Ch. 13 — Spread Spectrum

Chapter 13 of Wireless Communications (2nd ed. Draft) — Andrea Goldsmith

Chapter 14: Multiuser Systems

Abstract

This chapter develops the theoretical framework for allocating signaling dimensions and power among multiple users sharing a wireless medium, distinguishing between uplink (multiple access) and downlink (broadcast) architectures. It derives the fundamental capacity regions for both channels under additive white Gaussian noise (AWGN) and fading conditions, establishing that superposition coding with successive interference cancellation achieves the broadcast channel capacity while dirty paper coding optimizes the multiple-input multiple-output (MIMO) broadcast region. The chapter further examines practical access methods, including orthogonal and nonorthogonal code division, random access protocols like ALOHA and CSMA, and iterative power control algorithms designed to mitigate the near–far effect while minimizing transmit power.

Key Concepts

• Uplink and Downlink Architecture: The multiuser channel is categorized into the uplink (multiple access channel), where multiple transmitters send to a single receiver with individual power constraints, and the downlink (broadcast channel), where a single transmitter sends independent data streams to multiple receivers. In the downlink, all signals pass through the same channel to a given receiver, whereas uplink signals experience distinct channel gains, necessitating coordination for synchronization.
• Frequency-Division Multiple Access (FDMA): A channelization method where system bandwidth is divided into nonoverlapping frequency channels, each assigned to a different user. FDMA typically requires guard bands to mitigate adjacent channel interference and works well for continuous transmission applications like analog voice but suffers from frequency-selective fading in wideband systems.
• Time-Division Multiple Access (TDMA): A channelization method dividing the signal space along the time axis into nonoverlapping, cyclically repeating timeslots. While TDMA simplifies channel estimation during idle slots, it requires strict synchronization among uplink transmitters to prevent orthogonality loss due to multipath delays and varying propagation times.
• Space-Division Multiple Access (SDMA): A technique that utilizes the spatial dimension by assigning orthogonal channels based on user angle of arrival, implemented via directional antennas or sectorized arrays. SDMA increases system capacity by reusing resources across spatial sectors but requires dynamic adaptation as user locations change or relies on handoffs between sectors.
• Random Access and ALOHA: Protocols designed for bursty data traffic where users transmit packets without dedicated scheduling, leading to potential collisions. Pure ALOHA allows transmission at any time with low throughput (~18%), while Slotted ALOHA synchronizes users to slot boundaries, doubling maximum throughput to ~37% at the cost of synchronization overhead.
• Carrier-Sense Multiple Access (CSMA): An improvement over ALOHA where users sense the channel before transmission to detect busy states, significantly reducing collisions but introducing the hidden terminal problem in wireless environments where nodes cannot detect all other transmitting users.
• Iterative Power Control (Foschini–Miljanic): An algorithm that adjusts transmitter powers based on measured Signal-to-Interference-plus-Noise Ratio (SINR) to meet target thresholds while minimizing total power. The algorithm converges to the Pareto optimal power vector if the Perron–Frobenius eigenvalue of the interference matrix is less than unity.
• Superposition Coding: A multiresolution coding technique for the broadcast channel where users with better channel gains decode and subtract interference from users with worse gains. This allows the transmitter to send a composite signal that achieves the capacity region boundary, outperforming simple time or frequency division when channel gains are disparate.
• Dirty Paper Coding (DPC): A transmission strategy for MIMO broadcast channels where the transmitter non-causally precodes signals to “presubtract” interference known at the transmitter but unknown to the receiver. DPC achieves the optimal capacity region for the multiuser MIMO broadcast channel despite the lack of a known degraded ordering.
• Ergodic versus Outage Capacity: Metrics for fading channels where ergodic capacity averages the achievable rate over all fading states (suitable for delay-unconstrained traffic), while outage capacity defines fixed rates maintainable with a specific probability of failure in any fading state (suitable for delay-constrained traffic).

