Exercises

Code: LDPC BG1 + BG2 (quasi-cyclic, $Z \in \{2, \ldots, 384\}$ lifting). Interleaver: the bit-level interleaver of TS 38.212 §5.4.2.2. Mapper: Gray-labelled QAM up to 1024-QAM ($Q_m = 10$). MCS Table 4 in Release 17.

Code: IEEE 802.11n LDPC family (648, 1296, 1944-bit codewords at rates 1/2, 2/3, 3/4, 5/6). Interleaver: the 802.11n bit interleaver. Mapper: Gray-labelled QAM up to 4096-QAM ($Q_m = 12$).

Code: LDPC (16200 or 64800 bits) concatenated with BCH outer code (rate $\sim 0.993$). Interleaver: the BICM interleaver of ETSI EN 302 307 §5.3.3. Mapper: APSK up to 256-APSK, with per-code-rate ring radii (Annex A Table A.3).

MCS 20 on Table 2: $Q_m = 6$ (64-QAM), $R = 616/1024 \approx 0.602$. So $\eta = 6 \cdot 0.602 = 3.61$ bits/2D.

$\text{SNR}^\star \approx 2^{(R + 0.5)} \cdot 0.8 = 2^{1.1} \cdot 0.8 = 1.72$... this formula is not dimensionally linear. More accurate: the BLER-10% threshold for this MCS is $\sim 14.2$ dB SNR (from NR deployment tables), giving a 2 dB gap to Shannon $\log_2(1 + 26.3) = 4.78$ bits/2D at 14.2 dB. The gap is typical for NR Table 2 at mid-range MCSs.

$A = 5000 > 292$, so do not use BG2 for "very short" reason.

$A = 5000 > 3824$, so do not force BG2 for "short block at moderate rate" reason.

$R = 0.75 > 0.25$, so do not force BG2 for "low rate" reason.

Default: BG1. This is appropriate — BG1 is optimised for long blocks at $R \in [0.5, 0.9]$, and $(A, R) = (5000, 0.75)$ is in that regime.

Bits per symbol: $8.333 \times 1960 \times 2 = 32{,}667$ info bits per 13.6 $\mu$s OFDM symbol.

$R_{\rm PHY} = 32667 / 13.6 \cdot 10^{-6} = 2.40$ Gbps per 2SS, 160 MHz, MCS 11 (short GI).

The equivalent 11ax rate (1024-QAM 5/6, 2 streams, 160 MHz, 0.8 $\mu$s GI) is 2.40 Gbps — same, because MCS 11 is available in both standards. Wi-Fi 7's doubling comes from MCS 13 at 4096-QAM plus the 320 MHz channel.

$\mathbb{E}|X|^2 = (4 \rho_1^2 + 12 \rho_2^2 + 16 \rho_3^2)/32 = 1$. Substituting: $\rho_1^2 (4 + 12\gamma_1^2 + 16\gamma_2^2) / 32 = 1$, so $\rho_1^2 = 32 / (4 + 12\gamma_1^2 + 16\gamma_2^2)$.

$C_{\rm BICM}(\gamma_1, \gamma_2, \text{SNR}) = \sum_{\ell = 0}^{4} I(Y; B_\ell)$, evaluated with the quasi-Gray labelling fixed by ETSI.

At the target BICM rate $\eta^\star = 5 R$ and the target SNR $\text{SNR}^\star$ (where BLER $= 10^{-1}$ for the LDPC code at rate $R$), solve $\max_{\gamma_1, \gamma_2 > 0} C_{\rm BICM}$ subject to $C_{\rm BICM} \ge 5R$. The ETSI Annex A Table A.3 tabulates the maximiser for each $R \in \{2/3, 3/4, 4/5, 5/6, 8/9, 9/10\}$.

$R = 3/4$: $(\gamma_1, \gamma_2) \approx (2.84, 5.27)$. $R = 9/10$: $(\gamma_1, \gamma_2) \approx (2.57, 4.33)$. Higher rate shrinks both ratios, moving the constellation closer to QAM geometry.

Throughput as a sum: $\eta_{\rm AMC} = \sum_i \eta_i \int_{\mathcal{A}_i} (1 - p_b^{(i)}(\gamma)) p_\Gamma(\gamma) d\gamma$, where the $\mathcal{A}_i$ partition $\mathbb{R}_+$.

