Arquitetura para invasão de matrizes usando circuito divisor eficiente baseado no algoritmo Goldschmidt

Abstract

The matrix inversion calculation is present in several applications in the area of Signal Processing. Among these applications, the adaptive filtering, based on the algorithm of Affine Projections, includes the calculation of matrix inversion, which adds a high computational complexity. There are several algorithms for calculating matrix inversion. The complexity of the algorithm is associated with the size of the matrix, which varies according to the target application. This dissertation proposes the implementation in dedicated hardware of the analytical algorithm of matrix inversion. This algorithm is most appropriate for the implementation of a 2x2 size matrix, which is the appropriate size for an implementation of the algorithm of Affine Projections for several practical applications. In the matrix inversion block, the divisor circuit is that adds the highest computational complexity. Among the division algorithms from the literature, algorithms based on functional iterations are considered the fastest, because they are able to take advantage of high speed multipliers to converge in a quadratic form to a result. Among the algorithms based on functional iterations, Newton-Raphson and Goldschmidt algorithms are the most used algorithms. However, the Goldschmidt algorithm has been more used in applications that demand high processing speed, because unlike the Newton-Raphson algorithm, where the multiplications are dependent on each other, in the Goldschmidt algorithm the multiplications are performed in parallel. In this work, it is proposed the hardware implementation of an efficient divisor circuit based on the Goldschmidt algorithm. The divider circuit uses a radix-4 multiplier from the literature, which is more efficient in terms of power dissipation, when compared to the divider circuit using the multiplier from the synthesis tool. The proposed divider circuit increases the range of operating values by using the Q7.8 standard, which allows values between -127.99609375 and +127.99609375, rather than the original Goldschmidt divider, which supports a narrow range of values between 1 and 2. The main results show that the use of the proposed efficient Goldschmidt divider circuit makes the matrix inverter circuit with a lower power dissipation, which becomes an attractive for a future implementation of the complete affine projections algorithm in dedicated hardware.