Summary

Chapter 15 Summary: MIMO I — Channel Modeling and Capacity

Key Points

• 1.

  The MIMO channel matrix $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ encodes the input-output relationship $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$. Its rank determines the number of independent spatial sub-channels, and its condition number governs whether the channel favours spatial multiplexing or beamforming.

• 2.

  Physical MIMO models capture the spatial correlation present in real channels. The Kronecker model $\mathbf{H} = \mathbf{R}_{r}^{1/2}\mathbf{H}_{w}\mathbf{R}_{t}^{1/2}$ separates transmit and receive correlation; the Weichselberger model adds eigenmode coupling for greater accuracy. Keyhole channels ($\mathrm{rank} = 1$) demonstrate that multiplexing requires both multiple antennas AND rich scattering.

• 3.

  Deterministic MIMO capacity is achieved by SVD decomposition into parallel sub-channels with water-filling power allocation: $C = \sum_i \log_2(1 + p_i^* \sigma_i^2 / \sigma^2)$. At high SNR, equal power allocation is near-optimal; at low SNR, beamforming (concentrating power on the strongest mode) is optimal.

• 4.

  Ergodic MIMO capacity (Telatar's formula) averages over the random channel, with the Wishart eigenvalue distribution governing the capacity statistics. Without CSIT, isotropic transmission is optimal for i.i.d. Rayleigh channels. Outage capacity is the relevant metric for delay-constrained slow-fading scenarios.

• 5.

  Degrees of freedom equal $\min(n_t, n_r)$, meaning MIMO capacity scales linearly with the number of antennas on the smaller side --- without additional bandwidth or power. This linear scaling is the fundamental reason MIMO is deployed in every modern wireless standard.

• 6.

  The diversity-multiplexing tradeoff $d^*(r) = (n_t - r)(n_r - r)$ reveals that diversity and multiplexing are two endpoints of a continuous tradeoff. Full diversity $d^*(0) = n_t n_r$ and full multiplexing $r = \min(n_t, n_r)$ cannot be achieved simultaneously. DMT-optimal codes (e.g., Golden code) achieve the best possible tradeoff at every operating point.

• 7.

  Spatial correlation reduces capacity but is not purely detrimental: in massive MIMO, correlation creates low-rank structure that enables efficient beamforming, reduced feedback, and multi-user spatial separation. The practical impact of correlation depends on the antenna count, spacing, angular spread, and operating SNR.