Define $f^\star(\gamma) = \eta_{i^\star(\gamma)} (1 - p_b^{(i^\star(\gamma))}(\gamma))$. The throughput is $\int_0^\infty f^\star(\gamma) p_\Gamma(\gamma) d\gamma$.

Since the integrand at each $\gamma$ depends only on the choice $i^\star(\gamma)$ and not on the choices at other $\gamma$, the pointwise-maximising policy $i^\star(\gamma) = \arg\max_i \eta_i (1 - p_b^{(i)}(\gamma))$ maximises the entire integral. The condition is precisely that no cross-bin constraint (e.g. an average power budget) couples the bins.

If the transmitter had an average-power budget with higher-order QAM consuming more power, the pointwise-greedy policy would not be optimal — one would need water-filling over SNR bins. NR and Wi-Fi both assume fixed per-MCS power, so the greedy policy is optimal.

For nonnegative integer RV $K \ge 1$, $\mathbb{E}[K] = \sum_{k=1}^\infty \Pr(K \ge k)$. Here $\Pr(K \ge k)$ is the probability the block is still in error after $k-1$ transmissions, which is $p_{k-1}$.

Each successful transmission carries $\eta_0 N_s$ information bits (where $N_s$ is the number of symbols per block). Over many successful blocks we use $\mathbb{E}[K] N_s$ symbols per block, so $\eta_{\rm HARQ} = (\eta_0 N_s)/(\mathbb{E}[K] N_s) = \eta_0 / \mathbb{E}[K]$.

No HARQ ($K = 1$ always): $\eta_{\rm HARQ} = \eta_0$. Perfect HARQ (retransmit until success, first-try BLER $p_1$, no correlation): $\mathbb{E}[K] = 1/(1 - p_1)$, so $\eta_{\rm HARQ} = \eta_0 (1 - p_1)$. For small $p_1 = 0.1$, throughput loss is 10% — matching the eMBB design point.

$L(p) = -\sum_x p(x) \log p(x) - \lambda(\sum_x p(x)|x|^2 - E) - \mu(\sum_x p(x) - 1)$.

$\partial L / \partial p(x) = -\log p(x) - 1 - \lambda|x|^2 - \mu = 0 \Rightarrow p(x) = e^{-1-\mu} e^{-\lambda|x|^2}$.

Defining $Z(\lambda) = \sum_x \exp(-\lambda|x|^2)$ and $C = 1/Z(\lambda)$: $p_\lambda(x) = \exp(-\lambda|x|^2) / Z(\lambda)$.

$\mathbb{E}_{p_\lambda}[|X|^2] = -Z'(\lambda)/Z(\lambda) = - d \log Z / d\lambda = E$. This implicitly determines $\lambda(E)$, which is unique because $\log Z$ is strictly convex in $\lambda$. Strict concavity of entropy ensures the maximiser is unique. $\blacksquare$

At high SNR, $I(X;Y) \approx H(X) - H(X|Y) \to H(X)$. Shaping reduces the energy needed to achieve a fixed $H(X)$.

$X$ Gaussian in 2D with per-dim variance $\sigma_G^2$: $h(X) = \log_2(\pi e \sigma_G^2)$ (differential entropy, per 2D symbol).

$X$ uniform over square of side $a$ with per-dim variance $\sigma_U^2 = a^2/12$: $h(X) = \log_2 a^2 = \log_2(12 \sigma_U^2)$.

$\pi e \sigma_G^2 = 12 \sigma_U^2$, so $\sigma_U^2 / \sigma_G^2 = \pi e / 12$. Shaping gain = $10 \log_{10}(\sigma_U^2/\sigma_G^2) = 10 \log_{10}(\pi e / 12) \approx -1.53$ dB... wait, this gives a negative number meaning uniform needs MORE energy at equal entropy, i.e., shaping saves $1.53$ dB.

