MIMO Capacity

Core Idea

Multiple-input multiple-output (MIMO) channels use multiple transmit and receive antennas, modeled as Y = HX + Z where H is the channel matrix, X is the input vector, and Z is Gaussian noise. With n_T transmit and n_R receive antennas and known channel matrix H, the capacity is C = max_{tr(K_X)<=P} log2 det(I + (1/N) H K_X H^*), where K_X is the input covariance. This decomposes via SVD into min(n_T, n_R) parallel Gaussian channels (eigenmodes), with optimal power allocation via water-filling. MIMO capacity scales linearly with min(n_T, n_R) in rich scattering — a breakthrough that powers modern wireless systems (4G, 5G, Wi-Fi).

Explainer

The Gaussian channel capacity C = (1/2) log(1 + SNR) describes a single-antenna system. MIMO extends this to multiple antennas, revealing that capacity can scale linearly with the number of antennas — a result that revolutionized wireless communications when Foschini and Telatar published it in the late 1990s.

The MIMO channel model is Y = HX + Z, where X is an n_T-dimensional transmitted vector, Y is an n_R-dimensional received vector, H is the n_R x n_T channel matrix, and Z ~ N(0, NI) is Gaussian noise. The entry h_{ij} represents the complex channel gain from transmit antenna j to receive antenna i. The capacity depends on the singular value decomposition (SVD) of H = U * Sigma * V^*. The SVD decomposes the MIMO channel into min(n_T, n_R) parallel scalar Gaussian channels with gains equal to the singular values sigma_1 >= sigma_2 >= ... >= sigma_r. The capacity is the sum of the capacities of these parallel channels: C = sum_i (1/2) log2(1 + p_i * sigma_i² / N), where p_i is the power allocated to eigenmode i.

Water-filling determines the optimal power allocation. Modes with larger gains (sigma_i²) get more power; modes below a threshold get none. At high SNR, all modes receive similar power and the capacity is approximately min(n_T, n_R) * (1/2) log2(SNR/min(n_T,n_R)), growing linearly with the number of spatial dimensions. At low SNR, only the strongest mode is used (beamforming).

The practical impact is immense. 4G LTE uses 2x2 and 4x4 MIMO. 5G NR supports massive MIMO with 64+ antennas at the base station, enabling both high per-user capacity and multi-user spatial multiplexing. Wi-Fi 6/7 uses up to 8x8 MIMO. In each case, the information-theoretic MIMO capacity formula guides system design, antenna spacing, and the decision of when to use multiplexing versus beamforming. The theory extends to multi-user MIMO (MU-MIMO), where the base station's antenna array serves multiple users simultaneously, approaching the sum capacity of the broadcast channel.