Chapter Summary

Chapter Summary

Key Points

• 1.

  Diversity and multiplexing are two distinct asymptotic resources. On a block-fading MIMO channel at high SNR, the diversity gain $d^* = -\lim \log P_e / \log \text{SNR}$ measures reliability slope at fixed rate, and the multiplexing gain $r = \lim R(\text{SNR}) / \log_2 \text{SNR}$ measures rate slope at fixed reliability. Alamouti is full-diversity-zero-multiplexing $(r = 0, d = 2 n_r)$; V-BLAST is full-multiplexing-low-diversity $(r = \min(n_t, n_r), d$ small$)$.

• 2.

  The Zheng-Tse theorem binds the two resources. For an $n_t \times n_r$ i.i.d. Rayleigh channel with block length $L \ge n_t + n_r - 1$, the optimal tradeoff is $d^*(r) = (n_t - r)(n_r - r)$ at integer corners $r \in \{0, 1, \ldots, \min(n_t, n_r)\}$, linearly interpolated between (Thm. TZheng-Tse Diversity-Multiplexing Tradeoff). The proof combines an outage-exponent converse (Wishart large deviations) with a Gaussian random-coding achievability — the same pattern as Shannon's theorem, lifted to the exponential-equality regime.

• 3.

  Every unit of multiplexing costs diversity. The initial slope of $d^*(r)$ at $r = 0$ is $-(n_t + n_r - 1)$ (steep — the first unit is expensive); the final slope at $r = r_{\max}^-$ is $-(|n_t - n_r| + 1)$ (shallow — the last unit is cheap). Between corner points the cost decreases linearly in increments of $2$, a combinatorial consequence of the Wishart Vandermonde factor.

• 4.

  Classical codes are mostly DMT-sub-optimal. Alamouti achieves $d^*(0) = 2 n_r$ but drops off the DMT curve for any $r > 0$ (its operating trace is a chord from $(0, 2 n_r)$ to $(1, 0)$). V-BLAST-ZF gives $(n_r - n_t + 1)(1 - r/n_t)$ — a shallow chord below the DMT. V-BLAST-ML gives $n_r (1 - r/n_t)$ — better but still below. The DMT curve is achieved in full only by algebraically-structured codes: the Golden code on $2 \times 2$ (Belfiore-Rekaya-Viterbo 2005), the CDA / Perfect codes on $n \times n$ (Elia et al. 2006), and the LAST codes on arbitrary $n_t \times n_r$ (El Gamal-Caire-Damen 2004).

• 5.

  DMT is symmetric in $(n_t, n_r)$. The curve $d^*(r)$ is unchanged under $(n_t, n_r) \leftrightarrow (n_r, n_t)$ because the nonzero eigenvalues of $\mathbf{H}\mathbf{H}^{H}$ and $\mathbf{H}^{H}\mathbf{H}$ coincide. A $2 \times 4$ channel and a $4 \times 2$ channel have identical DMT curves, even though their physical uplink / downlink asymmetry differs significantly.

• 6.

  Rank adaptation = walking the DMT curve. LTE and 5G NR receivers report a rank indicator $k \in \{1, \ldots, \min(n_t, n_r, 8)\}$ to the base station; this is operationally the target multiplexing gain $r = k$, and the scheduler navigates the DMT curve $(k, (n_t - k) (n_r - k))$ in response to measured SNR. High SNR $\to$ high rank $\to$ more multiplexing; low SNR $\to$ low rank $\to$ more diversity.

• 7.

  Short block length truncates the DMT. For $L < n_t + n_r - 1$, the DMT curve is truncated at $r = L$ (Thm. TDMT Truncation for Short Block Length): no coding scheme can support multiplexing gain above the coherence-time block length. This matters at mmWave high mobility where coherence time can be $\sim 1$ OFDM symbol, capping effective $r_{\max}$ below $\min(n_t, n_r)$.

• 8.

  Full-rank correlation preserves DMT. For any Tx correlation matrix with $\det \mathbf{R}_t > 0$, the DMT exponent $d^*(r)$ is unchanged from the i.i.d. case (Thm. TDMT Invariance under Full-Rank Tx Correlation). Coding gain degrades by $(\det \mathbf{R}_t)^{-1/n_t}$ (Hadamard), typically $1$–$3$ dB in practical 5G NR scenarios. Rank-deficient correlation ($\det = 0$) does degrade the exponent — reducing the effective number of spatial degrees of freedom.

• 9.

  DMT is asymptotic; coding gain matters at finite SNR. Two codes with the same $d^*(r)$ can differ by several dB at moderate SNR (10–30 dB) due to different multiplicative constants in front of $\text{SNR}^{-d^*}$. The DMT is a first-order design criterion — it tells you which $(r, d)$ corner to target. Choosing between codes at the same corner requires finite-SNR analysis (determinant criterion, Monte Carlo) or the explicit coding-gain bounds of Chapter 13.