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The broadcast channel (BC) models one sender transmitting independent messages to multiple receivers through a shared medium. Unlike the MAC (many-to-one), the BC is one-to-many. For the degraded BC (where receiver 2's signal is a noisier version of receiver 1's), superposition coding achieves capacity: the sender layers messages at different power levels, and stronger receivers decode all layers while weaker receivers decode only their own. The Gaussian BC capacity region is fully characterized and equals the MAC capacity region by duality. The general (non-degraded) BC capacity region remains incompletely characterized, though Marton's inner bound is known to be tight in many cases.
The broadcast channel is the canonical downlink model: a base station sending different messages to multiple users, a satellite broadcasting different programs, or a server transmitting to multiple clients with different connection qualities. One transmitter, multiple receivers, each wanting their own message.
The simplest and best-understood case is the degraded broadcast channel, where the receivers can be ordered by quality — receiver 1 gets a strictly better version of the signal than receiver 2. For the Gaussian case, Y_1 = X + Z_1 and Y_2 = X + Z_2 with N_1 < N_2 (user 1 has less noise). The capacity-achieving strategy is superposition coding: encode user 2's message as a "cloud center" at high power, and encode user 1's message as a "cloud point" at lower power around this center. User 2 (weak) treats user 1's signal as noise and decodes the cloud center. User 1 (strong) first decodes user 2's message (the cloud center), subtracts it, then decodes their own message from the residual.
The capacity region for the two-user degraded Gaussian BC is: R_2 <= (1/2) log(1 + alpha*P/(( 1-alpha)*P + N_2)) and R_1 <= (1/2) log(1 + (1-alpha)*P/N_1), for alpha in [0,1]. As alpha varies, the tradeoff between user rates traces the capacity region boundary. Setting alpha = 1 gives all power to user 2 (R_1 = 0); setting alpha = 0 gives all power to user 1 (R_2 = 0). No TDMA or FDMA scheme can match the superposition coding boundary — orthogonal schemes are strictly suboptimal.
The general (non-degraded) broadcast channel is far harder. When receivers are not ordered by quality (e.g., one receiver is better at low frequencies, the other at high frequencies), superposition coding alone is insufficient. Marton's coding scheme, which uses correlated auxiliary random variables, provides the best known inner bound and is tight for several important classes. The general BC capacity region remains one of the major open problems in information theory, highlighting how multi-user problems can be vastly more complex than their single-user counterparts.