Ferkans — Interactive Telecom Tutor

The Satellite Renaissance

For decades, satellite communications were the backwater of wireless: narrowband, expensive, slow. That is changing. LEO constellations — SpaceX Starlink, Amazon Kuiper, OneWeb, Iridium Next — deploy thousands of satellites at 400-2000 km altitude, offering multi-hundred-Mbps links to ground UEs and global coverage (including remote, underserved regions). The technical challenges: extreme Doppler from orbital velocity, narrow beams requiring precise pointing, rapid satellite handover as UEs move between beams. OTFS, with its DD-native processing, addresses each of these challenges head-on — the CommIT contribution of Buzzi-Caire-Colavolpe (2024) is the quantitative case for LEO-OTFS.

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Definition:
LEO Satellite Parameters

Low Earth Orbit (LEO): satellites at altitude $h = 400$ - $2000$ km.

Orbital velocity $v_{\text{orbit}} = \sqrt{G M_\oplus / (R_\oplus + h)} \approx 7.5$ km/s for $h = 500$ km.
Orbital period: $T_\text{orb} = 2\pi \sqrt{(R_\oplus + h)^3 / (G M_\oplus)} \approx 95$ min.
Satellite pass duration (from UE perspective): $T_{\text{visibility}} \approx 5$ - $15$ minutes for a low-altitude LEO; depends on elevation angle geometry.
Elevation angle $\theta_{\text{elev}}$ : 10° (horizon) to 90° (zenith).
Slant range (UE to satellite): $\approx R_\oplus \sqrt{1 + h^2/R_\oplus^2 - 2\cos\theta_{\text{zenith}}} \approx 500$ - $2500$ km.

Frequency bands:

L-band (1.5 GHz): classical satellite (Iridium, Inmarsat). Low rate.
S-band (2 GHz): GPS, mobile satellite.
Ku/Ka-band (12-30 GHz): Starlink downlink. Moderate rate.
V-band (40-75 GHz): high-throughput LEO downlink. Gbps.

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The Geometric Picture

A LEO satellite sweeps from horizon to horizon in 5-15 minutes, passing through elevation angles $[10°, 90°]$ . The satellite's radial velocity (along the line-of-sight) varies continuously during the pass:

At horizon: $v_r = \pm v_{\text{orbit}} \sin\phi_{\text{rel}}$ where $\phi_{\text{rel}}$ is the angle between orbital track and LOS. For a direct overhead pass: near horizon $v_r \sim v_{\text{orbit}}$ ; at zenith $v_r = 0$ .
Continuous Doppler sweep from $+50$ kHz (at horizon, approaching) to $0$ kHz (at zenith) to $-50$ kHz (at horizon, receding).

This is the "hard" channel: Doppler is not a single tone but a time-varying spectrum. Classical Doppler compensation (LO tracking) is impossible — Doppler changes faster than a tracking loop can respond. OTFS's DD-domain processing is natively suited to this time-varying Doppler: each DD cell captures its own Doppler shift.

Theorem: LEO Channel Doppler Spread

For a LEO satellite at altitude $h$ , traveling at orbital velocity $v_{\text{orbit}}$ , the Doppler spread at a ground UE at elevation $\theta_{\text{elev}}$ is $\nu_{\text{LEO}}(\theta_{\text{elev}}) \;=\; \frac{v_{\text{orbit}} \cos \theta_{\text{elev}}}{c} \cdot f_0 \cdot \sin\phi_{\text{track}},$ where $\phi_{\text{track}}$ is the angle between orbital track and UE-satellite LOS.

Numerical (worst case: $\theta_{\text{elev}} = 10°$ , $\phi_{\text{track}} = 90°$ , $h = 500$ km, $f_0 = 28$ GHz): $\nu_{\text{LEO}} \;\approx\; (7.5 \times 10^3 / 3 \times 10^8) \cdot 28 \times 10^9 \cdot \cos 10° \cdot 1 \;\approx\; 690 \text{ kHz}.$ Over a pass: Doppler varies from $+690$ kHz at approach horizon to $-690$ kHz at receding horizon, passing through 0 at zenith. Rate of change: $d\nu/dt \sim 2 \nu_{\text{max}}/T_{\text{vis}} \sim 150$ Hz/s.

