Sub-Nyquist Sampling: Bridging Theory and Practice

Sampling theory encompasses all aspects related to the conversion of continuous-time signals to discrete streams of numbers. The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. In modern applications, an increasingly number of functions is being pushed forward to sophisticated software algorithms, leaving only those delicate finely-tuned tasks for the circuit level. In this paper, we review sampling strategies which target reduction of the ADC rate below Nyquist. Our survey covers classic works from the early 50's of the previous century through recent publications from the past several years. The prime focus is bridging theory and practice, that is to pinpoint the potential of sub-Nyquist strategies to emerge from the math to the hardware. In that spirit, we integrate contemporary theoretical viewpoints, which study signal modeling in a union of subspaces, together with a taste of practical aspects, namely how the avant-garde modalities boil down to concrete signal processing systems. Our hope is that this presentation style will attract the interest of both researchers and engineers in the hope of promoting the sub-Nyquist premise into practical applications, and encouraging further research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin

Turbo Bayesian Compressed Sensing

Compressed sensing (CS) theory specifies a new signal acquisition approach, potentially allowing the acquisition of signals at a much lower data rate than the Nyquist sampling rate. In CS, the signal is not directly acquired but reconstructed from a few measurements. One of the key problems in CS is how to recover the original signal from measurements in the presence of noise. This dissertation addresses signal reconstruction problems in CS. First, a feedback structure and signal recovery algorithm, orthogonal pruning pursuit (OPP), is proposed to exploit the prior knowledge to reconstruct the signal in the noise-free situation. To handle the noise, a noise-aware signal reconstruction algorithm based on Bayesian Compressed Sensing (BCS) is developed. Moreover, a novel Turbo Bayesian Compressed Sensing (TBCS) algorithm is developed for joint signal reconstruction by exploiting both spatial and temporal redundancy. Then, the TBCS algorithm is applied to a UWB positioning system for achieving mm-accuracy with low sampling rate ADCs. Finally, hardware implementation of BCS signal reconstruction on FPGAs and GPUs is investigated. Implementation on GPUs and FPGAs of parallel Cholesky decomposition, which is a key component of BCS, is explored. Simulation results on software and hardware have demonstrated that OPP and TBCS outperform previous approaches, with UWB positioning accuracy improved by 12.8x. The accelerated computation helps enable real-time application of this work

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling

Learning-Based Hardware Design for Data Acquisition Systems

This multidisciplinary research work aims to investigate the optimized information extraction from signals or data volumes and to develop tailored hardware implementations that trade-off the complexity of data acquisition with that of data processing, conceptually allowing radically new device designs. The mathematical results in classical Compressive Sampling (CS) support the paradigm of Analog-to-Information Conversion (AIC) as a replacement for conventional ADC technologies. The AICs simultaneously perform data acquisition and compression, seeking to directly sample signals for achieving specific tasks as opposed to acquiring a full signal only at the Nyquist rate to throw most of it away via compression. Our contention is that in order for CS to live up its name, both theory and practice must leverage concepts from learning. This work demonstrates our contention in hardware prototypes, with key trade-offs, for two different fields of application as edge and big-data computing. In the framework of edge-data computing, such as wearable and implantable ecosystems, the power budget is defined by the battery capacity, which generally limits the device performance and usability. This is more evident in very challenging field, such as medical monitoring, where high performance requirements are necessary for the device to process the information with high accuracy. Furthermore, in applications like implantable medical monitoring, the system performances have to merge the small area as well as the low-power requirements, in order to facilitate the implant bio-compatibility, avoiding the rejection from the human body. Based on our new mathematical foundations, we built different prototypes to get a neural signal acquisition chip that not only rigorously trades off its area, energy consumption, and the quality of its signal output, but also significantly outperforms the state-of-the-art in all aspects. In the framework of big-data and high-performance computation, such as in high-end servers application, the RF circuits meant to transmit data from chip-to-chip or chip-to-memory are defined by low power requirements, since the heat generated by the integrated circuits is partially distributed by the chip package. Hence, the overall system power budget is defined by its affordable cooling capacity. For this reason, application specific architectures and innovative techniques are used for low-power implementation. In this work, we have developed a single-ended multi-lane receiver for high speed I/O link in servers application. The receiver operates at 7 Gbps by learning inter-symbol interference and electromagnetic coupling noise in chip-to-chip communication systems. A learning-based approach allows a versatile receiver circuit which not only copes with large channel attenuation but also implements novel crosstalk reduction techniques, to allow single-ended multiple lines transmission, without sacrificing its overall bandwidth for a given area within the interconnect's data-path

