Prerequisites & Notation

Before You Begin

This chapter opens Part III — Space-Time Coding. From this chapter onwards, $\mathbf{H}$ genuinely denotes the MIMO channel matrix (not the Hadamard matrix of Part I or the parity-check matrix of coding-theoretic chapters). The treatment rests on three pillars built in earlier parts and in Book ITA:

the MIMO capacity formula of Telatar 1995/1999 (Book ITA Ch. 13.5),
the Chernoff bound + MGF-of- $\chi^2$ technique used for PEP in Ch. 2 (CM on AWGN) and Ch. 6 (BICM on fading), now applied to the $r$ -dimensional diversity branch structure, and
the quasi-static (block-fading) channel model — a fading realisation that is random but constant over the entire codeword.

Read Book ITA Ch. 13.5 if the i.i.d. Rayleigh MIMO capacity derivation feels unfamiliar. Read Ch. 5 of this book if the block-fading / Rayleigh-fading terminology is not second nature. The PEP derivation in §3 builds on Chs. 2 and 6 — if the Chernoff + MGF template is rusty, a quick pass through Ch. 2 §3 is recommended.

MIMO capacity formula $C = \mathbb{E}[\log\det(\mathbf{I} + (\text{SNR}/n_t)\mathbf{H}\mathbf{H}^{H})]$ (Review ch13)
Self-check: Can you derive the i.i.d. Rayleigh ergodic MIMO capacity via the SVD of $\mathbf{H}$ and water-filling over the squared singular values? Can you identify the high- $\text{SNR}$ prelog as $\min(n_t, n_r)$ ?
Chernoff bound and MGF of a central $\chi^2$ random variable(Review ch02)
Self-check: Given a sum $S = \sum_{i=1}^r |z_i|^2$ with $z_i \sim \mathcal{CN}(0, 1)$ i.i.d., can you write down the MGF $M_S(t) = (1-t)^{-r}$ for $t<1$ and use it to produce a Chernoff bound of the form $\Pr[S < \gamma] \le e^{-r\psi(\gamma)}$ ?
Pairwise error probability via the Q-function and MGF method(Review ch02)
Self-check: Can you derive the Q-function PEP on AWGN from the conditional Gaussian noise model, then average over Rayleigh fading to obtain the $(1 + \text{SNR}/4)^{-1}$ form at diversity one?
Rayleigh fading and block-fading channel models(Review ch05)
Self-check: Can you state the quasi-static (block-fading) model $\mathbf{y}_t = \mathbf{H}\mathbf{x}_t + \mathbf{w}_{t}$ for $t = 1, \ldots, T$ with $\mathbf{H}$ constant across the block, and contrast it with the fully-ergodic (fast-fading) model?
Singular value decomposition and eigenvalues of $\mathbf{A}\mathbf{A}^H$ (Review ch01)
Self-check: Given $\mathbf{A} \in \mathbb{C}^{m\times n}$ with $r = \mathrm{rank}(\mathbf{A})$ , can you relate the nonzero eigenvalues $\lambda_i(\mathbf{A}\mathbf{A}^H)$ to the squared singular values $\sigma_i^2(\mathbf{A})$ and state $\det(\mathbf{A}\mathbf{A}^H) = \prod_{i=1}^r \lambda_i$ when $m = r$ ?
Ergodic vs. outage capacity on a slowly varying fading channel(Review ch05)
Self-check: Can you explain why, on a block-fading channel whose coherence time exceeds the target codeword length, the meaningful capacity notion is $P_{\mathrm{out}}(R) \triangleq \Pr[C(\mathbf{H}) < R]$ rather than the ergodic average?

Notation for This Chapter

Symbols specific to this chapter's space-time code analysis. MIMO system- level notation ( $\mathbf{H}$ for the channel matrix, $n_t, n_r$ for antenna counts, $\text{SNR}$ for the total transmit SNR) is assumed from Book ITA Ch. 13.5 and the global notation table. We adopt the convention that the codeword matrix is $\mathbf{X} \in \mathbb{C}^{n_t \times T}$ (columns are time samples) and NOT $\mathbf{C}$ , to avoid collision with capacity.

Symbol	Meaning	Introduced
$\mathbf{X}, \hat{\mathbf{X}}$	Transmitted and (erroneous) decoded space-time codeword matrices, each in $\mathbb{C}^{n_t \times T}$	s03
$\boldsymbol{\Delta} = \mathbf{X} - \hat{\mathbf{X}}$	Codeword-pair error matrix, $\boldsymbol{\Delta} \in \mathbb{C}^{n_t \times T}$	s03
$T$	Space-time codeword block length (number of channel uses per codeword)	s03
$r$	Rank of the error matrix: $r = \mathrm{rank}(\boldsymbol{\Delta}) \le \min(n_t, T)$	s03
$\lambda_i(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$	The $i$ -th nonzero eigenvalue of the $n_t \times n_t$ PSD matrix $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$ , $i = 1, \ldots, r$	s03
$P(\mathbf{X} \to \hat{\mathbf{X}})$	Pairwise error probability of decoding $\hat{\mathbf{X}}$ when $\mathbf{X}$ was transmitted	s03
$d$	Diversity order; for a space-time code, $d = \min_{\mathbf{X}\ne\hat{\mathbf{X}}} r \cdot n_r$	s04
$\gamma_c$	Coding gain; for a full-rank code, $\gamma_c = (\min_{\mathbf{X}\ne\hat{\mathbf{X}}} \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H))^{1/n_t}$	s05
$P_{\mathrm{out}}(R)$	Outage probability at target rate $R$ , $P_{\mathrm{out}}(R) = \Pr[\log\det(\mathbf{I} + (\text{SNR}/n_t)\mathbf{H}\mathbf{H}^{H}) < R]$	s02
$C_\epsilon$	$\epsilon$ -outage capacity: largest $R$ such that $P_{\mathrm{out}}(R) \le \epsilon$	s02
$\eta$	Spectral efficiency (bits/s/Hz) of a space-time code, $\eta = \log_2\|\mathcal{X}\|/T$	s03
$\\|\cdot\\|_F$	Frobenius norm; $\\|\boldsymbol{\Delta}\\|_F^2 = \mathrm{tr}(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) = \sum_i \lambda_i(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$	s03

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