Ch. 14 — Multiuser Systems

Chapter 14 of Wireless Communications (2nd ed. Draft) — Andrea Goldsmith

Abstract

This chapter establishes the theoretical foundations for multiuser communication systems under fading conditions, focusing on the Multiple Access Channel (MAC) and Broadcast Channel (BC). It derives the ergodic and outage capacity regions for fading MAC systems, demonstrating that optimal sum-rate is achieved by scheduling the user with the best weighted channel gain. A central contribution is the development of the uplink-downlink duality principle, which maps the capacity regions of the BC to the MAC, facilitating the computation of optimal power allocations for computationally difficult BC problems. The chapter further quantifies multiuser diversity gains, showing that system throughput scales with the logarithm of the number of users, and extends these principles to MIMO architectures to analyze diversity and multiplexing trade-offs.

Key Concepts

• Fading MAC Capacity Regions: The capacity of a fading MAC is defined by the set of achievable rate vectors averaged over fading states (ergodic) or maintained in all states (outage/zero-outage). The ergodic capacity region is the union over all admissible power policies of the achievable rates. For sum-rate maximization, the optimal strategy under individual power constraints involves transmitting only the user with the largest weighted channel gain $g_k/{\lambda }_k$ in each fading state, where ${\lambda }_k$ relates to the user’s power constraint. When users have identical fading statistics, the user with the best instantaneous channel $g_k$ is selected.
• MIMO MAC Capacity Region: For multiple antenna systems, the capacity region is defined by the union of K-dimensional polyhedrons corresponding to spatial covariance matrices $Q_k$. Each covariance matrix set corresponds to a point on the boundary of the capacity region. The region is achieved via successive interference cancellation (SIC), where users are decoded and subtracted in a specific order. The corner points of the capacity region correspond to the order of decoding, similar to the single-antenna case but generalized to matrix-valued channel gains $H_k$.
• Uplink-Downlink Duality: A fundamental relationship exists between the Broadcast Channel (BC) and the Multiple Access Channel (MAC) where channel impulse responses and noise statistics are identical, and the BC sum power constraint equals the sum of individual MAC power constraints. The BC capacity region is equal to the union of dual MAC capacity regions over all possible power allocations summing to the BC power. Conversely, the MAC capacity region can be obtained as the intersection of scaled dual BC capacity regions. This duality allows complex BC optimization problems to be solved via simpler convex MAC optimization problems.
• MIMO Duality Extension: Duality extends to MIMO systems where the MIMO BC capacity region with power constraint $P$ and channel matrix $H$ equals the union of dual MIMO MAC capacity regions with channel gain matrix $H^H$ over individual power constraints summing to $P$. This relationship simplifies the computation of the MIMO BC capacity region, which is non-convex in the original formulation, by transforming it into the convex MIMO MAC optimization problem. The optimal transmission strategies for a point on the BC boundary are derived from the optimal strategies of the intersecting dual MAC region.
• Multiuser Diversity: This concept exploits independent fading across multiple users to improve system performance. By scheduling transmission to the user with the maximum instantaneous SNR $\gamma [i]={\max }_k{\gamma }_k[i]$, the system achieves an SNR gain roughly proportional to $\ln K$ in independent Rayleigh fading as $K$ grows. This selection diversity increases the throughput and reduces the probability of error compared to a single-user system, effectively mitigating fading effects as the number of users increases.
• Opportunistic Scheduling: Scheduling algorithms allocate channel resources to users based on current channel conditions to maximize throughput. While scheduling the best channel maximizes sum-rate, it may lead to unfairness or high delay for users with poor average SNRs. Proportional fair scheduling is introduced to balance throughput and fairness by selecting the user with the largest ratio of instantaneous rate $R_k[i]$ to average throughput $T_k[i]$. This algorithm ensures users with poor average throughput are prioritized until their throughput balances with the system.

