extended the applicability of the method to arbitrary input functions, that is, when H is considerably larger than V(φ), and the reconstruction is no longer exact. By deriving general error bounds for projectors, we are able to show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation (orthogonal projection), which is generally not accessible. We then perform a detailed analysis for the case in which the analysis filters are in $L _{ 2 }$ and determine all relevant bound constants explicitly. Finally, we use an interlaced sampling example to illustrate these various calculations

Signal Reconstruction From Nonuniform Samples Using Prolate Spheroidal Wave Functions: Theory and Application

Nonuniform sampling occurs in many applications due to imperfect sensors, mismatchedclocks or event-triggered phenomena. Indeed, natural images, biomedical responses andsensor network transmission have bursty structure so in order to obtain samples that correspondto the information content of the signal, one needs to collect more samples when thesignal changes fast and fewer samples otherwise which creates nonuniformly distibuted samples.On the other hand, with the advancements in the integrated circuit technology, smallscale and ultra low-power devices are available for several applications ranging from invasivebiomedical implants to environmental monitoring. However the advancements in the devicetechnologies also require data acquisition methods to be changed from the uniform (clockbased, synchronous) to nonuniform (clockless, asynchronous) processing. An important advancementis in the data reconstruction theorems from sub-Nyquist rate samples which wasrecently introduced as compressive sensing and that redenes the uncertainty principle. Inthis dissertation, we considered the problem of signal reconstruction from nonuniform samples.Our method is based on the Prolate Spheroidal Wave Functions (PSWF) which can beused in the reconstruction of time-limited and essentially band-limited signals from missingsamples, in event-driven sampling and in the case of asynchronous sigma delta modulation.We provide an implementable, general reconstruction framework for the issues relatedto reduction in the number of samples and estimation of nonuniform sample times. We alsoprovide a reconstruction method for level crossing sampling with regularization. Another way is to use projection onto convex sets (POCS) method. In this method we combinea time-frequency approach with the POCS iterative method and use PSWF for the reconstructionwhen there are missing samples. Additionally, we realize time decoding modulationfor an asynchronous sigma delta modulator which has potential applications in low-powerbiomedical implants

A preliminary study of a new sampling approach

Thesis (E.A.A.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2000.Includes bibliographical references (leaves 47-50).by Agus Budiyono.E.A.A

Sampling theory in shift-invariant spaces: generalizations

Roughly speaking sampling theory deals with determining whether we can or can not recover a continuous function from some discrete set of its values. The most important result and main pillar of this theory is the well-known Shannon’s sampling theorem which states that: If a signal f(t) contains no frequencies higher than 1/2 cycles per second, it is completely determined by giving its ordinates at a sequence of points spaced one second apart….A grandes rasgos la teoría de muestreo estudia el problema de recuperar una función continua a partir de un conjunto discreto de sus valores. El resultado más importante y pilar fundamental de esta teoría es el conocido teorema de muestreo de Shannon que afirma que: Si una señal f(t) no contiene frecuencias mayores que 1/2 ciclos por segundo entonces está completamente determinada por sus ordenadas en una sucesión de puntos espaciados en un segundo….Proyecto de investigación MTM2009–08345 del Ministerio de Ciencia e Innovación de España.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Luis Alberto Ibort Latre.- Secretario: Eugenio Hernández Rodríguez.- Vocal: Ole Christense

Energy-Efficient Time-Based Encoders and Digital Signal Processors in Continuous Time

Continuous-time (CT) data conversion and continuous-time digital signal processing (DSP) are an interesting alternative to conventional methods of signal conversion and processing. This alternative proposes time-based encoding that may not suffer from aliasing; shows superior spectral properties (e.g. no quantization noise floor); and enables time-based, event-driven, flexible signal processing using digital circuits, thus scaling well with technology. Despite these interesting features, this approach has so far been limited by the CT encoder, due to both its relatively poor energy efficiency and the constraints it imposes on the subsequent CT DSP. In this thesis, we present three principles that address these limitations and help improve the CT ADC/DSP system. First, an adaptive-resolution encoding scheme that achieves first-order reconstruction with simple circuitry is proposed. It is shown that for certain signals, the scheme can significantly reduce the number of samples generated per unit of time for a given accuracy compared to schemes based on zero-order-hold reconstruction, thus promising to lead to low dynamic power dissipation at the system level. Presented next is a novel time-based CT ADC architecture, and associated encoding scheme, that allows a compact, energy-efficient circuit implementation, and achieves first-order quantization error spectral shaping. The design of a test chip, implemented in a 0.65-V 28-nm FDSOI process, that includes this CT ADC and a 10-tap programmable FIR CT DSP to process its output is described. The system achieves 32 dB – 42 dB SNDR over a 10 MHz – 50 MHz bandwidth, occupies 0.093 mm2, and dissipates 15 µW–163 µW as the input amplitude goes from zero to full scale. Finally, an investigation into the possibility of CT encoding using voltage-controlled oscillators is undertaken, and it leads to a CT ADC/DSP system architecture composed primarily of asynchronous digital delays. The latter makes the system highly digital and technology-scaling-friendly and, hence, is particularly attractive from the point of view of technology migration. The design of a test chip, where this delay-based CT ADC/DSP system architecture is used to implement a 16-tap programmable FIR filter, in a 1.2-V 28-nm FDSOI process, is described. Simulations show that the system will achieve a 33 dB – 40 dB SNDR over a 600 MHz bandwidth, while dissipating 4 mW

Compressed Sensing in Multi-Signal Environments.

