Exercises

$r = R_E + h = 7.171 \times 10^6$ m. $v_{\text{sat}} = \sqrt{3.986 \cdot 10^{14} / 7.171 \cdot 10^6} \approx \sqrt{5.557 \cdot 10^7} \approx 7454$ m/s $\approx 7.45$ km/s.

$f_D \approx 7454 / (3 \cdot 10^8) \cdot 28 \cdot 10^9 \approx 695$ kHz. Two-way (uplink + downlink cumulative) is about $1.39$ MHz. $\blacksquare$

$R_E \sin 45^\circ = 4504$. Radicand: $6371^2 \cdot 0.5 + 2 \cdot 6371 \cdot 600 + 600^2 \approx 2.029 \cdot 10^7 + 7.645 \cdot 10^6 + 3.6 \cdot 10^5 \approx 2.830 \cdot 10^7$. Square root $\approx 5320$. Subtract $4504$: $d_{\text{slant}} \approx 816$ km.

$\tau_{\text{prop}} = 816 / (3 \cdot 10^5) \approx 2.72$ ms. $\blacksquare$

$\Delta f_D / \Delta f = 400 / 240000 \approx 1.67 \cdot 10^{-3}$.

$(\pi \cdot 1.67 \cdot 10^{-3} / \sqrt{3})^2 \approx (3.03 \cdot 10^{-3})^2 \approx 9.18 \cdot 10^{-6}$.

Inverting, $\text{SIR} \approx 1.09 \cdot 10^5$, which is $\approx 50.4$ dB. Plenty of margin for any practical MCS. $\blacksquare$

$10 \log_{10} 6 \approx 7.78$ dB.

$5 + 7.78 = 12.78$ dB. The cluster brings the link from marginal ($5$ dB, $\log_2(1 + 3.16) \approx 2.05$ bit/s/Hz) to comfortable ($12.78$ dB, $\log_2(1 + 19) \approx 4.32$ bit/s/Hz) on a single SNR measure. $\blacksquare$

At zenith, $d_{\text{slant}} = h = 550$ km. Wavelength $\lambda = 10.7$ mm.

$4\pi d / \lambda = 4\pi \cdot 5.5 \cdot 10^5 / 0.0107 \approx 6.46 \cdot 10^8$. FSPL $= 20\log_{10}(6.46 \cdot 10^8) \approx 176.2$ dB.

Using the formula: $d_{\text{slant}} = \sqrt{6371^2 \cdot 0.0302 + 2 \cdot 6371 \cdot 550 + 550^2} - 6371 \cdot 0.1736 \approx \sqrt{1225000 + 7008100 + 302500} - 1106 \approx 2932 - 1106 = 1826$ km.

$4\pi \cdot 1.826 \cdot 10^6 / 0.0107 \approx 2.14 \cdot 10^9$. FSPL $= 20\log_{10}(2.14 \cdot 10^9) \approx 186.6$ dB.

$186.6 - 176.2 = 10.4$ dB excess loss at low elevation. Combined with the rain-fade increase at low elevations (longer rain path), this is why operators enforce a minimum elevation in the $10^\circ$–$25^\circ$ range. $\blacksquare$

$\omega_{\text{orb}} = 7.59 / 6921 \approx 1.097 \cdot 10^{-3}$ rad/s.

$\arccos(5773/6921) = \arccos(0.8341) \approx 0.5867$ rad ($\approx 33.6^\circ$). $\theta_{\text{min}} = 25^\circ = 0.4363$ rad. $\alpha_{\max} = 0.5867 - 0.4363 = 0.1504$ rad.

$T_{\text{visible}} = 2 \cdot 0.1504 / (1.097 \cdot 10^{-3}) \approx 274$ s $\approx 4.57$ min. Consistent with the stated $4$ min (the stated $4$ min is a round value; the exact formula gives slightly more for a perfect zenith pass). $\blacksquare$

Single-link rain: mean $3$ dB, $\sigma = 2$ dB, $99$-th percentile $\approx 3 + 2.33 \cdot 2 = 7.66$ dB. Outage SNR: $15 - 7.66 = 7.34$ dB.

