Cognitive radios require spectral occupancy information in a given location, to avoid any interference with the existing licensed users. This is achieved by spectrum sensing. Existing narrowband, serial spectrum sensors are spectrally inefficient and power hungry. Wideband spectrum sensing increases the number of probable fre-quency candidates for cognitive radio. Wideband RF systems cannot use analog to digital converters (ADCs) for spectrum sensing without increasing the sampling rate and power consumption. The use of ADCs is limited because of the dynamic range of the signals that need to be sampled and the frequency of operation. In this work, we have presented a CMOS based area efficient, dedicated and scalable wideband parallel/serial spectrum sensor for cognitive radio. The key contributions of the thesis are: 1. An injection locked oscillator cascade (ILOC) for parallel LO synthesis. An area-efficient, wideband RF frequency synthesizer, which simultaneously gen-erates multiple local oscillator (LO) signals, is designed. It is suitable for parallel wideband RF spectrum sensing in cognitive radios. The frequency synthesizer consists of an injection locked oscillator cascade where all the LO signals are derived from a single reference oscillator. The ILOC is implemented in a 130-nm technology with an active area of 0.017 mm2. It generates 4 uni-formly spaced LO carrier frequencies from 500 MHz to 2 GHz. 2. A wideband, parallel RF spectrum sensor for cognitive radios has been de-signed. This spectrum sensor is designed to detect RF occupancy from 250 MHz to 5.25 GHz by using an array of CMOS receivers with envelope detec-tors. A parallel LO synthesizer is implemented as an ILOC. The simulated sensitivity is around -25 dBm for 250 MHz wide bandwidth. 3. A mitigation technique for harmonic downconversion in wideband spectrum sensors. The downconversion of radio frequency (RF) components around the harmonics of the local oscillator (LO), and its impact on the accuracy of white space detection using integrated spectrum sensors, is (are) studied. We propose an algorithm to mitigate the impact of harmonic Down conversion by utilizing multiple parallel downconverters in the system architecture. The proposed algorithm is validated on a test-board using commercially avail-able integrated circuits (IC) and a test-chip implemented in a 130-nm CMOS technology. The measured data shows that the impact of the harmonic down-conversion is closely related to the LO characteristics, and that much of it can be mitigated by the proposed technique. 4. A wideband spectrum sensor for narrowband energy detection. A wideband spectrum sensing system for cognitive radio is designed and implemented in a 130-nm RF mixed-mode CMOS technology. The system employs an I-Q downconverter, a pair of complex filters and a pair of envelope detectors for energy detection. The spectrum sensor works from 250 MHz to 3.25 GHz. The design makes use of the band pass nature of the complex filter to achieve two objectives : i) Separation of upper sideband (USB) and lower sideband (LSB) around the local oscillator (LO) signal and ii) Resolution of smaller bands within a large detection bandwidth. The measured sensitivity is close to -45 dBm for a single tone test over a bandwidth of 40 MHz. The measured Image reject ratio (IRR) is close to 30 dB. The overall sensing bandwidth is 3.5 GHz and the overall wideband detection bandwidth is 250 MHz which is partitioned into 40 MHz narrowband chunks with 8 such overlapping chunks.

