Chapter 11: Space-Time Block Codes

Learning Objectives

• Write the Alamouti codeword matrix $\mathbf{X}_A$ for $n_t = 2$ and prove that its orthogonality $\mathbf{X}_A\mathbf{X}_A^H = (|s_1|^2 + |s_2|^2)\mathbf{I}_2$ yields full diversity $2 n_r$ at rate 1 symbol per channel use with linear matched-filter decoding
• State the Tarokh-Jafarkhani-Calderbank construction of orthogonal space-time block codes (OSTBCs) via the Hurwitz-Radon-Eckmann family of dispersion matrices and prove that every OSTBC attains diversity $n_t n_r$
• Derive the Liang-Tarokh upper bound on the rate of complex OSTBCs and explain why rate $R_{\mathrm{OSTBC}} \le 3/4$ is unavoidable for $n_t > 2$
• Define Jafarkhani's quasi-orthogonal STBC (QOSTBC) and quantify the rate-diversity-complexity trade-off it implements for $n_t = 4$
• State the Hassibi-Hochwald linear dispersion code (LDC) framework and prove that LDCs subsume Alamouti, OSTBCs, QOSTBCs, and V-BLAST — and that they can achieve the ergodic MIMO capacity as the number $Q$ of dispersion matrices grows
• Compare Alamouti, OSTBC, QOSTBC, V-BLAST, and LDC on the axes (rate, diversity, decoder complexity) and identify which design is appropriate for each practical regime