Rozprawy doktorskie na temat „Gaussian inputs”

Dans cette thèse, les problèmes mixtes couteux sont abordés par le biais de processus gaussiens où les variables discrètes sont relaxées en variables latentes continues. L'espace continu est plus facilement exploité par les techniques classiques d'optimisation bayésienne que ne le serait un espace mixte. Les variables discrètes sont récupérées soit après l'optimisation continue, soit simultanément avec une contrainte supplémentaire de compatibilité continue-discrète qui est traitée avec des lagrangiens augmentés. Plusieurs implémentations possibles de ces optimiseurs mixtes bayésiens sont comparées. En particulier, la reformulation du problème avec des variables latentes continues est mise en concurrence avec des recherches travaillant directement dans l'espace mixte. Parmi les algorithmes impliquant des variables latentes et un lagrangien augmenté, une attention particulière est consacrée aux multiplicateurs de lagrange pour lesquels des techniques d'estimation locale et globale sont étudiées. Les comparaisons sont basées sur l'optimisation répétée de trois fonctions analytiques et sur une application mécanique concernant la conception d'une poutre. Une étude supplémentaire pour l'application d'une stratégie d'optimisation mixte proposée dans le domaine de l'auto-calibrage mixte est faite. Cette analyse s'inspire d'une application de quantification des radionucléides, qui définit une fonction inverse spécifique nécessitant l'étude de ses multiples propriétés dans le scenario continu. une proposition de différentes stratégies déterministes et bayésiennes a été faite en vue d'une définition complète dans un contexte de variables mixtes
In this thesis, costly mixed problems are approached through gaussian processes where the discrete variables are relaxed into continuous latent variables. the continuous space is more easily harvested by classical bayesian optimization techniques than a mixed space would. discrete variables are recovered either subsequently to the continuous optimization, or simultaneously with an additional continuous-discrete compatibility constraint that is handled with augmented lagrangians. several possible implementations of such bayesian mixed optimizers are compared. in particular, the reformulation of the problem with continuous latent variables is put in competition with searches working directly in the mixed space. among the algorithms involving latent variables and an augmented lagrangian, a particular attention is devoted to the lagrange multipliers for which a local and a global estimation techniques are studied. the comparisons are based on the repeated optimization of three analytical functions and a mechanical application regarding a beam design. an additional study for applying a proposed mixed optimization strategy in the field of mixed self-calibration is made. this analysis was inspired in an application in radionuclide quantification, which defined an specific inverse function that required the study of its multiple properties in the continuous scenario. a proposition of different deterministic and bayesian strategies was made towards a complete definition in a mixed variable setup

