A new fast filtering algorithm based on algebraic composition

This paper proposes a new type of time-domain direct-form fast filtering algorithm, which composes a sum of N/2 product-of-sum terms. The sum consists of the desired current output point, as well as the half partial results of the preceding and succeeding output points. After further algebraic manipulation, the required complexity per output point is 3N/4 multiplications and 3N/4+1/2 additions. This is about 25% reduction over the direct computation. The design technique can be extended to linear-phase filtering. In this case, the new algorithm only needs 3N/8+2 multiplications and N+10 additions, which is about 25% improvement over N/2 of the direct-form computation in multiplication complexity. The new algorithm can be also iteratively applied to a convolution operation for more complexity reduction. Since the new algorithm is also a direct-form type, its realization is regular and very suitable for ASIC design.