Compressive Sensing and Its Applications in Automotive Radar Systems

Die Entwicklung in Richtung zu autonomem Fahren verspricht, künftig einen sicheren Verkehr ohne tödliche Unfälle zu ermöglichen, indem menschliche Fahrer vollständig ersetzt werden. Dadurch entfällt der Faktor des menschlichen Fehlers, der aus Müdigkeit, Unachtsamkeit oder Alkoholeinfluss resultiert. Um jedoch eine breite Akzeptanz für autonome Fahrzeuge zu erreichen und es somit eines Tages vollständig umzusetzen, sind noch eine Vielzahl von Herausforderungen zu lösen. Da in einem autonomen Fahrzeug kein menschlicher Fahrer mehr in Notfällen eingreifen kann, müssen sich autonome Fahrzeuge auf leistungsfähige und robuste Sensorsysteme verlassen können, um in kritischen Situationen auch unter widrigen Bedingungen angemessen reagieren zu können. Daher ist die Entwicklung von Sensorsystemen erforderlich, die für Funktionalitäten jenseits der aktuellen advanced driver assistance systems eingesetzt werden können. Dies resultiert in neuen Anforderungen, die erfüllt werden müssen, um sichere und zuverlässige autonome Fahrzeuge zu realisieren, die weder Fahrzeuginsassen noch Passanten gefährden. Radarsysteme gehören zu den Schlüsselkomponenten unter der Vielzahl der verfügbaren Sensorsysteme, da sie im Gegensatz zu visuellen Sensoren von widrigen Wetter- und Umgebungsbedingungen kaum beeinträchtigt werden. Darüber hinaus liefern Radarsysteme zusätzliche Umgebungsinformationen wie Abstand, Winkel und relative Geschwindigkeit zwischen Sensor und reflektierenden Zielen. Die vorliegende Dissertation deckt im Wesentlichen zwei Hauptaspekte der Forschung und Entwicklung auf dem Gebiet der Radarsysteme im Automobilbereich ab. Ein Aspekt ist die Steigerung der Effizienz und Robustheit der Signalerfassung und -verarbeitung für die Radarperzeption. Der andere Aspekt ist die Beschleunigung der Validierung und Verifizierung von automated cyber-physical systems, die parallel zum Automatisierungsgrad auch eine höhere Komplexität aufweisen. Nach der Analyse zahlreicher möglicher Compressive Sensing Methoden, die im Bereich Fahrzeugradarsysteme angewendet werden können, wird ein rauschmoduliertes gepulstes Radarsystem vorgestellt, das kommerzielle Fahrzeugradarsysteme in seiner Robustheit gegenüber Rauschen übertrifft. Die Nachteile anderer gepulster Radarsysteme hinsichtlich des Signalerfassungsaufwands und der Laufzeit werden durch die Verwendung eines Compressive Sensing-Signalerfassungs- und Rekonstruktionsverfahrens in Kombination mit einer Rauschmodulation deutlich verringert. Mit Compressive Sensing konnte der Aufwand für die Signalerfassung um 70% reduziert werden, während gleichzeitig die Robustheit der Radarwahrnehmung auch für signal-to-noise-ratio-Pegel nahe oder unter Null erreicht wird. Mit einem validierten Radarsensormodell wurde das Rauschradarsystem emuliert und mit einem kommerziellen Fahrzeugradarsystem verglichen. Datengetriebene Wettermodelle wurden entwickelt und während der Simulation angewendet, um die Radarleistung unter widrigen Bedingungen zu bewerten. Während eine Besprühung mit Wasser die Radomdämpfung um 10 dB erhöht und Spritzwasser sogar um 20 dB, ergibt sich die eigentliche Begrenzung aus der Rauschzahl und Empfindlichkeit des Empfängers. Es konnte bewiesen werden, dass das vorgeschlagene Compressive Sensing Rauschradarsystem mit einer zusätzlichen Signaldämpfung von bis zu 60 dB umgehen kann und damit eine hohe Robustheit in ungünstigen Umwelt- und Wetterbedingungen aufweist. Neben der Robustheit wird auch die Interferenz berücksichtigt. Zum einen wird die erhöhte Störfestigkeit des Störradarsystems nachgewiesen. Auf der anderen Seite werden die Auswirkungen auf bestehende Fahrzeugradarsysteme bewertet und Strategien zur Minderung der Auswirkungen vorgestellt. Die Struktur der Arbeit ist folgende. Nach der Einführung der Grundlagen und Methoden für Fahrzeugradarsysteme werden die Theorie und Metriken hinter Compressive Sensing gezeigt. Darüber hinaus werden weitere Aspekte wie Umgebungsbedingungen, unterschiedliche Radararchitekturen und Interferenz erläutert. Der Stand der Technik gibt einen Überblick über Compressive Sensing-Ansätze und Implementierungen mit einem Fokus auf Radar. Darüber hinaus werden Aspekte von Fahrzeug- und Rauschradarsystemen behandelt. Der Hauptteil beginnt mit der Vorstellung verschiedener Ansätze zur Nutzung von Compressive Sensing für Fahrzeugradarsysteme, die in der Lage sind, die Erfassung und Wahrnehmung von Radarsignalen zu verbessern oder zu erweitern. Anschließend wird der Fokus auf ein Rauschradarsystem gelegt, das mit Compressive Sensing eine effiziente Signalerfassung und -rekonstruktion ermöglicht. Es wurde mit verschiedenen Compressive Sensing-Metriken analysiert und in einer Proof-of-Concept-Simulation bewertet. Mit einer Emulation des Rauschradarsystems wurde das Potential der Compressive Sensing Signalerfassung und -verarbeitung