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Formal Analysis of Arithmetic Circuits using Computer Algebra - Verification, Abstraction and Reverse Engineering
Despite a considerable progress in verification and abstraction of random and control logic, advances in formal verification of arithmetic designs have been lagging. This can be attributed mostly to the difficulty in an efficient modeling of arithmetic circuits and datapaths without resorting to computationally expensive Boolean methods, such as Binary Decision Diagrams (BDDs) and Boolean Satisfiability (SAT), that require “bit blasting”, i.e., flattening the design to a bit-level netlist. Approaches that rely on computer algebra and Satisfiability Modulo Theories (SMT) methods are either too abstract to handle the bit-level nature of arithmetic designs or require solving computationally expensive decision or satisfiability problems. The work proposed in this thesis aims at overcoming the limitations of analyzing arithmetic circuits, specifically at the post-synthesized phase. It addresses the verification, abstraction and reverse engineering problems of arithmetic circuits at an algebraic level, treating an arithmetic circuit and its specification as a properly constructed algebraic system. The proposed technique solves these problems by function extraction, i.e., by deriving arithmetic function computed by the circuit from its low-level circuit implementation using computer algebraic rewriting technique. The proposed techniques work on large integer arithmetic circuits and finite field arithmetic circuits, up to 512-bit wide containing millions of logic gates
Division-based versus general decomposition-based multiple-level logic synthesis
During the last decade, many different approaches have been proposed to solve the multiple-level synthesis problem with
different minimum functionally complete systems of primitive logic blocks. The most popular of them is the division-based
approach. However, modem microelectronic technology provides a large variety of building blocks which considerably
differ from those typically considered. The traditional methods are therefore not suitable for synthesis with many modem
building blocks. Furthermore, they often fail to find global optima for complex designs and leave unconsidered some
important design aspects. Some of their weaknesses can be eliminated without leaving the paradigm they are based on, other
ones are more fundamental. A paradigm which enables efficient exploitation of the opportunities created by the
microelectronic technology is the general decomposition paradigm. The aim of this paper is to analyze and compare the
general decomposition approach and the division-based approach. The most important advantages of the general
decomposition approach are its generality (any network of any building blocks can be considered) and totality (all important
design aspects can be considered) as well as handling the incompletely specified functions in a natural way. In many cases,
the general decomposition approach gives much better results than the traditional approaches
FPGA Implementation of Data Flow Graphs for Digital Signal Processing Applications
A rapid growth in digital signal processing applications has increased the requirement for high-speed digital systems. Multiprocessor systems are the best choice for these applications. A prior sequence of operations should be applied to the operations that described the nature of these applications before hardware implementation is produced. These operations should be scheduled and hardware allocated. This paper proposes a new scheduling technique for digital signal processing (DSP) applications has been represented by data flow graphs (DFGs). In addition, hardware allocation is implemented in the form of embedded system. A proposed scheduling technique also achieves the optimal scheduling of a DFG at design time. The optimality criteria considered in this algorithm are the maximum throughput within the available hardware resources. The maximum throughput is achieved by arranging the DFG nodes according to their inter-related data dependencies. Then, two nodes can be clustered into one compound task to reduce the overall execution time by minimizing the number of tasks to be executed that minimizing the number of cycles to execute them. Then each task is presented in form of instruction to be executed in the hardware system. A hardware system is composed of one or multiple homogenous pipelined processing elements and it is designed to meet the maximum-rate schedule. Two implementations are proposed of the system architecture according to the number of the processing elements, namely: the serial system and the parallel system. The serial system comprises one processing element where all tasks are processed sequentially, whilst the parallel system has four processing elements to execute tasks concurrently. These systems consist mainly of seven units: central shared memory, state table, multiway function unit buffer, execution array, processing element/s, instruction buffer and the address generation unit. The hardware components were built on an FPGA chip using Verilog HDL. In synthesis results, the parallel system has better system performance by 25.5% than the serial system. While the serial system requires smaller area size, which described by the number of slice registers and the number of the slice lookup tables (LUTs) than the parallel one. The relationship between the number of instructions that are executed in both systems, and the system area and the system performance that presented by system frequency, are studied. By increasing memories size in both systems, the system performance isn’t affected as in a serial system, and it is slightly decreased as the parallel system by 1.5% to 4.5%. In terms of the systems area, both serial system area and parallel system area are increased and in some cases are doubled. The proposed scheduling technique is shown to outperform the retaining technique, which we have chosen to compare with. The serial system has better performance by 19.3% higher system frequency than a retiming technique. And the parallel system also outperforms the retaining technique by 51.2% higher system frequency in synthesis results
Transient error mitigation by means of approximate logic circuits
Mención Internacional en el título de doctorThe technological advances in the manufacturing of electronic circuits have allowed to
greatly improve their performance, but they have also increased the sensitivity of electronic
devices to radiation-induced errors. Among them, the most common effects are
the SEEs, i.e., electrical perturbations provoked by the strike of high-energy particles,
which may modify the internal state of a memory element (SEU) or generate erroneous
transient pulses (SET), among other effects. These events pose a threat for the reliability
of electronic circuits, and therefore fault-tolerance techniques must be applied to
deal with them.
