NATURAL ALGORITHMS IN DIGITAL FILTER DESIGN

Digital filters are an important part of Digital Signal Processing (DSP), which plays vital roles within the modern world, but their design is a complex task requiring a great deal of specialised knowledge. An analysis of this design process is presented, which identifies opportunities for the application of optimisation. The Genetic Algorithm (GA) and Simulated Annealing are problem-independent and increasingly popular optimisation techniques. They do not require detailed prior knowledge of the nature of a problem, and are unaffected by a discontinuous search space, unlike traditional methods such as calculus and hill-climbing. Potential applications of these techniques to the filter design process are discussed, and presented with practical results. Investigations into the design of Frequency Sampling (FS) Finite Impulse Response (FIR) filters using a hybrid GA/hill-climber proved especially successful, improving on published results. An analysis of the search space for FS filters provided useful information on the performance of the optimisation technique. The ability of the GA to trade off a filter's performance with respect to several design criteria simultaneously, without intervention by the designer, is also investigated. Methods of simplifying the design process by using this technique are presented, together with an analysis of the difficulty of the non-linear FIR filter design problem from a GA perspective. This gave an insight into the fundamental nature of the optimisation problem, and also suggested future improvements. The results gained from these investigations allowed the framework for a potential 'intelligent' filter design system to be proposed, in which embedded expert knowledge, Artificial Intelligence techniques and traditional design methods work together. This could deliver a single tool capable of designing a wide range of filters with minimal human intervention, and of proposing solutions to incomplete problems. It could also provide the basis for the development of tools for other areas of DSP system design

Covariance and Gramian matrices in control and systems theory.

Covariance and Gramian matrices in control and systems theory and pattern recognition are studied in the context of reduction of dimensionality and hence complexity of large-scale systems. This is achieved by the removal of redundant or 'almost' redundant information contained in the covariance and Grarrdan matrices. The Karhunen-Loeve expansion (principal component analysis) and its extensions and the singular value decomposition of matrices provide the framework for the work presented in the thesis. The results given for linear dynamical systems are based on controllability and observability Gramians and some new developments in singular perturbational analysis are also presented

An exact and consistent adjoint method for high-fidelity discretization of the compressible flow equations

Methods and computing hardware advances have enabled accurate predictions of complex compressible turbulence phenomena, such as the generation of jet noise that motivates the present effort. However, limited understanding of underlying physical mechanisms restricts the utility of such predictions since they do not, by themselves, indicate a route to design improvement. Gradient-based optimization using adjoints can circumvent the flow complexity to guide designs. Such methods have enabled sensitivity analysis and active control of turbulence at engineering flow conditions by providing gradient information at computational cost comparable to that of simulating the flow. They accelerate convergence of numerical design optimization algorithms, though this is predicated on the availability of an accurate gradient of the discretized flow equations. This is challenging to obtain, since both the chaotic character of the turbulence and the typical use of discretizations near their resolution limits in order to efficiently represent its smaller scales will amplify any approximation errors made in the adjoint formulation. Formulating a practical exact adjoint that avoids such errors is especially challenging if it is to be compatible with state-of-the-art simulation methods used for the turbulent flow itself. Automatic differentiation (AD) can provide code to calculate a nominally exact adjoint, but existing general-purpose AD codes are inefficient to the point of being prohibitive for large-scale turbulence simulations. We analyze the compressible flow equations as discretized using the same high-order workhorse methods used for many high-fidelity compressible turbulence simulations, and formulate a practical space–time discrete-adjoint method without changing the basic discretization. A key step is the definition of a particular discrete analog of the continuous norm that defines our cost functional; our selection leads directly to an efficient Runge--Kutta-like scheme, with finite-difference spatial operators for the adjoint system. Its computational cost only modestly