NTN encompasses any 5G access node carried on a non-ground platform. The four principal host platforms differ in altitude, delay, and coverage footprint:

• LEO (Low Earth Orbit) — 300–1500 km altitude; fast-moving (~7.6 km/s); low delay (~2–5 ms one-way); coverage arc 1000–2000 km; requires large constellation for global coverage (e.g. Starlink ~550 km, OneWeb ~1200 km).
• MEO (Medium Earth Orbit) — 2000–35 786 km; intermediate delay (~30–60 ms); traditionally used for GNSS (GPS at 20 200 km); O3b mPower broadband constellation at 8062 km.
• GEO (Geostationary Earth Orbit) — exactly 35 786 km; stationary relative to Earth; large coverage footprint (~1/3 of Earth surface); high delay (~120 ms one-way); Doppler negligible; legacy satellite broadband standard.
• HAPS (High-Altitude Platform Station) — 17–22 km altitude (stratosphere); quasi-stationary; delay ~0.06 ms; coverage ~200 km diameter circle; powered by solar; bridges satellite and terrestrial worlds.

The 3GPP NTN work began in Rel-15 as a Study Item (TR 38.811) and culminated in the first normative standard in Rel-17 (TS 38.821). Rel-18 and Rel-19 extend NTN with IoT support, inter-satellite links, and mobility enhancements.

Orbital Velocity

For a circular orbit at altitude $h$ above Earth's surface, the centripetal acceleration must equal gravitational acceleration. Solving for tangential speed:

$$v_{orb} = \sqrt{\frac{GM}{R_E + h}}$$

where $GM = 3.986 \times 10^{14}\ \text{m}^3/\text{s}^2$ (Earth's gravitational parameter) and $R_E = 6371\ \text{km}$ (mean Earth radius).

Orbital Period

From Kepler's third law:

$$T_{orb} = 2\pi \sqrt{\frac{(R_E+h)^3}{GM}}$$

The short LEO visibility window (minutes per pass per satellite) drives the need for large constellations and fast handover procedures in Rel-18 NTN mobility enhancements.

The slant range $d_{slant}$ is the straight-line distance from the ground terminal (on Earth's surface) to the satellite at elevation angle $\theta_{el}$. Using the law of cosines on the Earth–satellite triangle:

$$d_{slant} = \sqrt{(R_E+h)^2 - R_E^2\cos^2\theta_{el}} \;-\; R_E\sin\theta_{el}$$

Special cases: at zenith ($\theta_{el}=90°$), $d_{slant}=h$; at horizon ($\theta_{el}=0°$), $d_{slant}=\sqrt{(R_E+h)^2-R_E^2}=\sqrt{h^2+2R_E h}$.

The spherical cap area visible from a satellite at altitude $h$ for terminals with minimum elevation angle $\theta_{el,min}$ is:

$$A_{coverage} = 2\pi R_E^2 \left(1 - \frac{R_E}{R_E+h}\cos\theta_{el,min}\right)$$

The coverage half-angle (Earth-central angle) $\psi_{max}$ satisfies:

$$\cos\psi_{max} = \frac{R_E}{R_E+h}\cos\theta_{el,min} - \text{NaN} \quad \Rightarrow \quad \psi_{max} = \arccos\!\left(\frac{R_E}{R_E+h}\cos\theta_{el,min}\right) - \theta_{el,min}$$

The minimum number of satellites $N_{sat}$ for continuous global coverage is approximately:

$$N_{sat} \geq \frac{2\pi R_E^2}{A_{coverage}} = \frac{1}{1 - \dfrac{R_E}{R_E+h}\cos\theta_{el,min}}$$

Practical constellations (e.g. Starlink phase-1: 1584 sats at 550 km, 53° inclination) deploy far more than the theoretical minimum to ensure multi-satellite visibility and capacity headroom.

$$\tau = \frac{d_{slant}}{c}$$

Reference values ($c = 2.998\times10^8$ m/s):

• LEO 550 km, zenith: $\tau = 550\,000/2.998\times10^8 = \mathbf{1.83\ \text{ms}}$
• LEO 550 km, 45° elevation: $\tau \approx 750\,000/c = \mathbf{2.50\ \text{ms}}$
• GEO 35 786 km, zenith: $\tau \approx \mathbf{119.3\ \text{ms}}$

The round-trip time (RTT) for service link only in a bent-pipe (transparent) architecture is:

$$RTT_{SL} = 2\tau$$

However the full end-to-end RTT in a bent-pipe system includes the feeder link (satellite $\leftrightarrow$ gateway) twice:

$$RTT_{e2e} = 2\tau_{SL} + 2\tau_{FL} + \tau_{GW}$$

NR HARQ round-trip timeout is nominally 8 ms (FR1, normal CP). LEO zenith barely fits; LEO at low elevation and all MEO/GEO require the Rel-17 HARQ disable mechanism (RRC signalling: harq-ProcessNumberCodeBlockGroupList set to 1 with extended K1/K2 offsets defined in TS 38.213 Table 9.2.3-1).

Model: circular orbit, sub-satellite-point pass directly overhead (maximum-Doppler track). $f_D(\theta_{el}) = (v_{orb}/c)\,f_c\,\cos\alpha$, where $\alpha$ is the angle between the satellite velocity vector and the UE–satellite direction. $\cos\alpha = \cos\theta_{el}\,\sin\phi_{az}$ integrated along the overpass geometry. For a direct-overhead pass: $f_D = (v_{orb}/c)\,f_c\,\cos\theta_{el}\,\text{sgn}(\dot\theta_{el})$.

Doppler Shift — Key Numbers

The maximum Doppler shift occurs when the satellite is at the horizon ($\theta_{el} = 0°$, satellite approaching). The exact formula is:

$$f_{D,max} = \frac{v_{orb}}{c} \cdot f_c$$

At LEO 550 km ($v_{orb}=7589$ m/s), the peak Doppler for various bands:

Without pre-compensation, LEO Doppler at C-band (88.5 kHz) is ~6× the 15 kHz subcarrier spacing — devastating for OFDM. The Rel-17 NTN solution requires either the satellite to pre-compensate (for the feeder-link common Doppler) or the UE to correct residual Doppler using its own position and the SIB19 ephemeris data. Residual Doppler after UE pre-compensation is typically <1 kHz (within 120 kHz SCS budget).

§2.1  Orbit Classification & Link-Budget Overview

Free-Space Path Loss (FSPL) is the dominant propagation impairment in NTN. For a slant range $d_{\rm km}$ (km) and carrier frequency $f_{\rm MHz}$ (MHz):

$$\text{FSPL} = 32.44 + 20\log_{10}(d_{\rm km}) + 20\log_{10}(f_{\rm MHz}) \quad [\text{dB}]$$

Example — LEO at $h = 550$ km zenith pass, $f = 3.5$ GHz ($f_{\rm MHz} = 3500$):

$$\text{FSPL} = 32.44 + 20\log_{10}(550) + 20\log_{10}(3500) = 32.44 + 54.81 + 70.88 = 158.1 \text{ dB}$$

Round-trip time (RTT) assumes two-way propagation over the zenith slant range $h$, neglecting atmospheric delay: $\text{RTT} = 2h/c$. Orbital velocity follows $v_{\rm orb} = \sqrt{GM/r}$ with $r = R_E + h$, $R_E = 6371$ km.