One of the fundamental problems in commutative algebra and algebraic geometry is to understand the nature of the solution space of a system of multivariate polynomial equations over a field k, such as real or complex numbers. An important algorithmic tool in this study is the notion of Groebner bases (Buchberger (1965)). Given a system of polynomial equations, f1= 0,..., fm = 0, Groebner basis is a “canonical" generating set of the ideal generated by f1,...., fm, that can answer, constructively, many questions in computational ideal theory. It generalizes several concepts of univariate polynomials like resultants to the multivariate case, and answers decisively the ideal membership problem. The dimension of the solution set of an ideal I called the affine variety, an important concept in algebraic geometry, is equal to the Krull dimension of the corresponding coordinate ring, k[x1,...,xn]/I. Groebner bases were first introduced to compute k-vector space bases of k[x1,....,xn]/I and use that to characterize zero-dimensional solution sets. Since then, Groebner basis techniques have provided a generic algorithmic framework for computations in control theory, cryptography, formal verification, robotics, etc, that involve multivariate polynomials over fields. The main aim of this thesis is to study problems related to computational ideal theory over Noetherian commutative rings (e.g: the ring of integers, Z, the polynomial ring over a field, k[y1,…., ym], etc) using the theory of Groebner bases. These problems surface in many domains including lattice based cryptography, control systems, system-on-chip design, etc. Although, formal and standard techniques are available for polynomial rings over fields, the presence of zero divisors and non units make developing similar techniques for polynomial rings over rings challenging. Given a polynomial ring over a Noetherian commutative ring, A and an ideal I in A[x1,..., xn], the first fundamental problem that we study is whether the residue class polynomial ring, A[x1,..., xn]/I is a free A-module or not. Note that when A=k, the answer is always ‘yes’ and the k-vector space basis of k[x1,..., xn]/I plays an important role in computational ideal theory over fields. In our work, we give a Groebner basis characterization for A[x1,...,xn]/I to have a free A-module representation w.r.t. a monomial ordering. For such A-algebras, we give an algorithm to compute its A-module basis. This extends the Macaulay-Buchberger basis theorem to polynomial rings over Noetherian commutative rings. These results help us develop a theory of border bases in A[x1,...,xn] when the residue class polynomial ring is finitely generated. The theory of border bases is handled as two separate cases: (i) A[x1,...,xn]/I is free and (ii) A[x1,...,xn]/I has torsion submodules. For the special case of A = Z, we show how short reduced Groebner bases and the characterization for a free A-module representation help identify the cases when Z[x1,...,xn]/I is isomorphic to ZN for some positive integer N. Ideals in such Z-algebras are called ideal lattices. These structures are interesting since this means we can use the algebraic structure, Z[x1,...,xn]/I as a representation for point lattices and extend all the computationally hard problems in point lattice theory to Z[x1,...,xn]/I . Univariate ideal lattices are widely used in lattice based cryptography for they are a more compact representation for lattices than matrices. In this thesis, we give a characterization for multivariate ideal lattices and construct collision resistant hash functions based on them using Groebner basis techniques. For the construction of hash functions, we define a worst case problem, shortest substitution problem w.r.t. an ideal in Z[x1,...,xn], and establish hardness results for this problem. Finally, we develop an approach to compute the Krull dimension of A[x1,...,xn]/I using Groebner bases, when A is a Noetherian integral domain. When A is a field, the Krull dimension of A[x1,...,xn]/I has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. We introduce the notion of combinatorial dimension of A[x1,...,xn]/I and give a Groebner basis method to compute it for residue class polynomial rings that have a free A-module representation w.r.t. a lexicographic ordering. For such A-algebras, we derive a relation between Krull dimension and combinatorial dimension of A[x1,...,xn]/I. For A-algebras that have a free A-module representation w.r.t. degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Groebner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such A-algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to A-algebras with a free A-module representation w.r.t. a degree compatible ordering as well.

L’objectif principal de cette thèse est de décliner et d’analyser l’algorithme kernel-LMS à noyau Gaussien dans trois cadres différents: celui des noyaux uniques et multiples, à valeurs réelles et à valeurs complexes, dans un contexte d’apprentissage distributé et coopératif dans les réseaux de capteurs. Plus précisement, ce travail s’intéresse à l’analyse du comportement en moyenne et en erreur quadratique de cas différents types d’algorithmes LMS à noyau. Les modèles analytiques de convergence obtenus sont validés par des simulations numérique. Tout d’abord, nous introduisons l’algorithme LMS, les espaces de Hilbert à noyau reproduisants, ainsi que les algorithmes de filtrage adaptatif à noyau existants. Puis, nous étudions analytiquement le comportement de l’algorithme LMS à noyau Gaussien dans le cas où les statistiques des éléments du dictionnaire ne répondent que partiellement aux statistiques des données d’entrée. Nous introduisons ensuite un algorithme LMS modifié à noyau basé sur une approche proximale. La stabilité de l’algorithme est également discutée. Ensuite, nous introduisons deux types d’algorithmes LMS à noyaux multiples. Nous nous concentrons en particulier sur l’analyse de convergence de l’un d’eux. Plus généralement, les caractéristiques des deux algorithmes LMS à noyaux multiples sont analysées théoriquement et confirmées par les simulations. L’algorithme LMS à noyau complexe augmenté est présenté et ses performances analysées. Enfin, nous proposons des stratégies de diffusion fonctionnelles dans les espaces de Hilbert à noyau reproduisant. La stabilité́ de cas de l’algorithme est étudiée. / The main objective of this thesis is to derive and analyze the Gaussian kernel least-mean-square (LMS) algorithm within three frameworks involving single and multiple kernels, real-valued and complex-valued, non-cooperative and cooperative distributed learning over networks. This work focuses on the stochastic behavior analysis of these kernel LMS algorithms in the mean and mean-square error sense. All the analyses are validated by numerical simulations. First, we review the basic LMS algorithm, reproducing kernel Hilbert space (RKHS), framework and state-of-the-art kernel adaptive filtering algorithms. Then, we study the convergence behavior of the Gaussian kernel LMS in the case where the statistics of the elements of the so-called dictionary only partially match the statistics of the input data. We introduced a modified kernel LMS algorithm based on forward-backward splitting to deal with $\ell_1$-norm regularization. The stability of the proposed algorithm is then discussed. After a review of two families of multikernel LMS algorithms, we focus on the convergence behavior of the multiple-input multikernel LMS algorithm. More generally, the characteristics of multikernel LMS algorithms are analyzed theoretically and confirmed by simulation results. Next, the augmented complex kernel LMS algorithm is introduced based on the framework of complex multikernel adaptive filtering. Then, we analyze the convergence behavior of algorithm in the mean-square error sense. Finally, in order to cope with the distributed estimation problems over networks, we derive functional diffusion strategies in RKHS. The stability of the algorithm in the mean sense is analyzed.

