Equivalence of Frequency-Domain Turbo Equalization Schemes for Single-Carrier Multiple-Input–Multiple-Output Systems

This paper investigates two typical frequency-domain turbo equalization (FDTE) schemes, i.e., frequency-domain equalization (FDE) with parallel interference cancelation (FDE-PIC) and FDE with frequency-domain decision feedback (FDE-FDDF) in single-carrier (SC) multiple-input-multiple-output (MIMO) systems. These two FDTE schemes are proven to be equivalent under the condition that we derived. We discover that the covariance matrix of the low-complexity FDE-PIC scheme and the correlation matrix of the FDE-FDDF scheme satisfy a unique relationship for constant-modulus modulations, and the original FDE-FDDF scheme suffers numerical instability for non-constant-modulus modulations because the estimation of the correlation matrices leads to the inversion of ill-conditioned matrices. By proposing a new estimation method of the correlation matrices, we guarantee that the non-constant-modulus FDE-FDDF scheme is numerically stable and that the relationship between these two FDTE schemes holds for both constant-modulus and non-constant-modulus modulations. Furthermore, we prove that the FDE-PIC and FDE-FDDF schemes are equivalent under this relationship. The equivalence of these two FDTE schemes is also verified through numerical simulations in terms of bit error rate (BER) at each turbo iteration and the estimation of the correlation matrices.