exceeds that of the flow equations. We confirm that its accuracy is limited only by computing precision, and we demonstrate it on the aeroacoustic control of a mixing layer with a challengingly broad range of turbulence scales. For comparison, the error from a corresponding discretization of the continuous-adjoint equations is quantified to potentially explain its limited success in past efforts to control jet noise. The differences are illuminating: the continuous-adjoint is shown to suffer from exponential error growth in (reverse) time even for the best-resolved largest turbulence scales. Though the gradient from our fully discrete adjoint is formally exact, it does include sensitivity to numerical solutions that are only an artifact of the discretization. These are typically saw-tooth type features, such as seen in under-resolved numerical simulations. Since these have no physical analog, for physical analysis or design of realistic actuators, such solutions are in a sense spurious. This has been addressed without sacrificing accuracy by redesigning the basic discretization to be dual-consistent, for which the discrete-adjoint is consistent with the adjoint of the continuous system, and thus, free from spurious numerical sensitivity modes. We extend our exact discrete-adjoint to a spatially dual-consistent discretization of the compressible flow equations and demonstrate its practical application for aeroacoustic control of a Mach 1.3 turbulent jet. The formulation admits a broad class of finite-difference schemes that satisfy a summation by-parts rule, and extends to multi-block curvilinear grids for efficient handling of complex geometries. The formulation is developed for several boundary conditions commonly used in simulation of free-shear and wall-bounded flows. In addition, the proposed discretization leads to superconvergent approximations of functionals, and can be tailored to achieve global conservation up to arbitrary orders of accuracy. We again confirm that the sensitivity gradient for turbulent jet noise computed using our dual-consistent method is only limited by computing precision

GNSS array-based acquisition: theory and implementation

This Dissertation addresses the signal acquisition problem using antenna arrays in the general framework of Global Navigation Satellite Systems (GNSS) receivers. The term GNSS classi es those navigation systems based on a constellation of satellites, which emit ranging signals useful for positioning. Although the American GPS is already available, which coexists with the renewed Russian Glonass, the forthcoming European contribution (Galileo) along with the Chinese Compass will be operative soon. Therefore, a variety of satellite constellations and signals will be available in the next years. GNSSs provide the necessary infrastructures for a myriad of applications and services that demand a robust and accurate positioning service. The positioning availability must be guaranteed all the time, specially in safety-critical and mission-critical services. Examining the threats against the service availability, it is important to take into account that all the present and the forthcoming GNSSs make use of Code Division Multiple Access (CDMA) techniques. The ranging signals are received with very low precorrelation signal-to-noise ratio (in the order of ���22 dB for a receiver operating at the Earth surface). Despite that the GNSS CDMA processing gain o ers limited protection against Radio Frequency interferences (RFI), an interference with a interference-to-signal power ratio that exceeds the processing gain can easily degrade receivers' performance or even deny completely the GNSS service, specially conventional receivers equipped with minimal or basic level of protection towards RFIs. As a consequence, RFIs (either intentional or unintentional) remain as the most important cause of performance degradation. A growing concern of this problem has appeared in recent times. Focusing our attention on the GNSS receiver, it is known that signal acquisition has the lowest sensitivity of the whole receiver operation, and, consequently, it becomes the performance bottleneck in the presence of interfering signals. A single-antenna receiver can make use of time and frequency diversity to mitigate interferences, even though the performance of these techniques is compromised in low SNR scenarios or in the presence of wideband interferences. On the other hand, antenna arrays receivers can bene t from spatial-domain processing, and thus mitigate the e ects of interfering signals. Spatial diversity has been traditionally applied to the signal tracking operation of GNSS receivers. However, initial tracking conditions depend on signal acquisition, and there are a number of scenarios in which the acquisition process can fail as stated before. Surprisingly, to the best of our knowledge, the application of antenna arrays to GNSS signal acquisition has not received much attention. This Thesis pursues a twofold objective: on the one hand, it proposes novel