Exercises

MMSE is Bayes-optimal under squared-error loss when (i) the channel $\mathbf{H}$ is jointly Gaussian with the observation $\mathbf{y}_p$ and (ii) the channel covariance $\boldsymbol{\Sigma}_{\ntn{ch}}$ and the noise covariance are known exactly.

Under these assumptions the conditional expectation $\mathbb{E}[\mathbf{H} \mid \mathbf{y}_p]$ is linear in $\mathbf{y}_p$ and equals the MMSE formula. The Bayes rule under squared-error loss is the conditional expectation, which equals the MMSE estimator. Neither non-Gaussianity nor imperfect covariance survive the derivation. $\blacksquare$

$\sum_{i=1}^{64} 2^{-(i-1)} = 2 - 2^{-63} \approx 2$. The distortion budget is $D = 0.2$.

We need $\sum_i \min(2^{-(i-1)}, \mu) = 0.2$. Try $\mu = 2^{-3} = 0.125$: eigenvalues below $\mu$ are $i \geq 4$, summing to $2^{-3} + 2^{-4} + \cdots \approx 0.25$ — slightly too much. Try $\mu = 2^{-4} = 0.0625$: eigenvalues below $\mu$ are $i \geq 5$, summing to $0.125$, plus 3 full low-index eigenvalues add nothing to the distortion. Total $\approx 0.125$, too little. Interpolate to $\mu \approx 0.08$ for $D = 0.2$.

$R(D) = \sum_{i: \lambda_i > \mu} \log_2(\lambda_i / \mu)$. For $\mu \approx 0.08$ the threshold is around $i = 4$, and $R(D) \approx \log_2(1 / 0.08) + \log_2(0.5 / 0.08) + \log_2(0.25/0.08) + \log_2(0.125/0.08)$ $\approx 3.64 + 2.64 + 1.64 + 0.64 \approx 8.6$ bits per channel instance. A CsiNet achieving $D = 0.2$ at this channel would thus be within a factor of 2-3 of the information-theoretic optimum at roughly 20-30 bits of actual feedback budget. $\blacksquare$

For typical urban/suburban multipath, the channel impulse response has only a handful of significant taps (roughly equal to the number of clusters in the environment). In the frequency domain the same channel looks "smooth" but not sparse; in the delay domain it is explicitly sparse.

The 2D conv layers in CsiNet have small kernels that can efficiently represent sparse + locally correlated signals. They cannot efficiently represent a "dense smooth" signal; a Transformer or MLP would handle that better.

Removing the IFFT preprocessing drops NMSE by 3-5 dB at the same bit budget on CDL-C channels. The convolutional inductive bias is mismatched to the frequency-domain representation; the network has to spend capacity learning the IFFT, and it does so imperfectly. Hard-coding the IFFT is a free gain and a textbook example of the model-based DL principle: use the physics, do not learn it from scratch. $\blacksquare$

The training distribution covers only vehicular velocities. Pedestrian UEs have much slower, more randomly-distributed beam transitions dominated by blocking and body shadowing rather than deterministic mobility. The LSTM's learned temporal features do not transfer.

Retrain on a mixture of 50 % pedestrian and 50 % vehicular traces. The mixed training teaches the network to handle both temporal scales. Expected top-5 accuracy after retraining: approximately 92 % on vehicular and 88 % on pedestrian — a few percent lower on vehicular (capacity sacrifice) but much more uniform.

Better: pass UE velocity as an auxiliary input to the LSTM. Now the same network handles both modes without sacrificing either. Velocity is a standard measurement the UE reports to the BS, so this is free at inference time. This is how the Rel-18 AI/ML study item recommends training sequence models for beam management. $\blacksquare$

"Maximize sum-rate" leads the RL agent to starve the weakest users — serving only the strongest users maximizes instantaneous sum-rate. The training reward looks excellent while the fairness metric (Jain's index, 95%-likely rate) collapses.

Reward $\sum_k \log R_k^{\text{avg}}$. The log penalizes starvation heavily. Sacrifices a small amount of peak sum-rate for a much better worst-user rate. Corresponds to the log-utility scheduler of Chapter 5.

Reward $\min_k R_k^{\text{avg}}$. The weakest user's rate becomes the entire objective. Strong fairness, but very large efficiency sacrifice (sum-rate can drop by 30-50 %). Best when URLLC-style minimum-rate guarantees matter.

Reward $\sum_k R_k^{1-\alpha}/(1-\alpha)$ with $\alpha \in (0, 1]$. Interpolates between sum-rate ($\alpha \to 0$) and max-min ($\alpha \to \infty$) via a single tunable parameter. Most flexible choice, lets the network designer dial the efficiency-fairness tradeoff post-training. Ranking by efficiency sacrifice: sum-rate (none) < alpha=0.5 < PF (log, alpha=1) < max-min. $\blacksquare$

Layer $k$ ($k = 1, \ldots, 5$): $\mathbf{x}^{k} = \mathcal{S}_{\lambda_k}(\mathbf{x}^{k-1} + \eta_k \mathbf{A}^H(\mathbf{y} - \mathbf{A}\mathbf{x}^{k-1}))$ with $\mathbf{x}^{0} = \mathbf{0}$. Final output is $\mathbf{x}^{5}$.

