Exercises

This exercise has no single correct answer — it depends on the chosen paper. The grading criteria are:

1. Contribution sentence should capture the novelty, not just the topic (e.g., "proposes an iterative algorithm for RIS phase optimization" not "studies RIS").
2. Metric should be specific: "ergodic sum rate" not just "performance."
3. Baseline count should be verifiable from the figure legend.
4. Relevance judgment should reference specific aspects (system model, antenna configuration, metric) rather than vague statements.

Total errors: $23 + 18 + 25 + 15 + 29 = 110$. Total bits: $5 \times 10^4 = 50{,}000$. $\hat{P}_e = 110 / 50{,}000 = 2.2 \times 10^{-3}$.

$\text{SE} = \sqrt{\hat{P}_e(1 - \hat{P}_e) / N} = \sqrt{2.2 \times 10^{-3} \times 0.9978 / 50{,}000} \approx 2.1 \times 10^{-4}$.

95% CI: $2.2 \times 10^{-3} \pm 4.1 \times 10^{-4}$, i.e., $[1.79 \times 10^{-3},\; 2.61 \times 10^{-3}]$.

The theoretical value $1.25 \times 10^{-3}$ lies outside the 95% CI. Possible explanations:

• The simulation has a bug (most likely: incorrect SNR normalization or using $E_s/N_0$ instead of $E_b/N_0$).
• The random number generator is not producing true Gaussian noise.
• Rounding in the SNR conversion.

This demonstrates how CIs can catch simulation errors: when the CI does not contain the known analytical answer, something is wrong.

The "unrealistic flag" column shows assumptions that should prompt skepticism during a paper review.

1. CSI asymmetry: NOMA uses perfect CSIT for power allocation and SIC ordering; OFDMA has no CSIT. With CSIT, OFDMA can perform frequency-selective scheduling (assigning each subcarrier to the user with the best channel), which dramatically improves throughput.

2. Power allocation asymmetry: NOMA presumably uses optimized power allocation (needed for SIC); OFDMA uses equal power. With CSIT, OFDMA can use water-filling across subcarriers.

3. Perfect SIC assumption: In practice, SIC suffers from error propagation, especially with imperfect CSI. This assumption inflates NOMA performance.

4. User pairing advantage: NOMA pairs users with different channel strengths on the same subcarrier. The comparison does not account for the scheduling gain that OFDMA would get from multi-user diversity.

• Give both schemes equal CSIT (e.g., MMSE-estimated channels with the same pilot overhead).
• Allow OFDMA to use frequency-selective scheduling and water-filling power allocation.
• Model imperfect SIC for NOMA with realistic error propagation.
• Compare at the same total power and same number of served users.
• Include an upper bound (e.g., DPC capacity) to gauge how close either scheme comes to optimal.

1. Channel normalization error: If $\mathbb{E}[\|\mathbf{h}_i\|^2] = N_t = 4$ instead of 1 (i.e., each entry has unit variance but the column norm is not normalized), the effective SNR is $N_t$ times higher than intended. For a 4x4 system this is a 6 dB shift — but if partially corrected, it might manifest as a small consistent bias.

Diagnosis: Print $\mathbb{E}[\|\mathbf{h}_i\|^2]$ from the simulation and verify it matches the intended normalization.

2. SNR definition mismatch: If the code defines $\text{SNR} = P / \sigma^2$ but the $x$-axis plots $E_b/N_0$, the conversion $E_b/N_0 = \text{SNR} / (\log_2 M \cdot R_c)$ might be slightly off (e.g., forgetting the code rate $R_c$ for uncoded transmission where $R_c = 1$, or mishandling MIMO degrees of freedom).

Diagnosis: Verify the BER at one SNR point analytically. For BPSK in AWGN (SISO), the BER should be exactly $Q(\sqrt{2 E_b/N_0})$.

3. Rounding in noise generation: Using randn with incorrect variance scaling (e.g., $\sigma^2/2$ per real dimension but applying it as $\sigma^2$) would reduce noise power and improve BER.

Diagnosis: Generate noise samples, compute their empirical variance, and compare to $\sigma^2$.

