Chapter 12: The Diversity-Multiplexing Tradeoff

Learning Objectives

• State the asymptotic definitions of diversity gain $d^*$ and multiplexing gain $r$ for a MIMO block-fading channel, and explain why they are the two natural high-SNR resources competing inside the outage-probability exponent
• Prove the Zheng-Tse theorem: for an $n_t \times n_r$ i.i.d. Rayleigh channel with block length $L \ge n_t + n_r - 1$, the diversity-multiplexing tradeoff curve is the piecewise-linear interpolation $d^*(r) = (n_t - r)(n_r - r)$, $r \in \{0, 1, \ldots, \min(n_t, n_r)\}$
• Interpret the DMT as the fundamental constraint every space-time code designer navigates: every unit of multiplexing gain costs diversity, and the cost function is quadratic at the endpoints and linear between corner points
• Classify the classical space-time codes by their $(r, d)$ operating points on the DMT curve: Alamouti at $(1, 2 n_r)$, V-BLAST-ZF at $(\min(n_t, n_r), n_r - n_t + 1)$, V-BLAST-ML at $(\min(n_t, n_r), n_r)$, and the Golden / CDA codes as DMT-optimal for all $r$
• Refine the basic DMT statement for short block length $L < n_t + n_r - 1$ (tradeoff truncation) and for spatially correlated fading (coding-gain reduction without DMT-exponent loss for full-rank correlation)
• Use the exponential-equality notation $\doteq$ fluently and distinguish DMT statements (asymptotic exponents) from finite-SNR statements (coding gains, moderate-SNR slopes)