An Optimized Implementation of GL Fractional-Order

An alternative implementation of of fractional derivative/integral defined by Grunwald-Letnikov definition (GL) is introduced. By using the average of GL binomial coefficients along a window size instead of the real values of coefficients. Such an implementation can help in implementing large window-sized approximated GL with fewer resources which will help obtaining less error on a small amount of resources. For example, the proposed optimization can implement a GL integral of fractional order (alpha = -0.5) to sin(t) limited by a window size L of 1024 by the same resources that are used to implement a window size of 10 with an absolute error of 0.138 instead of 0.512. Another way to optimize resources usage of the fixed window method is by implementing different fractional-order operator for systems that do not require certain fractional-order such as PIDs. As different fractional-order requires a different amount of resources for the same output error.