690 kHz Doppler is enormous by terrestrial standards. Comparison: automotive V2V at 300 km/h at 28 GHz is $\sim 7$ kHz — a factor of 100 less. No amount of 5G NR numerology can accommodate this; the Doppler is larger than any reasonable subcarrier spacing. OTFS, working in the DD domain, does not care — the channel is still sparse in DD (a single path at a specific $\nu$ for each satellite), just shifted to large $\nu$ values.

Proof

Doppler formula

$\nu = (v_r/c) \cdot f_0$ , radial velocity $v_r = v_{\text{orbit}} \cos\theta_{\text{elev}} \sin\phi_{\text{track}}$ .

Worst case

$\theta_{\text{elev}} = 10°$ (low horizon), $\phi_{\text{track}} = 90°$ (edge-on orbital pass). $v_r = 7.5 \times 10^3 \cdot 0.985 \cdot 1 = 7.4$ km/s. $\nu = 7.4 \times 10^3 / 3 \times 10^8 \cdot 28 \times 10^9 = 690$ kHz.

Time-variation

As satellite moves, $\theta_{\text{elev}}$ changes. $d\theta/dt \approx v/R \approx 0.1°$ /s. $d\nu/dt \sim \nu \cdot \sin\theta / R \sim 200$ Hz/s. $\blacksquare$

Definition:
LEO Propagation Channel

The LEO channel is nearly line of sight (LOS) in most scenarios:

Sky-clear links: $\sim 100\%$ LOS (no obstacles between UE and satellite).
Urban canyons: LOS $\sim 20$ - $60\%$ , rest are NLOS via reflection or blockage.
Rural/open: $\sim 95\%$ LOS.

Multipath: $P = 1$ - $3$ paths typical — mostly the LOS path, with occasional NLOS from nearby structures. Delay spread: $< 100$ ns in rural; $500$ ns in urban.

Path loss: large. At 500 km slant range and 28 GHz: $L = (4\pi d / \lambda)^2 = (4\pi \cdot 5 \times 10^5 / 0.01)^2 = 4 \times 10^{15}$ , or 156 dB. Link budget is the binding constraint.

Elevation-dependent: longer slant at low elevation means higher path loss and lower SNR. Practical operations exclude elevations below ~10°.

Example: LEO Link Budget at 28 GHz

A LEO satellite at $h = 500$ km serves a ground UE at elevation $45°$ on 28 GHz, with satellite Tx power 5 W (7 dBW), UE Rx antenna gain 10 dBi. Compute: (a) Slant range. (b) Free-space path loss. (c) Received signal power. (d) Received SNR at 100 MHz bandwidth.

Solution

Slant range

$d = h / \sin 45° = 500 / 0.707 = 707$ km.

Path loss

$L = (4\pi d/\lambda)^2 = (4\pi \cdot 7.07 \times 10^5 / 0.01)^2 = (8.9 \times 10^8)^2 = 7.9 \times 10^{17}$ = 179 dB.

Received signal

Satellite antenna gain: $G_t = 45$ dBi (phased array beam). Total: $P_r = P_t + G_t + G_r - L - L_{\text{atm}} = 7 + 45 + 10 - 179 - 3 = -120$ dBW = $-90$ dBm.

SNR

Noise floor at 100 MHz: $-174 + 80 + 3 = -91$ dBm (incl. 3 dB noise figure). SNR $= -90 - (-91) = 1$ dB. Very low.

Interpretation

LEO links are marginal at baseline 100 MHz bandwidth. Options: (a) Reduce bandwidth for higher SNR; (b) Use more powerful satellite TX; (c) Exploit antenna gain (UE array). For LEO- OTFS: typical operation is 20-40 MHz bandwidth with 15-20 dB SNR.

Doppler Sweep Over a LEO Pass

Plot LEO Doppler frequency as a function of elevation angle during a satellite pass (10° → 90° → 10°). Sliders: carrier frequency, orbital altitude.

