Reduction of Computational Complexity

Abstract

If a naive method is employed to implement the APA, the number of arithmetic operations per sample is approximately given by a polynomial $$\alpha p^{3} + \beta np^{2}$$ , where $$\alpha $$ and $$\beta $$ are positive coefficients, n is the filter length, and p is the projection order. This amount of computation is a heavy burden for real-time applications of the APA. The data used to update the coefficient vector in the APA have a time-shift property, i.e., the data used at time $$k-1$$ and those at time k are largely overlapped. In this chapter, we review two works on reduction of the computational complexity of the APA by exploiting this time-shift property: the fast affine projection (FAP) algorithm and the block exact fast affine projection (BEFAP) algorithm. The key point of the FAP algorithm is recursive computations of the quantities that appear in the update equation for the coefficient vector. The amount of computation for the FAP algorithm per sample is approximately given by a linear function of n and p. By blocking the input data and lower the frequency of updating the coefficient vector, one can reduce the computational complexity per sample. However, the behavior of this algorithm is different from the original APA. In the BEFAP algorithm, a correction term is introduced so that the output becomes exactly the same as the original APA. In the algorithm, the product of a Hankel matrix and a vector appears, which can be computed efficiently either by the convolution method using the FFT, or by a divide and conquer method. We see that the BEFAP algorithm further reduces the computational complexity compared with the FAP algorithm.