(a) Plots:

• Ergodic sum rate vs. SNR (all baselines)
• Per-user rate CDF at SNR = 10 dB
• BER vs. $E_b/N_0$ for QPSK

(b) Baselines:

• MRT precoding (lower complexity baseline)
• MMSE precoding (stronger baseline)
• Single-user bound: $K \log_2(1 + NP/(K\sigma^2))$
• Perfect CSI genie bound (for estimated-CSI case)

(c) MC justification:

• Rate: 1,000 realizations. With $N/K = 8$, channel hardening ensures low variance. Doubling to 2,000 changes the mean by less than 0.5%.
• BER: Adaptive, minimum 100 errors. At BER = $10^{-4}$ with QPSK ($2 \times 200 = 400$ bits/frame), need $10^6 / 400 = 2{,}500$ frames.

The noise is generated as np.random.randn(n_bits) + 1j * np.random.randn(n_bits), which has variance 2 (each component has variance 1). After scaling by np.sqrt(noise_power), the total noise power is $2 / \text{SNR}_{\text{lin}}$ instead of $1 / \text{SNR}_{\text{lin}}$.

The effective SNR is halved (3 dB lower than intended).

The BER curve is shifted right by 3 dB. For example, at a plotted $E_b/N_0 = 10$ dB, the actual simulation runs at 7 dB, producing a higher BER than expected.

Two valid fixes:

Fix 1: Scale noise correctly for complex baseband:

noise = (np.random.randn(n_bits)
         + 1j * np.random.randn(n_bits)) / np.sqrt(2)

Fix 2: Since BPSK is real-valued, use only real noise:

noise = np.random.randn(n_bits)
received = symbols + np.sqrt(noise_power) * noise

Fix 2 is preferred for BPSK because the imaginary noise component is irrelevant after taking the real part for detection.

Signal model: $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$ where $\mathbf{H} \in \mathbb{C}^{2 \times 2}$, $H_{ij} \sim \mathcal{CN}(0,1)$, $\mathbf{x} \in \{(\pm 1 \pm j)/\sqrt{2}\}^2$ (QPSK, unit energy), $\mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I})$.

Per-stream SNR: $\text{SNR} = 1/\sigma^2$ (since $\mathbb{E}[\|x_k\|^2] = 1$). $E_b/N_0 = \text{SNR} / \log_2(4) = 1/(2\sigma^2)$. So $\sigma^2 = 1/(2 \cdot 10^{E_b N_0[\text{dB}]/10})$.

MMSE filter: $\mathbf{G} = (\mathbf{H}^{H} \mathbf{H} + \sigma^2 \mathbf{I})^{-1} \mathbf{H}^{H}$. Estimated symbols: $\hat{\mathbf{x}} = \mathbf{G}\mathbf{y}$. Slice to nearest QPSK constellation point. Count bit errors using Gray-coded bit mapping.

At each SNR point, after accumulating $N_e$ errors from $N_{\text{total}}$ bits: $\hat{P}_e = N_e / N_{\text{total}}$, CI: $\hat{P}_e \pm 1.96\sqrt{\hat{P}_e(1-\hat{P}_e)/N_{\text{total}}}$.

The diversity order of a $2 \times 2$ system with 2 spatial streams and MMSE detection is $d = N_r - N_t + 1 = 1$. At high SNR, the BER should decay as $\text{BER} \propto \text{SNR}^{-1}$, i.e., a slope of $-1$ on a log-log plot (or $-10$ dB/decade on a log-linear BER vs. dB plot). Verify this by fitting a line to the high-SNR portion of the simulated curve.

The claim is highly suspicious. MMSE estimation is the optimal linear estimator for Gaussian channels, and for i.i.d. Rayleigh fading (which is Gaussian), MMSE is actually the optimal estimator (not just optimal among linear ones). A 5 dB gain over the optimal estimator is mathematically impossible unless:

• The MMSE baseline is implemented incorrectly.
• The comparison metric is not MSE (e.g., BER with mismatched detection).
• The "MMSE" baseline does not use the true channel statistics.

1. Training = testing distribution: The DL estimator is trained and tested on i.i.d. Rayleigh channels. This is the easiest possible setting and tells us nothing about real-world performance. The estimator may have simply memorized the distribution.