Les inondations en général affectent plus de personnes que tout autre catastrophe. Au cours de la dernière décennie du 20ème siècle, plus de 1.5 milliard de personnes ont été affectées. Afin d'atténuer l'impact de ce type de catastrophe, un effort scientifique significatif a été consacré à la constitution de codes de simulation numériques pour la gestion des risques. Les codes disponibles permettent désormais de modéliser correctement les événements d'inondation côtière à une résolution assez élevée. Malheureusement, leur utilisation est fortement limitée pour l'alerte précoce, avec une simulation de quelques heures de dynamique maritime prenant plusieurs heures à plusieurs jours de temps de calcul. Cette thèse fait partie du projet ANR RISCOPE, qui vise à remédier cette limitation en construisant des métamodèles pour substituer les codes hydrodynamiques coûteux en temps de calcul. En tant qu'exigence particulière de cette application, le métamodèle doit être capable de traiter des entrées fonctionnelles correspondant à des conditions maritimes variant dans le temps. À cette fin, nous nous sommes concentrés sur les métamodèles de processus Gaussiens, développés à l'origine pour des entrées scalaires, mais maintenant disponibles aussi pour des entrées fonctionnelles. La nature des entrées a donné lieu à un certain nombre de questions sur la bonne façon de les représenter dans le métamodèle: (i) quelles entrées fonctionnelles méritent d'être conservées en tant que prédicteurs, (ii) quelle méthode de réduction de dimension (e.g., B-splines, PCA, PLS) est idéale, (iii) quelle est une dimension de projection appropriée, et (iv) quelle est une distance adéquate pour mesurer les similitudes entre les points d'entrée fonctionnels dans la fonction de covariance. Certaines de ces caractéristiques - appelées ici paramètres structurels - du modèle et d'autres telles que la famille de covariance (e.g., Gaussien, Matérn 5/2) sont souvent arbitrairement choisies a priori. Comme nous l'avons montré à travers des expériences, ces décisions peuvent avoir un fort impact sur la capacité de prédiction du métamodèle. Ainsi, sans perdre de vue notre but de contribuer à l'amélioration de l'alerte précoce des inondations côtières, nous avons entrepris la construction d'une méthodologie efficace pour définir les paramètres structurels du modèle. Comme première solution, nous avons proposé une approche d'exploration basée sur la Méthodologie de Surface de Réponse. Elle a été utilisé efficacement pour configurer le métamodèle requis pour une fonction de test analytique, ainsi que pour une version simplifiée du code étudié dans RISCOPE. Bien que relativement simple, la méthodologie proposée a pu trouver des configurations de métamodèles de capacité de prédiction élevée avec des économies allant jusqu'à 76.7% et 38.7% du temps de calcul utilisé par une approche d'exploration exhaustive dans les deux cas étudiés. [...]
Currently, floods in general affect more people than any other hazard. In just the last decade of the 20th century, more than 1.5 billion were affected. In the seek to mitigate the impact of this type of hazard, strong scientific effort has been devoted to the constitution of computer codes that could be used as risk management tools. Available computer models now allow properly modelling coastal flooding events at a fairly high resolution. Unfortunately, their use is strongly prohibitive for early warning, with a simulation of few hours of maritime dynamics taking several hours to days of processing time, even on multi-processor clusters. This thesis is part of the ANR RISCOPE project, which aims at addressing this limitation by means of surrogate modeling of the hydrodynamic computer codes. As a particular requirement of this application, the metamodel should be able to deal with functional inputs corresponding to time varying maritime conditions. To this end, we focused on Gaussian process metamodels, originally developed for scalar inputs, but now available also for functional inputs. The nature of the inputs gave rise to a number of questions about the proper way to represent them in the metamodel: (i) which functional inputs are worth keeping as predictors, (ii) which dimension reduction method (e.g., B-splines, PCA, PLS) is ideal, (iii) which is a suitable projection dimension, and given our choice to work with Gaussian process metamodels, also the question of (iv) which is a convenient distance to measure similarities between functional input points within the kernel function. Some of these characteristics - hereon called structural parameters - of the model and some others such as the family of kernel (e.g., Gaussian, Matérn 5/2) are often arbitrarily chosen a priori. Sometimes, those are selected based on other studies. As one may intuit and has been shown by us through experiments, those decisions could have a strong impact on the prediction capability of the resulting model. Thus, without losing sight of our final goal of contributing to the improvement of coastal flooding early warning, we undertook the construction of an efficient methodology to set up the structural parameters of the model. As a first solution, we proposed an exploration approach based on the Response Surface Methodology. It was effectively used to tune the metamodel for an analytic toy function, as well as for a simplified version of the code studied in RISCOPE. While relatively simple, the proposed methodology was able to find metamodel configurations of high prediction capability with savings of up to 76.7% and 38.7% of the time spent by an exhaustive search approach in the analytic case and coastal flooding case, respectively. The solution found by our methodology was optimal in most cases. We developed later a second prototype based on Ant Colony Optimization (ACO). This new approach is more powerful in terms of solution time and flexibility in the features of the model allowed to be explored.[...]

