Many-User Multiple Access with Random User Activity: Achievability Bounds and Efficient Schemes

TL;DR

This work analyzes the Gaussian GMAC with random user activity in the many-user regime, where the number of users scales with code length. It develops two achievability bounds: a finite-length bound based on random Gaussian codebooks with ML decoding and a scalable asymptotic bound using spatially coupled Gaussian codebooks with AMP decoding. To bridge theory and practice, it proposes an efficient CDMA-type scheme using a spatially coupled signature matrix and AMP decoding, along with a novel state-evolution result for matrix-valued SC-AMP. The results together provide a rigorous framework to trade off active-user density and energy-per-bit for a fixed payload and target error, and they offer both tightBounds and practical algorithms with provable performance guarantees. These contributions advance understanding of multiuser decoding under sporadic activity and highlight the benefits of spatial coupling in high-dimensional random-access settings.

Abstract

We study the Gaussian multiple access channel with random user activity, in the regime where the number of users is proportional to the code length. The receiver may know some statistics about the number of active users, but does not know the exact number nor the identities of the active users. We derive two achievability bounds on the probabilities of missed detection, false alarm, and active user error, and propose an efficient CDMA-type scheme whose performance can be compared against these bounds. The first bound is a finite-length result based on Gaussian random codebooks and maximum-likelihood decoding. The second is an asymptotic bound, established using spatially coupled Gaussian codebooks and approximate message passing (AMP) decoding. These bounds can be used to compute an achievable tradeoff between the active user density and energy-per-bit, for a fixed user payload and target error rate. The efficient CDMA scheme uses a spatially coupled signature matrix and AMP decoding, and we give rigorous asymptotic guarantees on its error performance. Our analysis provides the first state evolution result for spatially coupled AMP with matrix-valued iterates, which may be of independent interest. Numerical experiments demonstrate the promising error performance of the CDMA scheme for both small and large user payloads, when compared with the two achievability bounds.

Figures (15)

• Figure 1: Three types of errors $\varepsilon_{\mathrm{MD}}, \varepsilon_{\mathrm{FA}}$ and $\varepsilon_{\mathrm{AUE}}$ in Theorem \ref{['thm:ngo_ach']} (thicker curves) and the error floors $\bar{\varepsilon}_{\mathrm{MD}}, \bar{\varepsilon}_{\mathrm{FA}}$ and $\bar{\varepsilon}_{\mathrm{AUE}}$ in Corollary \ref{['cor:error_floors']} (thinner horizontal lines) plotted against $E_b/N_0$, with different choices of decoding radii ${r_{l}}, {r_\mathrm{u}}$. Different colours correspond to different types of errors; different line styles correspond to different ${r_{l}}, {r_\mathrm{u}}$. Shaded regions: larger ${r_{l}}, {r_\mathrm{u}}$ cause higher errors due to noise overfitting. $n=2000,L=50, k=8$, $\alpha=0.5$, $p_{{K_{\mathrm{a}}}}=\text{Bin}(\alpha, L),$$P'$ is optimized over $(0,P)$, and $({\kappa_{l}},{\kappa_\mathrm{u}})$ are chosen so that $\mathbb{P}({K_{\mathrm{a}}}\notin [{\kappa_{l}}: {\kappa_\mathrm{u}}]) \le 10^{-13}$.
• Figure 2: Contour plots of $\max\{\bar{\varepsilon}_{\mathrm{MD}}, \bar{\varepsilon}_{\mathrm{FA}}\}+\bar{\varepsilon}_{\mathrm{AUE}}$ for different active user density $\mu_\mathrm{a}$ ($x$-axis) and normalized decoding radius $r/\mathrm{std}({K_{\mathrm{a}}})$ ($y$-axis) where ${r_{l}}={r_\mathrm{u}}=r$. $({\kappa_{l}},{\kappa_\mathrm{u}})$ are chosen to be the largest and smallest integers so that $\mathbb{P}({K_{\mathrm{a}}}\notin [{\kappa_{l}}: {\kappa_\mathrm{u}}]) \le \bar{p}$, where $\bar{p}$ is indicated below the subfigures. $L=600, k=6, \alpha=0.7$, $p_{{K_{\mathrm{a}}}}=\text{Bin}(\alpha, L)$, $P'=0.8P$, $r/\mathrm{std}({K_{\mathrm{a}}})=50$ corresponds to $r=L$.
• Figure 3: Contour plots of $\max\{\bar{\varepsilon}_{\mathrm{MD}}, \bar{\varepsilon}_{\mathrm{FA}}\}+\bar{\varepsilon}_{\mathrm{AUE}}$ for different active user density $\mu_\mathrm{a}$ ($x$-axis) and normalized decoding radius $r/\mathrm{std}({K_{\mathrm{a}}})$ ($y$-axis) where ${r_{l}}={r_\mathrm{u}}=r$. Subfigures use different $P'$, indicated below the subfigures. $L=600, k=6, \alpha=0.7$, $p_{{K_{\mathrm{a}}}}=\text{Bin}(\alpha, L)$, $r/\mathrm{std}({K_{\mathrm{a}}})=50$ corresponds to $r=L$, and $({\kappa_{l}},{\kappa_\mathrm{u}})$ are chosen so that $\mathbb{P}({K_{\mathrm{a}}}\notin [{\kappa_{l}}: {\kappa_\mathrm{u}}]) \le 10^{-4}$.
• Figure 4: Coding scheme for the proof of asymptotic achievability bounds. The joint message vector $\boldsymbol{x}= [\boldsymbol{x}_1^\top, \dots, \boldsymbol{x}_L^\top]^\top$ has $L$ sections corresponding to the $L$ users. Each section is drawn i.i.d. from $p_{\boldsymbol{\bar{x}}_{\sec}}$. The joint codebook matrix $\boldsymbol{A}=[\boldsymbol{A}_1, \dots, \boldsymbol{A}_L]$ has $L$ sections, with the $\ell$-th section storing the codewords of the $\ell$-th user as columns of the matrix.
• Figure 5: A spatially coupled design matrix $\boldsymbol{A}$ constructed using a base matrix $\boldsymbol{W}$ according to \ref{['eq:SC_design_Aij']}. The base matrix shown here is an $(\omega, \Lambda)$ base matrix (defined in Definition \ref{['def:ome_lamb_rho']}) with parameters $\omega=3, \Lambda=7$. The white parts of $\boldsymbol{A}$ and $\boldsymbol{W}$ correspond to zeros.

Theorems & Definitions (32)

• Theorem 1: Finite-length achievability bounds
• proof
• Corollary 1: Error floors
• proof
• Definition 1
• Lemma 1: Fixed points of state evolution
• proof : Proof of Lemma \ref{['lem:sparc_fixed_point']}
• Theorem 2: Asymptotic achievability bounds
• proof
• Corollary 2: Asymptotic achievability bound for $\alpha=1$