arraybased acquisition algorithms using a well-established statistical detection theory framework, and on the other hand demonstrates both their real-time implementation feasibility and their performance in realistic scenarios. The Dissertation starts with a brief introduction to GNSS receivers fundamentals, providing some details about the navigation signals structure and the receiver's architecture of both GPS and Galileo systems. It follows with an analysis of GNSS signal acquisition as a detection problem, using the Neyman-Pearson (NP) detection theory framework and the single-antenna acquisition signal model. The NP approach is used here to derive both the optimum detector (known as clairvoyant detector ) and the sov called Generalized Likelihood Ratio Test (GLRT) detector, which is the basis of almost all of the current state-of-the-art acquisition algorithms. Going further, a novel detector test statistic intended to jointly acquire a set of GNSS satellites is obtained, thus reducing both the acquisition time and the required computational resources. The eff ects of the front-end bandwidth in the acquisition are also taken into account. Then, the GLRT is extended to the array signal model to obtain an original detector which is able to mitigate temporally uncorrelated interferences even if the array is unstructured and moderately uncalibrated, thus becoming one of the main contributions of this Dissertation. The key statistical feature is the assumption of an arbitrary and unknown covariance noise matrix, which attempts to capture the statistical behavior of the interferences and other non-desirable signals, while exploiting the spatial dimension provided by antenna arrays. Closed form expressions for the detection and false alarm probabilities are provided. Performance and interference rejection capability are modeled and compared both to their theoretical bound. The proposed array-based acquisition algorithm is also compared to conventional acquisition techniques performed after blind null-steering beamformer approaches, such as the power minimization algorithm. Furthermore, the detector is analyzed under realistic conditions, accounting for the presence of errors in the covariance matrix estimation, residual Doppler and delay errors, and signal quantization e ects. Theoretical results are supported by Monte Carlo simulations. As another main contribution of this Dissertation, the second part of the work deals with the design and the implementation of a novel Field Programmable Gate Array (FPGA)-based GNSS real-time antenna-array receiver platform. The platform is intended to be used as a research tool tightly coupled with software de ned GNSS receivers. A complete signal reception chain including the antenna array and the multichannel phase-coherent RF front-end for the GPS L1/ Galileo E1 was designed, implemented and tested. The details of the digital processing section of the platform, such as the array signal statistics extraction modules, are also provided. The design trade-o s and the implementation complexities were carefully analyzed and taken into account. As a proof-of-concept, the problem of GNSS vulnerability to interferences was addressed using the presented platform. The array-based acquisition algorithms introduced in this Dissertation were implemented and tested under realistic conditions. The performance of the algorithms were compared to single antenna acquisition techniques, measured under strong in-band interference scenarios, including narrow/wide band interferers and communication signals. The platform was designed to demonstrate the implementation feasibility of novel array-based acquisition algorithms, leaving the rest of the receiver operations (mainly, tracking, navigation message decoding, code and phase observables, and basic Position, Velocity and Time (PVT) solution) to a Software De ned Radio (SDR) receiver running in a personal computer, processing in real-time the spatially- ltered signal sample stream coming from the platform using a Gigabit Ethernet bus data link. In the last part of this Dissertation, we close the loop by designing and implementing such software receiver. The proposed software receiver targets multi-constellation/multi-frequency architectures, pursuing the goals of e ciency, modularity, interoperability, and exibility demanded by user domains that require non-standard features, such as intermediate signals or data extraction and algorithms interchangeability. In this context, we introduce an open-source, real-time GNSS software de ned receiver (so-named GNSS-SDR) that contributes with several novel features such as the use of software design patterns and shared memory techniques to manage e ciently the data ow between receiver blocks, the use of hardware-accelerated instructions for time-consuming vector operations like carrier wipe-o and code correlation, and the availability to compile and run on multiple software platforms and hardware architectures. At this time of writing (April 2012), the receiver enjoys of a 