$(\eta_k, \lambda_k)_{k=1}^{5}$: 2 scalars per layer, 5 layers, total 10 trainable parameters. The measurement matrix $\mathbf{A}$ stays fixed (it is the pilot pattern, known).

Adding a trainable per-layer matrix $\mathbf{W}_k$ or a trainable measurement correction turns the network into a generic MLP and loses generalization. With 10 parameters total, the unfolded network can only improve the classical step sizes and thresholds — exactly the model-based DL sweet spot. Compare: a CsiNet with the same task has $\approx 10^5$ parameters. $\blacksquare$

PPO surrogate: $L^{\text{clip}}_t = \min(\rho_t \hat{A}_t, \text{clip}(\rho_t, 1-\epsilon, 1+\epsilon)\hat{A}_t)$. As $\epsilon \to \infty$, clip$(\rho_t, 1-\epsilon, 1+\epsilon) = \rho_t$, so both terms are equal and $L^{\text{clip}}_t = \rho_t \hat{A}_t$.

The expected value $\mathbb{E}[\rho_t \hat{A}_t]$ is the importance-weighted advantage. At $\phi = \phi_{\text{old}}$, $\rho_t = 1$ and the gradient with respect to $\phi$ recovers the vanilla policy gradient $\nabla_\phi \mathbb{E}[\pi_\phi(a \mid s) A^{\pi_{\text{old}}}(s,a)]$.

Without clipping, a single large positive advantage causes an arbitrarily large policy update, which pushes the new policy outside the trust region where the advantage estimate is valid. The training collapses and the reward trajectory becomes chaotic. The whole point of PPO is to keep the step small enough that the advantage estimate remains roughly correct. Empirically, $\epsilon \approx 0.2$ is the standard choice. $\blacksquare$

Set $(\eta_k, \lambda_k) = (\eta, \lambda)$ for all $k$, where $(\eta, \lambda)$ are the classical ISTA hyperparameters. The unfolded network is now literally $K$ applications of the classical iteration.

The loss $\mathcal{L}_0 = \|\mathbf{x}^{\star} - \hat{\mathbf{x}}^K_{\text{ISTA}}\|^2$ equals the classical-ISTA recovery error.

Any gradient-descent step with small enough learning rate decreases the loss: $\mathcal{L}_{t+1} \leq \mathcal{L}_t$. Therefore $\mathcal{L}_{\text{final}} \leq \mathcal{L}_0$ — the trained network is never worse than the classical $K$-iteration algorithm. In practice the strict inequality typically holds and gives 2-4 dB NMSE improvement. $\blacksquare$

Treat $\lambda(i) = C \cdot i^{-\alpha}$ as a continuous function of $i$. The water level $\mu$ determines the cutoff index $i^{\star}$ via $\lambda(i^{\star}) = \mu$, i.e. $i^{\star} = (C/\mu)^{1/\alpha}$.

The distortion in the discarded tail is $D = \int_{i^{\star}}^{\infty} \lambda(i) \, di = \frac{C}{\alpha - 1}(i^{\star})^{1 - \alpha}$. Substituting $i^{\star}$ gives $D \propto \mu^{1 - 1/\alpha}$ or equivalently $\mu \propto D^{\alpha/(\alpha-1)}$.

$R(D) = \int_1^{i^{\star}} \log_2(\lambda(i) / \mu) \, di = \int_1^{i^{\star}} (\log_2(C/\mu) - \alpha \log_2 i) \, di$. The dominant term is $i^{\star} \log_2(C/\mu) - \alpha i^{\star} \log_2 i^{\star}$, which simplifies to $R(D) \propto (C/\mu)^{1/\alpha} \log_2(C/\mu)$. Substituting $\mu \propto D^{\alpha/(\alpha-1)}$ gives $R(D) \sim D^{-1/(\alpha-1)} \log_2(1/D)$.

For $\alpha = 2$ (fast decay, typical of massive MIMO) $R(D) \sim D^{-1} \log(1/D)$ — the feedback overhead decays quickly as the distortion target relaxes. For $\alpha = 1.1$ (slow decay, rich multipath) $R(D) \sim D^{-10}$ — the feedback grows catastrophically as distortion tightens. Channels with strong angular structure (low rank, high $\alpha$) are intrinsically feedback-friendly; isotropic channels are not. This is the quantitative reason CsiNet helps for some channel types and not others. $\blacksquare$

A learned component (e.g. deep-unfolded ISTA channel estimator) runs in parallel with a classical fallback (e.g. sample-MMSE). A small monitor compares the learned output against a plausibility criterion; when the criterion fails the classical output is used instead. The classical branch guarantees worst-case correctness; the learned branch provides the gain when it is applicable.

For a channel estimator, the natural trigger is the residual norm $\rho = \|\mathbf{y}_p - \mathbf{S}_{i,k} \hat{\mathbf{H}}_{\text{learned}}\|^2 / \|\mathbf{y}_p\|^2$. On matched data this is small; on shifted data the learned network's output is mismatched to the measurement, which the residual detects.

