On FIR Filter Approximation of Fractional-Order Differentiators and Integrators

This paper considers finite-length impulse response (FIR) filter approximation of differentiators and integrators, collectively called differintegrators. The paper introduces and compares three different FIR filter structures for this purpose, all of which are optimized in the minimax sense using iterative reweighted l₁-norm minimization. One of the structures is the direct-form structure, but featuring equal-valued taps and zero-valued taps, the latter corresponding to sparse filters. The other two structures comprise two subfilters in parallel and cascade, respectively. In their basic forms, nothing is gained by realizing the filters in parallel or in cascade, instead of directly. However, as the paper will show, these forms enable substantial further complexity reductions, because they comprise symmetric and antisymmetric subfilters of different orders, and also features additional equal-valued and zero-valued taps. The cascade structure employs a structurally sparse filter. The additional sparsity, as well as tap equalities, are for all three structures found automatically in the design via the l₁-norm minimization. Design examples included reveal feasible multiplication complexity savings of more than 50% in comparison with regular (unconstrained) direct-form structures. In addition, an example shows that the proposed designs can even have lower complexity than existing infinite-length impulse response filter designs.