in einem realistischeren Szenario demonstriert. Die Entwicklung und Validierung des zugrunde liegenden Sensormodells wird ebenso dokumentiert wie die Entwicklung der datengetriebenen Wettermodelle. Nach der Betrachtung von Interferenz und der Koexistenz des Rauschradars mit kommerziellen Radarsystemen schließt ein letztes Kapitel mit Schlussfolgerungen und einem Ausblick die Arbeit ab.Developments towards autonomous driving promise to lead to safer traffic, where fatal accidents can be avoided after making human drivers obsolete and hence removing the factor of human error. However, to ensure the acceptance of automated driving and make it a reality one day, still a huge amount of challenges need to be solved. With having no human supervisors, automated vehicles have to rely on capable and robust sensor systems to ensure adequate reactions in critical situations, even during adverse conditions. Therefore, the development of sensor systems is required that can be applied for functionalities beyond current advanced driver assistance systems. New requirements need to be met in order to realize safe and reliable automated vehicles that do not harm passersby. Radar systems belong to the key components among the variety of sensor systems. Other than visual sensors, radar is less vulnerable towards adverse weather and environment conditions. In addition, radar provides complementary environment information such as target distance, angular position or relative velocity, too. The thesis ad hand covers basically two main aspects of research and development in the field of automotive radar systems. One aspect is to increase efficiency and robustness in signal acquisition and processing for radar perception. The other aspect is to accelerate validation and verification of automated cyber-physical systems that feature more complexity along with the level of automation. After analyzing a variety of possible Compressive Sensing methods for automotive radar systems, a noise modulated pulsed radar system is suggested in the thesis at hand, which outperforms commercial automotive radar systems in its robustness towards noise. Compared to other pulsed radar systems, their drawbacks regarding signal acquisition effort and computation run time are resolved by using noise modulation for implementing a Compressive Sensing signal acquisition and reconstruction method. Using Compressive Sensing, the effort in signal acquisition was reduced by 70%, while obtaining a radar perception robustness even for signal-to-noise-ratio levels close to or below zero. With a validated radar sensor model the noise radar was emulated and compared to a commercial automotive radar system. Data-driven weather models were developed and applied during simulation to evaluate radar performance in adverse conditions. While water sprinkles increase radome attenuation by 10 dB and splash water even by 20 dB, the actual limitation comes from noise figure and sensitivity of the receiver. The additional signal attenuation that can be handled by the proposed compressive sensing noise radar system proved to be even up to 60 dB, which ensures a high robustness of the receiver during adverse weather and environment conditions. Besides robustness, interference is also considered. On the one hand the increased robustness towards interference of the noise radar system is demonstrated. On the other hand, the impact on existing automotive radar systems is evaluated and strategies to mitigate the impact are presented. The structure of the thesis is the following. After introducing basic principles and methods for automotive radar systems, the theory and metrics of Compressive Sensing is presented. Furthermore some particular aspects are highlighted such as environmental conditions, different radar architectures and interference. The state of the art provides an overview on Compressive Sensing approaches and implementations with focus on radar. In addition, it covers automotive radar and noise radar related aspects. The main part starts with presenting different approaches on making use of Compressive Sensing for automotive radar systems, that are capable of either improving or extending radar signal acquisition and perception. Afterwards the focus is put on a noise radar system that uses Compressive Sensing for an efficient signal acquisition and reconstruction. It was analyzed using different Compressive Sensing metrics and evaluated in a proof-of-concept simulation. With an emulation of the noise radar system the feasibility of the Compressive Sensing signal acquisition and processing was demonstrated in a more realistic scenario. The development and validation of the underlying sensor model is documented as well as the development of the data-driven weather models. After considering interference and co-existence with commercial radar systems, a final chapter with conclusions and an outlook completes the work