FSPL values assume zenith (minimum slant range = altitude). Actual link margin must add atmospheric absorption ($\approx 0.4$ dB at 3.5 GHz), rain fade (climate-dependent), and pointing loss.

§2.2  Keplerian Orbital Elements & 3GPP SIB19 Ephemeris

A satellite orbit is fully described by six classical elements. These appear directly in TS 38.331 §6.3.2 as the orbitalElements IE within SIB19 / NTN-Config.

Orbital mechanics used by UE position prediction

Mean motion (rad/s) — angular speed for circular orbit:

$$n = \sqrt{\frac{GM}{a^3}}, \quad GM = 3.986\times10^{14} \text{ m}^3/\text{s}^2$$

Mean anomaly propagation:

$$M(t) = M_0 + n\,(t - t_0)$$

Kepler's equation — convert mean anomaly $M$ to eccentric anomaly $E$ (solved iteratively):

$$M = E - e\sin E$$

True anomaly $\nu$ and radius $r$ follow from $E$:

$$\tan\frac{\nu}{2} = \sqrt{\frac{1+e}{1-e}}\tan\frac{E}{2}, \qquad r = a(1 - e\cos E)$$

The UE uses these relations (with the satellite position broadcast via SIB19 epoch-Time) to compute Timing Advance pre-compensation and Doppler pre-correction locally — no real-time signalling required between each TA update.

3GPP reference: TS 38.331 §6.3.2 defines orbitalElements within NTN-Config. TS 38.821 §6.3 specifies UE-side satellite position and Timing Advance computation based on Keplerian propagation. Validity window is signalled via t0 and validityDuration (max 128 s in Rel-17).

§2.3  Walker-Delta Constellation Design

A Walker-Delta constellation is described by the notation $i°: T/P/F$ where:

• $T$ — total number of satellites,
• $P$ — number of equally-spaced orbital planes,
• $F$ — phasing parameter ($0 \le F < P$), controlling the relative anomaly offset between adjacent planes: $\Delta M = 360°F/T$.

Each plane carries $S = T/P$ satellites uniformly distributed.

Real-world constellations

Coverage geometry

A satellite at altitude $h$ above a spherical Earth ($R_E = 6371$ km) covers all ground points within an Earth central angle $\alpha$ from the sub-satellite point, where $\alpha$ is constrained by the minimum elevation angle $\theta_{\min}$:

$$\cos\alpha = \frac{R_E}{R_E + h}\cos\theta_{\min}$$

The instantaneous coverage area on Earth's surface is:

$$A_{\rm cov} = 2\pi R_E^2 (1 - \cos\alpha) \quad [\text{km}^2]$$

Minimum number of satellites for continuous global coverage at elevation $\theta_{\min}$:

$$N_{\min} \approx \frac{4\pi R_E^2}{A_{\rm cov}} = \frac{2}{1 - \cos\alpha}$$

For LEO at $h = 550$ km with $\theta_{\min} = 10°$: $\alpha \approx 58.7°$, giving $N_{\min} \approx 5$ satellites for single-coverage; practical constellations use 3–6× redundancy for reliability and capacity.

Visibility window duration

A ground observer at latitude $\varphi$ sees a satellite in a circular orbit at $h$ for an approximate contact time of:

$$T_{\rm vis} \approx \frac{T_{\rm orb}}{\pi} \arccos\!\left(\frac{\cos\alpha}{\cos\varphi}\right) \quad \text{(equatorial approximation)}$$

For Starlink at 550 km ($T_{\rm orb} = 95.5$ min, $\theta_{\min} = 10°$): $T_{\rm vis} \approx 8$–$12$ minutes per pass — consistent with observed 10-minute average contact windows.

§2.4  LEO vs. GEO — Detailed Parameter Comparison

The two most commercially significant orbits impose very different design constraints on NR-NTN. The table below quantifies the key parameters; formulas are given where the derivation adds clarity.

3GPP NTN adaptation summary: LEO requires UE-side Doppler pre-compensation (TS 38.213 §4.1) and extended TA; GEO requires HARQ-less operation (TS 38.300 §15.6) and larger timing advance fields. Both benefit from the satellite ephemeris broadcast in SIB19 (TS 38.331).

§2.5  HAPS — High Altitude Platform Stations

HAPS operates at approximately 20 km altitude in the stratosphere, using long-endurance aircraft, airships, or balloons. Because the platform is quasi-stationary ($v_{\rm HAPS} \approx 0$), Doppler shift is negligible and standard NR procedures require minimal adaptation — making HAPS the easiest NTN tier to integrate.

Key propagation parameters

HAPS footprint geometry

The coverage radius on the ground for a minimum elevation angle $\theta_{\min}$:

$$r_{\rm cov} = R_E \cdot \alpha = R_E \arccos\!\left(\frac{R_E\cos\theta_{\min}}{R_E + h}\right) \approx h / \tan\theta_{\min} \quad (h \ll R_E)$$

At $h = 20$ km, $\theta_{\min} = 10°$: $r_{\rm cov} \approx 20/\tan(10°) \approx 113$ km, giving a footprint diameter of $\approx 226$ km — comparable to a large metropolitan area.

Capacity & 3GPP standardisation (TR 38.865)

A HAPS with 2 m aperture at 26 GHz achieves $\approx 0.53°$ beamwidth — enabling hundreds of spot beams and tens of Gbps aggregate capacity. TR 38.865 (Rel-17) conclusions: standard NR waveform applies directly; TA within normal range (El > 10°); only adaptation needed is moving cell support and satellite-to-HAPS IAB backhaul.

Design insight: HAPS bridges the gap between terrestrial small cells and LEO satellites. Doppler impact is negligible ($f_d \ll$ subcarrier spacing even at $\Delta f = 15$ kHz), one-way delay is sub-millisecond, and FSPL is 36 dB lower than LEO at 550 km. HAPS is thus the NTN tier most amenable to 5G NR re-use with minimal specification change (TR 38.865, §6.2).

§2.6  LEO Overpass — Ground Track & Elevation Angle (Interactive)

The canvas below animates a simplified LEO satellite overpass for a 550 km circular orbit at 53° inclination passing overhead a ground station. The upper panel shows the projected ground track; the lower panel plots elevation angle vs. time. The satellite rises above the 10° mask angle for approximately 10 minutes.

Reading the plots: The upper canvas shows the satellite's projected ground latitude over time (blue = above 10° elevation, grey = below mask). The lower panel plots elevation angle directly; the dashed red line marks $\theta_{\min} = 10°$, and the green portion indicates the visible window ($\approx 10$ minutes for 550 km LEO). Peak elevation depends on the pass geometry — a zenith pass gives $\approx 90°$; an off-axis pass gives a lower peak with a longer but lower-quality window.