The most common fault-tolerance techniques are based in full replication (DWC or
TMR). These techniques are able to cover a wide range of failure mechanisms present
in electronic circuits. However, they suffer from high overheads in terms of area and
power consumption. For this reason, lighter alternatives are often sought at the expense
of slightly reducing reliability for the least critical circuit sections. In this context a new
paradigm of electronic design is emerging, known as approximate computing, which
is based on improving the circuit performance in change of slight modifications of the
intended functionality. This is an interesting approach for the design of lightweight
fault-tolerant solutions, which has not been yet studied in depth.
The main goal of this thesis consists in developing new lightweight fault-tolerant
techniques with partial replication, by means of approximate logic circuits. These
circuits can be designed with great flexibility. This way, the level of protection as
well as the overheads can be adjusted at will depending on the necessities of each
application. However, finding optimal approximate circuits for a given application is
still a challenge.
In this thesis a method for approximate circuit generation is proposed, denoted
as fault approximation, which consists in assigning constant logic values to specific
circuit lines. On the other hand, several criteria are developed to generate the most
suitable approximate circuits for each application, by using this fault approximation
mechanism. These criteria are based on the idea of approximating the least testable
sections of circuits, which allows reducing overheads while minimising the loss of reliability.
Therefore, in this thesis the selection of approximations is linked to testability
measures.
The first criterion for fault selection developed in this thesis uses static testability
measures. The approximations are generated from the results of a fault simulation of
the target circuit, and from a user-specified testability threshold. The amount of approximated
faults depends on the chosen threshold, which allows to generate approximate circuits with different performances. Although this approach was initially intended for
combinational circuits, an extension to sequential circuits has been performed as well,
by considering the flip-flops as both inputs and outputs of the combinational part of
the circuit. The experimental results show that this technique achieves a wide scalability,
and an acceptable trade-off between reliability versus overheads. In addition, its
computational complexity is very low.
However, the selection criterion based in static testability measures has some drawbacks.
Adjusting the performance of the generated approximate circuits by means of
the approximation threshold is not intuitive, and the static testability measures do not
take into account the changes as long as faults are approximated. Therefore, an alternative
criterion is proposed, which is based on dynamic testability measures. With this
criterion, the testability of each fault is computed by means of an implication-based
probability analysis. The probabilities are updated with each new approximated fault,
in such a way that on each iteration the most beneficial approximation is chosen, that
is, the fault with the lowest probability. In addition, the computed probabilities allow
to estimate the level of protection against faults that the generated approximate circuits
provide. Therefore, it is possible to generate circuits which stick to a target error rate.
By modifying this target, circuits with different performances can be obtained. The
experimental results show that this new approach is able to stick to the target error rate
with reasonably good precision. In addition, the approximate circuits generated with
this technique show better performance than with the approach based in static testability
measures. In addition, the fault implications have been reused too in order to
implement a new type of logic transformation, which consists in substituting functionally
similar nodes.
Once the fault selection criteria have been developed, they are applied to different
scenarios. First, an extension of the proposed techniques to FPGAs is performed,
taking into account the particularities of this kind of circuits. This approach has been
validated by means of radiation experiments, which show that a partial replication with
approximate circuits can be even more robust than a full replication approach, because
a smaller area reduces the probability of SEE occurrence. Besides, the proposed
techniques have been applied to a real application circuit as well, in particular to the
microprocessor ARM Cortex M0. A set of software benchmarks is used to generate
the required testability measures. Finally, a comparative study of the proposed approaches
with approximate circuit generation by means of evolutive techniques have
been performed. These approaches make use of a high computational capacity to generate
multiple circuits by trial-and-error, thus reducing the possibility of falling into
local minima. The experimental results demonstrate that the circuits generated with
evolutive approaches are slightly better in performance than the circuits generated with
the techniques here proposed, although with a much higher computational effort.
