Exercises

$l_{\max} = \lceil 2 \times 10^{-6} \cdot 2 \times 10^7\rceil = 40$. $k_{\max} = \lceil 400 \cdot 2.5 \times 10^{-3}\rceil = 1$.

$|\mathcal{G}_{\min}| = (l_{\max} + 1)(2 k_{\max} + 1) = 41 \cdot 3 = 123$ cells. With one pilot: 124 cells reserved.

$|\alpha|^2/\sigma^2 = 10^{2.0} = 100$. $-\ln(10^{-3}) = 6.91$. $\gamma_{\text{th}} = 6.91/100 = 0.069$. Relative to pilot (in dB): $10 \log_{10}(0.069) = -11.6$ dB.

Paths stronger than $-11.6$ dB relative to pilot (i.e., above 10% of pilot power) will be reliably detected with 0.1% false-alarm rate. Weaker paths may be missed.

$Y_{DD}[\ell_p + \ell_i, k_p + k_i] = \alpha h_i + W$ with $W \sim \mathcal{CN}(0, \sigma^2)$.

$\hat{h}_i = (\alpha h_i + W)/\alpha = h_i + W/\alpha$.

$\mathbb{E}[\hat{h}_i] = h_i + \mathbb{E}[W]/\alpha = h_i$. Unbiased. $\checkmark$

$\mathbb{E}|\hat{h}_i - h_i|^2 = \mathbb{E}|W|^2/|\alpha|^2 = \sigma^2/|\alpha|^2$. $\blacksquare$

$\hat{h}_i = (1/N_p)\sum Y_{DD}[\ell, k] P^*[\ell - \ell_i, k - k_i]$. Signal contribution: $\sqrt{\rho_p} h_i N_p / N_p = \sqrt{\rho_p} h_i$. Noise: sum of $N_p$ iid $\mathcal{CN}(0, \sigma^2)$ terms weighted by $P^*$ (unit modulus), divided by $N_p$.

$\mathrm{Var}((1/N_p)\sum W_i P^*_i) = \sigma^2/N_p$.

Normalize by $\sqrt{\rho_p}$: variance = $\sigma^2/(\rho_p N_p)$. With $N_p = MN$: variance = $\sigma^2/(\rho_p MN)$ — beats embedded pilot ($\sigma^2/|\alpha|^2$) when $\rho_p MN > |\alpha|^2$.

$l_{\max} = \tau_{\max} W = 10^{-2} \cdot 4 \times 10^7 = 4 \times 10^5$. We need $M > l_{\max}$, so $M = 5 \times 10^5$ is needed. $k_{\max} = f_D T = 3 \times 10^4 \cdot 2 \times 10^{-2} = 600$. We need $N > 2 k_{\max}$, so $N = 2048$ is adequate.

$|\mathcal{G}| = (l_{\max} + 1)(2 k_{\max} + 1) = 500{,}001 \cdot 1201 \approx 6 \times 10^8$. $MN = 5 \times 10^5 \cdot 2048 \approx 10^9$. Overhead: $\sim 60\%$ — prohibitive for embedded pilot.

With zero overhead, superimposed pilot is essential in LEO. This is a quantitative illustration of why Chapter 18 (LEO OTFS) uses superimposed pilots throughout.

With pilot at $(0, 0)$, the guard region is $\{(\ell, k) : 0 \leq \ell \leq l_{\max}, -k_{\max} \leq k \leq k_{\max}\}$. For $k < 0$, wrap around: cells $(\ell, N - k_{\max}), \ldots$.

Guard region: $\{(M/2 + \ell, N/2 + k)\}$ — same shape, just shifted. No wrap-around needed (assuming the rectangle fits entirely within the grid).

The path-detection output is the same either way — the DD convolution is toroidal, so position is irrelevant up to the modular shift that relates the guard-region coordinates to the path indices $(\ell_i, k_i)$.

Interior placement avoids modular wrap-around in the analysis, simplifying bookkeeping. Corner placement is equivalent but requires care with negative Doppler indices.

$(N/2) \cdot (M/2) = 7 \cdot 256 = 1792$ REs per frame reserved. $MN = 7168$. DMRS overhead: $1792/7168 = 25\%$.

$|\mathcal{G}| = 150$ on same grid. Overhead: $150/7168 = 2.1\%$.

DMRS/OTFS overhead ratio: $25/2.1 \approx 12\times$. For the same channel, OTFS delivers the same estimate quality with 12× less pilot burden.

$|\alpha|^2/\sigma^2 = 10^{1.5} = 31.6$. $a = \sqrt{2 \cdot 0.01 \cdot 31.6} = \sqrt{0.632} = 0.795$. $b = \sqrt{2 \cdot 0.1 \cdot 31.6} = \sqrt{6.32} = 2.51$.

$P_{\text{MD}} \approx Q(b - a) = Q(1.72) \approx 0.043$ ($\approx 4\%$ missed detection probability).

At 15 dB pilot SNR and threshold 0.1, paths at $-20$ dB relative to pilot are missed 4% of the time. To improve this, increase pilot boost or lower the threshold (at cost of higher false-alarm rate).

$p(Y | h_i) = \mathcal{CN}(\alpha h_i, \sigma^2)$. $\log p = -(Y - \alpha h_i)^*(Y - \alpha h_i)/\sigma^2 + \text{const}$.

$\partial^2 \log p / \partial h_i \partial h_i^* = -|\alpha|^2/\sigma^2$.

Fisher information: $|\alpha|^2/\sigma^2$. CRLB: $\sigma^2/|\alpha|^2$.

