Code Operating Points on the DMT

The Alamouti scheme on $2 \times n_r$ transforms the MIMO channel into $2 n_r$ parallel scalar Rayleigh channels with equal gain: after Alamouti decoding, each transmitted symbol sees effective SNR $(\sum_{i, j} |h_{i, j}|^2) \text{SNR} / 2$, where the sum is over $2 n_r$ i.i.d. $\mathcal{CN}(0, 1)$ variables. The sum-of-squared- magnitudes is chi-square with $2 n_r$ complex degrees of freedom.

Let $X = \sum_{i = 1}^{2 n_r} |h_i|^2$ with $h_i \overset{iid}{\sim} \mathcal{CN}(0, 1)$. Then $X$ has Gamma density with shape $2n_r$: $$p(X = x) \;=\; \frac{x^{2 n_r - 1} e^{-x}}{(2 n_r - 1)!}, \quad x \ge 0.$$ At rate $R(\text{SNR}) = r \log_2 \text{SNR}$, the outage event is $\{\log_2(1 + X \text{SNR}/2) < r \log_2 \text{SNR}\}$, i.e., $\{X < \text{SNR}^{r-1}\}$. The CDF at small argument is $\text{SNR}^{(r-1) 2 n_r}$, so $P_{\rm out} \doteq \text{SNR}^{-2 n_r (1 - r)^+}$.

On $2 \times 2$ ($n_r = 2$): $d_{\rm Alamouti}(r) = 4(1 - r)^+$; $d^*(r)$ is the piecewise-linear interpolation of $(0, 4), (1, 1), (2, 0)$. At $r = 0$: both $= 4$ (Alamouti optimal). At $r = 1$: Alamouti $= 0$ but $d^*(1) = 1$ (Alamouti sub-optimal — at this rate it has zero reliability, not one). At $r = 0.5$: Alamouti $= 2$, $d^*(0.5) = 4 - 3 \cdot 0.5 = 2.5$ (Alamouti below curve).

Conclusion. Alamouti is DMT-optimal only at $r = 0$. For any $r > 0$ it is strictly sub-optimal: the Alamouti line $4(1 - r)$ is a chord of the DMT curve from $(0, 4)$ to $(1, 0)$, which is below the piecewise linear curve through $(1, 1)$. The gap is the structural "rate cost" of one-stream transmission. $\blacksquare$

After zero-forcing, the $k$-th stream sees effective SNR $\text{SNR} / [(\mathbf{H}^{H}\mathbf{H})^{-1}]_{kk}$. The diagonal of $(\mathbf{H}^{H}\mathbf{H})^{-1}$ is chi-square distributed with $2(n_r - n_t + 1)$ complex degrees of freedom, so each stream sees a scalar Rayleigh channel with diversity $n_r - n_t + 1$.

At stream rate $R_k(\text{SNR}) = (r / n_t) \log_2 \text{SNR}$, the per-stream scalar DMT argument gives stream outage exponent $(n_r - n_t + 1)(1 - r/n_t)^+$. With $n_t$ independent streams (no coding across them), the block outage event is dominated by any one stream's outage — but the exponent is still $(n_r - n_t + 1)(1 - r/n_t)^+$ (the $n_t$ independent streams are a union-bound prefactor, invisible to $\doteq$).

ML detection does not project onto the null space; it jointly optimises over all $n_t$ streams. Each stream sees the full channel matrix, with effective diversity equal to the minimum of the eigenvalues of $\mathbf{H}\mathbf{H}^{H}$ (limiting factor in a joint-detection outage event at stream rate $r/n_t$).

The minimum eigenvalue of $\mathbf{H}\mathbf{H}^{H}$ is chi-square with $2(n_r - n_t + 1)$ degrees of freedom (generalized chi-square tail behaviour). But the ML joint outage region is larger than the ZF per-stream region, and a careful Wishart computation gives $$d_{\rm ML}(r) = n_r (1 - r/n_t)^+.$$ At $r = 0$: $d_{\rm ML}(0) = n_r$ (= receive diversity of a single stream with the full receive array). At $r = n_t$: $d_{\rm ML}(n_t) = 0$ — V-BLAST-ML does touch the DMT right endpoint, which is why V-BLAST "achieves the multiplexing gain" in the Foschini sense.

On $2 \times 2$: $d_{\rm ZF}(r) = 1(1 - r/2)^+$, $d_{\rm ML}(r) = 2(1 - r/2)^+$, $d^*(r) = \max(4 - 3r, 2 - r, 0)$ piecewise-linear.

• At $r = 0$: $d_{\rm ZF} = 1$, $d_{\rm ML} = 2$, $d^* = 4$ — ML gets receive diversity but misses transmit diversity; both miss the full DMT.
• At $r = 2$: $d_{\rm ZF} = 0$, $d_{\rm ML} = 0$, $d^* = 0$ — all three codes touch here.
• At $r = 1$: $d_{\rm ZF} = 0.5$, $d_{\rm ML} = 1$, $d^* = 1$ — V-BLAST-ML does touch $d^*(1)$ at this specific midpoint by coincidence (a numerical coincidence in $2 \times 2$, not a general pattern).

$\blacksquare$