Exercises

$v_{\text{orbit}} = \sqrt{GM/(R + h)} \approx 7.62$ km/s at 500 km.

At $\theta_{\text{elev}} = 10°$: $v_r = 7620 \cos 10° \approx 7510$ m/s.

$\nu = v_r/c \cdot f_0 = 7510/3 \times 10^8 \cdot 28 \times 10^9 = 701$ kHz.

OFDM ICI grows as $(\nu/\Delta f)^2$. At LEO Doppler, $\nu/\Delta f = 5.75$. ICI overwhelms signal.

SINR $\leq 3/(\pi^2 \cdot 5.75^2) = 3/(9.87 \cdot 33) = 0.0092$ = $-20$ dB. Negative. Unusable.

OFDM at LEO cannot function. OTFS achieves 20 dB SINR at same conditions.

Satellite moving toward UE at horizon: max Doppler $+690$ kHz.

Satellite directly overhead: radial velocity = 0. Doppler = 0.

Satellite moving away at horizon: Doppler $-690$ kHz.

Doppler sweeps $\sim 1.4$ MHz over 10-minute pass. Rate: $\sim 2$ kHz per second.

$d = 500/\sin 45° = 707$ km.

$L = (4\pi \cdot 707{,}000 / 0.01)^2 = 7.9 \times 10^{17}$ = 179 dB.

$P_r = 7 \text{ dBW} + 45 + 10 - 179 - 3 \text{ (atmos)} = -120$ dBW = $-90$ dBm.

Noise: $-174 + 80 + 3 = -91$ dBm. SNR = 1 dB. Very low. Need longer frame, larger array, or lower band for usable SNR.

$\Delta\nu = 1/T = 100$ Hz.

$N = 2 \cdot 690 \text{ kHz} / 100 \text{ Hz} = 13{,}800$ bins.

$M = W \cdot T = 20 \times 10^6 \cdot 0.01 = 200{,}000$? Wait: $M$ and $N$ relate via $MN = W T$, so $M = WT/N = 2 \times 10^5 / 13800 \approx 14$. Delay resolution: $\Delta\tau = 1/W = 50$ ns. 14 delay bins cover 700 ns — enough for LEO LOS (no significant multipath).

$MN = 14 \cdot 13800 \approx 2 \times 10^5$. Feasible.

$P_{\text{aggregate}} = p^S = 0.05^5 = 3.125 \times 10^{-7}$.

1 - $3.125 \times 10^{-7}$ = 99.99997% — "5 nines".

GEO single-sat: 99.5% (rain fades at Ka-band). LEO multi-sat OTFS: 99.9999%. $10^4\times$ better.

Macro-diversity delivers dramatic reliability improvements. Core advantage of LEO constellations over GEO.

Old satellite hand off → UE pauses → connects to new satellite. Pause: 0.1-1 second. Rate drop during pause.

UE maintains multiple concurrent satellite links. As one sets, its contribution diminishes gradually; as new rises, its contribution grows. Seamless.

Ground station coordinates which satellites serve which UE. As satellites change visibility, cluster updates. No UE- triggered handover.

Zero handover overhead. Continuous rate. Critical for URLLC and mobile applications.

Effective coverage area per LEO satellite (above 10° elevation): $\sim 5000$ km² effective footprint.

Total potential coverage: 6000 $\times$ 5000 km² = $3 \times 10^7$ km². Earth land surface: $\sim 1.5 \times 10^8$ km². Ratio: 20% simultaneously in coverage of any satellite.

Per-location probability of $S \geq 3$: depends on density. For Starlink: 80% of populated Earth has $S \geq 3$ (concentrating toward lower latitudes).

80% of 8 billion = 6.4 billion people potentially served by LEO-OTFS at $S \geq 3$. Including oceans/deserts: similar fraction of surface covered.

Per-satellite SNR: $\mathrm{SNR}_s = P_s/\sigma_w^2$. Assumes fixed per-sat SNR for simplicity (in practice varies with elevation).

Signal: $\sum_s e^{j\phi_s} \sqrt{P_s}$. Noise: $\sum_s w_s$ (sum of uncorrelated Gaussians). Signal magnitude: $\sqrt{S P_s}$ (coherent). Noise std: $\sqrt{S \sigma_w^2}$.

$\mathrm{SINR} = (S P_s)/(S \sigma_w^2) = P_s/\sigma_w^2$ = per-sat SNR. No gain if per-sat SNR is fixed!

The gain is in outage probability, not SNR: reliability improves as $(1-p)^S$. Rate: constant, but at $(1 - p^S)$ probability of availability.

If satellites have different path losses: best-AP SNR dominates. Effective SNR gain: $\sim \log_2 S$ for $S$ diverse paths.