§2 Key Takeaways

• FSPL scales as $20\log_{10}(d)$: GEO has 31.6 dB more path loss than LEO at 550 km, demanding larger apertures or lower throughput.
• Doppler is the LEO challenge: $\pm 89$ kHz at 3.5 GHz for 550 km LEO (vs < 0.4 kHz GEO) requires mandatory UE-side Doppler pre-compensation via SIB19 ephemeris.
• HARQ is the GEO challenge: 238 ms RTT makes standard 8-process HARQ infeasible; Rel-17 introduces HARQ-less mode with PDCP/RLC ARQ fallback.
• Walker-Delta gives systematic coverage: $T/P/F$ notation fully describes constellation geometry; $F$ controls inter-plane phasing to eliminate coverage gaps at target latitudes.
• HAPS is the NTN "easy mode": Sub-millisecond RTT, negligible Doppler, 122 dB FSPL at zenith — standard NR procedures apply almost unchanged (TR 38.865).
• Keplerian propagation underpins NTN TA pre-compensation: UE derives satellite position from SIB19 orbitalElements and computes TA/Doppler corrections locally for up to 128 s without network updates.

§3.1 TR 38.811 NTN Channel Model

Total Path Loss

The composite link budget in NTN accounts for free-space spreading, atmospheric absorptions, precipitation, ionospheric scintillation, and slow-fading shadow:

$$L_{total} = \text{FSPL} + L_{gas} + L_{rain} + L_{scint} + L_{shadow} \quad [\text{dB}]$$

Free-space path loss: $\text{FSPL} = 20\log_{10}(4\pi d f / c)$ where $d$ is slant range (km), $f$ carrier frequency (Hz). For LEO at 600 km altitude and FR1 (2 GHz), FSPL $\approx$ 164 dB. For GEO at 35 786 km, FSPL $\approx$ 189 dB at 20 GHz.

$L_{gas}$: O$_2$ + H$_2$O absorption; significant above 10 GHz. At 20 GHz zenith: ~0.5 dB; at low elevation angles (10°): ~3–5 dB. $L_{rain}$: dominant impairment at Ka-band; ITU-R P.838-3 model. At 30 GHz, 50 mm/hr rain: 10–15 dB for GEO. $L_{scint}$: tropospheric scintillation; Gaussian amplitude fluctuations, 0.5–2 dB at Ka-band low elevation. $L_{shadow}$: log-normal; standard deviation 1–4 dB depending on environment.

Rician Fading Channel

NTN links are LOS-dominant, especially at high elevation angles. The Rician model captures the strong specular component plus a diffuse scatter floor:

$$h(t) = \underbrace{\sqrt{\frac{K}{K+1}}\, e^{j\phi_0}\,\delta(\tau - \tau_0)\,e^{j2\pi f_d t}}_{\text{LOS component}} \;+\; \underbrace{\sqrt{\frac{1}{K+1}}\, h_{\text{scatter}}(t)}_{\text{NLOS scatter}}$$

where $K$ is the Rician K-factor (linear ratio, often quoted in dB), $\phi_0$ is the LOS path initial phase, $\tau_0$ is the LOS delay (e.g. $d/c$), and $f_d$ is the Doppler shift. The scatter term $h_{\text{scatter}}(t)$ follows a zero-mean complex Gaussian process with power spectral density shaped by the multipath Doppler spectrum.

As $K \to \infty$ the channel becomes pure AWGN (perfect LOS). At $K = 0$, it reduces to Rayleigh fading. In NTN, $K \gg 1$ dB for most scenarios.

K-Factor vs. Elevation Angle

TR 38.811 Table 6.6.2-2 provides elevation-dependent K-factor fits. The parameterisation used in simulations:

$$K_{dB} = K_0 + k_1 \cos\theta_{el} + k_2$$

where $\theta_{el}$ is the satellite elevation angle (degrees), and $(K_0, k_1, k_2)$ are environment-dependent coefficients. Representative values: at $\theta_{el} = 90°$ (zenith), $K \approx 10$ dB (strong LOS); at $\theta_{el} = 10°$ (near horizon), $K \approx 1$–2 dB (near-Rayleigh). Low-elevation links suffer increased shadow fading, diffraction, and multipath from terrain.

CDLNTN Delay Profiles

TR 38.811 defines four NTN-adapted CDL (Clustered Delay Line) models, parameterised by delay spread, K-factor, and mobility class:

All CDLNTN models include a large fixed delay offset $\tau_{fixed} = d_{min}/c$ (LEO: ~2–4 ms, GEO: ~120 ms) added before the relative cluster delays. The NR TA mechanism must pre-compensate this bulk delay so that UL arrives within the cyclic prefix window.

Doppler in NTN

Satellite orbital velocity combined with UE location determines the instantaneous radial velocity $v_r$: $$f_d = \frac{v_r}{c} \cdot f_c = \frac{\vec{v}_{sat} \cdot \hat{r}_{sat\to UE}}{c} \cdot f_c$$ For LEO at 550 km, orbital speed ~7.6 km/s, $f_c = 2$ GHz: max $f_d \approx 50$ kHz. For Ka-band (26.5 GHz): max $f_d \approx 660$ kHz. This far exceeds the subcarrier spacing ($\Delta f = 15$ kHz for NR SCS = 15 kHz) and requires explicit Doppler pre-compensation at the UE (TS 38.331 §5.7.2a).

§3.2 5G NR DL Physical Channels (TS 38.211)

The 5G NR DL air interface organises all signals into a resource grid of subcarriers × OFDM symbols. Physical channels carry coded data or control; reference signals enable channel estimation and synchronisation. NTN Rel-17 extensions modify timing offsets, search-window sizes, and feedback loops throughout.

4.1.1 PRACH — Physical Random Access Channel

The PRACH preamble is a Zadoff–Chu (ZC) sequence of length $L_{RA}$. The root sequence for index $u$ is:

$$x_u(n) = e^{-j\pi u\, n(n+1)/L_{RA}}, \quad n = 0, 1, \ldots, L_{RA}-1$$

$L_{RA} = 839$ for long formats (0–3, $\Delta f_{RA} = 1.25/5$ kHz); $L_{RA} = 139$ for short formats (A1–C2, $\Delta f_{RA} = 15$–$120$ kHz). Root index $u$ from TS 38.211 Table 6.3.3.1-3.

Cyclic Shift & Preamble Multiplexing

Multiple logical preambles are derived from a single root by applying cyclic shifts:

$$x_{u,v}(n) = x_u\!\bigl((n + C_v) \bmod L_{RA}\bigr), \quad C_v = v \cdot N_{CS}$$

where $N_{CS}$ is the RRC-configured cyclic-shift spacing. Number of preambles per root:

$$N_P = \left\lfloor \frac{L_{RA}}{N_{CS}} \right\rfloor$$

The ZCZ spans $N_{CS}-1$ samples; within it the cyclic auto-correlation of $x_u$ is zero, enabling per-UE timing discrimination. Maximum cell radius:

$$\boxed{d_{max} = (N_{CS} - 1)\cdot T_c \cdot c \;/\; 2}$$

$T_c \approx 0.509$ ns (5G NR basic unit). At $N_{CS}=13$: $d_{max}\approx 0.94$ km; $N_{CS}=138$: $d_{max}\approx 9.95$ km.

