Interblock-interference-free precoding using square-root decomposition for faster-than-Nyquist signaling

Precodings using square-root decomposition, including Cholesky and G-To-Minus-Half (GTMH) precodings, are promising for eliminating the Faster-Than-Nyquist (FTN)-induced Intersymbol-Interference (ISI). However, the existing precodings using square-root decomposition either ignore Interblock-Interference (IBI) or increase the signal power, deteriorating the Bit Error Rate (BER) performance for high-order modulations and severe ISI. To overcome these drawbacks, we adopt two approaches for constructing the circular ISI matrix. The first approach inserts a Cyclic Prefix/Suffix (CPS) after each precoded symbol block, while the second approach replaces the linear convolution of the FTN shaping and the matched filter by the circular convolution, resulting in the Circular FTN (CFTN). Using these two approaches, we propose three IBI-free precodings, i.e., CPS-Cholesky, CFTN-Cholesky and CFTN-GTMH precodings. Furthermore, employing QR decomposition shows that the GTMH and Cholesky precodings can be converted interchangeably. Thus, we demonstrate that the GTMH precoding is essentially equivalent to the Cholesky precoding. Simulation results indicate that the BER performance of three IBI-free precodings approaches Nyquist performance for moderate ISI. However, as ISI intensifies, the CPS-Cholesky scheme increases the transmit power, causing BER performance degradation. In contrast, the CFTN-Cholesky and CFTN-GTMH precodings maintain optimal BER performance even for severe ISI. Considering 128-amplitude phase shift keying with a code rate of 1/2, the BER loss of CFTN-Cholesky and CFTN-GTMH precodings for the ideal BER of 10-5 is approximately 0.002 dB and 0.005 dB when packing factor is 0.7 and roll-off factor is 0.3. To the best knowledge of the authors, this is the optimal performance achievable through precoding.