Compute-Forward Multiple Access for Gaussian Fast Fading Channels

TL;DR

This work extends CFMA to a two-user Gaussian fast fading MAC with CSIR, leveraging nested lattice codes and ambiguity decoding to recover linear combinations of codewords. It derives achievable-rate expressions for decoding two independent linear combinations and establishes a necessary-and-sufficient condition for ergodic sum capacity, showing capacity achievability depends critically on channel statistics. The paper also provides practical sufficient conditions, analyzes optimal coefficient choices, and demonstrates through numerical results that CFMA can closely approach the capacity boundary in many fading scenarios, while SIC remains a fallback for certain channel statistics. Overall, the results highlight how channel mean and variance shape the potential of low-complexity, lattice-based MAC strategies in fading environments with limited transmitter CSI.

Abstract

Compute-forward multiple access (CFMA) is a transmission strategy which allows the receiver in a multiple access channel (MAC) to first decode linear combinations of the transmitted signals and then solve for individual messages. Compared to existing MAC strategies such as joint decoding or successive interference cancellation (SIC), CFMA was shown to achieve the MAC capacity region for fixed channels under certain signal-to-noise (SNR) conditions without time-sharing using only single-user decoders. This paper studies the CFMA scheme for a two-user Gaussian fast fading MAC with channel state information only available at the receiver (CSIR). We develop appropriate lattice decoding schemes for the fading MAC and derive the achievable rate pairs for decoding linear combinations of codewords with any integer coefficients. We give a sufficient and necessary condition under which the proposed scheme can achieve the ergodic sum capacity. Furthermore, we investigate the impact of channel statistics on the capacity achievability of the CFMA scheme. In general, the sum capacity is achievable if the channel variance is small compared to the mean value of the channel strengths. Various numerical results are presented to illustrate the theoretical findings.

Figures (6)

• Figure 1: The achievable rate region and the capacity-achieving conditions for $\textbf{a} = (1,1), (1,2), (2,1)$. The choice of $\textbf{a} = (1,1)$ achieves the largest region of the capacity and has the largest range of $\gamma$ that can achieve the sum capacity.
• Figure 2: Symmetric Gaussian channel gains with $\mu=2,\sigma=0.5$. The left plot shows the majority part of the capacity region is achievable with $\textbf{a} = (1,1)$ and $\textbf{b} = (1,0)$ or $(0,1)$; The right plot shows the functions of $\gamma$ in Theorem \ref{['thm_sum_capacity_condition_expectation']} and Lemma \ref{['lem_Sum_Capacity_sufficient']}. We are interested in the $\gamma$'s which produce non-positive function values because with those choices of $\gamma$ the sum capacity is achievable.
• Figure 3: Symmetric Gaussian channel gains with $\mu=2$ and $\sigma=0,0.5,0.75.0.85$. The first subplot represents the fixed channel case. With the standard deviation increasing, the capacity becomes harder to achieve. In the last subplot, the achievable rate region falls off the boundary of the capacity boundary.
• Figure 4: Symmetric Gaussian channel gains. In this case, the channel gains of both users are i.i.d. With the mean and standard deviation shown in the right bottom region, the sum capacity is achievable with $\textbf{a} = (1,1)$ and $\textbf{b} = (1,0)$ or $(0,1)$. This indicates that it is good to have large mean and small variance. Asymptotically, to achieve the sum capacity, the mean is required to increase faster than the standard deviation. The right plot suggests that the sum capacity can be achieved if $\sigma^2 < 2\mu$.
• Figure 5: Asymmetric Gaussian channel gains with standard deviation equaling $0.25$. The standard deviations of both channel gains are fixed. In the most right top region (Region III), the sum capacity is achievable with $\gamma = \mu_1/\mu_2$. If we can choose the best $\gamma$, this region will become larger as Region IV + Region III. But the difference between them is small.

Theorems & Definitions (21)

• Definition 1: Ambiguity lattice decoder
• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1
• Remark 2
• Theorem 2
• proof
• Remark 3