Graded quantization for multiple description coding of compressive measurements

Compressed sensing (CS) is an emerging paradigm for acquisition of compressed representations of a sparse signal. Its low complexity is appealing for resource-constrained scenarios like sensor networks. However, such scenarios are often coupled with unreliable communication channels and providing robust transmission of the acquired data to a receiver is an issue. Multiple description coding (MDC) effectively combats channel losses for systems without feedback, thus raising the interest in developing MDC methods explicitly designed for the CS framework, and exploiting its properties. We propose a method called Graded Quantization (CS-GQ) that leverages the democratic property of compressive measurements to effectively implement MDC, and we provide methods to optimize its performance. A novel decoding algorithm based on the alternating directions method of multipliers is derived to reconstruct signals from a limited number of received descriptions. Simulations are performed to assess the performance of CS-GQ against other methods in presence of packet losses. The proposed method is successful at providing robust coding of CS measurements and outperforms other schemes for the considered test metrics

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

Subspace methods for portfolio design

Financial signal processing (FSP) is one of the emerging areas in the field of signal processing. It is comprised of mathematical finance and signal processing. Signal processing engineers consider speech, image, video, and price of a stock as signals of interest for the given application. The information that they will infer from raw data is different for each application. Financial engineers develop new solutions for financial problems using their knowledge base in signal processing. The goal of financial engineers is to process the harvested financial signal to get meaningful information for the purpose. Designing investment portfolios have always been at the center of finance. An investment portfolio is comprised of financial instruments such as stocks, bonds, futures, options, and others. It is designed based on risk limits and return expectations of investors and managed by portfolio managers. Modern Portfolio Theory (MPT) offers a mathematical method for portfolio optimization. It defines the risk as the standard deviation of the portfolio return and provides closed-form solution for the risk optimization problem where asset allocations are derived from. The risk and the return of an investment are the two inseparable performance metrics. Therefore, risk normalized return, called Sharpe ratio, is the most widely used performance metric for financial investments. Subspace methods have been one of the pillars of functional analysis and signal processing. They are used for portfolio design, regression analysis and noise filtering in finance applications. Each subspace has its unique characteristics that may serve requirements of a specific application. For still image and video compression applications, Discrete Cosine Transform (DCT) has been successfully employed in transform coding where Karhunen-Loeve Transform (KLT) is the optimum block transform. In this dissertation, a signal processing framework to design investment portfolios is proposed. Portfolio theory and subspace methods are investigated and jointly treated. First, KLT, also known as eigenanalysis or principal component analysis (PCA) of empirical correlation matrix for a random vector process that statistically represents asset returns in a basket of instruments, is investigated. Auto-regressive, order one, AR(1) discrete process is employed to approximate such an empirical correlation matrix. Eigenvector and eigenvalue kernels of AR(1) process are utilized for closed-form expressions of Sharpe ratios and market exposures of the resulting eigenportfolios. Their performances are evaluated and compared for various statistical scenarios. Then, a novel methodology to design subband/filterbank portfolios for a given empirical correlation matrix by using the theory of optimal filter banks is proposed. It is a natural extension of the celebrated eigenportfolios. Closed-form expressions for Sharpe ratios and market exposures of subband/filterbank portfolios are derived and compared with eigenportfolios. A simple and powerful new method using the rate-distortion theory to sparse eigen-subspaces, called Sparse KLT (SKLT), is developed. The method utilizes varying size mid-tread (zero-zone) pdf-optimized (Lloyd-Max) quantizers created for each eigenvector (or for the entire eigenmatrix) of a given eigen-subspace to achieve the desired cardinality reduction. The sparsity performance comparisons demonstrate the superiority of the proposed SKLT method over the popular sparse representation algorithms reported in the literature