Broadcast Channel Capacity

Core Idea

The capacity region of a broadcast channel (one sender, multiple receivers) is defined by the union of achievable rate tuples (R_1, R_2, ..., R_K) for reconstructing messages at each receiver. For degraded broadcast channels — where receivers can be ordered such that each receiver's signal is a degraded version of the prior — the capacity region is fully characterized by superposition coding: the sender transmits a superposition of independent messages at different power levels, strong receivers decode all layers (like successive interference cancellation at the transmitter side), and weak receivers decode only their own layer. For non-degraded broadcast channels, the capacity region is not fully known in general. Marton's coding scheme (which encodes using correlated auxiliary variables) provides the best known inner bound, tight for several important cases. The Gaussian broadcast channel's capacity region is known by a duality with the Gaussian MAC, and the region exhibits a tradeoff where increasing one user's rate decreases others' achievable rates.

Explainer

The broadcast channel (one sender, multiple receivers, one-to-many) is the canonical downlink: a base station to many users, a satellite to ground stations, a wireless access point to devices. Unlike the MAC where multiple independent senders cooperate via sequential decoding at the receiver, the broadcast channel has full sender control but must simultaneously satisfy the needs of receivers with potentially different channel qualities.

The key conceptual challenge is that the sender cannot separate transmissions by time or frequency without losing capacity — orthogonal access (TDMA, FDMA) is suboptimal. Instead, the sender must layer messages at different power levels so that the receiver capabilities determine what each can decode. This is superposition coding, introduced by Cover.

For the degraded Gaussian BC, where receiver 1 has noise N_1 < N_2 (better channel):

• Weak receiver: R_2 = (1/2) log2(1 + alpha*P / ((1-alpha)*P + N_2))
• Strong receiver: R_1 = (1/2) log2(1 + (1-alpha)*P / N_1)

The weak user's message is sent at high power alpha*P (the "cloud center"), and the strong user's message at lower power (the "cloud cloud"). The strong receiver decodes the weak receiver's message first (it dominates), subtracts it like SIC, then decodes their own. The weak receiver ignores the strong user's message entirely (treats it as noise). The parameter alpha in [0, 1] trades off rates: increasing alpha helps the weak user but hurts the strong user. The capacity region is the convex hull of these tradeoffs as alpha varies, which for the Gaussian BC is a closed two-dimensional region in the (R_1, R_2) plane.

The Gaussian BC capacity region is fully known and, remarkably, admits a duality with the Gaussian MAC: the capacity region is the same (up to a transformation of the power constraint). This duality was discovered by Bergmans and explained through water-filling arguments. The sum-rate on the MAC and BC are equal when power is allocated optimally.

For non-degraded broadcast channels, there is no natural ordering. Marton's coding scheme (1979) generalizes superposition coding by allowing the auxiliary variables U_1, U_2 (representing public information for each receiver) to be correlated. The sender encodes a message as a function of (U_1, U_2, X_1, X_2) where X_1, X_2 are the private messages. The receivers jointly decode the public information from (U_1, U_2) then extract private messages. The optimal choice of the correlation between U_1 and U_2 (and the conditional distributions of private messages) is complex, often solved via alternating optimization.

The capacity region of the general (non-degraded) broadcast channel remains not fully characterized for many cases, making it a frontier problem in network information theory. This gap between the known MAC and the incompletely understood BC, despite their mathematical similarity, illustrates how multi-user communication reveals surprising asymmetries. The broadcast channel is also the foundation for modern wireless downlink design: 5G NR uses NOMA-like strategies (superposition coding) to serve users with different channel conditions from a single transmitter, approaching information-theoretic limits set by BC capacity.