Sum of $4$ independent lognormals: mean $3$ dB, effective $\sigma = 2 / \sqrt{4} = 1$ dB. $99$-th percentile fade: $3 + 2.33 \cdot 1 = 5.33$ dB.

Coherent gain: $10 \log_{10} 4 = 6$ dB. Outage SNR: $15 + 6 - 5.33 = 15.67$ dB. Improvement over single-link: $15.67 - 7.34 = 8.33$ dB at matched outage probability. $\blacksquare$

$d_{\text{visibility}} = \sqrt{7008100 + 302500} \approx 2703$ km.

$\lambda_{\text{HO}} = 3 \cdot 7.59 / (15 \cdot 2703) \approx 5.62 \cdot 10^{-4}$ Hz.

$\lambda_{\text{HO}}^{-1} \approx 1780$ s $\approx 29.7$ min between cluster changes — consistent with the "soft reselection every several minutes" operating regime of Section 23.5. $\blacksquare$

$f_D = 7560 / (3 \cdot 10^8) \cdot 20 \cdot 10^9 = 504$ kHz.

$504 / 30 = 16.8$. The Doppler is almost $17$ subcarriers wide.

OFDM's orthogonality breaks down at ratios above about $0.1$ (ICI of $-20$ dB). A ratio of $16.8$ means the raw signal is indistinguishable from noise at the subcarrier level. Ephemeris-based pre-compensation reduces the residual to $O(100)$ Hz, a ratio of $\approx 3 \cdot 10^{-3}$, which gives $\approx 40$ dB SIR floor and restores the OFDM operating regime. $\blacksquare$

Triangle with vertices $O$ (Earth center), $T$ (terminal), $S$ (satellite). $|OT| = R_E$, $|OS| = R_E + h$, angle at $T$ between $TO$ and $TS$ is $90^\circ + \theta_{\text{el}}$ (elevation measured from the local horizontal, so the vertical to $O$ is at $90^\circ + \theta_{\text{el}}$ from the line-of-sight).

$\frac{R_E}{\sin\eta} = \frac{|TS|}{\sin(90^\circ + \theta_{\text{el}})}$ where $\eta$ is the angle at $S$ between $SO$ and $ST$. Since $\sin(90^\circ + \theta_{\text{el}}) = \cos\theta_{\text{el}}$, $\sin\eta = R_E \cos\theta_{\text{el}} / |TS|$, and $|TS| = d_{\text{slant}}$ from Definition DSlant Range and One-Way Delay.

The satellite velocity is tangent to its circular orbit, perpendicular to $SO$. The component along $ST$ is $v_{\text{sat}} \sin\eta = v_{\text{sat}} R_E \cos\theta_{\text{el}} / d_{\text{slant}}$. Dividing by $c$ and multiplying by $f_0$ gives the stated Doppler shift. $\blacksquare$

$5 \cdot 1 = 5$ Gb/s feeder-link load per user. Compared to single-satellite baseline ($1$ Gb/s), $5\times$ the load.

$1$ Gb/s feeder-link load per user; the master satellite distributes to the other $4$ over ISL at no additional feeder-link cost. The ISL total load is $4 \cdot 1 = 4$ Gb/s on the in-orbit mesh, but this is distinct from the feeder-link spectrum and scales with ISL capacity rather than gateway spectrum.

ISL-routed is $5\times$ more spectrum-efficient on the feeder side. This is why the 6G NTN research proposals (including the Buzzi-Caire-Colavolpe baseline) assume ISLs. Starlink v2.0 satellites have optical ISLs; OneWeb and earlier Starlink satellites do not. $\blacksquare$

In the delay-Doppler domain, a pure LOS channel with a single Doppler shift and a single delay becomes a single sparse tap, so OTFS equalization reduces to one multiplication per lattice point. OFDM, in contrast, suffers inter-carrier interference that grows with the Doppler-to-subcarrier-spacing ratio; on a fast-changing channel the ICI floor limits achievable SIR regardless of transmit power. OTFS therefore avoids both the OFDM ICI problem and the wide-beam CP budget problem in a single representation change.