In summary, several original fault mitigation techniques with approximate logic
circuits are proposed. These approaches are demonstrated in various scenarios, showing
that the scalability and adaptability to the requirements of each application are their
main virtuesLos avances tecnológicos en la fabricación de circuitos electrónicos han permitido mejorar
en gran medida sus prestaciones, pero también han incrementado la sensibilidad
de los mismos a los errores provocados por la radiación. Entre ellos, los más comunes
son los SEEs, perturbaciones eléctricas causadas por el impacto de partículas de alta
energía, que entre otros efectos pueden modificar el estado de los elementos de memoria
(SEU) o generar pulsos transitorios de valor erróneo (SET). Estos eventos suponen
un riesgo para la fiabilidad de los circuitos electrónicos, por lo que deben ser tratados
mediante técnicas de tolerancia a fallos.
Las técnicas de tolerancia a fallos más comunes se basan en la replicación completa
del circuito (DWC o TMR). Estas técnicas son capaces de cubrir una amplia variedad
de modos de fallo presentes en los circuitos electrónicos. Sin embargo, presentan un
elevado sobrecoste en área y consumo. Por ello, a menudo se buscan alternativas más
ligeras, aunque no tan efectivas, basadas en una replicación parcial. En este contexto
surge una nueva filosofía de diseño electrónico, conocida como computación aproximada,
basada en mejorar las prestaciones de un diseño a cambio de ligeras modificaciones
de la funcionalidad prevista. Es un enfoque atractivo y poco explorado para el diseño
de soluciones ligeras de tolerancia a fallos.
El objetivo de esta tesis consiste en desarrollar nuevas técnicas ligeras de tolerancia
a fallos por replicación parcial, mediante el uso de circuitos lógicos aproximados. Estos
circuitos se pueden diseñar con una gran flexibilidad. De este forma, tanto el nivel de
protección como el sobrecoste se pueden regular libremente en función de los requisitos
de cada aplicación. Sin embargo, encontrar los circuitos aproximados óptimos para
cada aplicación es actualmente un reto.
En la presente tesis se propone un método para generar circuitos aproximados, denominado
aproximación de fallos, consistente en asignar constantes lógicas a ciertas
líneas del circuito. Por otro lado, se desarrollan varios criterios de selección para, mediante
este mecanismo, generar los circuitos aproximados más adecuados para cada
aplicación. Estos criterios se basan en la idea de aproximar las secciones menos testables
del circuito, lo que permite reducir los sobrecostes minimizando la perdida de
fiabilidad. Por tanto, en esta tesis la selección de aproximaciones se realiza a partir de
medidas de testabilidad.
El primer criterio de selección de fallos desarrollado en la presente tesis hace uso de
medidas de testabilidad estáticas. Las aproximaciones se generan a partir de los resultados
de una simulación de fallos del circuito objetivo, y de un umbral de testabilidad
especificado por el usuario. La cantidad de fallos aproximados depende del umbral escogido, lo que permite generar circuitos aproximados con diferentes prestaciones.
Aunque inicialmente este método ha sido concebido para circuitos combinacionales,
también se ha realizado una extensión a circuitos secuenciales, considerando los biestables
como entradas y salidas de la parte combinacional del circuito. Los resultados
experimentales demuestran que esta técnica consigue una buena escalabilidad, y unas
prestaciones de coste frente a fiabilidad aceptables. Además, tiene un coste computacional
muy bajo.
Sin embargo, el criterio de selección basado en medidas estáticas presenta algunos
inconvenientes. No resulta intuitivo ajustar las prestaciones de los circuitos aproximados
a partir de un umbral de testabilidad, y las medidas estáticas no tienen en cuenta los
cambios producidos a medida que se van aproximando fallos. Por ello, se propone un
criterio alternativo de selección de fallos, basado en medidas de testabilidad dinámicas.
