In today¡¦s modern society, our latest up-to-date technology contains various types of multimedia applications. These applications don¡¦t necessarily have to be executed with the most precise accuracy. In short, they are fault tolerant. As a consequence, this thesis proposes a multi-precision iterative floating-point special function unit, which can be executed under different modes to meet the error requirements of each specific application, and also achieve power reduction during the process. In order to minimize the area of our design, we have developed two iterative architectures to implement the multi-precision floating point special function unit. The first proposed architecture can perform three kinds of operations, which include a reciprocal operation, a reciprocal square root operation, and last but not least, a logarithm operation. After deciding which function is to be performed, the user can choose four precision modes to execute the special function unit. According to each mode from lowest precision to highest, we name them the first mode, the second mode, the third mode, and the fourth mode. During implementation, a C model has also been designed to evaluate the maximum error of each mode by making comparisons with the most accurate software result, which is the 23 bit result. When the reciprocal function is chosen, and the user defines that application to be performed in full precision, the multi-precision special function operator needs to be executed twice, and it has the error rate of approximately 0.0001%. When less precision is required, we can choose from two intermediate modes, one offers 15 bit accuracy, and the other can guarantee a 12 bit precision. The former precision mode also required the hardware to be executed twice, but the latter only once. The 15 bit accuracy mode has an error rate around 0.01¢H, and the 12 bit mode has the error rate roughly around 0.05¢H. In addition, when visual effects or even audio effects are not our greatest concern, we provide a least accurate mode for the users to pick to execute the special function operator. This mode can maintain 8 bit accuracy, and has the error rate of approximately 0.8%. Other operations including the reciprocal square root, and the logarithm also have four precision modes to choose from. The reciprcocal square root operation can guarantee the same accuracy in each mode as the reciprocal operation, and their error rates are 0.004%, 0.01%, 0.06%, and 0.5% from the highest precision mode to the lowest one. The precisions the logarithm operation can guarantee from highest accuracy to the lowest one are 23, 16, 12, and 8 bits, respectively, and have error rates including 0.00003%, 0.002%, 0.06%, and 0.3%. These different precision choices are built in the proposed structure mainly to reduce the power consumption. The main concept is to pick a low precision mode in order shut down some components in our design. In addition to switching modes, we¡¦ve also added tri-state buffers in certain components as another means to decrease power. Through experimental results we¡¦ve discovered that the proposed architecture¡¦s affect on power reduction was not as we¡¦ve expected. Due to the integration of the Newton Raphson Method and the Piecewise Polynomial Approximation Method, our architecture¡¦s delay and area have largely increased, and causing a bad influence on saving power. As a consequence, we¡¥ve developed a second architecture to meet our demands. This architecture is mainly based on the Piecewise Polynomial Approximation Method. From this method, we¡¦ve implemented an iterative design which also supports three kinds of operations, the same as the first architecture. It also provides three precision modes for the user to choose. The lowest precision mode provides 8 bit accuracy. The second mode provides 14 bit accuracy, and the third mode, which is the most precise mode, can provide 22 bit accuracy. According to our C model, we can specify our maximum error rate in each function while executing under different modes. When the reciprocal function is executed, the largest error rate in from the lowest mode to the highest mode is 0.19%, 0.00006% and 0.000015% , and the error rate for reciprocal square root from lowest precision mode to the highest is 0.09%, 0.000022% and 0.000014%, and the error rate for the logarithm function is 0.33%, 0.000043% and 0.000015%, from the lowest to the highest. From experimental results we can discover that the newly proposed architecture is better in comparison with the traditional Piecewise Polynomial Approximation architecture. The proposed architecture has a smaller area, and a faster delay, and most important of all, it reduces power and energy affectively.

