Frequency-domain adaptive filtering in hypercomplex systems

In recent years, linear and nonlinear signal processing applications required the development of new multidimensional algorithms. Higher-dimensional algorithms include quaternion-valued filters. One of the drawbacks filter designers have to cope with is the increasing computational cost due to multidimensional processing. A strategy to reduce the computational complexity of long adaptive filters is to implement block algorithms and update the filter coefficients periodically. More efficient techniques embed frequency-domain processing in block algorithms with the use of the Fast Fourier Transform (FFT). Transform-domain adaptive filters in the quaternion field require quaternion-valued transforms. In this paper we also suggest a simple method to obtain a quaternionic DFT/FFT from a complex DFT/FFT. As an example, we propose the Overlap-Save Quaternion Frequency Domain algorithm.