Tato práce se zabývá úlohou odhadování prostorové funkce z hlediska regrese pomocí Gaussovských procesů (GPR) za současné nejistoty tréninkových pozic (pozic senzorů). Nejdříve je zde popsána teorie v pozadí GPR metody pracující se známými tréninkovými pozicemi. Tato teorie je poté aplikována při odvození výrazů prediktivní distribuce GPR v testovací pozici při uvážení nejistoty tréninkových pozic. Kvůli absenci analytického řešení těchto výrazů byly výrazy aproximovány pomocí metody Monte Carlo. U odvozené metody bylo demonstrováno zlepšení kvality odhadu prostorové funkce oproti standardnímu použití GPR metody a také oproti zjednodušenému řešení uvedenému v literatuře. Dále se práce zabývá možností použití metody GPR s nejistými tréninkovými pozicemi v~kombinaci s výrazy s dostupným analytickým řešením. Ukazuje se, že k dosažení těchto výrazů je třeba zavést značné předpoklady, což má od počátku za následek nepřesnost prediktivní distribuce. Také se ukazuje, že výsledná metoda používá standardní výrazy GPR v~kombinaci s upravenou kovarianční funkcí. Simulace dokazují, že tato metoda produkuje velmi podobné odhady jako základní GPR metoda uvažující známé tréninkové pozice. Na druhou stranu prediktivní variance (nejistota odhadu) je u této metody zvýšena, což je žádaný efekt uvážení nejistoty tréninkových pozic.

We consider a wireless communication scenario with two transmit-receive pairs where each of the transmitters has a message for its corresponding receiver and only one of the receivers face interference from the undesired transmitter. In our research, we focused on devising optimal ways to manage this undesired interference and characterize the best communication rates for both transmit-receive pairs. Currently, this problem of interference is dealt with by restricting the two communications in di erent frequency or time bands. We explore the possibility of achieving better rates by allowing them to operate in the same band. Such channels were identi ed about 4 decades ago, but the maximum rate of communication when the transmitters have a power constraint is still unknown. In this work, we characterize the best rates for this channel under a reasonable practical constraint of using Gaussian signals at both the transmitters.

With a rapid development of computing power, computer experiments have gained popularity in various scientific fields, like cosmology, ecology and engineering. However, some computer experiments for complex processes are still computationally demanding. A surrogate model or emulator, is often employed as a fast substitute for the simulator. Meanwhile, a common challenge in computer experiments and related fields is to efficiently explore the input space using a small number of samples, i.e., the experimental design problem. This dissertation focuses on the design problem under Gaussian process surrogates. The first work demonstrates empirically that space-filling designs disappoint when the model hyperparameterization is unknown, and must be estimated from data observed at the chosen design sites. A purely random design is shown to be superior to higher-powered alternatives in many cases. Thereafter, a new family of distance-based designs are proposed and their superior performance is illustrated in both static (one-shot design) and sequential settings. The second contribution is motivated by an agent-based model(ABM) of delta smelt conservation. The ABM is developed to assist in a study of delta smelt life cycles and to understand sensitivities to myriad natural variables and human interventions. However, the input space is high-dimensional, running the simulator is time-consuming, and its outputs change nonlinearly in both mean and variance. A batch sequential design scheme is proposed, generalizing one-at-a-time variance-based active learning, as a means of keeping multi-core cluster nodes fully engaged with expensive runs. The acquisition strategy is carefully engineered to favor selection of replicates which boost statistical and computational efficiencies. Design performance is illustrated on a range of toy examples before embarking on a smelt simulation campaign and downstream high-fidelity input sensitivity analysis.
Doctor of Philosophy
With a rapid development of computing power, computer experiments have gained popularity in various scientific fields, like cosmology, ecology and engineering. However, some computer experiments for complex processes are still computationally demanding. Thus, a statistical model built upon input-output observations, i.e., a so-called surrogate model or emulator, is needed as a fast substitute for the simulator. Design of experiments, i.e., how to select samples from the input space under budget constraints, is also worth studying. This dissertation focuses on the design problem under Gaussian process (GP) surrogates. The first work demonstrates empirically that commonly-used space-filling designs disappoint when the model hyperparameterization is unknown, and must be estimated from data observed at the chosen design sites. Thereafter, a new family of distance-based designs are proposed and their superior performance is illustrated in both static (design points are allocated at one shot) and sequential settings (data are sampled sequentially). The second contribution is motivated by a stochastic computer simulator of delta smelt conservation. This simulator is developed to assist in a study of delta smelt life cycles and to understand sensitivities to myriad natural variables and human interventions. However, the input space is high-dimensional, running the simulator is time-consuming, and its outputs change nonlinearly in both mean and variance. An innovative batch sequential design method is proposed, generalizing one-at-a-time sequential design to one-batch-at-a-time scheme with the goal of parallel computing. The criterion for subsequent data acquisition is carefully engineered to favor selection of replicates which boost statistical and computational efficiencies. The design performance is illustrated on a range of toy examples before embarking on a smelt simulation campaign and downstream input sensitivity analysis.

