The study of the spatial aspects of multipath in wireless communications environments is an increasingly important addition to the study of the temporal aspects in the search for ways to increase the utilization of the available wireless channel capacity. Traditionally, multipath has been viewed as an encumbrance in wireless communications, two of the major impairments being signal fading and intersymbol interference. However, recently the potential advantages of the diversity offered by multipath rich environments in multiuser communications have been recognised. Space time coding, for example, is a recent technique which relies on a rich scattering environment to create many practically uncorrelated signal transmission channels. Most often, statistical models have been used to describe the multipath environments in such applications. This approach has met with reasonable success but is limited when the statistical nature of a field is not easily determined or is not readily described by a known distribution.¶ Our primary aim in this thesis is to probe further into the nature of multipath environments in order to gain a greater understanding of their characteristics and diversity potential. We highlight the shortcomings of beamforming in a multipath multiuser access environment. We show that the ability of a beamformer to resolve two or more signals in angle directly limits its achievable capacity.¶ We test the probity of multipath as a source of spatial diversity, the limiting case of which is co-located users. We introduce the concept of separability to define the fundamental limits of a receiver to extract the signal of a desired user from interfering users’ signals and noise. We consider the separability performances of the minimum mean square error (MMSE), decorrelating (DEC) and matched filter (MF) detectors as we bring the positions of a desired and an interfering user closer together. We show that both the MMSE and DEC detectors are able to achieve acceptable levels of separability with the users as close as λ/10.¶ In seeking a better understanding of the nature of multipath fields themselves, we take two approaches. In the first we take a path oriented approach. The effects on the variation of the field power of the relative values of parameters such as amplitude and propagation direction are considered for a two path field. The results are applied to a theoretical analysis of the behaviour of linear detectors in multipath fields. This approach is insightful for fields with small numbers of multipaths, but quickly becomes mathematically complex.¶ In a more general approach, we take a field oriented view, seeking to quantify the complexity of arbitrary fields. We find that a multipath field has an intrinsic dimensionality of (πe)R/λ≈8.54R/λ, for a field in a two dimensional circular region, increasing only linearly with the radius R of the region. This result implies that there is no such thing as an arbitrarily complicated multipath field. That is, a field generated by any number of nearfield and farfield, specular and diffuse multipath reflections is no more complicated than a field generated by a limited number of plane waves. As such, there are limits on how rich multipath can be. This result has significant implications including means: i) to determine a parsimonious parameterization for arbitrary multipath fields and ii) of synthesizing arbitrary multipath fields with arbitrarily located nearfield or farfield, spatially discrete or continuous sources. The theoretical results are corroborated by examples of multipath field analysis and synthesis.

The paper deals with the problem of finding an optimal one-time rebalancing strategy for the Bachelier model, and makes some remarks for the similar problem within Black-Scholes model. The problem is studied on finite time interval under mean-square criterion of optimality. The methods of the paper are based on the results for optimal stopping problem and standard mean-square criterion. The solution of the problem, considered in the paper, let us interpret how and - that is more important for us -when investor should rebalance the portfolio, if he wants to hedge it in the best way.

Orthogonal Frequency Division Multiplexing (OFDM) techniques have been used in many wireless communication systems to improve the system capacity and achieve high data-rate. It possesses good spectral efficiency and robustness against interferences. The OFDM system has been adopted in many communication standards, such as the 802.11a/g standards for the high-speed WLAN, HIPERLAN2, and IEEE 802.16 standard, and meanwhile, it is also employed in the European DAB and DVB systems. To avoid the inter-block interference (IBI), usually, in the transmitter of OFDM systems the redundancy with sufficient length is introduced, it allows us to overcome the IBI problem, due to highly dispersive channel. Many redundancy insertion methods have been proposed in the literatures, there are cyclic prefix (CP), zero padding (ZP) and the pseudorandom postfix (PRP). Under such system we have still to know the correct channel state information for equalizing the noisy block signal. Especially, in time-varying channel, the incorrect channel state information may introduce serious inter-symbol interference (ISI), if the channel estimation could not perform correctly. In this thesis, the PRP-OFDM system is considered. According to the PRP-OFDM scheme, the redundancy with pseudorandom postfix (PRP) approach is employed to make semi-blind channel estimation with order-one statistics of the received signal. But these statistic characteristics may not be available under time-varying channel. Hence, in this thesis, we propose a modified PRP-OFDM system with assistant training sequences, which is equipped with minimum mean-square-error equalizer and utilize Kalman filter algorithm to implement time-varying channel estimation. To do so, we first model time-varying channel estimation problem with a dynamic system, and adopt the Kalman filter algorithm to estimate the true channel coefficients. Unfortunately, since most parameters in dynamic system are random and could not to be known in advance. We need to apply effective estimation schemes to estimate the statistics of true parameters for implementing the Kalman filter algorithm. When the channel state information is known, MMSE equalizer follows to suppress the inter-symbol interference (ISI). Moreover, after making decision the binary data can be used to re-modulate PRP-OFDM symbol and to be re-used in Kalman filter to obtain more accurate CSI to improve the effectiveness of the equalizer. Via computer simulations, we verify that desired performance in terms of bit error rate (BER), can be achieved compared with the CP-OFDM systems.