The use of Hilbert space theory became an important tool for ergodic theoreticians ever since John von Neumann proved the fundamental Mean Ergodic theorem in Hilbert space. Recurrence is one of the corner stones in the study of dynamical systems. In this dissertation some extended ideas besides those of the basic, well-known recurrence results are investigated. Hilbert space theory proves to be a very useful approach towards the solution of multiple recurrence problems in ergodic theory. Another very important use of Hilbert space theory became evident only relatively recently, when it was realized that non-commutative dynamical systems become accessible to the ergodic theorist through the important Gelfand-Naimark-Segal (GNS) representation of C*-algebras as Hilbert spaces. Through this construction we are enabled to invoke the rich catalogue of Hilbert space ergodic results to approach the more general, and usually more involved, non-commutative extensions of classical ergodic-theoretical results. In order to make this text self-contained, the basic, standard, ergodic-theoretical results are included in this text. In many instances Hilbert space counterparts of these basic results are also stated and proved. Chapters 1 and 2 are devoted to the introduction of these basic ergodic-theoretical results such as an introduction to the idea of measure-theoretic dynamical systems, citing some basic examples, Poincairé’s recurrence, the ergodic theorems of Von Neumann and Birkhoff, ergodicity, mixing and weakly mixing. In Chapter 2 several rudimentary results, which are the basic tools used in proofs, are also given. In Chapter 3 we show how a Hilbert space result, i.e. a variant of a result by Van der Corput for uniformly distributed sequences modulo 1, is used to simplify the proofs of some multiple recurrence problems. First we use it to simplify and clarify the proof of a multiple recurrence result by Furstenberg, and also to extend that result to a more general case, using the same Van der Corput lemma. This may be considered the main result of this thesis, since it supplies an original proof of this result. The Van der Corput lemma helps to simplify many of the tedious terms that are found in Furstenberg’s proof. In Chapter 4 we list and discuss a few important results where classical (commutative) ergodic results were extended to the non-commutative case. As stated before, these extensions are mainly due to the accessibility of Hilbert space theory through the GNS construction. The main result in this section is a result proved by Niculescu, Ströh and Zsidó, which is proved here using a similar Van der Corput lemma as in the commutative case. Although we prove a special case of the theorem by Niculescu, Ströh and Zsidó, the same method (Van der Corput) can be used to prove the generalized result. Copyright 2004, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. Please cite as follows: Beters, FJC 2004, A Hilbert space approach to multiple recurrence in ergodic theory, MSc dissertation, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-02222006-104936 / > / Dissertation (MSc (Applied Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