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This thesis describes theory and algorithms for use with Radial Basis Functions (RBFs), emphasising techniques motivated by three particular application areas. In Part I, we apply RBFs to the problem of interpolating to integral data. While the potential of using RBFs for this purpose has been established in an abstract theoretical context, their use has been lacking an easy to check sufficient condition for finding appropriate parent basic functions, and explicit methods for deriving integral basic functions from them. We present both these components here, as well as explicit formulations for line segments in two dimensions and balls in three and five dimensions. We also apply these results to real-world track data. In Part II, we apply Hermite and pointwise RBFs to the problem of surface reconstruction. RBFs are used for this purpose by representing the surface implicitly as the zero level set of a function in 3D space. We develop a multilevel piecewise technique based on scattered spherical subdomains, which requires the creation of algorithms for constructing sphere coverings with desirable properties and for blending smoothly between levels. The surface reconstruction method we develop scales very well to large datasets and is very amenable to parallelisation, while retaining global-approximation-like features such as hole filling. Our serial implementation can build an implicit surface representation which interpolates at over 42 million points in around 45 minutes. In Part III, we apply RBFs to the problem of animation control in the area of motion synthesis---controlling an animated character whose motion is entirely the result of simulated physics. While the simulation is quite well understood, controlling the character by means of forces produced by virtual actuators or muscles remains a very difficult challenge. Here, we investigate the possibility of speeding up the optimisation process underlying most animation control methods by approximating the physics simulator with RBFs.

[pt] Erdal Arıkan introduziu os códigos polares em 2009. Trata-se de uma nova classe de códigos de correção de erros capaz de atingir o limite de Shannon. Usando decodificação de cancelamento sucessivo em lista, concatenada por verificação de redundância cíclica e a construções rápida de código, os códigos polares tornaram-se um código de correção de erros atraente e de alto desempenho para uso prático. Recentemente, códigos polares foram adotados para o padrão de geração 5th para sistemas celulares, mais especificamente para as informações de controle dos canais reverso e direto para os serviços de comunicação eMBB. No entanto, os códigos polares são limitados a comprimentos de bloco a potências de dois, devido a um produto Kronecker recursivo do kernel polarizador 2x2. Para aplicações práticas, é necessário fornecer técnicas de construção de código polar de comprimento flexível. Outro aspecto a ser analisado é o obtenção de uma técnica de construção de códigos polares de baixa complexidade e que tenha um ótimo desempenho em canal de ruído aditivo gaussiano branco, principalmente para blocos longos, inspirada na otimização da construção da aproximação gaussiana. Outro aspecto relevante é o poder de decodificação paralela do decodificador de propagação de crenças. Esta é uma alternativa para atender aos novos critérios de velocidade e latência previstos para o padrão de próxima geração para sistemas celulares. No entanto, ele precisa de melhorias de desempenho para tornar-se operacionalmente viável, tanto para 5G quanto para as gerações futuras. Nesta tese, três aspectos dos códigos polares são abordados: a construção de códigos com comprimentos arbitrários que visam maximizar a flexibilidade e eficiência dos códigos polares, o aprimoramento do método de construção por métodos gaussianos aproximação e a decodificação de códigos usando um algoritmo adaptativo de propagação de crenças reponderadas, bem como analisar quaisquer compromissos que afetem o desempenho da correção de erros. / [en] Erdal Arikan introduced the polar codes in 2009. This is a new class of error correction codes capable of reaching the Shannon limit. Using cyclic redundancy check concatenated list successive cancellation decoding and fast code constructs, polar codes have become an attractive, high-performance error correction code for practical use. Recently, polar codes have been adopted for the 5th generation standard for cellular systems, more specifically for the uplink and downlink control information for the extended Mobile Broadband (eMBB) communication services. However, polar codes are limited to block lengths to powers of two, due to a recursive Kronecker product of the 2x2 polarizing kernel. For practical applications, it is necessary to provide flexible length polar code construction techniques. Another aspect analyzed is the development of a technique of construction of polar codes of low complexity and that has an optimum performance on additive white Gaussian noise channels, mainly for long blocks, inspired by the optimization of the Gaussian approximation construction. Another relevant aspect is the parallel decoding power of the belief propagation decoder. This is an alternative to achieve the new speed and latency criteria foreseen for the next generation standard for cellular systems. However, it needs performance improvements to become operationally viable, both for 5G and for future generations. In this thesis, three aspects of polar codes are addressed: the construction of codes with arbitrary lengths that are intended for maximizing the flexibility and efficiency of polar codes, the improvement of the construction method by Gaussian approximation and the decoding of codes using an adaptive reweighted belief propagation algorithm, as well as the analysis of trade-offs affecting error correction performance.