Key Equations and Algorithms

• Uplink SINR Constraint: The Signal-to-Interference-plus-Noise Ratio for the $k$-th user in an uplink system is constrained by ${\gamma }_k\le \frac{P_k{g}_k}{{\sum }_{j\ne k}\rho P_j{g}_j+N_0}$. This expression relates the transmit power $P_k$, channel gain $g_k$, and interference reduction factor $\rho$ to the target SINR ${\gamma }_k$, forming the basis for feasibility analysis.
• Perron–Frobenius Power Control Condition: Feasibility of meeting SINR targets is determined by $\rho (\mathbf{F})<1$, where $\mathbf{F}$ is the normalized interference matrix with elements $F_{kj}=\frac{{\gamma }_k{g}_j}{g_k}$ for $j\ne k$ and zero otherwise. This eigenvalue condition ensures that a positive power vector $\mathbf{P}>0$ exists to satisfy $(\mathbf{I}-\mathbf{F})\mathbf{P}\ge \mathbf{u}$.
• Broadcast Channel Rate Region (Superposition): The capacity region for a degraded broadcast channel with effective noises $n_1\le n_2$ is defined by rate pairs $R_1\le B{\log }_2(1+\frac{P_1}{n_{1}B})$ and $R_2\le B{\log }_2(1+\frac{P_2}{n_{2}B+P_1\frac{n_2}{n_1}})$. This region is achieved via superposition coding, allowing user 1 to decode user 2’s message and cancel it before decoding its own.
• Broadcast Channel Sum-Rate Capacity: The maximum aggregate throughput is achieved by allocating all power $P$ to the user with the minimum effective noise $n_{\min }$, yielding $C_{SR}=B{\log }_2(1+\frac{P}{n_{\min }B})$. This result implies that for sum-throughput maximization in a broadcast setting, resources are dedicated to the strongest channel state rather than shared.
• Multiple Access Channel (MAC) Capacity Region: The achievable rate region for two users in AWGN is the convex hull of constraints $R_1\le C_1$, $R_2\le C_2$, and $R_1+R_2\le B{\log }_2(1+\frac{g_1{P}_1+g_2{P}_2}{N_0})$. Unlike the broadcast channel, the MAC sum-rate is maximized when all users transmit at full power, as individual power constraints do not compete for a shared pool in the same manner.
• Foschini–Miljanic Iterative Algorithm: The power update rule for user $k$ at step $i+1$ is $P_k^{(i+1)}=\frac{{\gamma }_k}{SIN{R}_k^{(i)}}{P}_k^{(i)}$. This simple per-user procedure converges to the minimum power solution if the target SINRs are feasible, requiring only local SINR measurements and feedback commands.
• Slotted ALOHA Throughput: The relationship between throughput $T$ and offered load $L$ is given by $T=L{e}^{-L}$. This function peaks at $L=1$ with a maximum throughput $T\approx 0.37$, illustrating the fundamental limit on efficiency for unscheduled random access under Poisson traffic assumptions.
• Ergodic Capacity with Fading: The capacity region of a fading broadcast channel is the convex hull of the union of capacity regions over all fading states $\mathbf{n}[i]$, optimized via a water-filling power policy $P_k(\mathbf{n})$. The optimal strategy allocates more power to users with instantaneous favorable channel gains, exploiting multiuser diversity.
• Dirty Paper Coding (DPC) Rate Vector: For a permutation $\pi$ of users, the achievable rate for user $k$ is $R_k=\log \det (\mathbf{I}+{\sum }_{j=k}^K{\mathbf{H}}_k{\mathbf{Q}}_{\pi (j)}{\mathbf{H}}_k^H)-\log \det (\mathbf{I}+{\sum }_{j=k+1}^K{\mathbf{H}}_k{\mathbf{Q}}_{\pi (j)}{\mathbf{H}}_k^H)$. This formula assumes interference from predecessors is non-causally cancelled at the transmitter, rendering it invisible to the receiver.

Key Claims and Findings

• Superposition coding combined with successive interference cancellation achieves the full capacity region of the additive white Gaussian noise broadcast channel, outperforming time-division and frequency-division methods whenever user channel gains are unequal.
• The sum-rate capacity of the broadcast channel is achieved by assigning all available power to the single user with the strongest channel gain, whereas the multiple access channel sum-rate capacity requires all users to transmit at their maximum individual power limits.
• In the absence of perfect channel state information at the transmitter, the broadcast channel capacity region is unknown for general non-degraded cases, but dirty paper coding provides the exact capacity region for MIMO broadcast channels when the transmitter has non-causal interference knowledge.
• Power control in uplink systems is essential to counteract the near–far effect, where users at different distances would otherwise experience vastly different received powers, rendering far users inaudible to the base station.
• Random access protocols like Slotted ALOHA are limited to a maximum throughput of 37% of the channel capacity due to collisions, necessitating scheduling or more sophisticated protocols like CSMA for data-heavy applications.
• Fading capacity regions are strictly larger than static capacity regions when ergodic capacity is considered, because adaptive power and rate allocation allow the system to exploit favorable channel states over time (multiuser diversity).
• The zero-outage capacity region for Rayleigh fading channels is often zero or severely limited because maintaining a fixed rate in deep fades requires infinite power, unlike ergodic capacity which averages performance.
• In CDMA systems with nonorthogonal codes, the number of supported users is not hard-constrained by bandwidth but by the interference level, allowing more than $G$ (processing gain) users at the cost of degraded signal-to-interference ratios.

Terminology

• Broadcast Channel (BC): A downlink multiuser channel model where one transmitter sends independent information to multiple receivers, characterized by a capacity region rather than a single rate value.
• Multiple Access Channel (MAC): An uplink multiuser channel model where multiple transmitters send information to a single receiver, subject to individual power constraints per user.
• Degraded Channel: A specific condition in multiuser channels where the signals received by users can be ordered such that one receiver’s channel is effectively a noisier version of another’s, simplifying capacity calculation.
• Near–Far Effect: An interference phenomenon in CDMA uplinks where a strong signal from a nearby user masks the weak signal of a distant user, necessitating power control to equalize received powers.
• Superposition Coding: A coding scheme where the transmitter sends a composite signal consisting of layered information, allowing receivers with better channel quality to decode higher layers intended for users with worse quality.
• Dirty Paper Coding (DPC): A precoding technique for MIMO channels that allows the transmitter to “write on dirty paper” by pre-canceling known interference, achieving the same capacity as if the interference did not exist.
• Ergodic Capacity: The long-term average rate achievable over a fading channel, assuming the coding block length is long enough to experience all fading states.
• Outage Capacity: A capacity metric for delay-constrained systems defined as the maximum rate that can be maintained with a probability of failure (outage) less than a specified threshold.
• Perron–Frobenius Eigenvalue: The largest real eigenvalue of a nonnegative irreducible matrix, used in power control analysis to determine if a set of SINR targets is achievable within the system’s power constraints.
• Multiuser Diversity: The performance gain arising from scheduling transmissions to users when their channel conditions are at their peak, exploiting the statistical independence of fading channels.