As part of this project and in preparation for future experimental studies of gas-phase ion-molecule reactions, extensive modification and characterization of the crossed molecular beam machine in the Department of Chemistry, University of Canterbury has been carried out. This instrument has been configured and some preliminary testing completed to enable the future study of gas-phase ion-molecule collisions of H⁺₃ and Y⁻ (Y = F, Cl, Br) with dipole-oriented CZ₃X (Z = H, F and X = F, Cl, Br). Theoretical calculations (ab initio and density functional theory) are reported on previously experimentally characterized Na + CH₃NO₂, Na + CH₃NC, and K + CH₃NC systems, and several other systems of relevance. All gas-phase experimental and theoretical studies have the common theme of studying collision orientation dependence of reaction under singlecollision conditions. Experimental measurements, theoretical simulations and calculations are also reported on some selected ferrous (Fe²⁺) high-spin (S=2) crystals, in an attempt to resolve microscopic contributions of two fundamental macroscopic tensor properties: the electric-field gradient (efg); and the mean square displacement (msd) in the case when more than one symmetry related site of low local point-group symmetry contributes to the same quadrupole doublet. These determinations have been made using the nuclear spectroscopic technique of Mössbauer spectroscopy, and complemented with X-ray crystallographic measurements.

A number of topics related to Fourier imaging are investigated. Relationships between the magnitude of errors in the amplitude and phase of the Fourier transform of images and the mean square error in reconstructed images are derived. The differing effects of amplitude and phase errors are evaluated, and "equivalent" amplitude and phase errors are derived. A model of the probability density function of the Fourier amplitudes of images is derived. The fundamental basis of phase dominance is studied and quantitated. Inconsistencies in published counter-examples of phase dominance are highlighted. The key characteristics of natural images that lead to their observed power spectral behaviour with spatial frequency are determined.

This thesis is concerned with distributed estimation in a wireless sensor network (WSN) with analog transmission. For a scenario in which a large number of sensors are deployed under a limited bandwidth constraint, a semi-orthogonal multiple-access channelization (MAC) approach is proposed to provide transmission of observations from K sensors to a fusion center (FC) via N orthogonal channels, where K≥N. The proposed semi-orthogonal MAC can be implemented with either fixed sensor grouping or adaptive sensor grouping. The mean squared error (MSE) is adopted as the performance criterion and it is first studied under equal power allocation. The MSE can be expressed in terms of two indicators: the channel noise suppression capability and the observation noise suppression capability. The fixed version of the semi-orthogonal MAC is shown to have the same channel noise suppression capability and two times the observation noise suppression capability when compared to the orthogonal MAC under the same bandwidth resource. For the adaptive version, the performance improvement of the semi-orthogonal MAC over the orthogonal MAC is even more significant. In fact, the semi-orthogonal MAC with adaptive sensor grouping is shown to perform very close to that of the hybrid MAC, while requiring a much smaller amount of feedback. Another contribution of this thesis is an analysis of the behavior of the average MSE in terms of the number of sensors, namely the scaling law, under equal power allocation. It is shown that the proposed semi-orthogonal MAC with adaptive sensor grouping can achieve the optimal scaling law of the analog WSN studied in this thesis. Finally, improved power allocations for the proposed semi-orthogonal MAC are investigated. First, the improved power allocations in each sensor group for different scenarios are provided. Then an optimal solution of power allocation among sensor groups is obtained by the convex optimization theory, and shown to outperform equal power allocation. The issue of balancing between the performance improvement and extra feedback required by the improved power allocation is also thoroughly discussed.

