Frequency-domain adaptive filtering: from real to hypercomplex signal processing

Frequency-domain adaptive filters (FDAFs) have been widely used over the years, but they are still matter of research due to their powerful capabilities that differentiate them from the whole family of time-domain adaptive filters. This paper aims at providing an overview on FDAFs through a unifying framework that can be used for the derivation of the most popular algorithms of the FDAF family and enables the processing of a wide variety of signals, from real-valued ones to complex- and hypercomplex-valued signals. In particular, we focus on a recent class of FDAFs in the quaternion domain and we show how to derive it from the described framework. Moreover, we evaluate the application of the derived quaternion FDAF to the processing of 3D audio signals. Experimental results show the effectiveness of the proposed adaptive filter in estimating the inverse of a multidimensional acoustic impulse response.