2. MMSE baseline likely wrong: For i.i.d. Rayleigh with known noise variance, MMSE estimation has a closed form. If the implementation uses LS estimation as the "MMSE" baseline, the 5 dB gap is explained by the LS-to-MMSE gap, not by the neural network.

3. No complexity comparison: Deep learning estimators require significant computation (matrix multiplications through many layers). Without a complexity comparison, the practical value is unclear.

4. No generalization test: What happens when the channel model changes (e.g., correlated fading, Rician)?

5. Perfect noise variance knowledge: In practice, noise variance must also be estimated, which affects MMSE performance.

As a reviewer, request:

• MSE learning curves showing training vs. validation loss
• Comparison with correctly implemented MMSE using true channel covariance
• Testing on a different channel model (generalization)
• Complexity table (flops per estimation, inference time)
• Performance with estimated (not perfect) noise variance

Original noise: $n \sim \mathcal{N}(0, \sigma^2)$, PDF $f_0(n)$. Shifted noise: $n \sim \mathcal{N}(\mu^*, \sigma^2)$, PDF $f_1(n)$.

IS weight: $w(n) = \frac{f_0(n)}{f_1(n)} = \exp\!\left(-\frac{n^2}{2\sigma^2} + \frac{(n-\mu^*)^2}{2\sigma^2}\right) = \exp\!\left(\frac{-2n\mu^* + (\mu^*)^2}{2\sigma^2}\right)$.

With $\mu^* = -\sqrt{2E_b/N_0}$ and $\sigma^2 = N_0/2$: $w(n) = \exp\!\left(\frac{2n\sqrt{2E_b/N_0} + 2E_b/N_0}{N_0}\right)$.

$\mathbb{E}_{f_1}[\mathbf{1}_{\text{error}} \cdot w(n)] = \int \mathbf{1}_{\text{error}}(n) \frac{f_0(n)}{f_1(n)} f_1(n)\, dn = \int \mathbf{1}_{\text{error}}(n) f_0(n)\, dn = P_e$.

The IS estimator $\hat{P}_e^{\text{IS}} = \frac{1}{N}\sum_{i=1}^N \mathbf{1}_{\text{error}}(n_i) w(n_i)$ is unbiased.

With the optimal IS shift, nearly every sample produces an error (the noise is biased to the wrong decision region). The variance of the IS estimator is approximately $P_e^2 / N$, so for 10% relative error: $N_{\text{IS}} \approx 100$.

Brute force: $N_{\text{BF}} \approx 100 / P_e = 10^9$.

Speedup: $10^9 / 100 = 10^7 \times$. This is why importance sampling is essential for simulating coded systems at operational BER levels.

• SNR definition (per-antenna? total? $E_b/N_0$?)
• SNR range and step size
• Channel model details (i.i.d.? correlated? path loss?)
• CSI assumption (perfect? estimated? what estimator?)
• Number of Monte Carlo realizations ("sufficient" is vague)
• Noise variance normalization
• Modulation and coding scheme
• Frame/block length
• Power constraint type (total? per-antenna?)
• Algorithm initialization and stopping criterion
• Baseline implementation details
• Computational platform and software version
• Random seed or number of independent runs

"The simulation setup lacks critical details for reproducibility. Specifically: (1) The SNR definition is not stated — is this total transmit power to noise ratio, per-antenna SNR, or $E_b/N_0$? (2) 'Sufficient channel realizations' is not quantitative; please state the exact number and provide 95% confidence intervals on all curves. (3) The channel model says 'Rayleigh fading' but does not specify correlation, path loss, or whether the channel is frequency-flat or frequency-selective. (4) The CSI assumption is not stated. (5) The baselines are not described in enough detail to re-implement. I recommend adding a simulation parameter table (see IEEE JSAC guidelines) and consider releasing the simulation code."

1. All papers must include a simulation parameter table listing at minimum: antenna configuration, channel model, CSI assumption, SNR definition, modulation/coding, and MC trial count.
2. SNR must be defined by an explicit equation.
3. Number of Monte Carlo realizations must be stated numerically.
4. 95% confidence intervals must be shown on all stochastic curves.
5. Baseline implementations must be described or cited with specific equation/algorithm numbers.
6. Computational complexity must be compared in FLOPs or runtime.
7. Authors are encouraged to provide code via a public repository, with a "Reproducibility" section in the paper.