This study presents the results of an extraction of the 2nd-order nonlinear responses from model test data. Emphasis is given on the effects of assumptions made for the Gaussian and non-Gaussian input on the estimation of the 2nd-order response, employing the quadratic Volterra model. The effects of sea severity and data length on the estimation of response are also investigated at the same time. The data sets used in this study are surge forces on a fixed barge, a surge motion of a compliant mini TLP (Tension Leg Platform), and surge forces on a fixed and truncated column. Sea states are used from rough sea (Hs=3m) to high sea (Hs=9m) for a barge case, very rough sea (Hs=3.9m) for a mini TLP, and phenomenal sea (Hs=15m) for a truncated column. After the estimation of the response functions, the outputs are reconstructed and the 2nd order nonlinear responses are extracted with all the QTF distributed in the entire bifrequency domain. The reconstituted time series are compared with the experiment in both the time and frequency domains. For the effects of data length on the estimation of the response functions, 3, 15, and 40- hour data were investigated for a barge, but 3-hour data was used for a mini TLP and a fixed and truncated column due to lack of long data. The effects of sea severity on the estimation of the response functions are found in both methods. The non-Gaussian method for estimation is more affected by data length than the Gaussian method.

L'analyse de sensibilité est un outil puissant qui permet d'analyser des modèles mathématiques et des codes de calculs. Elle révèle les variables d'entrées les plus influentes sur la variable de sortie, en leur affectant une valeur appelée "indice de sensibilité". Dans ce cadre, les effets de Shapley, récemment définis par Owen, permettent de gérer des variables d'entrées dépendantes. Cependant, l'estimation de ces indices ne peut se faire que dans deux cadres très particuliers : lorsque la loi du vecteur d'entrée est connue ou lorsque les entrées sont gaussiennes et le modèle est linéaire. Cette thèse se divise en deux parties. Dans la première partie, l'objectif est d'étendre les estimateurs des effets de Shapley lorsque seul un échantillon des entrées est disponible et leur loi est inconnue. Dans la deuxième partie porte sur le cas linéaire gaussien. Le problème de la grande dimension est abordé et des solutions sont proposées lorsque les variables forment des groupes indépendants. Enfin, l'étude montre comment les effets de Shapley du cadre linéaire gaussien peuvent estimer ceux d'un cadre plus général
Sensitivity analysis is a powerful tool to study mathematical models and computer codes. It reveals the most impacting input variables on the output variable, by assigning values to the the inputs, that we call "sensitivity indices". In this setting, the Shapley effects, recently defined by Owen, enable to handle dependent input variables. However, one can only estimate these indices in two particular cases: when the distribution of the input vector is known or when the inputs are Gaussian and when the model is linear. This thesis can be divided into two parts. First, the aim is to extend the estimation of the Shapley effects when only a sample of the inputs is available and their distribution is unknown. The second part focuses on the linear Gaussian framework. The high-dimensional problem is emphasized and solutions are suggested when there are independent groups of variables. Finally, it is shown how the values of the Shapley effects in the linear Gaussian framework can estimate of the Shapley effects in more general settings