Technological advances and the ability to build cheap high performance sensors make it possible to deploy tens or even hundreds of sensors to acquire information about a common phenomenon of interest. The increasing number of sensors allows us to acquire ever more detailed information about the underlying scene that was not possible before. This, however, directly translates to increasing amounts of data that needs to be acquired, transmitted, and processed. The amount of data can be overwhelming, especially in applications that involve high-resolution signals such as images or videos. Compressed sensing (CS) is a novel acquisition and reconstruction scheme that is particularly useful in scenarios when high resolution signals are difficult or expensive to encode. When applying CS in a multi-signal scenario, there are several aspects that need to be considered such as the sensing matrix, the joint signal model, and the reconstruction algorithm. The purpose of this dissertation is to provide a complete treatment of these aspects in various multi-signal environments. Specific applications include video, multi-view imaging, and structural health monitoring systems. For each application, we propose a novel joint signal model that accurately captures the joint signal structure, and we tailor the reconstruction algorithm to each signal model to successfully recover the signals of interest.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/98007/1/jaeypark_1.pd

ИНТЕЛЛЕКТУАЛЬНЫЙ числовым программным ДЛЯ MIMD-компьютер

For most scientific and engineering problems simulated on computers the solving of problems of the computational mathematics with approximately given initial data constitutes an intermediate or a final stage. Basic problems of the computational mathematics include the investigating and solving of linear algebraic systems, evaluating of eigenvalues and eigenvectors of matrices, the solving of systems of non-linear equations, numerical integration of initial- value problems for systems of ordinary differential equations.Для більшості наукових та інженерних задач моделювання на ЕОМ рішення задач обчислювальної математики з наближено заданими вихідними даними складає проміжний або остаточний етап. Основні проблеми обчислювальної математики відносяться дослідження і рішення лінійних алгебраїчних систем оцінки власних значень і власних векторів матриць, рішення систем нелінійних рівнянь, чисельного інтегрування початково задач для систем звичайних диференціальних рівнянь.Для большинства научных и инженерных задач моделирования на ЭВМ решение задач вычислительной математики с приближенно заданным исходным данным составляет промежуточный или окончательный этап. Основные проблемы вычислительной математики относятся исследования и решения линейных алгебраических систем оценки собственных значений и собственных векторов матриц, решение систем нелинейных уравнений, численного интегрирования начально задач для систем обыкновенных дифференциальных уравнений

Joint Estimation of Attenuation and Scatter for Tomographic Imaging with the Broken Ray Transform

The single-scatter approximation is fundamental for many tomographic imaging problems. This class broadly includes x-ray scattering imaging and optical scatter imaging for certain media. In all cases, noisy measurements are affected by both local events and nonlocal attenuation. Related applications typically focus on reconstructing one of two images: scatter density or total attenuation. However, both images are media specific. Both images are useful for object identification. Knowledge of one image aides estimation of the other, especially when estimating images from noisy data.Joint image recovery has been demonstrated analytically in the context of the broken ray transform (BRT) for attenuation and scatter-density images. The BRT summarizes the nonlocal affects of attenuation in single-scatter measurement geometries. We find BRT analysis particularly interesting as joint image recovery has been demonstrated analytically using only two scatter angles. Limiting observations to two scatter angles is significant because it supports joint reconstruction in two dimensions for anisotropic scatter modalities (e.g. Bragg, Compton). However, all analytic inversion strategies share two fundamental assumptions limiting their utility: nonzero scatter everywhere, and a deterministic data model.There are two themes to our work. First, we consider the BRT in a purely deterministic setting. We are the first to recognize the BRT as a linear shift-invariant operator. This linear-systems perspective motivates frequency-domain analysis both of the data and operator. Frequency-domain representations provide new insights on the operator and a common framework for contrasting recent inversion formulas. New algorithms are presented for regularized inversion of the BRT in addition to fast forward and adjoint operators. Second, we incorporate the BRT in a stochastic data model. Approximating the detectors as photon counting processes, we model the data as Poisson distributed. Our iterative algorithm, alternating scatter and attenuation image updates, guarantees monotonic reduction of the regularized log-likelihood function of the data. We are the first to consider joint image estimation from noisy data. Our results demonstrate a significant improvement over analytic methods for data sets with missing data (regions with zero scatter). In addition to joint image estimation, our approach can be specialized for single image estimation. With known attenuation, we can improve the quality of scatter image estimates. Similarly, with known scatter, we can improve the quality of attenuation image estimates.Through analysis and simulations, we highlight challenges for attenuation image estimation from BRT data, and ambiguity in the joint image recovery problem. Performance will vary with scaling of the problem. Total attenuation, detected counts, and scatter angle all affect the quality of image estimates. We are the first to incorporate both scatter density and attenuation in noisy data models. Our results demonstrate the benefits of accounting for both images, and should inform design of future measurement systems