Chapter 10: MIMO Channel Capacity and Code Design Criteria

Learning Objectives

• Recall the $n_r \times n_t$ i.i.d. Rayleigh MIMO ergodic capacity formula $C = \mathbb{E}[\log\det(\mathbf{I}_{n_r} + (\text{SNR}/n_t)\mathbf{H}\mathbf{H}^{H})]$ from Book ITA Ch. 13.5 and identify the high-SNR multiplexing scaling $\min(n_t, n_r)\log_2\text{SNR}$
• State the quasi-static (block-fading) channel model and distinguish the ergodic and outage notions of capacity; define the outage probability $P_{\mathrm{out}}(R) = \Pr[\log\det(\mathbf{I} + (\text{SNR}/n_t)\mathbf{H}\mathbf{H}^{H}) < R]$ and the $\epsilon$-outage capacity $C_\epsilon$
• Derive the central space-time-code PEP upper bound $P(\mathbf{X} \to \hat{\mathbf{X}}) \leq \prod_{i=1}^{r} (1 + \tfrac{\text{SNR}}{4n_t}\lambda_i(\boldsymbol{\Delta}\boldsymbol{\Delta}^H))^{-n_r}$ from the Chernoff-bound MGF machinery
• State and prove the rank criterion (Tarokh-Seshadri-Calderbank 1998): the diversity order of a space-time code is $r \cdot n_r$ where $r = \min_{\mathbf{X} \ne \hat{\mathbf{X}}} \mathrm{rank}(\boldsymbol{\Delta})$; maximum diversity $n_t n_r$ requires $\boldsymbol{\Delta}$ full rank for every codeword pair
• State and prove the determinant criterion: given full-rank codes, the coding gain is the minimum determinant $\min \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ — doubling it shifts the BER curve by $3n_r/n_t$ dB
• Reason about the rank-vs-determinant trade-off, the asymptotic (high-SNR) nature of the criteria, and their role as a forward guide to Chs. 11 (STBCs) and 12 (DMT)