In wireless communications systems, the transmission channel between the transmitter and the receiver antennas is one of the main sources of disruption for the signal. The multicarrier modulations, such as the orthogonal frequency division multiplexing (OFDM), are very robust against the multipath effect, and allow to recover the transmitted signal with a low error rate, when they are combined with a channel encoding. The channel estimation then plays a key role in the performance of the communications systems. In this PhD thesis, we study techniques based on least square (LS) and minimum mean square error (MMSE) estimators. The MMSE is optimal, but is much more complex than LS, and requires the a priori knowledge of the second order moment of the channel and the noise. In this presentation, two methods that allow to reach a performance close to the one of LMMSE while getting around its drawback are investigated. In another way, a third part of the presentation investigates the errors of estimation due to the interpolations.

This thesis investigates optimality of heuristic forecasting. According to Goldstein a Gigerenzer (2009), heuristics can be viewed as predictive models, whose simplicity is exploiting the bias-variance trade-off. Economic agents learning in the context of rational expectations (Marcet a Sargent 1989) employ, on the contrary, complex models of the whole economy. Both of these approaches can be perceived as an optimal response complexity of the prediction task and availability of observations. This work introduces a straightforward extension to the standard model of decision making under uncertainty, where agents utility depends on accuracy of their predictions and where model complexity is moderated by regularization parameter. Results of Monte Carlo simulations reveal that in complicated environments, where few observations are at disposal, it is beneficial to construct simple models resembling heuristics. Unbiased models are preferred in more convenient conditions.

Realised Stochastic Volatility model of Koopman and Scharth (2011) is applied to the five stocks listed on NYSE in this thesis. Aim of this thesis is to investigate the effect of speeding up the trade data processing by skipping the cleaning rule requiring the quote data. The framework of the Realised Stochastic Volatility model allows the realised measures to be biased estimates of the integrated volatility, which further supports this approach. The number of errors in recorded trades has decreased significantly during the past years. Different sample lengths were used to construct one day-ahead forecasts of realised measures to examine the forecast precision sensitivity to the rolling window length. Use of the longest window length does not lead to the lowest mean square error. The dominance of the Realised Stochastic Volatility model in terms of the lowest mean square errors of one day-ahead out-of-sample forecasts has been confirmed.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Adaptive filtering is applied as solution for many problems in engineer. There are many techniques to improve adaptive filtering as kernel methods and, in addiction, it is used a pretuned dictionary. In this context, here is presented the KSIG algorithm, the kernel version of Sigmoide, where is used the kernel, to decrease the error, and the non-linear and even cost function to increase the convergence speed. Here it is described also, the KSIG with a pretuned dictionary, to reduce the size of the data set used to calculate the filter output, which is a kernel method consequence . The KSIG and KSIG with pre-tuned dictionary theoretical efficiency is one result of their convergence proof, which evidence that the algorithms converge in average. The learning curves, which are results of some experiments, show that when KSIG and KLMS algorithms are compared, the first converges faster, in less iterations, than the second, in the version with and without pre-tuned dictionary of both algorithms. / A filtragem adaptativa é aplicada na solução de diversos problemas da engenharia. Há muitas alternativas para melhorá-la, uma delas é o uso de kernel e, em adição, o uso de um dicionário pré-definido de dados. Neste contexto, este trabalho apresenta o KSIG, a versão em kernel do algoritmo Sigmoide, um algoritmo que otimiza o erro do filtro pelo emprego de uma função de custo par e não linear. Ademais, é apresentada a versão do KSIG com dicionário de dados pré-definido, visando redução do grande número de dados utilizados para obtenção da saída decorrente do uso da técnica com kernel. A eficiência teórica do KSIG e de sua versão com dicionário pré-definido é um resultado presente nas provas de convergência construídas para ambos os algoritmos, as quais demonstraram que estes convergem em média. Já as curvas de aprendizagem obtidas nas simulações computacionais dos experimentos realizados demonstraram que o KSIG quando comparado ao KLMS, em diferentes problemas de filtragem adaptativa, apresenta convergência mais rápida, em menos iterações, tanto nas versões sem tanto com dicionário pré-definido de ambos os algoritmos.