MRT: $\mathbf{W} = \mathbf{H}$, normalized. $\text{SINR}_k^{\text{MRT}} = \frac{p_k |\mathbf{h}_k^H \mathbf{w}_{k}|^2} {\sum_{j \neq k} p_j |\mathbf{h}_k^H \mathbf{w}_{j}|^2 + \sigma^2}$. Complexity: $O(NK)$.

ZF: $\mathbf{W} = \mathbf{H}(\mathbf{H}^{H} \mathbf{H})^{-1}$, normalized. $\text{SINR}_k^{\text{ZF}} = p_k / [\mathbf{H}^{H} \mathbf{H})^{-1}]_{kk}$. Complexity: $O(NK^2 + K^3)$.

MMSE: $\mathbf{W} = \mathbf{H}(\mathbf{H}^{H} \mathbf{H} + \alpha \mathbf{I})^{-1}$, $\alpha = K\sigma^2/P$. Complexity: $O(NK^2 + K^3)$.

At $N = 16$ (only $2\times$ overprovisioning), ZF and MMSE significantly outperform MRT at high SNR because MRT's inter-user interference floor limits the sum rate. MMSE outperforms ZF at low SNR by trading off interference suppression for noise enhancement. At $N = 128$ ($16\times$ overprovisioning), all three precoders converge because the channel vectors become nearly orthogonal (channel hardening and favorable propagation). The 95% CIs are narrow (< 1% relative) due to channel hardening at $N = 128$, but wider at $N = 16$. This confirms the massive MIMO benefit: simple MRT is near-optimal when the antenna-to-user ratio is large, avoiding the $O(K^3)$ matrix inversion of ZF/MMSE.

Per-antenna symbol energy: $E_s = P_{\text{total}} / (N_t \cdot \Delta f) = P_{\text{total}} T_s / N_t$.

With $\text{SNR} = P_{\text{total}} / \sigma^2 = P_{\text{total}} / (N_0 / T_s)$: $E_s / N_0 = \text{SNR} / N_t$.

$E_b / N_0 = E_s / (N_0 \cdot R \cdot \log_2 M) = \text{SNR} / (N_t \cdot R \cdot \log_2 M)$.

With $N_t = 4$, $R = 1/2$, $\log_2 M = 4$: $E_b/N_0 = \text{SNR} / (4 \times 0.5 \times 4) = \text{SNR} / 8$.

At $\text{SNR} = 20$ dB ($= 100$ linear): $E_b/N_0 = 100/8 = 12.5$ ($11.0$ dB).

The paper's 20 dB SNR is equivalent to only 11 dB $E_b/N_0$. $\blacksquare$

ZF completely nulls inter-user interference but at the cost of noise enhancement (amplifying noise to satisfy the zero-forcing constraint). MMSE minimizes the total MSE by balancing interference suppression against noise amplification. At low SNR, the noise dominates, and ZF's noise enhancement is costly. MMSE adapts: at low SNR it behaves like MRT (ignores interference, maximizes received power), at high SNR it behaves like ZF.

$\mathbf{W}_{\text{MMSE}} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H} + \alpha\mathbf{I})^{-1}$$where$\alpha = K\sigma^2/P$is the regularization parameter. When$\alpha = 0$: reduces to ZF. When$\alpha \to \infty$: reduces to MRT (up to scaling).

ZF and MMSE become equivalent as $\text{SNR} \to \infty$ (i.e., $\alpha \to 0$), or when $N_t \gg K$ (massive MIMO regime where $\mathbf{H}\mathbf{H}^{H} \approx N_t \mathbf{I}$ by channel hardening). $\blacksquare$

(a) Per-antenna transmit SNR — the power is divided by $N_t$ to give the SNR per antenna element. No channel gain is included.

(b) Instantaneous total receive SNR — includes the channel gain $\|\mathbf{h}\|^2$. This is a random variable that depends on the fading realization.

(c) Energy per bit to noise density — normalized by both the modulation order and code rate. The standard for comparing different modulation/coding combinations on a common basis. $\blacksquare$