OFDM is the 5G NR waveform, and the ecosystem — modem ASICs, channel estimation procedures, standards compliance, test and measurement equipment — is all OFDM-based. Deploying OTFS would require a parallel non-standard modem stack, which no operator has been willing to fund at scale. 3GPP has not adopted OTFS as of Release 18, so LEO deployments through 2025+ will use OFDM with aggressive ephemeris pre-compensation and narrow spot beams. $\blacksquare$

With matched filtering $\mathbf{v}_{m} = \sqrt{P_m/N_t} \mathbf{H}_{m}$, the contribution to the received signal from satellite $m$ is $\sqrt{\beta_{m}} \mathbf{H}_{m}^{H} \mathbf{v}_{m} = \sqrt{\beta_{m}} \sqrt{P_m/N_t} \mathbf{H}_{m}^{H} \mathbf{H}_{m} = \sqrt{\beta_{m} P_m N_t}$.

Under phase-coherent combining, the total amplitude is $A = \sum_m \sqrt{\beta_{m} P_m N_t}$. With equal power $P_m = P_t/M$, $A = \sqrt{N_t P_t/M} \sum_m \sqrt{\beta_{m}}$.

$|A|^2 = (N_t P_t/M) (\sum_m \sqrt{\beta_{m}})^2 \geq (N_t P_t/M) \sum_m \beta_{m}$ by Jensen's, with equality when all $\beta_{m}$ are equal.

$\text{SNR}^{\text{coh}} = |A|^2 / \sigma^2 = (N_t/M) (P_t/\sigma^2) (\sum_m \sqrt{\beta_{m}})^2$. Restricting to the equal-gain case gives the stated $(N_t/M)(P_t/ \sigma^2) \sum_m \beta_{m}$ form. $\blacksquare$

$\text{SINR}_{\text{single}} \approx (N_t - K) \cdot \text{SNR} / K = 56 \cdot 10^{1.2} / 8 \approx 56 \cdot 15.85 / 8 \approx 111 \approx 20.5$ dB.

The cluster acts as an $M N_t = 256$-element virtual array. $\text{SINR}_{\text{cluster}} \approx (M N_t - K) \cdot \text{SNR} / K = 248 \cdot 15.85 / 8 \approx 491 \approx 26.9$ dB. Gain: $6.4$ dB over the single-satellite baseline, close to the $10\log_{10} 4 = 6$ dB ideal.

Per user: $\log_2(1 + 491) \approx 8.94$ bit/s/Hz for the cluster vs $\log_2(1 + 111) \approx 6.80$ bit/s/Hz for the single satellite. At $100$ MHz bandwidth, the cluster gives $\approx 894$ Mb/s/user vs $\approx 680$ Mb/s/user for single-satellite.

The ZF model is optimistic (it assumes perfect CSI and no pilot contamination); Release 17 NTN actually uses quantized codebook-based precoding with much lower spectral efficiency. The key point is that the cluster gain over single-satellite is approximately $M^{1/2}$ to $M$ depending on the interference regime, and the qualitative advantage is robust. $\blacksquare$

Beam center (zenith of illuminated spot): $d_{\text{center}} = 600$ km. Beam edge: $d_{\text{edge}} = \sqrt{600^2 + 100^2} \approx 608.3$ km.

$\Delta d = 8.3$ km. $\Delta \tau = 8.3 / (3 \cdot 10^5) \approx 27.6$ $\mu$s.

FR2 CP $\approx 0.58$ $\mu$s. The differential delay ($27.6$ $\mu$s) is $\approx 48\times$ larger than the CP. Even with a narrow $100$ km beam, the CP is inadequate for a common waveform across the footprint.

