There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper gives an overview of the majorization-minimization (MM) algorithmic framework, which can provide guidance in deriving problem-driven algorithms with low computational cost. A general introduction of MM is presented, including a description of the basic principle and its convergence results. The extensions, acceleration schemes, and connection to other algorithmic frameworks are also covered. To bridge the gap between theory and practice, upperbounds for a large number of basic functions, derived based on the Taylor expansion, convexity, and special inequalities, are provided as ingredients for constructing surrogate functions. With the pre-requisites established, the way of applying MM to solving specific problems is elaborated by a wide range of applications in signal processing, communications, and machine learning.

Majorization-Minimization Algorithms in Signal Processing, Communications, and Machine Learning

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Spotlight Synthetic Aperture Radar Signal Processing Algorithms

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. However, deriving efficient strategies that jointly bring into play the primal and dual problems is a more recent idea that has generated many important new contributions in recent years. These novel developments are grounded in the recent advances in convex analysis, discrete optimization, parallel processing, and nonsmooth optimization with an emphasis on sparsity issues. In this article, we aim to present the principles of primal?dual approaches while providing an overview of the numerical methods that have been proposed in different contexts. Last but not least, primal?dual methods lead to algorithms that are easily parallelizable. Today, such parallel algorithms are becoming increasingly important for efficiently handling high-dimensional problems.

https://hal.archives-ouvertes.fr/hal-01010437/document

Playing with Duality: An overview of recent primal?dual approaches for solving large-scale optimization problems

This paper reviews recent progress of portable short-range noncontact microwave radar systems for motion detection, positioning, and imaging applications. With the continuous advancements of modern semiconductor technologies and embedded computing, many functionalities that could only be achieved by bulky radar systems in the past are now integrated into portable devices with integrated circuit chips and printed circuits boards. These portable solutions are able to provide high motion detection sensitivity, excellent signal-to-noise ratio, and satisfactory range detection capability. Assisted by on-board signal processing algorithms, they can play important roles in various areas, such as health and elderly care, veterinary monitoring, human-computer interaction, structural monitoring, indoor tracking, and wind engineering. This paper reviews some system architectures and practical implementations for typical wireless sensing applications. It also discusses potential future developments for the next-generation portable smart radar systems.

A Review on Recent Progress of Portable Short-Range Noncontact Microwave Radar Systems

In this paper, a multichannel recursive least-squares (RLS) algorithm is proposed for the identification of multiple-input single-output (MISO) systems. As compared to the conventional (multichannel) RLS algorithm, the proposed solution exploits a Kronecker product decomposition of the global impulse response, together with low-rank approximations. The gain is twofold, in terms of both performance and complexity. Simulations performed in the context of stereophonic acoustic echo cancellation indicate the appealing features of this algorithm.

A Multichannel Recursive Least-Squares Algorithm Based on a Kronecker Product Decomposition

In applications involving system identification problems, some characteristics of the impulse response of the system to be identified are usually exploited to design adaptive algorithms with improved performance. In this context, this paper focuses on the identification of systems that own intrinsic symmetric or antisymmetric properties, which can be further formulated by using a combination of bilinear forms. Based on such an approach, the least-mean-square (LMS) and normalized LMS (NLMS) algorithms with symmetric/antisymmetric properties (termed here LMS-SAS and NLMS-SAS) are proposed. Simulation results are shown confirming the improved convergence speed achieved by the proposed algorithms as compared to the conventional LMS and NLMS counterparts for different operating scenarios.

LMS and NLMS Algorithms for the Identification of Impulse Responses with Intrinsic Symmetric or Antisymmetric Properties

The stereophonic systems create the illusion of audio directionality using independent channels with a standard configuration consisting of two microphones and two speakers. The basic associated mathematical model used to correct the acoustic echo that affects the system has two-input/ two-output real random variables with four adaptive filters. An important simplification to this scheme is the widely linear (WL) model which uses a single-input/ single-output system with complex valued random variables (CRVs) and only one adaptive filter. In order to demonstrate the functionality, the recursive least-squares (RLS) adaptive algorithm is used. The RLS is conveniently combined with the dichotomous coordinate descent (DCD) algorithm in order to greatly reduce the matrix inversion arithmetic complexity and an aggregate of two filters (with transfer logic - TL) is used to considerably improve performance in high disturbance situations such as double talk.

A robust dual-path DCD-RLS algorithm for stereophonic acoustic echo cancellation

The paper addresses a classical problem of spectral analysis, that of high accuracy estimation of line-spectra, based on zero-padded Discrete Fourier Transform (DFT), transposed in the modern approach of sparse vectors estimation. Based on the analysis of the specific dictionary, some conclusions are drawn concerning the possibility to obtain high accuracy and resolution, by using a limited temporal analysis window. The issue of uniqueness is discussed. Two algorithms are presented, based on Matching Pursuit (MP) and Orthogonal Matching Pursuit (OMP).

Refining accuracy of the spectral lines estimation by a sparsity based approach

Solving a high-dimension system identification problem could involve significant challenges in terms of complexity and accuracy of the solution. Due to the large parameter space, a decomposition-based approach fits very well in this context. This was the idea behind the recently developed iterative Wiener filter for multilinear forms, which reformulates the problem using a combination of shorter filters. Nevertheless, there are inherent limitations related to the Wiener solution, while the least-mean-square (LMS) adaptive filter would represent a more practical alternative. Consequently, in this paper, we develop LMS-based algorithms for multilinear forms, in the context of a multiple-input/single-output system identification problem. Simulation results indicate the good performance of the proposed algorithms, especially in terms of their fast convergence features.

Silviu Ciochina

Papers

A Multichannel Recursive Least-Squares Algorithm Based on a Kronecker Product Decomposition

LMS and NLMS Algorithms for the Identification of Impulse Responses with Intrinsic Symmetric or Antisymmetric Properties

A robust dual-path DCD-RLS algorithm for stereophonic acoustic echo cancellation

Refining accuracy of the spectral lines estimation by a sparsity based approach

LMS Algorithms for Multilinear Forms