Chapter Summary

Chapter Summary

Key Points

• 1.

  MIMO capacity (Telatar 1999 review). On an $n_t \times n_r$ i.i.d. Rayleigh MIMO channel with CSIR and isotropic input, the ergodic capacity is $C = \mathbb{E}[\log\det(\mathbf{I}_{n_r} + (\text{SNR}/n_t)\mathbf{H}\mathbf{H}^{H})]$ (Thm. TErgodic MIMO Capacity (Telatar 1999)), with high-SNR prelog $\min(n_t, n_r)\log_2\text{SNR}$. A $4\times 4$ system at $10$ dB achieves $\sim 10.9$ bits/channel use — $3.2\times$ the SISO rate.

• 2.

  Block-fading and outage capacity. On a quasi-static channel, the capacity $C(\mathbf{H}) = \log\det(\mathbf{I} + (\text{SNR}/n_t) \mathbf{H}\mathbf{H}^{H})$ is random and the operational benchmark is the outage capacity $C_\epsilon$ (Def. $\epsilon$-Outage Capacity" data-ref-type="definition">DOutage Probability and $\epsilon$-Outage Capacity), the $\epsilon$-quantile of this random variable. At $10$ dB and $\epsilon = 0.1$, the $2 \times 2$ Rayleigh outage capacity is $\sim 3.9$ bits — about $68\%$ of the ergodic $5.7$ bits. At high SNR, $P_{\mathrm{out}}(r\log_2\text{SNR}) \doteq \text{SNR}^{-d^\star(r)}$ (Thm. $P_{\mathrm{out}}(R)$ (DMT Preview)" data-ref-type="theorem">THigh-SNR Scaling of $P_{\mathrm{out}}(R)$ (DMT Preview), the Zheng-Tse DMT, with $d^\star(0) = n_t n_r$).

• 3.

  Space-time PEP bound. On an i.i.d. Rayleigh block-fading channel, the pairwise error probability under ML decoding satisfies $P(\mathbf{X}\to\hat{\mathbf{X}}) \le \prod_{i=1}^r (1 + (\text{SNR}/4n_t)\lambda_i(\boldsymbol{\Delta}\boldsymbol{\Delta}^H))^{-n_r}$ (Thm. TSpace-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999)). The derivation reuses the Chernoff

  • MGF-of-$\chi^2$ template of Chs. 2 and 6, now applied to the $r$-dimensional eigen-structure of $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$. At high SNR, this simplifies to $(\text{SNR}/4n_t)^{-r n_r} \prod_i\lambda_i^{-n_r}$.
• 4.

  Rank criterion. The diversity order of a space-time code is $d = r_{\min} n_r$, where $r_{\min} = \min_{\mathbf{X}\ne \hat{\mathbf{X}}} \mathrm{rank}(\boldsymbol{\Delta})$ (Thm. TRank Criterion (Tarokh-Seshadri-Calderbank 1998), Tarokh-Seshadri-Calderbank 1998). Maximum diversity $n_t n_r$ requires full rank for every codeword pair. This is the designer's first objective: without full rank, no amount of coding-gain optimisation can recover the lost slope. A rank-$1$ code (e.g., spatial repetition) achieves only $n_r$ diversity — half of the maximum at $n_t = 2$.

• 5.

  Determinant criterion. Among full-rank codes, the coding gain is $\gamma_c = \det_{\min}(\mathcal{C})^{1/n_t}$ (Thm. TDeterminant Criterion (Tarokh-Seshadri-Calderbank 1998)), with $\det_{\min} = \min \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ over codeword pairs. Doubling $\det_{\min}$ shifts the BER curve left by $3 n_r/n_t$ dB. Alamouti achieves $\det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) = (|e_1|^2 + |e_2|^2)^2$ — constant-determinant on every one- symbol-error pair.

• 6.

  Asymptotic honesty. The rank and determinant criteria are high-SNR asymptotic statements derived from a union bound that is loose at low SNR (often meaningless there). They are the correct tool for code design at target BERs in the $10^{-3}$–$10^{-6}$ range, where the dominant PEP pair drives the error probability. At very low SNR the bound breaks down and Monte-Carlo simulation is required.

• 7.

  The link to 2026 standards. LTE TM3/TM4 and 5G NR codebook precoding enforce the rank criterion at the layer level: the PMI codebooks are rank-preserving. Transmit diversity modes (SFBC = OFDM Alamouti) are full-rank by construction and guarantee $n_t n_r$ diversity. The theoretical framework of 1998 (Tarokh-Seshadri- Calderbank, Alamouti, Guey-Fitz-Bell-Kuo) maps directly onto the contemporary 3GPP design language.

• 8.

  Forward pointers within Part III. Ch. 11 constructs full-rank codes with clean determinant structure (Alamouti, OSTBCs, linear dispersion codes). Ch. 12 generalises the rank criterion's fixed-rate diversity to the entire Zheng-Tse $(r, d^\star(r))$ tradeoff. Ch. 13 provides the CommIT contribution by Elia-Kumar- Pawar-Kumar-Caire (2006): DMT-optimal cyclic-division-algebra constructions whose minimum determinant is bounded away from zero — the ultimate realisation of "rank + determinant" discipline.