NTN Doppler Challenge

LEO 550 km radial velocity $v_r\approx7.6$ km/s → Doppler at 3.5 GHz: $f_d\approx88.7$ kHz. Preamble energy shifts by $\Delta k = \text{round}(f_d/\Delta f_{RA})$ bins. For format 0 ($\Delta f_{RA}=1.25$ kHz): $\Delta k \approx 71$ — outside ZCZ if $N_{CS}\le71$. Detection becomes a 2-D $(\tau,\nu)$ search (§4.2). NTN solution (Rel-17): CP > 2 ms formats, UE Doppler pre-compensation via SIB19 ephemeris, $N_{CS} > 72$ mandated for LEO.

4.1.2 PUSCH — Physical UL Shared Channel

User-plane data + piggybacked UCI. Two waveform modes:

• DFT-s-OFDM (SC-FDMA): M-point DFT pre-coding + OFDM mapping. PAPR $\approx5.5$ dB — preferred for UL coverage/PA backoff.
• CP-OFDM: standard OFDM, PAPR $\approx9$–$11$ dB. Required for rank $>1$ (DFT-s-OFDM is single-layer only).

LDPC channel coding (BG1: large/high-rate, BG2: small/low-rate). TBS from TS 38.214 MCS tables. Scheduling grant: DCI 0_0 (fallback) or 0_1 (full) on PDCCH.

NTN: Extended K2 Offset

Terrestrial $K_2\in\{0,1,2,3\}$ slots; for GEO RTT $\approx600$ ms, Rel-17 extends $K_2$ to hundreds of slots. Combined with UE-autonomous HARQ (disabled feedback or extended timers), the protocol stack functions across multi-hundred-ms round trips:

$$\text{PUSCH slot} = n_{\text{PDCCH}} + K_2$$

4.1.3 PUCCH — Physical UL Control Channel

Carries UCI: HARQ-ACK, SR, CSI (CQI/PMI/RI). Five formats (TS 38.211 §6.3.2):

Format 0 sequence (group $u$, seq $v$, cyclic shift $m_{cs}$):

$$r_{u,v}^{(\alpha)}(n) = e^{j\alpha n}\, \bar{r}_{u,v}(n), \quad \alpha = 2\pi m_{cs}/12$$

HARQ 1/0 and SR are encoded in $m_{cs}$ — no explicit bits; gNB infers by correlation peak.

NTN: Extended K1

PUCCH for PDSCH slot $n$: slot $n+K_1$. Terrestrial $K_1\le8$; NTN Rel-17 extends to 73 slots (LEO RTT $\approx7.3$ ms, $\mu=0$). GEO: HARQ disabled entirely (one-shot).

4.1.4 SRS — Sounding Reference Signal

UL channel sounding across configurable comb-2/4 subcarriers:

$$\hat{H}_{UL}(k) = \frac{Y_{SRS}(k)}{X_{SRS}(k)}, \quad k \in \mathcal{P}_{SRS}$$

$X_{SRS}$: known ZC sequence; $\mathcal{P}_{SRS}$: comb-2 (every 2nd SC) or comb-4.

SRS uses: TDD reciprocity (DL BF weights), UL MCS selection, angle-of-arrival positioning.

NTN SRS Staleness

One-way LEO delay $\approx 1.83$ ms at El=90°; channel ages by RTT/2 before gNB uses the estimate. Mitigation: linear Doppler-rate prediction:

$$\hat{H}_{UL}^{pred}(k,\, t+\Delta t) = \hat{H}_{UL}(k,t)\, e^{j2\pi f_d \Delta t}, \quad \Delta t = \text{RTT}/2$$

Feasible because LEO Doppler evolution is ephemeris-predictable.

4.2.1 NTN Initial Access Sequence

In terrestrial NR the UE assumes negligible propagation delay during initial access. In NTN the UE must actively compensate for $\tau_{prop}$ and Doppler before sending the first uplink message. The augmented 8-step flow:

1. SSB detection: PSS (length-127 m-seq) for timing, SSS (Gold code) for cell-ID, PBCH for MIB.
2. SIB1 + SIB19: SIB19 carries ephemeris, TAcommon, feeder-link compensation.
3. Propagation delay: $\tau_{prop} = d_{slant}/c$, $\quad d_{slant} = \sqrt{h^2+2Rh(1-\sin El)}/\sin El$
4. TX timing pre-adjust: $T_{TX} = T_{ref} - \tau_{prop}$ — signal arrives at gNB at $T_{ref}$.
5. Doppler pre-compensation: UE shifts TX frequency by $-f_d$ (from ephemeris $v_r$); residual error $\sigma_{f_d}\approx 2$ Hz for $\sigma_{eph}=100$ m.
6. PRACH TX: gNB performs 2-D $(\tau,\nu)$ search (§4.2.2).
7. RAR (msg2): TA command + TC-RNTI + UL grant.
8. Msg3/Msg4: RRCSetupRequest → RRCSetup/Reject + contention resolution.

TAcommon in SIB19 pre-offsets all UE TAs by the satellite-to-cell-centre delay, shrinking per-UE TA command range.

4.2.2 2-D PRACH Detection (Delay–Doppler Search)

The gNB PRACH detector correlates received signal $y[n]$ against all preamble hypotheses over a grid of delay $\tau$ and Doppler $\nu$ offsets:

$$\hat{R}(\tau,\nu) = \sum_{n=0}^{L_{RA}-1} y[n+\tau]\, x_u^*[n]\, e^{-j2\pi\nu n/L_{RA}}$$

Detection decision:

$$(\hat{\tau},\hat{\nu}) = \arg\max_{\tau,\nu}\; \bigl|\hat{R}(\tau,\nu)\bigr|^2 > \gamma_{th}$$

Search grid: delay $\tau\in[0,N_{CP}+N_{ZCZ}]$, Doppler $\nu\in[-\nu_{max},+\nu_{max}]$ ($\nu_{max}=\lceil f_{d,max}/\Delta f_{RA}\rceil$ bins; $\pm71$ for LEO at 3.5 GHz format 0). Load $O(N_\tau N_\nu L_{RA})$; FFT-based cyclic convolution per Doppler bin reduces this to $O(N_\tau N_\nu L_{RA}\log L_{RA})$.

False Alarm vs. Detection

$|\hat{R}(\tau,\nu)|^2$ is chi-squared distributed. False-alarm probability across $N_\tau N_\nu$ cells:

$$P_{FA} = 1 - \bigl(1 - e^{-\gamma_{th}/\sigma^2}\bigr)^{N_\tau N_\nu}$$

For $P_{FA}=10^{-3}$, $N_\tau N_\nu=28\,600$: $\gamma_{th}/\sigma^2\approx12.4$ dB — $\approx3$ dB penalty vs. terrestrial from the wider search window.

4.3.1 Gaseous Absorption (ITU-R P.676-12)

The atmosphere attenuates radio waves primarily via molecular resonance lines of oxygen (O₂) and water vapour (H₂O). At ground level the specific attenuation (dB/km) is:

$$\gamma_{total}(f) = \gamma_{O_2}(f) + \gamma_{H_2O}(f) \quad \text{[dB/km]}$$

Key lines (dry air, 1013 hPa, 15°C): O₂ at 60 GHz ($\approx15$ dB/km) and 118.75 GHz; H₂O at 22.235 GHz ($\approx0.18$ dB/km, $\rho_w=7.5$ g/m³) and 183.31 GHz.

