An Investigation of the Compressed Sensing Phase in Unsourced Multiple Access

TL;DR

This work addresses energy-efficient unsourced multiple access (UMA) for IoT by evaluating the compressed sensing (CS) phase with sparse greedy recovery. It models a two-part message where the first part is CS-encoded via a common sensing matrix and the second part carries remaining bits, and analyzes four greedy CS algorithms (OMP, gOMP, CoSaMP, SP) in terms of minimum transmit power, minimum channel uses, and ROC for a target detection probability $p_d=0.999$. The results show that OMP and gOMP are the most competitive, providing substantial margins over benchmarks and enabling practical design choices such as using around $N_p=2000$ channel uses (or $N_p=1000$ when $K_a\le100$) and manageable energy partitioning between CS and the second phase. These findings offer actionable guidelines for deploying CS-based UMA in IoT networks and highlight the robustness of greedy CS decoders when the active-user count is not precisely known.

Abstract

A vast population of low-cost low-power transmitters sporadically sending small amounts of data over a common wireless medium is one of the main scenarios for Internet of things (IoT) data communications. At the medium access, the use of grant-free solutions may be preferred to reduce overhead even at the cost of multiple-access interference. Unsourced multiple access (UMA) has been recently established as relevant framework for energy efficient grant-free protocols. The use of a compressed sensing (CS) transmission phase is key in one of the two main classes of UMA protocols, yet little attention has been posed to sparse greedy algorithms as orthogonal matching pursuit (OMP) and its variants. We analyze their performance and provide relevant guidance on how to optimally setup the CS phase. Minimum average transmission power and minimum number of channel uses are investigated together with the performance in terms of receiver operating characteristic (ROC). Interestingly, we show how the basic OMP and generalized OMP (gOMP) are the most competitive algorithms in their class.

Figures (4)

• Figure 1: Considered transmitter architecture. The message of every user is split in two parts. The first part entails the first $b_{p}$ bits of the message and are encoded with CS while, the second part is composed by remaining $b_{d}$ bits. The $b_{d}$ bits are also encoded and side information is embedded before transmission. The side information can be in the form of a user-specific spreading sequence as in Truhcachev2020Zheng2020 or a user-specific permutation as in Pradhan2022 and, it is indicated by the first $b_{p}$ bits.
• Figure 2: Active users vs. minimum $P_{1}$ that guarantees the target recovery probability $p_d=99.9\%$. We compare the four greedy sparse recovery algorithms OMP, gOMP, CoSaMP and SP. For OMP and gOMP we also provide to the CS algorithm a sparsity level in excess by $10\%$ (dashed lines). The black solid line represents the maximum $P_{1}$ that needs to be achieved by the CS phase so that the overall per-user error probability does not exceed $5\%$ and assuming $\alpha=1$. The black dashed line is the same but with a $1$ dB backoff. The black dotted line instead assumes $\alpha=1.5$ and no backoff.
• Figure 3: Active users vs. minimum $N_p$ that guarantees the target recovery probability $p_d=99.9\%$, under the assumption of equal energy, i.e. $\alpha=1$ and $\frac{E_b}{N_0}$ from the achievability bound in Polyanskiy2017. We compare the four greedy sparse recovery algorithms OMP, gOMP, CoSaMP and SP. For OMP also provide to the CS algorithm a sparsity level in excess by $10\%$ (dashed line).
• Figure 4: ROC for OMP and gOMP and with $K_{a} \in \left\{50,100,150\right\}$. $p_d$ is the detection probability while $p_f$ is the probability that any of the $N_p-K_{a}$ columns of $\bm A$ is marked as active even if it was not selected by any user. We fix $N_p=2000$ channel uses, $\alpha=1$ and, $P_{1} = \frac{E_b}{N_0}\frac{b}{N_p}$ (see eq. \ref{['eq:Eb_N0']}), where $\frac{E_b}{N_0}$ is computed according to Polyanskiy2017 for the corresponding number of active users.

Theorems & Definitions (2)

• Remark 1
• Remark 2