Prerequisites & Notation

Before You Begin

This chapter constructs the central family of linear space-time codes — Alamouti, OSTBCs, QOSTBCs, LDCs — from the design criteria of Chapter 10. The reader should be comfortable with the quasi-static MIMO channel model, the rank and determinant criteria, and the basic matched-filter / ML-decoder analysis of Chapter 10. A working knowledge of the Hurwitz-Radon theorem on sums-of-squares identities is useful but will be reviewed where needed.

Quasi-static MIMO channel model $\mathbf{Y} = \mathbf{H}\mathbf{X} + \mathbf{w}$ over a coherence block of length $T$ (Review ch10)
Self-check: Can you write the block-wise MIMO model with codeword matrix $\mathbf{X} \in \mathbb{C}^{n_t \times T}$ , channel $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ , and noise $\mathbf{w} \in \mathbb{C}^{n_r \times T}$ with i.i.d. $\mathcal{CN}(0, \sigma^2)$ entries?
Rank and determinant criteria for space-time codes(Review ch10)
Self-check: Can you state the pairwise error probability bound $P_e(\mathbf{X}\to\hat{\mathbf{X}}) \le \prod_{i=1}^{r} (1 + \text{SNR}\lambda_i(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)/(4 n_t))^{-n_r}$ where $\boldsymbol{\Delta} = \mathbf{X} - \hat{\mathbf{X}}$ , and identify the diversity order as $r \cdot n_r$ where $r = \mathrm{rank}(\boldsymbol{\Delta})$ ?
Matched-filter and zero-forcing receivers for MIMO(Review ch10)
Self-check: Can you write the sufficient statistic $\mathbf{H}^{H} \mathbf{y}$ and explain in what sense it is sufficient for ML detection over $\mathbf{X}$ when $\mathbf{X}$ has an orthogonal-column structure?
Maximal-ratio combining (MRC) for $1 \times n_r$ SIMO(Review ch10)
Self-check: Can you state that an MRC receiver with $n_r$ receive antennas sees effective SNR $\text{SNR}\sum_{k=1}^{n_r}|h_k|^2$ — i.e., $n_r$ -fold diversity and an $n_r$ -fold array gain? (This is the benchmark Alamouti compares against.)
V-BLAST (vertical Bell-Labs layered space-time) architecture(Review ch10)
Self-check: Can you state that uncoded V-BLAST with successive interference cancellation achieves full multiplexing $\min(n_t, n_r)$ symbols per channel use but diversity only $n_r - n_t + 1$ per layer? It is the rate-maximising, diversity-minimising counterpoint to Alamouti.
Hurwitz-Radon number $\rho(n)$ and sums-of-squares identities
Self-check: Do you recall that the Hurwitz-Radon theorem limits the number of mutually anti-commuting orthogonal real matrices of order $n$ , and that this underlies the non-existence of rate-1 complex OSTBCs for $n_t > 2$ ? A brief review is given in RReview: The Hurwitz-Radon Sum-of-Squares Identity.

Notation for This Chapter

Symbols specific to space-time block codes. Chapter 10 MIMO notation (channel $\mathbf{H}$ , noise $\mathbf{w}$ , SNR $\text{SNR}$ , $n_t, n_r$ ) continues to apply and is not repeated here.

Symbol	Meaning	Introduced
$T$	Space-time block length (number of channel uses over which one STBC codeword is transmitted)	s01
$\mathbf{X} \in \mathbb{C}^{n_t \times T}$	Space-time codeword matrix: rows index transmit antennas, columns index time slots	s01
$\mathbf{X}_A$	Alamouti codeword matrix, $\mathbf{X}_A = \begin{pmatrix} s_1 & -s_2^* \\ s_2 & s_1^* \end{pmatrix}$	s01
$\boldsymbol{\Delta} = \mathbf{X} - \hat{\mathbf{X}}$	Error matrix between two codewords (used in rank/determinant PEP analysis)	s01
$K$	Number of complex information symbols carried by one STBC codeword	s02
$R_{\mathrm{OSTBC}} = K / T$	Rate of a space-time block code, in complex symbols per channel use	s02
$\mathbf{A}_q, \mathbf{B}_q$	Real dispersion matrices for the real and imaginary parts of the $q$ -th information symbol in an LDC / OSTBC expansion	s02
$Q$	Number of dispersion matrices in a linear dispersion code, equal to the number of real dimensions $2K$ spanned by the codebook	s05
$\mathcal{H}(n)$	Hurwitz-Radon number of $n$ , i.e., the maximum number of mutually anti-commuting orthogonal $n\times n$ real matrices plus one	s03
$\mathcal{O}$	Orthogonal design / Hurwitz-Radon family used to build OSTBCs	s02
$d_{\min}^2(\boldsymbol{\Delta})$	Squared minimum Frobenius distance over the codebook, $\min_{\mathbf{X}\ne\hat{\mathbf{X}}}\\|\boldsymbol{\Delta}\\|_F^2$	s01
$\mathbf{I}_n$	$n \times n$ identity matrix	s01

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