$d = 35{,}786$ km. $L = (4\pi \cdot d/\lambda)^2$ at 28 GHz: $(4\pi \cdot 3.6 \times 10^7 / 0.01)^2 = 6 \times 10^{21}$ = 218 dB.

$d = 700$ km (45° elevation): $L = 179$ dB. 39 dB less than GEO.

For same Tx power: 39 dB better SNR for LEO. Rate: $\log_2(1 + \mathrm{SNR}_{\text{LEO}})$ vs $\log_2(1 + \mathrm{SNR}_{\text{GEO}})$. At 20 dB LEO SNR: 7 bits/s/Hz. At -19 dB GEO SNR: essentially 0.

LEO can add $\sim 7$ dB via $S = 5$ combining. GEO: no diversity. Total advantage: 40+ dB in effective SNR, 6-7 bits/s/Hz.

$\nu(t) = v_{\text{orbit}} / c \cdot f_0 \cdot \cos\theta_{\text{elev}}(t)$ where $\cos\theta_{\text{elev}}(t) = (R + h)\sin\omega t / d(t)$.

$\theta_{\text{elev}} = 90°$. $\cos = 0$. $\nu = 0$. $d\theta/dt$ at zenith: maximum.

Over a 10-minute pass ($2 \cdot 10^{-3}$ rad): $d\nu/dt = \nu_{\max}/T_{\text{pass}/2} = 690000 / 300 = 2.3$ kHz/s.

2.3 kHz/s Doppler drift is easily tracked by OTFS DD processing. Per-frame Doppler shift: 23 Hz (at 10 ms frame). Well within $\Delta\nu = 100$ Hz per-bin resolution.

Starlink orbits have inclination 53°. Satellites spend most time in middle latitudes. Polar coverage: few satellites simultaneously, quick transits.

At Antarctica (80°S): $S = 2$-$3$ satellites on average. Polar orbit (90° inclination) would be needed for high-lat coverage — Iridium Next has this.

$S = 2$: aggregate SNR $\sim$ single-sat + 3 dB diversity. Rate $\sim 30-40$ Mbps (vs 60-80 Mbps equatorial). Still dramatically better than GEO (where polar is zero coverage above 82°).

Antarctic research stations: LEO-OTFS is the only broadband option. Starlink + Iridium NEXT combined coverage.

Per-satellite: 100 Mbps → $10^8$ bps. Aggregate 10,000 sats: $10^{12}$ bps = 1 Tbps.

Each satellite's beam covers ~5000 km² with 50 UEs. Spatial reuse: 50 UEs/beam × 1000 beams/sat = 50,000 UEs/sat.

10,000 sats × 50,000 UEs/sat = $5 \times 10^8$ served UEs. Half a billion users.

1 Tbps / 500M users = 2 Mbps average. Modest but sufficient for basic services.

For 1B users at 50 Mbps average: need 50 Tbps aggregate, = 500,000 satellites. Not feasible with today's launches but within 2035+ horizon.

A LEO satellite at 550 km: visibility from any ground location for 5-10 minutes per ~95-minute orbit. $5/95 \approx 5\%$ of time.

For 99.99% availability with 95% per-sat: $S \geq \log(0.0001)/\log(0.05) = 3$.

Need $S = 3$ satellites simultaneously visible. Given per-sat 5% visibility: total satellites $N = S / 0.05 = 60$ (at any given time). Over full sky: $\sim 6000$ sats needed.

Starlink: 6000 satellites. Matches the minimum for 99.99% global availability. Not coincidentally the deployed scale.

Initial NTN support for OFDM. Starlink Direct-to-Cell. Doppler pre-compensation.

Enhanced NTN. Better pre-comp. OFDM still dominant.

LEO-specific optimizations. Cell reconfiguration.

3GPP evaluates OTFS for NTN. Performance comparisons. Standards work begins.

OTFS becomes standard for LEO-NTN. First OTFS-capable satellites launched.

Full 6G deployment. OTFS universal.

UE software-upgradable from OFDM to OTFS (hardware supports both). Satellites launch OTFS-enabled from 2028.

Single-plane constellation: satellites uniformly spread along orbit. Between satellites: coverage gap.

Multiple orbital planes, each with satellites. Multiple planes simultaneously overhead. Starlink: 72 planes × 22 sats/plane.

At any given ground location: always multiple planes visible. No single-plane gap.

Handover between planes is hard (satellites in different orbits). But within each plane: soft handover (§3).

For 99.99% global coverage: 3+ orbital planes per latitude band. Starlink, Kuiper: several planes. Iridium Next: 6 planes for polar coverage.