Slant path ($h_{O_2}\approx6$ km, $h_{H_2O}\approx2$ km scale heights):

$$L_{eff} = \frac{h_{scale}}{\sin(El)}, \quad A_{gas}(f,El) = \gamma_{O_2}\cdot L_{eff,O_2} + \gamma_{H_2O}\cdot L_{eff,H_2O}$$

At El=10°, path multiplier $\approx5.8\times$; low-elevation links see substantially higher gaseous loss.

4.3.2 Rain Attenuation (ITU-R P.838-3)

Rain dominates above ~10 GHz. Specific attenuation (dB/km):

$$\gamma_R = k(f)\; R_p^{\,\alpha(f)} \quad \text{[dB/km]}$$

$R_p$ = rain rate (mm/h) at $p\%$ annual exceedance (ITU-R P.837); $k$, $\alpha$ from P.838-3. Total attenuation:

$$A_{rain} = \gamma_R \cdot r \cdot L_{slant}, \quad L_{slant} = \frac{h_R - h_s}{\sin(El)}$$

$h_R\approx4.5$ km (0°C isotherm). Path reduction $r\approx0.8$ (non-uniform rain distribution).

Worked Example: Ka-band 26 GHz, El = 45°

Given: $f = 26$ GHz, $El = 45°$, $R_p = 25$ mm/h (European climate, $p = 0.01\%$), site altitude $h_s = 0$ km, rain height $h_R = 4.5$ km, path reduction $r = 0.8$.

$$L_{slant} = \frac{4.5 - 0}{\sin 45°} = \frac{4.5}{0.7071} \approx 6.36 \text{ km}$$ $$\gamma_R = k_H \cdot R_p^{\,\alpha_H} = 0.19 \times 25^{1.015}$$

Computing $25^{1.015}$: $\ln(25) = 3.219$, $1.015 \times 3.219 = 3.267$, $e^{3.267} = 26.2$.

$$\gamma_R = 0.19 \times 26.2 = 4.98 \text{ dB/km}$$ $$A_{rain} = 4.98 \times 0.8 \times 6.36 \approx 25.3 \text{ dB}$$

25 dB fade at 0.01% ($\approx53$ min/year) is severe; at $p=0.1\%$ rate drops to $\sim15$ mm/h → $A_{rain}\approx14$ dB. Ka-band NTN requires ACM + site diversity. Full P.618-14 adds 0°C isotherm correction and ice-crystal enhancement below El=5°.

4.3.3 Ionospheric Effects (ITU-R P.531)

Ionised plasma 60–1000 km altitude. Effects scale with TEC (1 TECU = $10^{16}$ el/m²; typical 10–100 daytime, storm peaks >500 TECU).

Faraday Rotation

In the presence of the Earth's magnetic field, the plane of linear polarisation rotates by angle $\Omega_F$ (Faraday rotation):

$$\Omega_F \approx \frac{2.36 \times 10^{-4} \cdot \text{TEC}}{f^2} \quad \text{[rad]}$$

with $f$ in Hz, TEC in TECU. Impact at selected bands:

Circular polarisation (RHCP/LHCP) is immune to Faraday rotation — hence mandatory for satellite systems below ~4 GHz. FR1 NTN standard requires circular pol. at the satellite antenna; linear UE incurs up to 3 dB polarisation loss.

Group Delay Dispersion

Dispersive medium; for $f\gg f_p$ (plasma frequency):

$$\tau_{group}(f) = \frac{40.3\,\text{TEC}}{c\,f^2} \quad \text{[s]}, \qquad \Delta\tau \approx \frac{80.6\,\text{TEC}}{c\,f_c^3}\cdot B \quad \text{(over BW }B\text{)}$$

L-band 29.9 ns → 8.97 m range error; significant for precision NTN positioning. Dual-frequency removes the effect since $\tau_{group}\propto f^{-2}$. Ka-band (>26 GHz): negligible.

Scintillation

Electron density irregularities (±20° magnetic equator, auroral zones) cause rapid amplitude/phase fluctuations. Scintillation index $S_4$:

$$S_4 = \sqrt{\frac{\langle I^2\rangle - \langle I\rangle^2}{\langle I\rangle^2}}$$

$S_4>0.6$ = strong; peak L-band fades >20 dB. Intensity $\propto f^{-1.5}$: Ka-band sees $\approx400\times$ less variance than L-band.

Estimate slant-path attenuation. Models: P.676 Annex 2 (gaseous), P.838-3 (rain), P.531 (iono).

Complete NTN C/N₀ budget accumulates all propagation losses:

$$\frac{C}{N_0} = EIRP_{UE} - L_{fs} - A_{gas} - A_{rain} - A_{ion} + \frac{G_{sat}}{T_{sys}} - k_B$$

$k_B = -228.6$ dBW/(Hz·K). $L_{fs}$ = free-space path loss.

• ACM: QPSK 1/3 to 256-QAM 9/10 gives >20 dB dynamic range — essential for Ka/FR2 rain margin.
• ULPC: UE ramps TX power (+23/+26 dBm class 3/1) to compensate fade.
• Site diversity: two stations >10–20 km apart; simultaneous intense rain probability very low.
• Frequency diversity: Ku+Ka redundant links; switch to lower-attenuation band in heavy rain.
• Beamforming gain: LEO phased arrays 30–40 dBi partially offsets atmospheric loss.

§5.1 Doppler Physics and ICI Analysis

A satellite in LEO moves at orbital velocity $v_{orb} = \sqrt{GM/r}$, where $GM = 3.986 \times 10^{14}$ m³/s² and $r = R_E + h$ is the geocentric radius. Only the radial component of that velocity toward the UE produces a Doppler shift.

Orbital Velocity & Radial Doppler

Let $\psi$ be the angle between the satellite velocity vector and the UE line-of-sight. The radial velocity is $v_r = v_{orb}\cos\psi$ and the Doppler shift is:

$$f_d = \frac{v_r}{c}\,f_c = \frac{v_{orb}\cos\psi}{c}\,f_c$$

Maximum Doppler occurs near pass-over ($\psi \to 0$):

$$f_{d,\max} = \frac{v_{orb}}{c}\,f_c \approx \frac{7589}{3\times10^8}\,f_c$$

At 550 km LEO: $v_{orb} = 7589$ m/s, $r = 6921$ km. At $f_c = 3.5$ GHz: $f_{d,\max} = 88.5$ kHz. At the horizon ($\psi = 90°$), $f_d = 0$.

Doppler Rate (Chirp Rate)

Because $\psi$ changes continuously during a pass, $f_d$ changes at the Doppler rate:

$$\dot{f}_d = \frac{df_d}{dt} = \frac{f_c\,v_{orb}^2}{c\cdot r}$$

At 550 km, 3.5 GHz: $\dot{f}_d \approx 97.1$ Hz/s. This chirp is small within a single OFDM symbol but significant over a frame (10 ms → ~1 Hz accumulated shift) and must be tracked in long integrations.