Per-user timing advance pre-compensation is mandatory, as in 3GPP TR 38.821 §6.6. Each terminal aligns its own frame reference to its own slant range, so the net ISI within its decoded stream is zero. The differential delay only matters for broadcast channels, which use a much longer CP or are sent at a low modulation order robust to ISI. $\blacksquare$

$M N_t = 4 \cdot 64 = 256$ virtual array elements.

$4 \cdot 200 = 800$ W total across $10$ users, or $80$ W/user served.

$200$ W across $10$ users = $20$ W/user.

The cluster costs $4\times$ more DC power per user for a spectral-efficiency gain of roughly $\log_2(M) \approx 2$ bits/s/Hz at high SNR. The rate-per-watt is thus lower in the cluster, but the reliability (outage at $99\%$) is an order of magnitude better. The trade-off is a classic "burn power for reliability" design choice, and operators typically use the cluster mode only during rain events or low-elevation passes, reverting to single-satellite during favourable conditions. $\blacksquare$

$J(M) = \log_2(1 + M \cdot \text{SNR}) - \lambda M$.

$dJ/dM = (\text{SNR} / \ln 2) / (1 + M \cdot \text{SNR}) - \lambda = 0$.

$1 + M^\star \cdot \text{SNR} = (\text{SNR})/(\lambda \ln 2)$, so $M^\star = 1/(\lambda \ln 2) - 1/\text{SNR}$. For $\lambda$ small (feeder link is cheap) and SNR moderate, $M^\star$ is large; as $\lambda$ grows, $M^\star$ decreases.

For $\text{SNR} = 10$ (10 dB) and $\lambda = 0.1$ (each additional cluster member costs $0.1$ bits/s/Hz of distribution overhead), $M^\star \approx 1/(0.1 \cdot 0.693) - 1/10 = 14.4 - 0.1 \approx 14.3$, i.e. $M^\star \approx 14$. For $\lambda = 0.3$, $M^\star \approx 4.7$, i.e. $M^\star \approx 5$. The paper's typical value of $M^\star \approx 3$–$6$ corresponds to the heavy-feeder-link regime $\lambda \approx 0.3$–$0.5$. $\blacksquare$

Define a composite metric combining (i) estimated rain attenuation $\hat{L}_{\text{rain}}$ from the terminal's Ka-band SNR history, (ii) current minimum elevation $\theta_{\text{el}}^{\min}$ across the visible set, and (iii) $99\%$-outage margin $M_{\text{margin}}$. Trigger cluster mode when $\hat{L}_{\text{rain}} > 2$ dB or $\theta_{\text{el}}^{\min} < 15^\circ$.

Evaluate the trigger every $100$ ms (about $10$ OFDM symbol frames). A quicker interval wastes ISL bandwidth; a slower one misses rain-fade onset.

When switching from single-sat to cluster, the master satellite gradually ramps up the cluster members over one cluster-change window ($\approx 10$ ms). When switching back, the departing satellites fade out their contribution. The terminal sees a smooth amplitude change, not a hard re-acquisition.

• Always single-sat wastes the macro-diversity gain in adverse conditions; users experience rain-fade outages.
• Always cluster wastes $M\times$ feeder-link spectrum during clear-sky zenith passes when no diversity is needed.
• Adaptive spends feeder-link resources only when they buy reliability, saving $\approx 75\%$ of the always-cluster feeder-link load while preserving $>90\%$ of the reliability gain.

Hysteresis is needed to avoid rapid mode oscillation ($5$-second dwell time minimum). ISL capacity must still support peak cluster loading across all simultaneously-in-cluster users, not just the average. Operators can combine adaptive clustering with asymmetric cluster weights (few "full" members, many "diversity-only" members) for finer control. This is roughly the operating mode envisioned in the Buzzi-Caire-Colavolpe paper's Section VI concluding discussion. $\blacksquare$

Let $a_m = \sqrt{\beta_{m} P_m N_t}$. The perturbed amplitude is $\tilde{A} = \sum_m a_m e^{j \epsilon_m}$. Expected value: $\mathbb{E}[\tilde{A}] = \sum_m a_m \mathbb{E}[e^{j \epsilon_m}] = \sum_m a_m e^{-\sigma_\epsilon^2/2} = e^{-\sigma_\epsilon^2/2} \sum_m a_m$ using the characteristic function of a Gaussian.