Quantum mechanics is the physics of the very small. Quantum computers are devices that utilize the power of quantum mechanics for their computational primitives. Associated to each quantum system is an abstract space known as the Hilbert space. A subspace of the Hilbert space is known as a quantum code. Quantum codes allow to protect the computational state of a quantum computer against decoherence errors. The well-known classes of quantum codes are stabilizer or additive codes, non-additive codes and Clifford codes. This thesis aims at demonstrating a general approach to the construction of the various classes of quantum codes. The framework utilized is the Fourier transform over finite groups. The thesis is divided into four chapters. The first chapter is an introduction to basic quantum mechanics, quantum computation and quantum noise. It lays the foundation for an understanding of quantum error correction theory in the next chapter. The second chapter introduces the basic theory behind quantum error correction. Also, the various classes and constructions of active quantum error-control codes are introduced. The third chapter introduces the Fourier transform over finite groups, and shows how it may be used to construct all the known classes of quantum codes, as well as a class of quantum codes as yet unpublished in the literature. The transform domain approach was originally introduced in (Arvind et al., 2002). In that paper, not all the classes of quantum codes were introduced. We elaborate on this work to introduce the other classes of quantum codes, along with a new class of codes, codes from idempotents in the transform domain. The fourth chapter details the computer programs that were used to generate and test for the various code classes. Code was written in the GAP (Groups, Algorithms, Programming) computer algebra package. The fifth and final chapter concludes, with possible directions for future work. References cited in the thesis are attached at the end of the thesis.

In this work we study two aspects of image security: improper usage and illegal access of images. In the first part we present our results on steganalysis – protection against improper usage of images. In the second part we present our results on image encryption – protection against illegal access of images. Steganography is the collective name for methodologies that allow the creation of invisible –hence secret– channels for information transfer. Steganalysis, the counter to steganography, is a collection of approaches that attempt to detect and quantify the presence of hidden messages in cover media. First we present our studies on stego-images using features developed for data stream classification towards making some qualitative assessments about the effect of steganography on the lower order bit planes(LSB) of images. These features are effective in classifying different data streams. Using these features, we study the randomness properties of image and stego-image LSB streams and observe that data stream analysis techniques are inadequate for steganalysis purposes. This provides motivation to arrive at steganalytic techniques that go beyond the LSB properties. We then present our steganalytic approach which takes into account such properties. In one such approach, we perform steganalysis from the point of view of quantifying the effect of perturbations caused by mild image processing operations–zoom-in/out, rotation, distortions–on stego-images. We show that this approach works both in detecting and estimating the presence of stego-contents for a particularly difficult steganographic technique known as LSB matching steganography. Next, we present our results on our image encryption techniques. Encryption approaches which are used in the context of text data are usually unsuited for the purposes of encrypting images(and multimedia objects) in general. The reasons are: unlike text, the volume to be encrypted could be huge for images and leads to increased computational requirements; encryption used for text renders images incompressible thereby resulting in poor use of bandwidth. These issues are overcome by designing image encryption approaches that obfuscate the image by intelligently re-ordering the pixels or encrypt only parts of a given image in attempts to render them imperceptible. The obfuscated image or the partially encrypted image is still amenable to compression. Efficient image encryption schemes ensure that the obfuscation is not compromised by the inherent correlations present in the image. Also they ensure that the unencrypted portions of the image do not provide information about the encrypted parts. In this work we present two approaches for efficient image encryption. First, we utilize the correlation preserving properties of the Hilbert space-filling-curves to reorder images in such a way that the image is obfuscated perceptually. This process does not compromise on the compressibility of the output image. We show experimentally that our approach leads to both perceptual security and perceptual encryption. We then show that the space-filling curve based approach also leads to more efficient partial encryption of images wherein only the salient parts of the image are encrypted thereby reducing the encryption load. In our second approach, we show that Singular Value Decomposition(SVD) of images is useful from the point of image encryption by way of mismatching the unitary matrices resulting from the decomposition of images. It is seen that the images that result due to the mismatching operations are perceptually secure.

