Tensor-Based Modulation on the Unit Circle: A Coding Perspective

TL;DR

Simulations in single-user AWGN and multi-user non-coherent multi-antenna fading channels demonstrate strong robustness and interference resilience, establishing TBM as a scalable, algebraically structured modulation-coding scheme bridging tensor representations and modern coding theory.

Abstract

Tensor-based modulation (TBM) provides a multi-linear spreading framework for blind multi-user separation in unsourced random access. In this paper, we show that TBM is a coded modulation built on a non-binary linear block code over $\mathbb{Z}_M$, whose symbols are mapped to $M$-PSK modulation, defining a geometrically uniform signal space code. We explicitly derive this generator matrix, characterize its rank deficiency, and show that reference symbols for tensor identifiability correspond to code shortening, producing a quasi-systematic or a systematic code, depending on the number of considered reference symbols for the TBM. Simulations in single-user AWGN and multi-user non-coherent multi-antenna fading channels demonstrate strong robustness and interference resilience, establishing TBM as a scalable, algebraically structured modulation-coding scheme bridging tensor representations and modern coding theory.

Figures (6)

• Figure 1: Two equivalent interpretations of PSK-TBM: MLS, and coded modulation.
• Figure 2: Equivalent systematic encoding of TBM-PSK.
• Figure 3: Factor graph representation of the proposed code construction for $\left(T_1, T_2, T_3\right) =(4,2,2)$, Case 1. Note that the node $c_1$ is not connected to the graph. It can be considered as a pilot symbol for the non-coherent setting.
• Figure 4: Rate vs. minimum SNR required to achieve PER=$10^{-2}$.
• Figure 5: PER comparison for $T = 3200$, AWGN channel.