Con este criterio, la testabilidad de cada fallo se calcula mediante un análisis de probabilidades
basado en implicaciones. Las probabilidades se actualizan con cada nuevo
fallo aproximado, de forma que en cada iteración se elige la aproximación más favorable,
es decir, el fallo con menor probabilidad. Además, las probabilidades calculadas
permiten estimar la protección frente a fallos que ofrecen los circuitos aproximados
generados, por lo que es posible generar circuitos que se ajusten a una tasa de fallos
objetivo. Modificando esta tasa se obtienen circuitos aproximados con diferentes prestaciones.
Los resultados experimentales muestran que este método es capaz de ajustarse
razonablemente bien a la tasa de fallos objetivo. Además, los circuitos generados
con esta técnica muestran mejores prestaciones que con el método basado en medidas
estáticas. También se han aprovechado las implicaciones de fallos para implementar
un nuevo tipo de transformación lógica, consistente en sustituir nodos funcionalmente
similares.
Una vez desarrollados los criterios de selección de fallos, se aplican a distintos
campos. En primer lugar, se hace una extensión de las técnicas propuestas para FPGAs,
teniendo en cuenta las particularidades de este tipo de circuitos. Esta técnica se ha validado
mediante experimentos de radiación, los cuales demuestran que una replicación
parcial con circuitos aproximados puede ser incluso más robusta que una replicación
completa, ya que un área más pequeña reduce la probabilidad de SEEs. Por otro lado,
también se han aplicado las técnicas propuestas en esta tesis a un circuito de aplicación
real, el microprocesador ARM Cortex M0, utilizando un conjunto de benchmarks
software para generar las medidas de testabilidad necesarias. Por ´último, se realiza un
estudio comparativo de las técnicas desarrolladas con la generación de circuitos aproximados
mediante técnicas evolutivas. Estas técnicas hacen uso de una gran capacidad
de cálculo para generar múltiples circuitos mediante ensayo y error, reduciendo la posibilidad
de caer en algún mínimo local. Los resultados confirman que, en efecto, los
circuitos generados mediante técnicas evolutivas son ligeramente mejores en prestaciones
que con las técnicas aquí propuestas, pero con un coste computacional mucho
mayor.
En definitiva, se proponen varias técnicas originales de mitigación de fallos mediante
circuitos aproximados. Se demuestra que estas técnicas tienen diversas aplicaciones,
haciendo de la flexibilidad y adaptabilidad a los requisitos de cada aplicación
sus principales virtudes.Programa Oficial de Doctorado en Ingeniería Eléctrica, Electrónica y AutomáticaPresidente: Raoul Velazco.- Secretario: Almudena Lindoso Muñoz.- Vocal: Jaume Segura Fuste
Analysis of Hardware Descriptions
The design process for integrated circuits requires a lot of analysis of circuit descriptions. An important class of analyses determines how easy it will be to determine if a physical component suffers from any manufacturing errors. As circuit complexities grow rapidly, the problem of testing circuits also becomes increasingly difficult. This thesis explores the potential for analysing a recent high level hardware description language called Ruby. In particular, we are interested in performing testability analyses of Ruby circuit descriptions. Ruby is ammenable to algebraic manipulation, so we have sought transformations that improve testability while preserving behaviour. The analysis of Ruby descriptions is performed by adapting a technique called abstract interpretation. This has been used successfully to analyse functional programs. This technique is most applicable where the analysis to be captured operates over structures isomorphic to the structure of the circuit. Many digital systems analysis tools require the circuit description to be given in some special form. This can lead to inconsistency between representations, and involves additional work converting between representations. We propose using the original description medium, in this case Ruby, for performing analyses. A related technique, called non-standard interpretation, is shown to be very useful for capturing many circuit analyses. An implementation of a system that performs non-standard interpretation forms the central part of the work. This allows Ruby descriptions to be analysed using alternative interpretations such test pattern generation and circuit layout interpretations. This system follows a similar approach to Boute's system semantics work and O'Donnell's work on Hydra. However, we have allowed a larger class of interpretations to be captured and offer a richer description language. The implementation presented here is constructed to allow a large degree of code sharing between different analyses. Several analyses have been implemented including simulation, test pattern generation and circuit layout. Non-standard interpretation provides a good framework for implementing these analyses. A general model for making non-standard interpretations is presented. Combining forms that combine two interpretations to produce a new interpretation are also introduced. This allows complex circuit analyses to be decomposed in a modular manner into smaller circuit analyses which can be built independently
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