Exercises

$n_t^* = \min(2, 4, 3) = 2$.

$n_t^*(1 - n_t^*/T) = 2(1 - 2/6) = 2 \cdot 2/3 \approx 1.33$.

Coherent pre-log = $\min(n_t, n_r) = 2$. Non-coherent loses 33% of the high-SNR degrees of freedom.

$\rho = 10$; $C = \frac{1}{2}\log_2(1 + 10) = 1.73$ bits/use.

$V = 10 \cdot 12 / (2 \cdot 121) \cdot (\log_2 e)^2 = 0.496 \cdot 2.081 = 1.03$.

$R^* \approx 1.73 - \sqrt{1.03/300} \cdot 4.75 = 1.73 - 0.278 = 1.45$ bits/use. 16% below Shannon.

$P_{\rm opt} = (10^{-5}/(2 \cdot 10^{-3}))^{1/3} = (5 \cdot 10^{-3})^{1/3} \approx 0.171$ mW $\approx -7.7$ dBm.

Plug into formula: $\mathrm{SNR}^* = P_{\rm opt}/(3 P_{\rm ASE}/2) \approx 11{,}400$ ($\sim 40.6$ dB).

$\log_2(1 + 11400) \approx 13.5$ bits/symbol.

The learned constellation geometry is optimised for the specific training HPA parameters. Different $p$ and IBO change the distortion function, so the compressed constellation is no longer matched.

The autoencoder learned to REDUCE outer points — a general principle that still helps on any nonlinear HPA, but with smaller gain when the specific nonlinearity doesn't match.

Train with domain randomisation (sample $p \in [1.2, 2.5]$ and IBO $\in [1.5, 3.5]$ dB during training) to make the learned constellation robust across the deployment envelope.

$X \sim \mathcal{N}(0, \rho)$, $Y = X + Z$, $Z \sim \mathcal{N}(0, 1)$. $f(Y|X) = \phi(Y - X; 0, 1)$, $f(Y) = \phi(Y; 0, 1+\rho)$.

$i(X;Y) = \log f(Y|X) - \log f(Y) = -\frac{1}{2}\log(1) - \frac{(Y-X)^2}{2} + \frac{1}{2}\log(1+\rho) + \frac{Y^2}{2(1+\rho)}$.

Direct computation of $\mathrm{Var}(i(X;Y))$ using $X, Z$ Gaussian: $V = \rho(\rho + 2) / (2(1+\rho)^2)$ (in nats). Multiply by $(\log_2 e)^2$ to convert to bits$^2$.

$n_t^* = \min(2, 2, \lfloor 4/2 \rfloor) = 2$.

Non-coherent: $2 (1 - 2/4) = 1$. Coherent: $2$.

Half the DoF are lost for $T = 2 n_t$. To recover the coherent rate, the system would need longer coherence time ($T \gg 2 n_t$).

The normal approximation matches the meta-converse upper bound up to $O(\log n/n)$. At $n \to \infty$: $R^*(n, \epsilon) = C - \sqrt{V/n} Q^{-1}(\epsilon) + \frac{1}{2n}\log n + O(1/n)$.

At $n = 100$, the $O(\log n/n)$ term is $\approx 0.023$ bits/use — comparable to the SNR-dependent $\sqrt{V/n} Q^{-1}$ term. More accurate bounds (Polyanskiy-Wang 2010 "$\kappa\beta$ achievability") retain this correction.

For rigorous URLLC design, use the explicit meta-converse bound or a tight saddle-point approximation. Normal approx is quick but can be off by 0.1-0.2 bits/use.

Shannon's AWGN theorem assumes LINEAR channel with MEMORYLESS Gaussian noise. Kerr nonlinearity violates linearity; dispersion violates memorylessness.

The GN model (Essiambre 2010) approximates the nonlinearity as ADDITIVE Gaussian noise with power proportional to $P^3$. Under this approximation, Shannon's $\log(1 + \mathrm{SNR}_{\rm eff})$ applies — but the effective SNR PEAKS and then decreases.

At higher launch powers, the GN model fails: nonlinear interference is NOT Gaussian, and capacity is likely higher than GN predicts (digital back-propagation exploits this). The true capacity is an open research question.

At 5 dB training SNR, the autoencoder may learn a constellation with SLIGHTLY reduced minimum distance (at low SNR, fewer points in dense regions matter).

Hand-designed 16-QAM benefits from the 5 dB SNR boost fully. The autoencoder's trained-at-low-SNR constellation may have suboptimal minimum distance, so the BER gain at 10 dB is less than at 5 dB.

Typically: the autoencoder's AWGN gain DIMINISHES as the deployment SNR diverges from the training SNR. For best robustness, train over a range of SNRs. This is "SNR-robust autoencoder training" (Cammerer et al. 2019).