ICI in CP-OFDM

A residual frequency offset $f_{CFO}$ normalized to subcarrier spacing $\Delta f$ gives $\epsilon = f_{CFO}/\Delta f$. The SINR due to inter-carrier interference (ICI) is:

$$\text{SINR}_{ICI} = \frac{|H_0|^2}{\displaystyle\sum_{k\neq 0}|H_k|^2 + \sigma_n^2/\sigma_s^2}$$

For a rectangular window (standard CP-OFDM) the ICI power fraction has the closed form:

$$\text{ICI fraction} = 1 - \text{sinc}^2(\epsilon) = 1 - \frac{\sin^2(\pi\epsilon)}{(\pi\epsilon)^2}$$

For small $\epsilon$ this simplifies to ICI $\approx \pi^2\epsilon^2/3$, giving an SINR ceiling (ICI floor):

$$\text{SINR}_{floor} = \frac{3}{\pi^2\,\epsilon^2}$$

The table below shows all NR numerologies at LEO 550 km, $f_c = 3.5$ GHz, before any pre-compensation.

After UE pre-compensation the residual $\epsilon$ drops to <0.01 for μ=1, making SINR_floor >40 dB — well clear of practical link budgets.

§5.2 Doppler Pre-Compensation Architecture (3GPP TS 38.821)

Rel-17 assigns Doppler pre-compensation primarily to the UE. The UE uses GNSS and satellite ephemeris broadcast in SIB19 to predict $f_d$ and shift its transmit frequency before the signal leaves the handset.

UE-Side Pre-Compensation: Step-by-Step

1. Read SIB19 ephemeris: satellite position vector $\vec{r}_{sat}(t)$, velocity $\vec{v}_{sat}(t)$, and epoch $t_0$.
2. GNSS self-position: UE determines $\vec{r}_{UE}$ (3 m accuracy typical).
3. Relative radial velocity: $v_r = (\vec{v}_{sat} - \vec{v}_{UE})\cdot\hat{u}$, where $\hat{u}$ is the unit vector from UE to satellite. $\vec{v}_{UE}$ is included for rapidly moving UEs (e.g., aircraft at ~250 m/s).
4. Frequency offset: $\Delta f = -v_r \cdot f_c / c$ (negative sign: UE pre-shifts to cancel incoming Doppler).
5. Pre-shifted TX frequency: $f_{TX} = f_c + \Delta f$.

Residual Doppler Budget

Two sources of residual Doppler remain after pre-compensation:

1. GNSS position error: $\sigma_{pos} = 3$ m induces velocity error $\sigma_v = \sigma_{pos} \cdot \omega_{orb}$ where $\omega_{orb} = v_{orb}/r$. At 550 km: $\omega_{orb} = 7589/6921000 = 1.10\times10^{-3}$ rad/s. $\sigma_v \approx 3.3$ mm/s. Residual Doppler: $\Delta f_{res} = 3.3\times10^{-3} \times 3.5\times10^9 / 3\times10^8 \approx 0.039$ Hz. Completely negligible.

2. Stale ephemeris (dominant residual): Doppler rate error $= \dot{f}_d \cdot \Delta t$. For $\Delta t = 10$ s stale: $\Delta f_{res} = 97.1 \times 10 = 971$ Hz. For μ=1 ($\Delta f = 30$ kHz): $\epsilon_{res} = 0.032$ — SINR floor = +34 dB. Acceptable. SIB19 is broadcast every few seconds, keeping $\Delta t \ll 10$ s in practice.

§5.3 CFO Estimation in OFDM — CP-Based and Pilot-Based

CP-Based Estimator (Moose 1994 / Van de Beek 1997)

The cyclic prefix is a copy of the last $N_{CP}$ samples of each OFDM symbol. In the presence of normalized CFO $\epsilon = f_{CFO}/\Delta f$, the received signal is:

$$r[n] = e^{j2\pi\epsilon n/N}\cdot h[n]*s[n] + w[n]$$

The phase difference between the CP and its tail copy is $2\pi\epsilon$ per sample. Maximum-likelihood estimator:

$$\hat{\epsilon}_{CP} = \frac{1}{2\pi}\angle\sum_{n=0}^{N_{CP}-1}r[n+N]\,r^*[n]$$

Limitation: the $\angle(\cdot)$ operation has range $(-\pi, +\pi]$, so $|\hat{\epsilon}| \leq 0.5$ (i.e., $|f_{CFO}| \leq \Delta f/2$). For NTN with $f_d$ up to 88.5 kHz and μ=1 ($\Delta f = 30$ kHz), $\epsilon \approx 2.95$ — far outside CP-estimator range. CP estimation alone is insufficient for NTN before pre-compensation.

Pilot-Based Coarse CFO Estimator

Using two consecutive OFDM symbols ($l=0,1$) with identical pilot sets $\mathcal{P}$:

$$\hat{\epsilon}_{pilot} = \frac{N}{2\pi}\angle\sum_{k\in\mathcal{P}} Y_1(k)\,Y_0^*(k)$$

Estimation range: $|\hat{\epsilon}_{pilot}| \leq N/2$ (far larger than CP estimator). For N=2048, μ=1: range = $\pm1024 \times \Delta f = \pm30.7$ MHz — covers all NTN scenarios.

Estimation variance (AWGN channel, $|\mathcal{P}|$ pilots):

$$\sigma^2_{\hat{\epsilon}} \approx \frac{3}{2\pi^2\,|\mathcal{P}|\,\text{SNR}}$$

At SNR = 10 dB, $|\mathcal{P}|=12$ (one RB): $\sigma_{\hat{\epsilon}} \approx 0.030$ → frequency error $\approx 0.9$ kHz at μ=1. Sufficient for residual correction after UE pre-compensation.

Frequency-Domain Phase Correction

After estimating $\hat{\epsilon}$, the frequency-domain correction is:

$$\tilde{Y}(k) = Y(k)\cdot e^{-j2\pi\hat{\epsilon}k/N}$$

This is a per-subcarrier phase rotation. Note: this corrects the residual within the OFDM symbol; time-domain pre-multiplication $r[n] \cdot e^{-j2\pi\hat{\epsilon}n/N}$ is equivalent and is usually applied before the FFT.

Two-Stage NTN CFO Correction

§5.4 NTN-Specific: Satellite-gNB Frequency Compensation

While the UE handles most Doppler pre-compensation, the gNB must still manage residual offsets on both uplink (UL) and downlink (DL) paths.

gNB UL Residual Estimation

• Coarse (PRACH): gNB detects preamble at bin $\hat{k}_{PRACH}$; frequency offset estimate = $\hat{k}_{PRACH} \times \Delta f_{RA}$ (1-bin resolution, $\Delta f_{RA}$ = PRACH SCS).
• Fine (PUSCH DMRS): LS frequency offset estimator across consecutive DMRS symbols; sub-SCS accuracy achievable at moderate SNR.
• DL correction signal: gNB applies $\Delta f_{gNB} = -\hat{f}_{d,UL}$ either locally (regenerative payload) or signals the feeder-link transparent satellite to apply a frequency shift.

Rel-17 baseline: NTN UE pre-compensation reduces UL CFO to within $\pm\Delta f/2$, so gNB residual correction is a standard fractional-bin correction.