Key Equations and Algorithms

• Fading MAC Ergodic Capacity Union: ${\mathcal{C}}_{ergodic}={\bigcup }_{P\in {\mathcal{F}}_{MAC}}\left\{(R_1,\dots ,R_K):R_k=\int \dots \right\}$. This equation defines the ergodic capacity region as the union of achievable rate vectors averaged over all fading states $g$ under all valid power policies $P$ satisfying individual average power constraints ${\bar{P}}_k$.
• Sum-Rate Optimal Policy: User $k$ transmits if $g_k/{\lambda }_k={\max }_j(g_j/{\lambda }_j)$. This condition identifies the user who should transmit in a specific fading state $g$ to maximize the system sum-rate, where ${\lambda }_k$ are Lagrange multipliers associated with power constraints. If power constraints and fading distributions are symmetric, the user with the highest $g_k$ transmits.
• MIMO MAC Capacity Region: $R_S\le {\log }_2\det \left(I+{\sum }_{k\in S}{H}_k{Q}_k{H}_k^H\right)$ for all $S\subseteq \{1,\dots ,K\}$. This inequality characterizes the achievable rates for subsets of users $S$ in a MIMO MAC, defined by the channel matrices $H_k$ and spatial covariance matrices $Q_k$ under power constraints.
• BC-MAC Duality Relationship: ${\mathcal{C}}_{BC}(P,g)={\bigcup }_{\sum P_k=P}{\mathcal{C}}_{MAC}(P_1,\dots ,P_K;g)$. This expresses the BC capacity region as the union of dual MAC capacity regions where individual user power constraints $P_k$ sum to the total BC power $P$. It enables solving BC problems by optimizing over power splits in the dual MAC.
• Proportional Fair Scheduling Update: $T_k[i+1]=(1-\frac{1}{t_c})T_k[i]+\frac{1}{t_c}{R}_k[i]\mathbb{I}(k=k^{\ast })$. This algorithm updates the average throughput $T_k[i]$ for a user selected in time slot $i$ (where $k^{\ast }$ is the scheduled user). The parameter $t_c$ controls the trade-off between system throughput and fairness latency.
• Multiuser Diversity SNR Gain: ${\gamma }_{max}\approx {\gamma }_{single}\ln K$. This relationship describes the asymptotic behavior of the maximum SNR among $K$ i.i.d. Rayleigh fading users, showing that the system SNR grows logarithmically with the number of users, providing significant diversity gains.

Key Claims and Findings

• Single-User Transmission Optimality: The sum-rate capacity of the fading MAC is achieved by a policy that allows only one user to transmit in any given fading state. This is optimal because adaptive power allocation assigns all system resources to the user with the best weighted channel gain, commensurate with channel quality.
• Duality Simplifies BC Optimization: The capacity region and optimal transmission strategy of the Broadcast Channel can be derived from its dual Multiple Access Channel. This is particularly valuable for MIMO BCs, where direct optimization is difficult, whereas the dual MIMO MAC problem is a standard convex optimization that is computationally tractable.
• MIMO Sum-Rate Scaling: The sum-rate capacity gain in MIMO Broadcast Channels increases roughly linearly with the minimum of the number of transmit antennas and the total number of receive antennas associated with all users. Consequently, adding antennas to user terminals yields diminishing returns when the base station has many user terminals.
• Fairness Throughput Trade-off: Proportional fair scheduling reduces the maximum achievable system throughput compared to pure opportunistic scheduling but ensures that users with poor average channel qualities access the system. The latency is controlled by the filter parameter $t_c$, which balances the flexibility of the scheduler against the delay constraints.
• MIMO Multiuser Diversity Gains: In MIMO multiuser systems, diversity benefits arise not only from selecting users with good channel quality but also from the spatial separation provided by independent fading directions. This allows simple, suboptimal transmitter and receiver techniques to achieve near-optimal performance as the number of users increases.

Terminology

• Ergodic Capacity Region: The set of achievable rate vectors averaged over all fading states for a Multiuser Channel. It characterizes performance when delay is not constrained and allows transmission to adapt to time-varying channel states.
• Outage Capacity Region: The maximum rate vector that can be maintained in all fading states with a nonzero probability of outage. It defines the reliability constraints for services requiring guaranteed minimum rates.
• Power Policy: A function that maps a fading state $g$ to a set of transmit powers $P_k(g)$ for each user. The policy determines how power is allocated across users and time to satisfy average power constraints while maximizing capacity.
• Zero-Outage Capacity: The set of rates that can be simultaneously achieved for all users in all fading states while strictly meeting average power constraints. No transmission is suspended in any fading state under this definition.
• Successive Decoding: A reception technique where user signals are decoded sequentially and subtracted from the received signal to remove interference. In the MAC, this process achieves the corner points of the capacity region polyhedron.
• Uplink-Downlink Duality: A relationship connecting the capacity regions of the MAC (uplink) and BC (downlink) such that specific power allocation and channel scaling transformations map one region to the other. This allows algorithms developed for one channel to be adapted for the dual.
• Multiuser Diversity: The performance improvement gained by scheduling transmissions to users with the best instantaneous channel conditions among a large set of users. This technique leverages the statistical independence of user fading to improve system resources utilization.
• Proportional Fair Scheduling: A resource allocation algorithm that maximizes the sum of the logarithms of user throughputs. It selects the user with the highest ratio of instantaneous rate to average throughput, balancing spectral efficiency with user fairness.