Local site conditions are known to influence ground motion during earthquake events and increase the severity of damage. Data from earthquakes are useful to study the response but they are available only from active regions. Ubiquitous ambient vibrations on the other hand offer a more practical approach to quantify site responses. This thesis explores the use of various methods for obtaining site responses. The primary area of study is the Kachchh rift basin, NW India, a Mesozoic rift that features significant lateral variations in surface geology and has experienced ground responses during 1819 and 2001 earthquakes. The Mw 7.6, 2001 event was followed by hundreds of aftershocks, which were recorded by temporary networks. In this study we have used earthquake signals as well as ambient vibrations to understand site response in various parts of the basin. In addition we have collected data from a few sites from the Indo-Gangetic plains and Kathmandu valley, both affected by large earthquakes, 1934 the M ~ 8 (Bihar) and 2015, Mw 7.8 (Nepal). Velocity and acceleration records from a network of eight stations in the Kachchh Rift were used to evaluate site responses using Standard Spectral Ratio (SSR) and Horizontal to Vertical spectral ratio (HVSR-E) methods. Ambient vibrations were analyzed following Nakamura’s H/V method (HVSR-AV), for data collected from 110 sites that represent different field conditions within the Kachchh Rift. Fundamental resonance frequency (f0) varied between 0.12 – 2.30 Hz, while the amplification factor (A0) was in the range of 2.0 – 9.1. We found that higher A0 and liquefaction index (Kg) values were mostly associated with higher liquefaction potential. Using a close network of stations, we studied the role of site response in damage to the Bhuj city that suffered maximum damage in 2001; our results suggest that site response was not a significant factor. Studies based on passive data were complemented by Multi-channel Analysis of Surface Waves (MASW) to map shear wave velocities of the various subsurface units up to depths of 10m (Vs10) and 30m (Vs30). Our results imply average Vs could be a good proxy to characterize site amplifications where sediment thicknesses are shallow. Power law relationship between f0 and thickness (h) suggest a strong positive correlation (r = 0.89) adding credence to HVSR-AV method, making it a cost-effective alternative to MASW to infer site conditions. Further, to understand the influence of topography on site effects, we analyzed data from hills, valleys and their edges, both from the Kachchh rift and Kathmandu valley. Sites on the edges of valleys showed multiple, fuzzy peaks in the low frequency range (< 1 Hz) and broad peaks attributable to sites prone to higher damage. Spectrograms generated through Huang-Hilbert Transforms (HHT) suggested focusing of energy in narrow frequency bands on the edges, while valleys tend to scatter energy over wide frequencies. Although our current results are based on limited observations, we recognize spectral analysis as a powerful tool to quantify site effects in regions with significant topography. It is known that coseismic liquefaction could lead to nonlinear behavior wherein the near-surface soil layer loses its shear strength, causing a reduction of its fundamental resonance frequency. We used data from selected sites of coseismic liquefaction to highlight the significance of nonlinear effects in site response. Earthquake signals and ambient vibrations from Umedpur, a region that experienced intense liquefaction during 2001 were used in this analysis. Here we followed an empirical decomposition method based on HHT and signals were decomposed as many intrinsic mode functions (IMFs) that showed characteristic peaks for events of various values of PGAs. Thus, the first IMF for events with relatively higher PGAs (0.03g) showed distinct peaks for the S wave coda part, which were not noted for those with lower PGA (0.01g). These observations in a region of coseismic liquefaction are useful in developing models for quantifying nonlinear behavior. In conclusion, site response studies using different types of data and processing techniques in regions affected by recent earthquakes brings out the scope and limitations of each of these sets of data and techniques. This study suggests that ambient vibrations provide reasonable estimates of site response and can be reliably used in regions where earthquake data are not available.

The aim of the master’s thesis is to acquaint the reader with Discrete Fourier Transform DFT and its introduction to spectral analysis descrete time signals. Furthermore it describes efficient computation of the DFT using the algorithm Fast Fourier Transform FFT and effects that appear in the DFT spectral analysis, the effect of the length of the DFT, the effect of the length of the segment of the analysed signal and the leakage. Next is explained the issue of the complex representation of real bandpass signals in the baseband, called the complex envelope of signals. This represenation uses the Hilbert Transform and the analytic signal. The thesis also describes the application library Real Time Toolbox of the software Matlab/Simulink, that uses the data acquition card AD622 allows to connect and process real signals in real time. In Matlab/Simulink with use this facilities is create the laboratory lesson, that simulating the complex represenation of real bandpass signals in real time.

The thesis is focused on the study of detection of QRS complex in time a frequency domain. The aim is to implement selected methods and their comparison to assess the effectiveness of QRS complex.

The present final thesis discusses the transmission of a narrowly directional parametric sound beam through an amplitude-modulated ultrasonic wave, utilizing the effect of auto-demodulation in a nonlinear medium and ensuring the subsequent processing of the input signal for the parametric sound transmitter. Emphasis is placed on the mathematical tools that relate to parametric sound transmission in a nonlinear medium. The basic part of the thesis describes a parametric speaker and the associated amplitude modulation techniques, which constitute a major prerequisite for the processing of the transmitted signal. In the following section, the author then analyzes the computational intensity of these techniques, considering applicable hardware approaches. Finally, the fabrication and practical use of the proposed solution are discussed, including the measurement of typical parameters such as the spatial radiation characteristics, total harmonic distortion, and transmission channel bandwidth.