Complete CFO Correction Loop — Block Diagram

The feedback path (gNB → DL comp signal → UE correction) has RTT latency ≈ 25 ms for LEO. For GEO (RTT ≈ 600 ms) the open-loop UE pre-compensation must be accurate enough to be self-sufficient.

Feeder-Link Doppler: Regenerative vs Transparent Payloads

§5.5 Doppler-Induced Timing Error

A Doppler shift $f_d$ on a carrier $f_c$ corresponds to a fractional change in apparent signal speed. This creates an apparent time scaling that manifests as a slow timing drift:

$$\frac{d\tau}{dt} = -\frac{f_d}{f_c} = -\frac{v_r}{c}$$

At LEO 550 km: $d\tau/dt = -7589/(3\times10^8) = -25.3$ ns/s.

The key conclusion is that Doppler-induced timing drift is slow: it does not cause intra-slot timing errors but does accumulate over seconds. The standard NR TA maintenance mechanism (periodic MAC CE TA commands) is adequate for LEO if the TA timer is set appropriately.

Interactive Doppler Pass Simulator

Adjust parameters to see how Doppler evolves during a satellite overpass. Blue: raw Doppler; dashed amber: residual after pre-compensation (0.5%).

§6.1 Standard 5G NR Timing Advance (TS 38.213 §4.2)

Timing advance (TA) adjusts the UE's transmit timing so that uplink signals arrive at the gNB within the correct receive window. Without TA adjustment, propagation delay would cause UL frames to arrive late, colliding with adjacent slots.

TA Fundamental Equation

The TA value is encoded as an integer $N_{TA}$. The actual time advance applied is:

$$T_A = N_{TA} \times \kappa \times T_c \quad \text{where} \quad \kappa = 16$$

$T_c$ is the basic time unit in NR:

$$T_c = \frac{1}{480000 \times 4096} = 5.087 \times 10^{-10}\text{ s} \approx 0.509\text{ ns}$$

Thus one TA step = $16 T_c \approx 8.14$ ns, corresponding to $\Delta d = c \times 8.14\text{ ns}/2 \approx 1.22$ m in round-trip range.

Terrestrial TA Range

For standard NR (terrestrial), $N_{TA} \in [0, 3846]$:

$$T_{A,\max}^{terr} = 3846 \times 16 \times T_c = 31.3\,\mu\text{s}$$

This corresponds to a one-way range of $c \times 31.3\,\mu\text{s} / 2 \approx 4.7$ km — adequate for terrestrial cells but grossly insufficient for satellite links where one-way delays reach milliseconds.

TA MAC CE Format (TS 38.321 §6.1.3.6)

For differential TA, $T_A = 31$ means no change; $T_A = 32$ advances by $16T_c$; $T_A = 30$ retards by $16T_c$. Maximum differential step = $\pm 31 \times 16T_c \approx \pm 252$ ns.

§6.2 NTN Extended Timing Advance (TS 38.213 / TS 38.821, Rel-17)

Satellite propagation delays are orders of magnitude larger than terrestrial. Rel-17 introduces an extended TA framework with two components: a cell-common part broadcast via SIB19, and a UE-specific differential correction via MAC CE.

$$T_{A,NTN} = T_{A,common} + T_{A,UE\text{-}specific}$$

TA Components Explained

• $T_{A,common}$: Accounts for the satellite altitude and the reference UE position within the cell. Broadcast in SIB19. All UEs in the cell use this as a baseline. Equivalently: $T_{A,common} = 2 d_{ref}/c$ where $d_{ref}$ is the slant range from the reference point to the satellite.
• $T_{A,UE\text{-}specific}$: Per-UE differential correction based on actual UE position vs reference position. UE computes and pre-applies this; gNB sends MAC CE only for residual after pre-compensation.

Maximum TA Values by Orbit (3GPP TS 38.821 §6.1.3)

GEO requires split TA architecture: UE pre-applies based on GNSS+ephemeris, gNB only corrects residual. Full closed-loop TA is impractical for GEO.

NTN TA Pre-Compensation by UE (Rel-17, TS 38.331 §6.3.2)

1. UE reads ServingCellTAInfo from SIB19: contains reference TA $T_{A,ref}$ at epoch $t_0$.
2. UE computes current slant range $d_{slant}(t)$ from GNSS + ephemeris propagation.
3. Differential TA: $\Delta T_A = 2 d_{slant}(t)/c - T_{A,ref}$.
4. UE pre-applies advance: $T_{A,pre} = T_{A,ref} + \Delta T_A = 2 d_{slant}(t)/c$.
5. gNB sends MAC CE TA command only for the small residual (typically <1 μs).

§6.3 RACH-Based TA Acquisition (TS 38.213 §4.2.3)

Before any scheduled UL transmission, the UE must acquire an initial TA via the Random Access Channel (RACH). The gNB measures the arrival timing of the PRACH preamble to infer the round-trip delay.

Initial TA Acquisition Procedure

1. UE transmits PRACH preamble at slot $n_0$ (applying any available open-loop pre-compensation from SIB19 / GNSS if configured).
2. gNB detects the preamble and measures its arrival time offset $\hat{\tau}$ relative to the expected window center.
3. gNB computes integer TA: $N_{TA} = \lfloor\hat{\tau} / T_c\rfloor$, then quantizes to units of $16T_c$.
4. gNB transmits RAR (Random Access Response, Msg2) containing the 12-bit TA field.
5. UE applies TA: advances TX timing by $N_{TA} \times 16 T_c$ from the next uplink slot.

ZCZ-Based TA Resolution

PRACH preambles use Zadoff-Chu sequences. The zero-correlation zone (ZCZ) width determines the maximum unambiguous TA range per preamble:

$$\text{ZCZ width} = (N_{CS} - 1)\text{ samples at PRACH SCS}$$ $$\text{TA range per preamble} = \frac{N_{CS}-1}{L_{RA}} \times \frac{1}{\Delta f_{RA}}$$

For long preamble ($L_{RA}=839$), $N_{CS}=138$ (unrestricted set, TS 38.211 Table 6.3.3.1-5): ZCZ covers ≈ $138/839 \times 1/1250 \approx 131$ μs → $\approx19.7$ km one-way range.

NTN challenge: At LEO 550 km with cell radius ~1000 km, the TA range spread within a cell can be several ms. NTN uses $N_{CS}=1023$ or new PRACH formats with extended preambles to handle one-way delays up to 5.44 ms (e.g., LEO at 10° elevation).

§6.4 Timing Advance State Machine

The 3GPP TA state machine governs when the UE's UL timing is considered valid and what happens when the TA timer expires.

timeAlignmentTimer (TAT) for NTN

For GEO, the timeAlignmentTimer is set to infinity in the RRC configuration (TS 38.321 §5.2): UL is always considered aligned as long as the UE maintains GNSS-based pre-compensation. This avoids the pathological case of TAT expiry forcing a RACH attempt with 600 ms RTT.

