Chebyshev approximation technique: analysis and applications

Abstract

Approximation techniques have always been of great benefit to engineering and other applications which involve real-life data which can sometimes be of high complexity to solve exactly. In many situations, exact solutions to complex problems may be challenging or impossible to obtain, making approximation techniques necessary for making informed decisions. In this study, Chebyshev approximation technique is thoroughly investigated along with its convergence rate and accuracy. An FPGA implementation of this approximation technique is discussed and analyzed. This implementation will be compared to other implementations of approximation techniques such as Taylor series with parameters such as accuracy, speed, and design size taken into consideration. Applications of this FPGA implementation were also discussed and shown such as approximating the sigmoid function for machine learning in an efficient manner. In comparison with other methods, Chebyshev approximation is preferred for its fast convergence rate, global approximation capability, and robustness in approximating functions with rapid variations. Chebyshev approximation is also known to be well-suited for implementation in hardware due to its simplicity in computation and high accuracy. This study proved the adequacy of the Chebyshev approximation and its accelerated FPGA implementation for various applications including machine learning, filter design, DSP, and many other practical applications in engineering and science.