2-dimensional Distance Root Mean Square (DRMS) error lower than 2 meters for a GPS L1 C/A scenario with 8 satellites in lock and a Horizontal Dilution Of Precision (HDOP) of 1.2.Esta tesis aborda el problema de la adquisición de la señal usando arrays de antenas en el marco general de los receptores de Sistemas Globales de Navegación por Satélite (GNSS). El término GNSS engloba aquellos sistemas de navegación basados en una constelación de satélites que emiten señales útiles para el posicionamiento. Aunque el GPS americano ya está disponible, coexistiendo con el renovado sistema ruso GLONASS, actualmente se está realizando un gran esfuerzo para que la contribución europea (Galileo), junto con el nuevo sistema chino Compass, estén operativos en breve. Por lo tanto, una gran variedad de constelaciones de satélites y señales estarán disponibles en los próximos años. Estos sistemas proporcionan las infraestructuras necesarias para una multitud de aplicaciones y servicios que demandan un servicio de posicionamiento confiable y preciso. La disponibilidad de posicionamiento se debe garantizar en todo momento, especialmente en los servicios críticos para la seguridad de las personas y los bienes. Cuando examinamos las amenazas de la disponibilidad del servicio que ofrecen los GNSSs, es importante tener en cuenta que todos los sistemas presentes y los sistemas futuros ya planificados hacen uso de técnicas de multiplexación por división de código (CDMA). Las señales transmitidas por los satélites son recibidas con una relación señal-ruido (SNR) muy baja, medida antes de la correlación (del orden de -22 dB para un receptor ubicado en la superficie de la tierra). A pesar de que la ganancia de procesado CDMA ofrece una protección inherente contra las interferencias de radiofrecuencia (RFI), esta protección es limitada. Una interferencia con una relación de potencia de interferencia a potencia de la señal que excede la ganancia de procesado puede degradar el rendimiento de los receptores o incluso negar por completo el servicio GNSS. Este riesgo es especialmente importante en receptores convencionales equipados con un nivel mínimo o básico de protección frente las RFIs. Como consecuencia, las RFIs (ya sean intencionadas o no intencionadas), se identifican como la causa más importante de la degradación del rendimiento en GNSS. El problema esta causando una preocupación creciente en los últimos tiempos, ya que cada vez hay más servicios que dependen de los GNSSs Si centramos la atención en el receptor GNSS, es conocido que la adquisición de la señal tiene la menor sensibilidad de todas las operaciones del receptor, y, en consecuencia, se convierte en el factor limitador en la presencia de señales interferentes. Un receptor de una sola antena puede hacer uso de la diversidad en tiempo y frecuencia para mitigar las interferencias, aunque el rendimiento de estas técnicas se ve comprometido en escenarios con baja SNR o en presencia de interferencias de banda ancha. Por otro lado, los receptores basados en múltiples antenas se pueden beneficiar del procesado espacial, y por lo tanto mitigar los efectos de las señales interferentes. La diversidad espacial se ha aplicado tradicionalmente a la operación de tracking de la señal en receptores GNSS. Sin embargo, las condiciones iniciales del tracking dependen del resultado de la adquisición de la señal, y como hemos visto antes, hay un número de situaciones en las que el proceso de adquisición puede fallar. En base a nuestro grado de conocimiento, la aplicación de los arrays de antenas a la adquisición de la señal GNSS no ha recibido mucha atención, sorprendentemente. El objetivo de esta tesis doctoral es doble: por un lado, proponer nuevos algoritmos para la adquisición basados en arrays de antenas, usando como marco la teoría de la detección de señal estadística, y por otro lado, demostrar la viabilidad de su implementación y ejecución en tiempo real, así como su medir su rendimiento en escenarios realistas. La tesis comienza con una breve introducción a los fundamentos de los receptores GNSS, proporcionando algunos detalles sobre la estructura de las señales de navegación y la arquitectura del receptor aplicada a los sistemas GPS y Galileo. Continua con el análisis de la adquisición GNSS como un problema de detección, aplicando la teoría del detector Neyman-Pearson (NP) y el modelo de señal de una única antena. El marco teórico del detector NP se utiliza aquí para derivar tanto el detector óptimo (conocido como detector clarividente) como la denominada Prueba Generalizada de la Razón de Verosimilitud (en inglés, Generalized Likelihood Ratio Test (GLRT)), que forma la base de prácticamente todos los algoritmos de adquisición del estado del arte actual. Yendo más lejos, proponemos un nuevo detector diseñado para adquirir simultáneamente un conjunto de satélites, por lo tanto, obtiene una reducción del tiempo de adquisición y de los recursos computacionales necesarios en el proceso, respecto a las técnicas convencionales. El efecto del ancho de banda del receptor