Construction of radiotherapy treatments for cancer is a laborious and time consuming task. At the same time, when presented with a treatment plan, an oncologist can quickly judge whether or not it is suitable. This means that the problem of constructing these treatment plans is well suited for automation. This thesis investigates a novel way of automatic treatment planning. The treatment planning system this pipeline is constructed for provides dose mimicking functionality with probability density functions of dose–volume histograms (DVHs) and spatial dose as inputs. Therefore this will be the output of the pipeline. The input is historically treated patient scans, segmentations and spatial doses. The approach involves three modules which are individually replaceable with little to no impact on the remaining two modules. The modules are: an autoencoder as a feature extractor to concretise important features of a patient segmentation, a distance optimisation step to learn a distance in the previously constructed feature space and, finally, a probabilistic spatial dose estimation module using sparse pseudo-input Gaussian processes trained on voxel features. Although performance evaluation in terms of clinical plan quality was beyond the scope of this thesis, numerical results show that the proposed pipeline is successful in capturing salient features of patient geometry as well as predicting reasonable probability distributions for DVH and spatial dose. Its loosely connected nature also gives hope that some parts of the pipeline can be utilised in future work.
Skapandet av strålbehandlingsplaner för cancer är en tidskrävande uppgift. Samtidigt kan en onkolog snabbt fatta beslut om en given plan är acceptabel eller ej. Detta innebär att uppgiften att skapa strålplaner är väl lämpad för automatisering. Denna uppsats undersöker en ny metod för att automatiskt generera strålbehandlingsplaner. Planeringssystemet denna metod utvecklats för innehåller funktionalitet för dosrekonstruktion som accepterar sannolikhetsfördelningar för dos–volymhistogram (DVH) och dos som input. Därför kommer detta att vara utdatan för den konstruerade metoden. Metoden är uppbyggd av tre beståndsdelar som är individuellt utbytbara med liten eller ingen påverkan på de övriga delarna. Delarna är: ett sätt att konstruera en vektor av kännetecken av en patients segmentering, en distansoptimering för att skapa en distans i den tidigare konstruerade känneteckensrymden, och slutligen en skattning av sannolikhetsfördelningar med Gaussiska processer tränade på voxelkännetecken. Trots att utvärdering av prestandan i termer av klinisk plankvalitet var bortom räckvidden för detta projekt uppnåddes positiva resultat. De estimerade sannolikhetsfördelningarna uppvisar goda karaktärer för både DVHer och doser. Den löst sammankopplade strukturen av metoden gör det dessutom möjligt att delar av projektet kan användas i framtida arbeten.

This thesis aims to contribute to the developments of wireless communication systems. The work generally consists of three parts: the first part is a discussion on general digital communication systems, the second part focuses on wireless channel modelling and fading mitigation techniques, and in the third part we discuss the possible application of advanced digital signal processing, especially time-frequency representation and blind source separation, to wireless communication systems. The first part considers general digital communication systems which will be incorporated in later parts. Today's wireless communication system is a subbranch of a general digital communication system that employs various techniques of A/D (Analog to Digital) conversion, source coding, error correction, coding, modulation, and synchronization, signal detection in noise, channel estimation, and equalization. We study and develop the digital communication algorithms to enhance the performance of wireless communication systems. In the Second Part we focus on wireless channel modelling and fading mitigation techniques. A modified Jakes' method is developed for Rayleigh fading channels. We investigate the level-crossing rate (LCR), the average duration of fades (ADF), the probability density function (PDF), the cumulative distribution function (CDF) and the autocorrelation functions (ACF) of this model. The simulated results are verified against the analytical Clarke's channel model. We also construct frequency-selective geometrical-based hyperbolically distributed scatterers (GBHDS) for a macro-cell mobile environment with the proper statistical characteristics. The modified Clarke's model and the GBHDS model may be readily expanded to a MIMO channel model thus we study the MIMO fading channel, specifically we model the MIMO channel in the angular domain. A detailed analysis of Gauss-Markov approximation of the fading channel is also given. Two fading mitigation techniques are investigated: Orthogonal Frequency Division Multiplexing (OFDM) and spatial diversity. In the Third Part, we devote ourselves to the exciting fields of Time-Frequency Analysis and Blind Source Separation and investigate the application of these powerful Digital Signal Processing (DSP) tools to improve the performance of wireless communication systems.

