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

Abstract

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.

Improper Signaling Versus Time-Sharing in the Two-User Gaussian Interference Channel With TIN

“…The achievable rate with possibly improper signals and symbol extensions can be expressed in an equivalent 2T × 2T real-valued MIMO system [32] with transmit covariance matrices Q 1 and Q 2 . We have [52]…”

On the Superiority of Improper Gaussian Signaling in Wireless Interference MIMO Scenarios

“…, and the augmented complex representation, which works with the complex vector and its complex conjugate: [40]. Both representations are mathematically equivalent in the sense that it is equivalent to work with one or the other, since they are related by the following bijective transformation:…”

Reconsidering Linear Transmit Signal Processing in 1-Bit Quantized Multi-User MISO Systems

“…To obtain a SL solution we use the fact that the SL structure in Def. 2 is maintained under multiplication, addition, transposition and inversion as shown in [42] and [55]. Therefore, if we initialize our algorithm with a SL matrix, i.e., P (0) =P (0) with the structure from Def.…”

“…Moreover, we introduce Widely-Linear (WL) notation (see e.g., [40]- [42]) to accommodate our analysis of whether proper signaling is optimal in quantized MU-MISO systems. To this end, we introduce the following definitions:…”

“…First, we initialize our algorithm with a random full rank matrixP (0) and calculate the initial scaling factor β (0) using the function g * (P). This function finds the optimal scaling factor for a given precoder matrix and will be introduced in Appendix A (see (42)). We use an initial constant step-size of γ = 10, but we allow for backtracking in Step 9, where we Algorithm 1 Gradient Projection Algorithm to Solve for the Higher-Rank, WL TxWFQ-Π 1: Initialization: 2:P (0) , β (0) ← g * (P (0) ), γ = 10 and n = 0 3: repeat 4: if ε (n+1) ≤ ε (n) then…”