Compute $\rho$ on a held-out matched validation set and set the threshold at the 95th percentile. On deployment, any input with $\rho$ above the threshold triggers the fallback. This yields roughly 5 % fallback rate on matched data (controlled false-positive) and much higher fallback rate on shifted data (where it is supposed to trigger). Monitor the fallback rate continuously: if it drifts above 20 %, the network needs retraining. $\blacksquare$

1. Linear channel estimation in a Gaussian regime. MMSE is Bayes-optimal; no network can beat it.
2. Downlink precoder design for a known channel. The closed-form RZF / water-filling formula is optimal.
3. LDPC decoding. Belief propagation has decades of analysis and very strong performance. DL decoders have never beaten BP on a well-designed code.

1. Beam prediction in mobility. Rich temporal structure, no clean rule-based model, 3GPP Rel-18 candidate.
2. Scenario classification (indoor vs outdoor vs LoS vs NLoS). Pure pattern recognition, no analytical form, huge ML literature to draw on.
3. Channel compression for CSI feedback in fixed deployments. Per-scenario training is acceptable, and the rate-distortion gain over Type II is real. $\blacksquare$

CsiNet worst case (CDL-D or similar shift): $-11$ dB. OAMP-Net worst case: $-14$ dB. The model-based approach wins by 3 dB on the scenario it was not trained on.

CsiNet: approximately $(-18 + (-11)\cdot 3)/4 = -12.75$ dB (one scenario matched, three shifted). OAMP-Net: approximately $(-15 + (-14)\cdot 3)/4 = -14.25$ dB. OAMP-Net wins on average too.

CsiNet to match OAMP-Net would need to be retrained separately for each deployment, i.e. four separate trained models plus a model-distribution mechanism to swap them. OAMP-Net needs one trained model. Operational cost heavily favors OAMP-Net.

Deploy OAMP-Net. It gives up $3$ dB peak efficiency on matched CDL-C to gain $3$ dB worst-case efficiency and eliminate the per-scenario retraining operational overhead. This is a textbook case of the 6G@UT/Huawei position: model-based DL beats data-driven DL on every commercial deployment metric. The CsiNet gain is only visible in academic papers, not in field trials. $\blacksquare$

Let $C_{\text{full}} = |\mathcal{B}|$ measurements (exhaustive). LSTM + fallback overhead per slot: $C_{\text{LSTM}} = k + (1 - p_k) \cdot |\mathcal{B}|$.

Saving $= 1 - C_{\text{LSTM}} / C_{\text{full}} = 1 - k / |\mathcal{B}| - (1 - p_k) = p_k - k / |\mathcal{B}|$.

For $|\mathcal{B}| = 64$, $k = 5$, $p_k = 0.95$: Saving $= 0.95 - 5/64 \approx 0.95 - 0.078 = 0.872$, i.e. 87.2 % overhead reduction — an 8x improvement, consistent with Example 25.3. For $p_k = 0.90$: Saving $\approx 82 \%$, still very useful. If $p_k$ drops to 0.5 (weak predictor): Saving $\approx 42 \%$, marginal. This is why the top-5 accuracy is such a useful headline metric: it directly determines the deployment value. $\blacksquare$

Real PA nonlinearity, oscillator phase noise, and IQ imbalance add structured noise that the simulator typically omits. Mitigation: Inject empirical hardware noise models into the simulator using measurements from real RRUs during data generation.

Simulator assumes actions take effect instantly; real scheduling decisions have 1-2 slot processing delays and variable latency. Mitigation: Add a Markovian delay model to the simulator so the learned policy accounts for non-instantaneous control.

Real UE traffic is bursty, heavy-tailed, and correlated across users; simulators use independent Poisson or constant arrivals. Mitigation: Replay real traffic traces into the simulator instead of using synthetic models, and use domain randomization over multiple traffic profiles during training. $\blacksquare$

Attention: $2 N_f^2 d = 2 \cdot 4096 \cdot 128 = 1.05 \cdot 10^6$ MACs. Feedforward block (2 layers, 4x expansion): $8 N_f d^2 = 8 \cdot 64 \cdot 16384 = 8.4 \cdot 10^6$ MACs. Total per attention layer: approximately $9.5$ MMAC.

3 conv layers, $3\times 3$ kernels, 32 channels, spatial size $N_f \times N_t = 64 \times 64$: $3 \cdot 9 \cdot 32 \cdot 32 \cdot 4096 \approx 1.1$ MMAC per layer, roughly $3.5$ MMAC total for 3 layers.

At a 5 ms CSI update rate, the 1 GMAC/s NPU budget provides 5 MMAC per inference. CsiNet at 3.5 MMAC fits comfortably. Transformer-CSI at 9.5 MMAC does not fit — it would need either a larger NPU, a reduced update rate (10 ms instead of 5 ms), or architectural compression (distillation to a smaller network). In the current generation of handset NPUs the CsiNet-class models are deployable; Transformer-CSI is not yet. This is why CsiNet remains the 3GPP reference despite Transformers' better NMSE on paper. $\blacksquare$