Achieving a higher transmission rate had always been a goal in the field of communications. Having a two-way channel in which two nodes transmit and receive data at the same time, is an important tool to achieve this goal. A two-way channel is the first step from point-to-point communication channel toward multi-user networks. In its ideal form, we can transmit data two times faster by using a perfect two-way channel. However, the area of two-way channels had not been of interest of researchers during the past years and number of articles on this area is considerably low comparing to other types of multi-user communication networks, such as multiple-access channel, broadcast channel and interference channel. On the other hand, use of analog-to-digital converters (ADC) is a must in modern systems to enable us to analyze data faster; nevertheless, presence of ADC add some other difficulties to the system. In this thesis, different scenarios about two-way channel are studied. The Shannon's model of two-way channel and his inner and outer bounds on the capacity of this channel are presented. For the Gaussian Two-Way Channel with quantized output, in which the ambient noise has a Gaussian distribution, the expression of Shannon's inner bound for both Gaussian and discrete inputs are derived. The best uniform quantizer to obtain the maximum achievable rate for Gaussian input is found numerically. Then we will evaluate the additive noise model for the quantizer from an information theoretic point of view. For the discrete input, the method of rotating one input with respect to other one is employed to enlarge the achievable rate region. At last, two scenarios will be studied in which, minimizing the power of interference, does not necessarily maximizes the transmission rate.

In the application of cycle-to-cycle control to manufacturing processes, the model of the process reduces to a gain matrix and a pure delay. For a general multiple input – multiple output process, this matrix shows the degree of influence each input has on each output. For a system of high order, determining this gain matrix requires excessive numbers of experiments to be performed, and thus a simplified, but non-ideal form for the gain matrix must be developed. In this paper, the model takes the form of a Gaussian distribution with experimentally determined standard deviation and scaling coefficients. Discrete die sheet metal forming, a multiple input-multiple output process with high dimensionality, is chosen as a test application. Results of the prediction capabilities of the Gaussian model, as well as those of two previously existing models, are presented. It is shown that the Gaussian distribution model does the best job of predicting the output for a given input. The model’s invariance over a set of different formed parts is also presented. However, as shown in the companion paper on cycle-to-cycle control, the errors inherent in this model will cause non-ideal performance of the resulting control system. However, this model appears to be the best form for this problem, given the limit of minimal experimentation.
Singapore-MIT Alliance (SMA)

Despite the impressive performance of deep neural networks (DNNs) on numerous vision tasks, they still exhibit yet-to-understand uncouth behaviours. One puzzling behaviour is the reaction of DNNs to various noise attacks, where it has been shown that there exist small adversarial noise that can result in a severe degradation in the performance of DNNs. To rigorously treat this, we derive exact analytic expressions for the first and second moments (mean and variance) of a small piecewise linear (PL) network with a single rectified linear unit (ReLU) layer subject to general Gaussian input. We experimentally show that these expressions are tight under simple linearizations of deeper PL-DNNs, especially popular architectures in the literature (e.g. LeNet and AlexNet). Extensive experiments on image classification show that these expressions can be used to study the behaviour of the output mean of the logits for each class, the inter-class confusion and the pixel-level spatial noise sensitivity of the network. Moreover, we show how these expressions can be used to systematically construct targeted and non-targeted adversarial attacks. Then, we proposed a special estimator DNN, named mixture of linearizations (MoL), and derived the analytic expressions for its output mean and variance, as well. We employed these expressions to train the model to be particularly robust against Gaussian attacks without the need for data augmentation. Upon training this network on a loss that is consolidated with the derived output probabilistic moments, the network is not only robust under very high variance Gaussian attacks but is also as robust as networks that are trained with 20 fold data augmentation.

