Block-Skew-Circulant Matrices in Complex-Valued Signal Processing

Two main lines of approach can be identified in the recent literature on improper signals and widely linear operations. The augmented complex formulation based on the signal and its complex conjugate is considered as more insightful since it leads to convenient mathematical formulations for many considered problems. Moreover, it allows an easy distinction between proper and improper signals as well as between linear and widely linear operations. On the other hand, the composite real representation using the real and imaginary parts of the signal is closer to the actual implementation, and it allows to readily reuse results that have originally been derived for real-valued signals or proper complex signals. In this work, we aim at getting the best of both worlds by introducing mathematical tools that make the composite real representation more powerful and elegant. The proposed approach relies on a decomposition of real matrices into a block-skew-circulant and a block-Hankel-skew-circulant component. By means of various application examples from the field of signal processing for communications, we demonstrate the usefulness of the proposed framework.