Smooshed Zero Constellation for Binary Modulation on Conjugate-Reciprocal Zeros

In this study, we introduce smooshed binary modulation on conjugate-reciprocal zeros (SBMOCZ) to mitigate timing offset (TO)-induced zero rotation in binary modulation on conjugate-reciprocal zeros (BMOCZ)-based orthogonal frequency division multiplexing (OFDM). In our approach, we modify Huffman BMOCZ with zero smooshing to create a structured gap in the zero constellation which rotates under a TO. By exploiting the auto-correlation properties of SBMOCZ, we develop a low-complexity rotation estimator at the receiver that can be implemented using a single discrete Fourier transform (DFT). We theoretically analyze the properties of SBMOCZ polynomials and derive closed-form expressions for the aperiodic auto-correlation function (AACF) and peak-to-average power ratio (PAPR). To learn the smooshed zero constellation parameters, we introduce two machine learning (ML)-based methods. The first uses a soft-output from the direct zero-testing (DiZeT) decoder with hinge loss, while the second jointly trains a neural network (NN)-based decoder robust to fading and the constellation parameters. Our numerical simulations demonstrate that SBMOCZ remains effective under TO impairments without relying on affine cyclically permutable codes (ACPCs) and improves bit error rate (BER) and block error rate (BLER) at no additional overhead.