A wireless relay network consists of multiple source nodes, multiple destination nodes, and possibly many relay nodes in between to facilitate its transmission. It is clear that the performance of such networks highly depends on information for- warding strategies adopted at the relay nodes. This dissertation studies a particular information forwarding strategy called compute-and-forward. Compute-and-forward is a novel paradigm that tries to incorporate the idea of network coding within the physical layer and hence is often referred to as physical layer network coding. The main idea is to exploit the superposition nature of the wireless medium to directly compute or decode functions of transmitted signals at intermediate relays in a net- work. Thus, the coding performed at the physical layer serves the purpose of error correction as well as permits recovery of functions of transmitted signals. For the bidirectional relaying problem with Gaussian channels, it has been shown by Wilson et al. and Nam et al. that the compute-and-forward paradigm is asymptotically optimal and achieves the capacity region to within 1 bit; however, similar results beyond the memoryless case are still lacking. This is mainly because channels with memory would destroy the lattice structure that is most crucial for the compute-and-forward paradigm. Hence, how to extend compute-and-forward to such channels has been a challenging issue. This motivates this study of the extension of compute-and-forward to channels with memory, such as inter-symbol interference. The bidirectional relaying problem with parallel Gaussian channels is also studied, which is a relevant model for the Gaussian bidirectional channel with inter-symbol interference and that with multiple-input multiple-output channels. Motivated by the recent success of linear finite-field deterministic model, we first investigate the corresponding deterministic parallel bidirectional relay channel and fully characterize its capacity region. Two compute-and-forward schemes are then proposed for the Gaussian model and the capacity region is approximately characterized to within a constant gap. The design of coding schemes for the compute-and-forward paradigm with low decoding complexity is then considered. Based on the separation-based framework proposed previously by Tunali et al., this study proposes a family of constellations that are suitable for the compute-and-forward paradigm. Moreover, by using Chinese remainder theorem, it is shown that the proposed constellations are isomorphic to product fields and therefore can be put into a multilevel coding framework. This study then proposes multilevel coding for the proposed constellations and uses multistage decoding to further reduce decoding complexity.

Multiple users compete for a common resource like bandwidth to communicate data in interference networks. Existing approaches in dealing with interference limit the rate of communication due to paucity of shared resources. This limitation in the rate gets more glaring as the number of users in the network increases. For example, existing wireless systems either choose to orthogonalize the users (for example, Frequency Division Multiple Access (FDMA) systems or Code Division Multiple Access (CDMA) systems) or treat interference as Gaussian noise at the receivers. It is well known that these approaches are sub-optimal in general. Orthogonalization of users limit the number of available interference-free channels (known as degrees of freedom, abbreviated as DoF) and treating interference as noise means that the receiver cannot make use of the structure in the interfering signals. This motivates the need to analyze alternate transmit and decoding schemes in interference networks. This thesis mainly analyzes transmit schemes that use linear precoding for various configurations of interference networks with some practical constraints imposed by the use of finite input constellations, propagation delays, and channel state availability at the transmitters. The main contributions of this thesis are listed below. Achievable rates using precoding with finite constellation inputs in Gaussian Interference Channels (GIC) is analyzed. A metric for finding the approximate angle of rotation to maximally enlarge the Constellation Constrained (CC) capacity of two-user Gaussian Strong Interference Channel (GSIC) is proposed. Even as the Gaussian alphabet FDMA rate curve touches the capacity curve of the GSIC, with both the users using the same finite constellation, we show that the CC FDMA rate curve lies strictly inside the CC capacity curve at high powers. For a K-user MIMO GIC, a set of necessary and sufficient conditions on the precoders under which the mutual information between between relevant transmit-receive pairs saturate like in the single user case is derived. Gradient-ascent based algorithms to optimize the sum-rate achieved by precoding with finite constellation inputs and treating interference as noise are proposed. For a class of Gaussian interference networks with general message demands, identified as symmetrically connected interference networks, the expected sumspectral efficiency (in bits/sec/Hz) is shown to grow linearly with the number of transmitters at finite SNR, using a time-domain Interference Alignment (IA) scheme in the presence of line of sight (LOS) channels. For a 2×2 MIMO X-Network with M antennas at each node, we identify spacetime block codes that could be coupled with an appropriate precoding scheme to achieve the maximum possible sum-DoF of 4M 3 , for M = 3, 4. The proposed schemes are shown to achieve a diversity gain of M with SNR-independent finite constellation inputs. The proposed schemes have lower CSIT requirements compared to existing schemes. This thesis also makes an attempt to guarantee a minimum throughput when the zero-interference conditions cannot be satisfied in a wireline network with three unicast sessions with delays, using Precoding Based Network Alignment (PBNA). Three different PBNA schemes namely PBNA with time-varying local encoding coefficients (LECs), PBNA using transform approach and time-invariant LECs, and PBNA using transform approach and block time-varying LECs are proposed and their feasibility conditions analyzed.