$d^*(r) = (n_t - r)(n_r - r)$ for integer $r \in [0, \min(n_t, n_r)]$.

Zheng-Tse (2002) show that for $T \ge n_t + n_r - 1$, the non-coherent DMT matches the coherent DMT: no penalty.

When the coherence block is long enough, the receiver can "estimate" the channel essentially for free by sacrificing $n_t$ of the $T$ slots to pilots. The remaining $T - n_t$ slots achieve the coherent rate. If $T \ge n_t + n_r$, the pilot overhead becomes negligible in the high-SNR DMT exponent. $\blacksquare$

Assume 80 km/span. Full-link $\eta_{\rm NL}^{\rm full} = N_{\rm span} \cdot \eta_{\rm NL}$, $P_{\rm ASE}^{\rm full} = N_{\rm span} \cdot P_{\rm ASE}$.

PAS-shaped 64-QAM: ~5.5 bits/symbol/pol. Dual-pol: 11 bits/symbol. Need $\mathrm{SNR}^* = 2^{5.5} - 1 \approx 44$ at peak.

$P_{\rm opt} = (P_{\rm ASE}^{\rm full}/(2\eta_{\rm NL}^{\rm full}))^{1/3}$ is INDEPENDENT of $N_{\rm span}$ (both scale linearly). But $\mathrm{SNR}^* = P_{\rm opt}/(3 P_{\rm ASE}^{\rm full}/2) \propto 1/N_{\rm span}^{2/3}$.

Target $\mathrm{SNR}^* = 44$ requires $N_{\rm span} \approx [11400/44]^{3/2} = 260^{3/2} \approx 4200$. But each span is 80 km, so reach $\approx 80$ km (1 span) — 800G is metro only under this crude model. (Real systems with DBP reach 200-400 km.)

Minimising $\|\hat{s} - s\|^2$ over the encoder/decoder pair produces Gaussian-like signalling — close to Shannon's capacity but SOFT bit decisions instead of hard detection.

Minimising $-\sum_i p_i \log q_i$ over symbols gives a DISCRETE constellation (because one-hot labels guide the output toward argmax behaviour). The learned constellation resembles QAM in shape with Gray labelling.

For COMMUNICATION with a hard-decision output, cross-entropy gives the "right" geometry (discrete constellation with Gray). For SOFT-decoded channel coding, MSE can be competitive. In practice, CE is preferred.

The pre-log gap is $n_t^* \cdot n_t^*/T = (n_t^*)^2/T$ — a loss of $(n_t^*)^2/T$ bits/channel use per $\log\text{SNR}$ at high SNR.

In addition, non-coherent decoding incurs a $O(1)$ offset in the log-rate due to the random decoder not knowing which Grassmannian direction the signal takes.

$\DeltaC = (n_t^*)^2/T \cdot \log\text{SNR} + O(1)$. As $T \to \infty$, this vanishes — non-coherent converges to coherent. For small $T$, the penalty is significant.

For non-coherent STC, finite-blocklength URLLC, autoencoder codes, and optical fibre, the design criteria are still being formulated. We have BOUNDS (Polyanskiy, Marzetta-Hochwald, nonlinear Shannon peak) but not CONSTRUCTIVE theorems.

A mature design chapter for each area would include: (1) a formal design criterion (like rank+det for STC); (2) an explicit construction (like CDA for DMT); (3) a deployed example. Today we have (1) in some areas, (2) partially, and (3) only for PAS in optical.

Ch 22 is a research survey BECAUSE the field is still being built. The book ends here precisely where the reader would need to contribute to close the loop.

PROBABILISTICALLY-SHAPED CDA CODES: extend Ch 13's CDA framework (DMT-optimal, non-vanishing determinant) to support probabilistic input distributions (Ch 19 PAS). The key question: does the NVD property extend when symbol probabilities are non-uniform?

Requires: (a) algebraic analysis of CDA codeword determinants under non-uniform symbol distributions; (b) analogue of the approximate-universality theorem for shaped inputs; (c) finite- SNR performance comparisons with hand-designed shaped constellations.

A shaped CDA code would unify the MIMO and single-carrier paradigms: DMT-optimal (like Golden code) + near-capacity (like PAS). This would redefine MCS design for high-SNR MIMO (Wi-Fi 7, 5G mmWave, optical PAS).

Other natural combinations: (i) LAST codes for URLLC (short blocks + DMT-optimality); (ii) autoencoder-learned CDA initialisation (neural search over the CDA manifold); (iii) ARQ- LAST for non-terrestrial networks (long RTT URLLC + MIMO).