LS variance $= \sigma^2/|\alpha|^2 =$ CRLB. LS is efficient for this problem — no estimator does better than LS for a single complex Gaussian observation. The CRLB confirms that the MSE bottleneck is the pilot SNR, not the estimator design.

Path 1 observed at $(\ell_p + 5, k_p + 0)$. Path 2 observed at $(\ell_p + 5, k_p + 3)$. Distinct cells of the $Y_{DD}$ grid.

The DD-domain noise $W_{DD}$ is i.i.d. $\mathcal{CN}(0, \sigma^2)$ across cells (Chapter 4). Cell $(5, 0)$ and cell $(5, 3)$ have independent noise samples.

$\hat{h}_1 = h_1 + W_1/\alpha$, $\hat{h}_2 = h_2 + W_2/\alpha$, with $W_1, W_2$ independent. Therefore $\hat{h}_1, \hat{h}_2$ are independent. $\blacksquare$

Independence justifies the simple per-path MSE sum in Theorem TMSE of Embedded Pilot Channel Estimation: no covariance cross-term inflates the aggregate error.

$\hat{h}^{(1)} = \hat{h} + (\text{data interf.} + \text{noise})/(\rho_p MN)$. Data interference magnitude: $(1 - \rho_p) \mathbb{E}|X|^2$.

$\hat{X}^{(1)} = \text{detect}(Y - \sqrt{\rho_p} \hat{h}^{(1)} \star\star P)$. Detection error is bounded (at moderate SNR the bit-error is small).

Subtract $\sqrt{1-\rho_p} \hat{X}^{(1)} \star\star \hat{h}^{(1)}$ from $Y$. The residual approaches pilot-only-plus-noise: variance $\sigma^2/(\rho_p MN)$.

Iterations are effectively "denoising" the channel estimate by removing data interference. After 2-3 iterations, the estimator variance is within 1 dB of the noise-only bound (see Mohammadi et al. 2023, §IV).

For distinct Zadoff-Chu roots, $|\text{cross-corr}| = 1/\sqrt{MN}$. For $MN = 1024$: $1/32 = 0.031$.

$10\log_{10}(0.031) = -15$ dB.

Pilot contamination is 15 dB below signal — negligible for channel estimation. Zadoff-Chu's scaling $1/\sqrt{MN}$ is the reason cell-free OTFS with superimposed pilots scales cleanly to many users: contamination decreases as the grid grows.

Each $h_i$ observed at a distinct guard-region cell with noise variance $\sigma^2$. The joint Fisher matrix is diagonal: $F = (|\alpha|^2/\sigma^2) I_P$.

$\mathrm{Cov}(\hat{h}) \succeq F^{-1} = (\sigma^2/|\alpha|^2) I_P$.

LS estimator: $\hat{h}_i = Y/\alpha$ for each $i$. Independent across $i$, each with variance $\sigma^2/|\alpha|^2$. Diagonal covariance $\propto I_P$. Matches CRLB exactly.

In the noise-limited regime, embedded-pilot LS is optimal — no estimator does better. This supports the choice of the simple per-cell threshold detector over more sophisticated schemes. $\blacksquare$

Embedded pilot for coarse path detection (small guard, low overhead) + superimposed ZC for refinement at detected path positions.

Coarse detection: $\gamma_{\text{th}}$ set high enough for 1% missed detection. Refinement: LS over the superimposed pattern within a small neighborhood of each detected path, giving MSE $\sigma^2/(\rho_p \cdot |\mathcal{N}|)$ where $|\mathcal{N}|$ is the refinement neighborhood size.

Choose embedded overhead $\eta$ to reliably detect all paths (e.g., 1% overhead). Choose $\rho_p = 0.02$ for refinement (2% of power to pilot, 98% to data). Combined MSE: $\sim 50\%$ lower than pure superimposed, at cost of only 1% extra overhead. Hybrid design is preferred when the deployment is latency-sensitive (fewer detection iterations) but cannot tolerate large pilot overhead (cell-free).

A single-cell pilot with amplitude $\alpha$ gives maximum time-domain PAPR (all the pilot energy is concentrated in a Dirac, which after OFDM modulation produces a Kronecker-delta waveform with very high peaks).

Use a Chu sequence $P[\ell, k] = e^{j\pi r \ell(\ell+1)/M}$ across one OFDM symbol of the DD grid. After ISFFT, this produces a constant-modulus TF-grid pattern, which has low PAPR.

The Chu pilot spreads over many DD cells — it acts more like a superimposed pilot. For embedded use, compromise: use a Chu sequence across a small pilot region, with PAPR similar to data. Zero-overhead superimposed pilots automatically have data-like PAPR.

Optimal PAPR-vs-detectability pilot designs remain a research topic — see Hsieh et al. (2022) for initial results. For standardized OTFS (Chapter 19), the Zadoff-Chu family is the likely choice because of its constant modulus and well-studied auto-correlation properties.

$\mathrm{MSE}_{\text{OTFS}} = P/\mathrm{SNR}_p = 8/316 = 0.025$ (−16 dB).

Imperfect channel estimate adds effective noise. If the channel coefficient has MSE $\epsilon$, the effective SNR for data detection is reduced by approximately $1/(1 + \epsilon) \approx 1 - \epsilon$. Penalty in dB: $-10\log_{10}(1 - 0.025) \approx 0.1$ dB.

At 25 dB pilot SNR, the channel-estimation penalty at the detector is $< 0.2$ dB — below typical implementation tolerances. This is why OTFS is robust: the pilot boost keeps estimation error well below the data-SNR floor.