Sparse channel estimation based on compressed sensing theory for UWB systems

Català: L'estimació de canal en receptors wireless esdevé un factor determinant a l'hora de incrementar les prestacions dels sistemes sense fils per tal de satisfer les exigències cada vegades més elevades dels consumidors en quant a velocitats de transmissió i qualitat. En aquesta tesi es proposa explotar la "sparsity" que mostren els canals wireless per tal de millorar els clàssics sistemes d'estimació de canal mitjançant les noves teòries de Compressed Sensing. Així doncs, es proposa un nou model freqüencial de senyal on el canal i un nou algoritme de reconstrucció de senyals sparse que redueix la probabilitat de detecció de falsos camins de propagació millorant d'aquesta manera l'estimació de temps d'arribada.Castellano: En los últimos años, la revolución inalámbrica se ha convertido en una realidad. Wi-fi está en todas partes, impactando significativamente en nuestro estilo de vida. Sin embargo, las comunicaciones inalámbricas nunca tendrán las condiciones de propagación igual que los cables debido a las duras condiciones de la propagación inalámbricas. El canal de radio móvil se caracteriza por la recepción múltiple, eso es que la señal recibida no sólo contiene una camino de propagación, sino también un gran número de ondas reflejadas. Estas ondas reflejadas interfieren con la onda directa, lo que provoca una degradación significativa del rendimiento del enlace. Un sistema inalámbrico debe estar diseñado de tal manera que el efecto adverso del desvanecimiento multicamino sea reducido al mínimo. Afortunadamente, el multipath puede ser visto como diversidad de información dependiendo de la cantidad de Channel State Information (CSI) disponible para el sistema. Sin embargo, en la práctica CSI rara vez se dispone a priori y debe ser estimado. Por otro lado, un canal inalámbrico a menudo puede ser modelado como un canal sparse, en la que el retraso de propagación puede ser muy grande, pero el número de caminos de propagación es normalmente muy pequeño. El conocimiento previo de la sparsity del canal se puede utilizar eficazmente para mejorar la estimación de canal utilizando la nueva teoría de Compressed Sensing (CS). CS se origina en la idea de que no es necesario invertir una gran cantidad de energía en la observación de las entradas de una señal sparse porque la mayoría de ellas será cero. Por lo tanto, CS proporciona un marco sólido para la reducción del número de medidas necesarias para resumir señales sparse. La estimación de canal sparse se centra en este trabajo en Ultra-Wideband (UWB) porque la gran resolución temporal que proporcionan las señales UWB se traduce en un número muy grande de componentes multipath que se pueden resolver. Por lo tanto, UWB mitiga significativamente la distorsión de trayectoria múltiple y proporciona la diversidad multicamino. Esta diversidad junto con la resolución temporal de las señales UWB crear un problema de estimación de canal muy interesante. En esta tesis se estudia el uso de CS en la estimación de canal altamente sparse por medio de un nuevo enfoque de estimación basado en el modelo de frecuencial de la señal UWB. También se propone un nuevo algoritmo llamado extended Orthogonal Matching Pursuit (eOMP) basado en los mismos principios que el clásico OMP, con el fin de mejorar algunas de sus característica.English: In recent years, the wireless revolution has become a reality. Wireless is everywhere having significant impact on our lifestyle. However, wireless will never have the same propagation conditions as wires due to the harsh conditions of the wireless propagation. The mobile radio channel is characterized by multipath reception, that is the signal offered to the receiver contains not only a direct line-of-sight radio wave, but also a large number of reflected radio waves. These reflected waves interfere with the direct wave, which causes significant degradation of the performance of the link. A wireless system has to be designed in such way that the adverse effect of multipath fading is minimized. Fortunately, multipath can be seen as a blessing depending on the amount of Channel State Information (CSI) available to the system. However, in practise CSI is seldom available a priori and needs to be estimated. On the other hand, a wireless channel can often be modeled as a sparse channel in which the delay spread could be very large, but the number of significant paths is normally very small. The prior knowledge of the channel sparseness can be effectively use to improve the channel estimation using the novel Compressed Sensing (CS) theory. CS originates from the idea that is not necessary to invest a lot of power into observing the entries of a sparse signal because most of them will be zero. Therefore, CS provides a robust framework for reducing the number of measurement required to summarize sparse signals. The sparse channel estimation here is focused on Ultra-WideBand (UWB) systems because the very fine time resolution of the UWB signal results in a very large number of resolvable multipath components. Consequently, UWB significantly mitigates multipath distortion and provides path diversity. The rich multipath coupled with the fine time resolution of the UWB signals create a challenging sparse channel estimation problem. This Master Thesis examines the use of CS in the estimation of highly sparse channel by means of a new sparse channel estimation approach based on the frequency domain model of the UWB signal. It is also proposed a new greedy algorithm named extended Orthogonal Matching Pursuit (eOMP) based on the same principles than classical Orthogonal Matching Pursuit (OMP) in order to improve some OMP characteristics. Simulation results show that the new eOMP provides lower false path detection probability compared with classical OMP, which also leads to a better TOA estimation without significant degradation of the channel estimation. Simulation results will also show that the new frequency domain sparse channel model outperforms other models presented in the literature