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

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

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

Theoretical and experimental studies of multiple-input/multiple-output (MIMO) radio channels are presented. A simple stochastic MIMO model channel has been developed. This model uses the correlation matrices at the mobile station (MS) and base station (BS) so that results of the numerous single-input/multiple-output studies that have been published in the literature can be used as input parameters. The model is simplified to the narrowband channels. The validation of the model is based upon data collected in both picocell and microcell environments. The stochastic model has also been used to investigate the capacity of MIMO radio channels, considering two different power allocation strategies, water filling and uniform and two different antenna topologies, 4/spl times/4 and 2/spl times/4. Space diversity used at both ends of the MIMO radio link is shown to be an efficient technique in picocell environments, achieving capacities within 14 b/s/Hz and 16 b/s/Hz in 80% of the cases for a 4/spl times/4 antenna configuration implementing water filling at a SNR of 20 dB.

/pdf/a-stochastic-mimo-radio-channel-model-with-experimental-4fgkgc22dj.pdf

A stochastic MIMO radio channel model with experimental validation

This study investigates the application potential of the SAGE (space-alternating generalized expectation-maximization) algorithm to jointly estimate the relative delay, incidence azimuth, Doppler frequency, and complex amplitude of impinging waves in mobile radio environments The performance, ie, high-resolution ability, accuracy, and convergence rate of the scheme, is assessed in synthetic and real macro- and pico-cellular channels The results indicate that the scheme overcomes the resolution limitation inherent to classical techniques like the Fourier or beam-forming methods In particular, it is shown that waves which exhibit an arbitrarily small difference in azimuth can be easily separated as long as their delays or Doppler frequencies differ by a fraction of the intrinsic resolution of the measurement equipment Two waves are claimed to be separated when the mean-squared estimation errors (MSEEs) of the estimates of their parameters are close to the corresponding Cramer-Rao lower bounds (CRLBs) derived in a scenario where only a single wave is impinging The adverb easily means that the MSEEs rapidly approach the CLRBs, ie, within less than 20 iteration cycles Convergence of the log-likelihood sequence is achieved after approximately ten iteration cycles when the scheme is applied in real channels In this use, the estimated dominant waves can be related to a scatterer/reflector in the propagation environment The investigations demonstrate that the SAGE algorithm is a powerful high-resolution tool that can be successfully applied for parameter extraction from extensive channel measurement data, especially for the purpose of channel modeling

Channel parameter estimation in mobile radio environments using the SAGE algorithm

A simple statistical model of azimuthal and temporal dispersion in mobile radio channels is proposed. The model includes the probability density function (PDF) of the delay and azimuth of the impinging waves as well as their expected power conditioned on the delay and azimuth. The statistical properties are extracted from macrocellular measurements conducted in a variety of urban environments. It is found that in typical urban environments the power azimuth spectrum (PAS) is accurately described by a Laplacian function, while a Gaussian PDF matches the azimuth PDF. Moreover, the power delay spectrum (PDS) and the delay PDF are accurately modeled by an exponential decaying function. In bad urban environments, channel dispersion is better characterized by a multicluster model, where the PAS and PDS are modeled as a sum of Laplacian functions and exponential decaying functions, respectively.

http://www2.elo.utfsm.cl/~ipd465/Papers%20y%20apuntes%20varios/A%20Stochastic%20Model,%20Ped%20Mog%20Fleury.pdf

A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments

We present analytical results relating to first- and second-moment characterization of direction dispersion and spatial selectivity in the radio channel as well as to the duality between these two effects. Dispersion in direction can be characterized either by the direction power spectrum or by a family of spatial Doppler power spectra at the reception site. Two measures called the direction spread and the maximum spatial Doppler spread are introduced which describe the extent of channel dispersion in direction and in spatial Doppler frequency, respectively. Both measures are analogous to the delay and Doppler spreads, which are commonly employed to describe the extent of dispersion in delay and Doppler frequency, respectively. The relationships between the two approaches for characterizing spatial dispersion and especially between the direction and maximum spatial Doppler spreads are analytically established. The coherence distance at a certain level summarizes certain features of space selectivity in the radio channel which impact on the performance of communication systems. Two uncertainty relations between the direction spread and the coherence distance as well as between the maximum spatial Doppler spread and the coherence distance express the duality between direction dispersion and space selectivity. These relations are analogous to those established previously between the delay spread and the coherence bandwidth and between the Doppler spread and the coherence time. Examples relevant to mobile communications in the case where the waves propagate only horizontally illustrate the theoretical results. An application of these results to the design of uniform linear antenna arrays is also discussed.

First- and second-order characterization of direction dispersion and space selectivity in the radio channel

This paper present measurement results obtained with a testbed equipped with an array antenna. The investigations focus on the power azimuth spectrum of the channel in urban and rural areas, for various base station antenna heights. The power azimuth spectrum is well modeled with a Laplacian function. The local azimuth spread is found to range from 1/spl deg/ to 25/spl deg/ depending on the environment and antenna height. The azimuth spread significantly increases as the antenna height is reduced. An analytical expression for the spatial correlation function is derived based on the Laplacian model. Finally, the power azimuth spectrums impact on a conventional beamformer ability to suppress interfering users is investigated.

Spatial channel characteristics in outdoor environments and their impact on BS antenna system performance

From extensive 2D channel measurements it is found that the shape of the power azimuth spectrum at the base station in outdoor environments is accurately described by a truncated Laplacian function in both rural and urban environments.

Bernard Henri Fleury

Papers

Channel parameter estimation in mobile radio environments using the SAGE algorithm

A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments

First- and second-order characterization of direction dispersion and space selectivity in the radio channel

Spatial channel characteristics in outdoor environments and their impact on BS antenna system performance

Power azimuth spectrum in outdoor environments