también se ha tenido en cuenta en los análisis. A continuación, el detector GLRT se extiende al modelo de señal de array de antenas para obtener un detector nuevo que es capaz de mitigar interferencias no correladas temporalmente, incluso utilizando arrays no estructurados y moderadamente descalibrados, convirtiéndose así en una de las principales aportaciones de esta tesis. La clave del detector es asumir una matriz de covarianza de ruido arbitraria y desconocida en el modelo de señal, que trata de captar el comportamiento estadístico de las interferencias y otras señales no deseadas, mientras que utiliza la dimensión espacial proporcionada por los arrays de antenas. Se han derivado las expresiones que modelan las probabilidades teóricas de detección y falsa alarma. El rendimiento del detector y su capacidad de rechazo a interferencias se han modelado y comparado con su límite teórico. El algoritmo propuesto también ha sido comparado con técnicas de adquisición convencionales, ejecutadas utilizando la salida de conformadores de haz que utilizan algoritmos de filtrado de interferencias, como el algoritmo de minimización de la potencia. Además, el detector se ha analizado bajo condiciones realistas, representadas con la presencia de errores en la estimación de covarianzas, errores residuales en la estimación del Doppler y el retardo de señal, y los efectos de la cuantificación. Los resultados teóricos se apoyan en simulaciones de Monte Carlo. Como otra contribución principal de esta tesis, la segunda parte del trabajo trata sobre el diseño y la implementación de una nueva plataforma para receptores GNSS en tiempo real basados en array de antenas que utiliza la tecnología de matriz programable de puertas lógicas (en ingles Field Programmable Gate Array (FPGA)). La plataforma está destinada a ser utilizada como una herramienta de investigación estrechamente acoplada con receptores GNSS definidos por software. Se ha diseñado, implementado y verificado la cadena completa de recepción, incluyendo el array de antenas y el front-end multi-canal para las señales GPS L1 y Galileo E1. El documento explica en detalle el procesado de señal que se realiza, como por ejemplo, la implementación del módulo de extracción de estadísticas de la señal. Los compromisos de diseño y las complejidades derivadas han sido cuidadosamente analizadas y tenidas en cuenta. La plataforma ha sido utilizada como prueba de concepto para solucionar el problema presentado de la vulnerabilidad del GNSS a las interferencias. Los algoritmos de adquisición introducidos en esta tesis se han implementado y probado en condiciones realistas. El rendimiento de los algoritmos se comparó con las técnicas de adquisición basadas en una sola antena. Se han realizado pruebas en escenarios que contienen interferencias dentro de la banda GNSS, incluyendo interferencias de banda estrecha y banda ancha y señales de comunicación. La plataforma fue diseñada para demostrar la viabilidad de la implementación de nuevos algoritmos de adquisición basados en array de antenas, dejando el resto de las operaciones del receptor (principalmente, los módulos de tracking, decodificación del mensaje de navegación, los observables de código y fase, y la solución básica de Posición, Velocidad y Tiempo (PVT)) a un receptor basado en el concepto de Radio Definida por Software (SDR), el cual se ejecuta en un ordenador personal. El receptor procesa en tiempo real las muestras de la señal filltradas espacialmente, transmitidas usando el bus de datos Gigabit Ethernet. En la última parte de esta Tesis, cerramos ciclo diseñando e implementando completamente este receptor basado en software. El receptor propuesto está dirigido a las arquitecturas de multi-constalación GNSS y multi-frecuencia, persiguiendo los objetivos de eficiencia, modularidad, interoperabilidad y flexibilidad demandada por los usuarios que requieren características no estándar, tales como la extracción de señales intermedias o de datos y intercambio de algoritmos. En este contexto, se presenta un receptor de código abierto que puede trabajar en tiempo real, llamado GNSS-SDR, que contribuye con varias características nuevas. Entre ellas destacan el uso de patrones de diseño de software y técnicas de memoria compartida para administrar de manera eficiente el uso de datos entre los bloques del receptor, el uso de la aceleración por hardware para las operaciones vectoriales más costosas, como la eliminación de la frecuencia Doppler y la correlación de código, y la disponibilidad para compilar y ejecutar el receptor en múltiples plataformas de software y arquitecturas de hardware. A fecha de la escritura de esta Tesis (abril de 2012), el receptor obtiene un rendimiento basado en la medida de la raíz cuadrada del error cuadrático medio en la distancia bidimensional (en inglés, 2-dimensional Distance Root Mean Square (DRMS) error) menor de 2 metros para un escenario GPS L1 C/A con 8 satélites visibles y una dilución de la precisión horizontal (en inglés, Horizontal Dilution Of Precision (HDOP)) de 1.2