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

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Table Of Integrals Series And Products

Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ) error term combined with a sparseness-inducing regularization term. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution, and compressed sensing are a few well-known examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is de-emphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance.

/pdf/gradient-projection-for-sparse-reconstruction-application-to-jxqvrma0ll.pdf

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case.

In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large.

The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

http://cvssp-data.eps.surrey.ac.uk/Personal/AdrianHilton/Publications_files/survey_techrep.pdf

A survey of advances in vision-based human motion capture and analysis

Spatial modulation (SM) is a recently developed transmission technique that uses multiple antennas. The basic idea is to map a block of information bits to two information carrying units: 1) a symbol that was chosen from a constellation diagram and 2) a unique transmit antenna number that was chosen from a set of transmit antennas. The use of the transmit antenna number as an information-bearing unit increases the overall spectral efficiency by the base-two logarithm of the number of transmit antennas. At the receiver, a maximum receive ratio combining algorithm is used to retrieve the transmitted block of information bits. Here, we apply SM to orthogonal frequency division multiplexing (OFDM) transmission. We develop an analytical approach for symbol error ratio (SER) analysis of the SM algorithm in independent identically distributed (i.i.d.) Rayleigh channels. The analytical and simulation results closely match. The performance and the receiver complexity of the SM-OFDM technique are compared to those of the vertical Bell Labs layered space-time (V-BLAST-OFDM) and Alamouti-OFDM algorithms. V-BLAST uses minimum mean square error (MMSE) detection with ordered successive interference cancellation. The combined effect of spatial correlation, mutual antenna coupling, and Rician fading on both coded and uncoded systems are presented. It is shown that, for the same spectral efficiency, SM results in a reduction of around 90% in receiver complexity as compared to V-BLAST and nearly the same receiver complexity as Alamouti. In addition, we show that SM achieves better performance in all studied channel conditions, as compared with other techniques. It is also shown to efficiently work for any configuration of transmit and receive antennas, even for the case of fewer receive antennas than transmit antennas.

Spatial Modulation

In recent years, interest in optical wireless (OW) as a promising complementary technology for RF technology has gained new momentum fueled by significant deployments in solid state lighting technology. This article aims at reviewing and summarizing recent advancements in OW communication, with the main focus on indoor deployment scenarios. This includes a discussion of challenges, potential applications, state of the art, and prospects. Related issues covered in this article are duplex transmission, multiple access, MAC protocols, and link capacity improvements.

Indoor optical wireless communication: potential and state-of-the-art

What is index modulation (IM)? This is an interesting question that we have started to hear more and more frequently over the past few years. The aim of this paper is to answer this question in a comprehensive manner by covering not only the basic principles and emerging variants of IM, but also reviewing the most recent as well as promising advances in this field toward the application scenarios foreseen in next-generation wireless networks. More specifically, we investigate three forms of IM: spatial modulation, channel modulation and orthogonal frequency division multiplexing (OFDM) with IM, which consider the transmit antennas of a multiple-input multiple-output system, the radio frequency mirrors (parasitic elements) mounted at a transmit antenna and the subcarriers of an OFDM system for IM techniques, respectively. We present the up-to-date advances in these three promising frontiers and discuss possible future research directions for IM-based schemes toward low-complexity, spectrum- and energy-efficient next-generation wireless networks.

Index Modulation Techniques for Next-Generation Wireless Networks

The multiplexing gain of multiple antenna transmission strongly depends on transmit and receive antenna spacing, transmit antenna synchronization, and the algorithm used to eliminate interchannel interference (ICI) at the receiver. In this paper, a new transmission approach, called spatial modulation, that entirely avoids ICI and requires no synchronization between the transmitting antennas while maintaining high spectral efficiency is presented. A block of information bits is mapped into a constellation point in the signal and the spatial domain, i.e. into the location of a particular antenna. The receiver estimates the transmitted signal and the transmit antenna number and uses the two information to de-map the block of information bits. For this purpose, a novel transmit antenna number detection algorithm called iterative-maximum ratio combining (i-MRC) is presented. Spatial modulation is used to transmit different number of information bits and i-MRC is used to estimate both the transmitted signal and the transmit antenna number. The results are compared to ideal V-BLAST (vertical-Bell Lab layered space-time) and to MRC. Spatial modulation outperforms MRC. The (bit-error-ratio) BER performance and the achieved spectral efficiency is comparable to V-BLAST. However, spatial modulation results in a vast reduction in receiver complexity.

Spatial Modulation - A New Low Complexity Spectral Efficiency Enhancing Technique

In this paper, a generalised technique for spatial modulation (SM) is presented. Generalised spatial modulation (GSM) overcomes in a novel fashion the constraint in SM that the number of transmit antennas has to be a power of two. In GSM, a block of information bits is mapped to a constellation symbol and a spatial symbol. The spatial symbol is a combination of transmit antennas activated at each instance. The actual combination of active transmit antennas depends on the random incoming data stream. This is unlike SM where only a single transmit antenna is activated at each instance. GSM increases the overall spectral efficiency by base-two logarithm of the number of antenna combinations. This reduces the number of transmit antennas needed for the same spectral efficiency. The performance of GSM is analysed in this paper, and an upper bound on the bit-error-ratio (BER) performance is derived. In addition, an algorithm to optimise the antenna combination selection is proposed. Finally, the performance of GSM is validated through Monte Carlo simulations. The results are compared with traditional SM. It is shown that for the same spectral efficiency GSM performs nearly the same as SM, but with a significant reduction in the number of transmit antennas.

Raed Mesleh

Papers

Spatial Modulation

Indoor optical wireless communication: potential and state-of-the-art

Index Modulation Techniques for Next-Generation Wireless Networks

Spatial Modulation - A New Low Complexity Spectral Efficiency Enhancing Technique

Generalised spatial modulation