$\mathbb{E}[|\tilde{A}|^2] = \mathbb{E}[\sum_{m,m'} a_m a_{m'} e^{j(\epsilon_m - \epsilon_{m'})}]$. Split into $m = m'$ (contributing $\sum_m a_m^2$) and $m \neq m'$ (contributing $\sum_{m \neq m'} a_m a_{m'} \mathbb{E}[e^{j(\epsilon_m - \epsilon_{m'})}] = (|\sum_m a_m|^2 - \sum_m a_m^2) e^{-\sigma_\epsilon^2}$).

$\mathbb{E}[|\tilde{A}|^2] = \sum_m a_m^2 (1 - e^{-\sigma_\epsilon^2}) + |\sum_m a_m|^2 e^{-\sigma_\epsilon^2}$. For equal path losses $a_m = a$, this is $M a^2 (1 - e^{-\sigma_\epsilon^2}) + M^2 a^2 e^{-\sigma_\epsilon^2}$. The ratio to the ideal $M^2 a^2$ is $e^{-\sigma_\epsilon^2} + (1 - e^{-\sigma_\epsilon^2})/M$.

Setting the ratio $= 0.9$: for $M = 4$, $e^{-\sigma_\epsilon^2} + (1 - e^{-\sigma_\epsilon^2})/4 = 0.9$. Let $x = e^{-\sigma_\epsilon^2}$. Then $0.75 x = 0.65$, $x = 0.867$, $\sigma_\epsilon^2 = 0.143$, $\sigma_\epsilon \approx 0.378$ rad $\approx 21.6^\circ$. Converting to time synchronization at $f_0 = 28$ GHz: $\Delta t = \sigma_\epsilon / (2 \pi f_0) \approx 2.1$ ps. The satellites must be synchronized to picosecond precision to capture $90\%$ of the ideal coherent gain. This is achievable with GNSS-disciplined atomic clocks plus ephemeris correction — and is one of the harder engineering challenges of cell-free LEO.

Realistic deployments will likely operate at $\sigma_\epsilon \approx 0.2$–$0.3$ rad ($12$–$17^\circ$), capturing $\approx 95\%$ of the ideal gain and losing $\approx 0.2$ dB to phase jitter — well below the $\approx 6$ dB headline gain. $\blacksquare$

Starlink terminals track two visible satellites simultaneously, but only one is actively serving at any instant; the second is on stand-by for handover. The Buzzi-Caire-Colavolpe cell-free mode has $M > 1$ satellites simultaneously transmitting. This is the biggest operational difference and the one most likely to change first as ISL capacity matures.

Starlink uses per-satellite narrow spot beams with frequency reuse 1/4 or similar. Cell-free mode removes the concept of a beam boundary — every cluster member covers the user regardless of its nominal beam direction, and inter-beam interference becomes a cluster-wide MU-MIMO problem. This requires significant upgrades to the on-board digital beamformer.

Starlink does a hard every-2–15-minute handover with brief service interruption. Cell-free mode does soft cluster reselection with no interruption. This is a user-experience difference rather than a throughput difference.

Difference 1 (single-sat vs cluster) is the most likely first move, because it builds directly on the ISL capacity Starlink already has (v2.0 onwards) and does not require new on-board digital beamformer hardware. The easiest first step would be a two- satellite selection-diversity mode (ground selects the currently-best of two visible satellites with no coherent combining), followed by two-satellite coherent combining once synchronization is validated. Moving to $M = 4$ and beyond is a longer-term 6G NTN evolution that benefits from the research programme the Buzzi-Caire-Colavolpe paper is part of. $\blacksquare$