Generalized approximate message passing equalization for multi-carrier faster-than-nyquist signaling

Multi-carrier faster-than-Nyquist (MFTN) signaling constitutes a promising spectrally efficient non-orthogonal physical layer waveform. In this correspondence, we propose a pair of appealingly low-complexity generalized approximated message passing (GAMP)-based frequency-domain (FD) equalization (FDE) algorithms for MFTN systems operating in multipath channels. To mitigate the ill-conditioning of the resultant equivalent channel matrix, we construct block circulant interference matrices by inserting a few cyclic postfixes, followed by truncating the duration of the inherent two-dimensional interferences. Based on the decomposition of the block circulant matrices, we develop a novel FD received signal model using the two-dimensional fast Fourier transform for mitigating the colored noise imposed by the non-orthogonal matched filter. Moreover, we derive a GAMP-based FDE algorithm and its refined version, where the latter relies on approximations for circumventing the emergence of the ill-conditioned matrices. Our simulation results demonstrate that, for a fixed spectral efficiency, MFTN signaling can significantly improve the bit error rate (BER) performance by jointly optimizing the time- and frequency-domain packing factors. Compared to its Nyquist-signaling counterpart, MFTN signaling employing the proposed refined GAMP equalizer achieves about 39% higher transmission rate at a negligible BER performance degradation.