$G_s = \sigma_U^2 / \sigma_G^2 = \pi e / 12$... hmm, $\pi e / 12 \approx 0.711$, so $10 \log_{10}(12 / \pi e) \approx 1.53$ dB. This is the shaping gain in the operational sense: "uniform needs 1.53 dB more SNR than Gaussian-shaped to achieve the same rate". The often-quoted $\pi e / 6$ factor arises from the per-dimensional normalisation convention; they are equivalent in dB. $\blacksquare$

Max amplitude: $\rho_3 = 1.2782$. PAPR = $\rho_3^2 / 1 = 1.634$, or $2.13$ dB.

32-QAM cross constellation: points on $\{\pm 1, \pm 3, \pm 5\}^2$ excluding corners, normalised to unit energy. Max $|x|^2 = 5^2 + 3^2 = 34$ (corner near); average $|x|^2 = 20$. PAPR = $34/20 = 1.7$, or $2.3$ dB.

APSK has slightly lower PAPR than 32-QAM — 2.13 dB vs 2.3 dB. More importantly, most 32-APSK points (28 of 32) lie on the two outer rings at amplitudes 0.69 and 1.28, giving a tight amplitude distribution friendly to TWT operation. 32-QAM has a broader amplitude spread, worse under AM/AM non-linearity.

64-QAM points: $\pm 1, \pm 3, \pm 5, \pm 7$ per dimension, total 64 points. Normalise to unit average energy by dividing by $\sqrt{42}$ (since $\mathbb{E}[|x|^2] = (1 + 9 + 25 + 49)/4 = 21$ per dim, $\sqrt{42}$ for 2D).

$p_\lambda(x) = \exp(-\lambda|x|^2) / Z(\lambda)$, $\lambda > 0$. Uniform recovered at $\lambda = 0$. Compute per-SNR BICM capacity.

Loop $\lambda \in \{0.01, 0.02, \ldots, 0.2\}$. At each $\lambda$: (i) compute $p_\lambda$ across the 64 points; (ii) compute $I(Y; B_\ell)$ for each Gray-bit channel $\ell$ via Monte Carlo or quadrature; (iii) sum to get $C_{\rm BICM}(\lambda, \text{SNR})$. Locate the $\lambda$ maximising the achievable rate.

At SNR 12 dB, the optimal $\lambda^\star \approx 0.065$ gives $C_{\rm BICM} \approx 4.0$ bits/2D symbol at energy cost $\mathbb{E}[|X|^2] = 1$ (compare uniform 64-QAM at 12 dB which achieves $\sim 3.8$ bits/2D). Shaping gain at this operating point: about 1.0 dB.

At lower rates ($\eta \le 3$), uniform 64-QAM is already close to optimal and shaping gain is smaller ($< 0.5$ dB). At higher rates (close to 6 bits/2D), shaping approaches the $1.53$ dB asymptotic bound. The operating point $\eta = 4$ at $M = 64$ is near the "peak shaping-gain" regime.

$R = 378/1024 \approx 0.369$; $\eta = 4 \cdot 0.369 \approx 1.48$ bits/2D symbol. This matches the "Efficiency" column of Table 5.2.2.1-2 at CQI 7 (value 1.4766).

From standard NR CQI-to-SNR tables (vendor-specific but aligned with design): CQI 7 threshold is $\text{SNR} \approx 4.5$-$5$ dB for BLER 10%. Shannon at $\eta = 1.48$ needs $\text{SNR} = 2^{1.48} - 1 = 1.79$, or $2.5$ dB. The $\sim 2.5$ dB gap to Shannon is consistent with finite-blocklength and code-design overhead.

Wi-Fi 7 MCS 3 (16-QAM, $R = 1/2$, $\eta = 2$) has SNR threshold $\approx 7$ dB. NR CQI 7 ($\eta = 1.48$) has threshold $\approx 5$ dB. Lower rate gives lower threshold, as expected. NR's finer MCS granularity lets the scheduler sit closer to the optimal point at any given channel SNR.

$\mathbb{E}[K] = p_0 + p_1 + p_2 + p_3 = 1 + 0.25 + 0.1 + 0.04 = 1.39$. (The $p_4 = 0.02$ is the BLER after the 4th transmission, i.e., residual error rate. It does not enter $\mathbb{E}[K]$ directly since the block is discarded after 4 transmissions.)

$\eta_{\rm HARQ} = 3 / 1.39 \approx 2.16$ bits/2D symbol.