Contemporary research in digital signal processing (DSP) is focused on issues of computational complexity, very high data rate and large quantum of data. Thus, the success in newer applications and areas hinge on handling these issues. Conventional ways to address these challenges are to develop newer structures like Multirate signal processing, Multiple Input Multiple Output(MIMO), bandpass sampling, compressed domain sensing etc. In the implementation domain, the approach is to look at floating point over fixed point representation and / or longer wordlength etc., related to number representations and computations. Of these, a simple approach is to look at number representation, perhaps with a simple integer. This automatically guarantees accuracy and zero quantization error as well as longer wordlength. Thus, it is necessary and interesting to explore viable DSP alternatives that can reduce complexity and yet match the required performance. The main aim of this work is to highlight the importance, use and analysis of integer sequences. Firstly, the thesis explores the use of integer sequences as windowing functions. The results of these investigations show that integer sequences and their convolution, indeed, outperform many of the classical real valued window functions in terms of mainlobe width, sidelobe attenuation etc. Secondly, the thesis proposes techniques to approximate discrete Gaussian distribution using integer sequences. The key idea is to convolve symmetrized integer sequences and examine the resulting profiles. These profiles are found to approximate discrete Gaussian distribution with a mean square error of the order of 10−8 or less. While looking at integer sequences to approximate discrete Gaussian, Fibonacci sequence was found to exhibit some interesting properties. The third part of the thesis proves certain fascinating optimal probabilistic limit properties (mean and variance) of Fibonacci sequence. The thesis also provides complete generalization of these properties to probability distributions generated by second order linear recurrence relation with integer coefficients and any kth order linear recurrence relation with unit coefficients. In addition to the above, the thesis also throws light on possible architectural implications of using integer sequences in DSP applications and ideas for further exploration.

This thesis addresses the design of low-complexity coding schemes for wireless relay networks and multiple access channels. The first part of the thesis is on wireless relay networks and the second part is on multiple access channels. Distributed space-time coding is a well known technique to achieve spatial diversity in wireless networks wherein, several geographically separated nodes assist a source node to distributively transmit a space-time block code (STBC) to the destination. Such STBCs are referred to as Distributed STBCs (DSTBCs). In the first part of the thesis, we focus on designing full diversity DSTBCs with some nice properties which make them amenable for implementation in practice. Towards that end, a class of full diversity DST-BCs referred to as Co-ordinate Interleaved DSTBCs (CIDSTBCs) are proposed for relay networks with two-antenna relays. To construct CIDSTBCs, a technique called co-ordinate vector interleaving is introduced wherein, the received signals at different antennas of the relay are processed in a combined fashion. Compared to the schemes where the received signals at different antennas of the relay are processed independently, we show that CIDSTBCs provide coding gain which comes in with negligible increase in the processing complexity at the relays. Subsequently, we design single-symbol ML decodable (SSD) DSTBCs for relay networks with single-antenna nodes. In particular, two classes of SSD DSTBCs referred to as (i) Semi-orthogonal SSD Precoded DSTBCs and (ii) Training-Symbol Embedded (TSE) SSD DSTBCs are proposed. A detailed analysis on the maximal rate of such DSTBCs is presented and explicit DSTBCs achieving the maximal rate are proposed. It is shown that the proposed codes have higher rates than the existing SSD DSTBCs. In the second part, we study two-user Gaussian Multiple Access Channels (GMAC). Capacity regions of two-user GMAC are well known. Though, capacity regions of such channels provide insights into the achievable rate pairs in an information theoretic sense, they fail to provide information on the achievable rate pairs when we consider finitary restrictions on the input alphabets and analyze some real world practical signal constellations like QAM and PSK signal sets. Hence, we study the capacity aspects of two-user GMAC with finite input alphabets. In particular, Constellation Constrained (CC) capacity regions of two-user SISO-GMAC are computed for several orthogonal and non-orthogonal multiple access schemes (abbreviated as O-MA and NO-MA schemes respectively). It is first shown that NO-MA schemes strictly offer larger capacity regions than the O-MA schemes for finite input alphabets. Subsequently, for NO-MA schemes, code pairs based on Trellis Coded Modulation (TCM) are proposed such that any rate pair on the CC capacity region can be approached. Finally, we consider a two-user Multiple-Input Multiple-Output (MIMO) fading MAC and design STBC pairs such that ML decoding complexity is reduced.