Parameters

f_0

(GHz)28

h

(km)500

Pass geometry

Theorem: LEO Satellite Visibility Time

A LEO satellite at altitude $h$ above a ground UE with minimum useful elevation $\theta_{\min}$ is visible for a duration $T_{\text{visibility}} \;=\; \frac{2 R_\oplus (\theta_{\max} - \theta_{\min})}{v_{\text{orbit}}},$ where $\theta_{\max} = 90°$ for an overhead pass.

Numerical ( $h = 500$ km, $\theta_{\min} = 10°$ ):

Overhead pass: $T_{\text{visibility}} \approx 10$ minutes.
Oblique pass: $\approx 5$ - $8$ minutes.
Low-visibility pass: $\approx 1$ - $2$ minutes.

Consequence: A UE must handover between satellites every $\sim 5$ - $10$ minutes during continuous operation. The constellation must be dense enough that a fresh satellite appears in the sky before the current one sets — typical Starlink density: $\sim 5$ simultaneous-visible satellites over any ground location.

Unlike geostationary satellites (which remain fixed in the sky), LEO satellites sweep past. Continuous service requires multiple satellites — a constellation. The handover between satellites is frequent but each handover is "soft" (gradual signal change) if multiple satellites are simultaneously visible.

Proof

Geometry

Satellite traces an arc above the UE. From elevation $\theta_{\min}$ (rising) to $\theta_{\min}$ (setting) through $\theta_{\max}$ .

Arc length

Arc length = $R_\oplus \cdot 2\pi \cdot (\theta_{\max} - \theta_{\min})/360°$ . In angular terms: $2(\theta_{\max} - \theta_{\min})$ subtended from UE.

Time

Time = arc length / orbital speed. $\blacksquare$

🔧Engineering Note

Current and Future LEO Constellations

Major LEO constellations (2024):

SpaceX Starlink: 6000+ satellites at 550 km altitude. Capacity: 10 Tbps aggregate. Services Ka/V-band downlink to ~4 million customers. Terrestrial terminals with phased arrays.
Amazon Kuiper: 3236 satellites planned, 2024-2030 rollout. Target: competitor to Starlink.
OneWeb: 648 satellites at 1200 km. Ku-band. Enterprise- focused.
Iridium Next: 66 satellites at 780 km. L-band. Low-rate but global coverage (polar).
Planet / small-sat swarms: hundreds of imaging satellites. Non-communications.

Upcoming (2025+):

AST SpaceMobile: direct-to-smartphone LEO service.
China Guowang: Chinese LEO constellation.
Various national LEO programs (UK, India, Brazil).

Total LEO satellites in 2028: projected $\sim 20{,}000$ . This is the environment where OTFS-enabled LEO comms will live.

Practical Constraints

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2024: ~7k active LEO satellites globally
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2028: projected 20k+
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Primary users: broadband (Starlink), IoT (Iridium)
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6G integration: 2028+ with OTFS

Common Mistake: Antenna Pointing Is Hard

Mistake:

Treating LEO antenna pointing as solved. Ground UEs must point phased-array antennas at a moving satellite with sub-degree accuracy — challenging with inexpensive hardware.

Correction:

Use GPS-based tracking for the UE antenna: UE knows its own position from GPS, tracks satellite's position from ephemeris data (broadcast by satellite). Combined with coarse mechanical steering (or phased-array electronic steering), sub-degree pointing achievable with $\sim \$500$ consumer hardware. Starlink terminals use this approach. For mobile UEs (cars, ships): additional IMU integration. Design for the worst case: $\sim 1°$ pointing error, which translates to $\sim 1$ dB signal loss at typical array sizes.

Why This Matters: Connection: Telecom Ch 26 Satellite Communications

Telecom Chapter 26 provides the classical satellite communications background: link budgets, GEO vs MEO vs LEO, regulatory bands, TT&C protocols. This chapter builds on that foundation, extending to the modern LEO regime (constellations of thousands) with OTFS as the enabling modulation. The CommIT Buzzi-Caire-Colavolpe contribution is the quantitative bridge — showing how OTFS's DD-domain processing solves the high-Doppler problem that Telecom Ch 26's classical modulations cannot.

The LEO Satellite Channel