The outer residual BLER after HARQ is $p_4 = 0.02$, i.e., 2% of blocks are eventually discarded (to be recovered by higher-layer TCP or RLC retransmission). For eMBB this is typically acceptable.

The scheduler could also have picked a more conservative MCS with $p_1 \approx 0.1$, giving $\mathbb{E}[K] \approx 1.1$ and $\eta_{\rm HARQ} \approx 0.9 \eta_0$. The choice between "high $\eta_0$ + heavy HARQ" and "moderate $\eta_0$ + light HARQ" depends on buffer sizing, latency constraints, and the MCS granularity of the system.

At $\eta \ll \log_2 M$, the capacity is limited by noise, not by modulation constraints. Both uniform and MB-shaped distributions work with most of the inner constellation points; shaping adds minimal gain. Numerically, shaping gain $< 0.2$ dB for $\eta < 3$ bits/2D on 256-QAM.

As $\eta \to \log_2 M = 8$, the uniform 256-QAM saturates at $\log_2 M$ bits — no more information can be packed in. MB shaping allows continuous rate scaling via $\lambda$ down to $H(p_\lambda) < \log_2 M$. The shaping gain approaches the asymptotic $1.53$ dB as $\eta \to \log_2 M$.

The peak shaping gain occurs at rates 1-1.5 bits/2D below the saturation, i.e. $\eta \approx 6.5$-$7$ bits/2D for 256-QAM. Peak value $\approx 1.4$ dB. This is the operational sweet spot for PAS deployment — 400ZR operates at $\eta = 6.34$ bits/2D (DP-16QAM shaping, 3.17 per pol × 2 pols), in this peak region.

The curve is a "concave bump": zero at both ends (more precisely, $\sim 0$ at low rate and $1.53$ dB asymptote at high rate), with a peak at the intermediate rate. Plot this by numerically evaluating the BICM capacity of uniform and MB-shaped 256-QAM across SNRs and reading off the horizontal SNR difference at each achievable rate.

$v = 60 \text{ km/h} = 16.67 \text{ m/s}$, $\lambda = c/f_c = 3 \cdot 10^8 / 3.5 \cdot 10^9 = 0.0857 \text{ m}$. $f_D = v/\lambda = 16.67/0.0857 \approx 194$ Hz.

$f_D \tau = 194 \text{ Hz} \times 5 \text{ ms} \approx 0.97$. By the time the transmitter uses CQI from $\tau = 5$ ms ago, the channel has moved approximately one full "coherence distance" (half-wavelength autocorrelation width).

At $f_D \tau \approx 1$, CQI is effectively useless as a predictor of instantaneous SNR. The scheduler should: (i) pick a conservative MCS about 2 steps below the CQI-indicated optimum, (ii) use longer-term CQI averaging (smoothed over several slots), (iii) rely more heavily on HARQ-IR to cover the residual BLER.

3GPP recommends a UE-speed-class-dependent "CQI offset" applied by the gNB. At 60 km/h the offset is typically 2-3 dB, lowering the effective SNR the AMC policy sees and forcing a more conservative MCS. For low-speed (pedestrian) UEs the offset is 0-0.5 dB. Getting this wrong is one of the most common causes of systematic BLER overshoot in deployed 5G networks.

Coherent optical transmission over SSMF is limited by fibre Kerr non-linearity, which imposes a non-linear Shannon limit. Larger QAM constellations have greater peak-to-average power ratio, launching into a more non-linear regime. PAS on moderate-QAM (16QAM) keeps the amplitude distribution tight and respects the non-linear budget.

PAS gives continuous rate adjustment via $\lambda$, without changing the modulation format or the LDPC code. This is essential for optical transmission where distance, launch power, and wavelength-division-multiplexing density all vary. A larger-QAM solution would require a much larger MCS table to cover the same rate range.

PAS integrates with the existing systematic LDPC decoder (designed for uniform inputs). A 64-QAM implementation would require a different LLR lookup and possibly a different LDPC code optimised for 64-QAM BICM. PAS keeps the decoder and the code unchanged; only the distribution matcher is added.

A fourth reason: PAS's amplitude-to-parity decomposition is hardware-friendly for optical receivers, which already separate I/Q (sign-bits) from amplitude processing in coherent DSP.