§6.5 NTN TA: Closed-Loop vs Open-Loop

RTT Latency Analysis

For closed-loop correction, the key question is how much TA drift accumulates during the feedback RTT:

$$\Delta\tau_{drift} = \frac{d\tau}{dt} \times RTT = \frac{v_r}{c} \times RTT$$

Closed-loop correction is acceptable for TA drift for all orbits. The real constraint for GEO is not drift accuracy but the inefficiency of waiting 600 ms for each TA correction, justifying the open-loop approach.

§6.6 Interaction with HARQ Timing

HARQ (Hybrid ARQ) requires the gNB to transmit a feedback ACK/NACK within K1 slots of the PDSCH. In standard NR, K1 ≥ 1 slot. For NTN, the propagation RTT dominates, dramatically increasing the required K1 and hence the number of HARQ processes needed to keep the pipeline full.

NTN K1 Minimum Calculation

$$K1_{min} = \left\lceil \frac{RTT_{total}}{T_{slot}} \right\rceil + K1_{processing}$$

For bent-pipe LEO at μ=1 ($T_{slot} = 0.5$ ms): RTT = 25 ms → $\lceil 25/0.5 \rceil = 50$ propagation slots. Adding processing time ($K1_{proc} = 5$ slots): $K1_{min} = 55$ slots.

The number of HARQ processes needed to keep the UL pipeline full (one new TB per slot while awaiting feedback):

$$N_{HARQ} = K1_{min} + 1 = 56 \quad \text{vs 16 in standard NR}$$

HARQ Requirements by Orbit and Numerology

3GPP Solutions for NTN HARQ (Rel-17)

Interactive TA & HARQ Timeline Calculator

Adjust altitude and elevation to compute slant range, required TA, K1 minimum, and HARQ processes needed.

Summary: NTN HARQ Design Constraints

Base-Graph Architecture

5G NR uses Quasi-Cyclic LDPC (QC-LDPC) codes with two base graphs (BGs). Each BG entry is either a circulant permutation matrix of size $Z_c \times Z_c$ or a zero block.

The lifting size $Z_c$ belongs to one of eight sets $\{2,4,8,...\}$ × scaling factors, giving 51 distinct values in $\{2, 3, 4, 6, 8, 9, 10, 12, \ldots, 384\}$. Lifted code-word length: $N = Z_c \times N_{\text{bg,cols}}$. Number of information bits after shortening: $K = k_b \times Z_c$. Effective code rate: $R = K / (N - 2Z_c)$ (two parity columns are punctured).

HARQ Circular Buffer & Redundancy Versions

The full rate-matched sequence is written into a circular buffer of length $N_{cb} \leq N$. Each retransmission starts at offset $k_0(rv)$:

$$k_0(rv) = \begin{cases} 0, & rv = 0 \\[4pt] \left\lfloor \dfrac{17N_{cb}}{66} \right\rceil \times Z_c, & rv = 1 \\[6pt] \left\lfloor \dfrac{33N_{cb}}{66} \right\rceil \times Z_c, & rv = 2 \\[6pt] \left\lfloor \dfrac{56N_{cb}}{66} \right\rceil \times Z_c, & rv = 3 \end{cases}$$

where $\lfloor \cdot \rceil$ denotes nearest integer. RV=0 carries all systematic bits; RV=1,2,3 are increasingly parity-heavy, enabling incremental redundancy (IR-HARQ).

LLR combining: the receiver accumulates soft values from all transmissions in the same buffer position before decoding:

$$L_{\text{total}}[n] = \sum_{i=1}^{N_{tx}} L_i[n]$$

The effective code rate after $N_{tx}$ retransmissions, each contributing $E$ coded bits:

$$R_{\text{eff}} = \frac{K}{N_{tx} \times E}$$

NTN relevance: RTT ≈ 600 ms (GEO) means each HARQ round-trip costs one very long slot; 3GPP Rel-17 disables synchronous HARQ for NTN and instead uses a larger HARQ process count (up to 32) with extended HARQ-RTT timers (TS 38.321 §5.4.2).

LDPC Capacity Distance

Shannon capacity at $R = 1/2$: channel capacity $C = 1$ bit/s/Hz implies $E_b/N_0|_{\text{Shannon}} = 0$ dB. 5G NR LDPC (BG1, $K = 8448$, $R = 0.5$) achieves BER $10^{-5}$ at $E_b/N_0 \approx 1.5$ dB — within ~1.5 dB of the Shannon limit.

$$\text{Capacity gap} \approx 1.5 \text{ dB at } R=0.5, \; K=8448$$

For NTN power-limited links (EIRP-constrained UE), operating near the capacity limit at low-order modulation is the preferred working point.

Construction & Channel Polarization

Polar codes of length $N = 2^n$ are used for PDCCH, PUCCH format 2/3/4, and PBCH. The generator matrix is built by $n$-fold Kronecker product of the kernel $\mathbf{F} = \begin{pmatrix}1&0\\1&1\end{pmatrix}$, permuted by a bit-reversal matrix $\mathbf{B}_N$:

$$\mathbf{G}_N = \mathbf{B}_N \, \mathbf{F}^{\otimes n}$$

Channel polarization: as $N \to \infty$, synthetic sub-channels $\{W_N^{(i)}\}_{i=0}^{N-1}$ polarize — a fraction approaches noiseless, the rest become pure noise. Information bits are placed on the $K$ most reliable sub-channels; frozen bits (set to 0) fill the rest.

Sub-channel reliability is quantified by the Bhattacharyya parameter $Z_N^{(i)}$ (lower = more reliable):

$$Z_N^{(i)} = \sqrt{1 - \left(1 - 2Z_{N/2}^{(2i)}\right)^2} \quad \text{(split rule for "bad" channel)}$$

In practice, NR uses pre-computed reliability sequences defined in TS 38.212 §5.3.1.

CRC-Aided Successive Cancellation List (CA-SCL)

Successive cancellation (SC) decoding has $O(N \log N)$ complexity but poor performance at short block lengths. NR uses CA-SCL with list size $L = 8$:

1. Maintain $L$ decoding paths simultaneously.
2. At each bit position split each path into two (frozen bit = prune).
3. Keep the $L$ most-probable paths by path metric.
4. At the end, select the path that passes the appended CRC.

The outer CRC detects false-alarm paths and provides early termination. Typical CRC lengths: 24 bits (PBCH/PDCCH), 11 bits (small UCI).

PDCCH aggregation levels: AL $\in \{1,2,4,8,16\}$ CCEs; each CCE = 6 REGs = 54 REs. Higher AL gives more coding protection for NTN edge-of-coverage UEs.

Gray-Coded Constellations — BER Formulae

All BER expressions below assume AWGN, Gray coding, and optimal ML detection.

$\pi/2$-BPSK ($Q_m = 1$): each symbol rotated by $\pi/2$ from the previous, so the transmit sequence never visits the origin. PAPR $\approx 0$ dB (constant-envelope for SC-FDMA). Used in NTN UL for power-limited UEs.

$$P_b^{\text{BPSK}} = Q\!\left(\sqrt{2E_b/N_0}\right)$$

QPSK ($Q_m = 2$): minimum Euclidean distance $d_{\min} = \sqrt{2}$ (normalised to unit average power):

$$P_b^{\text{QPSK}} = Q\!\left(\sqrt{2E_b/N_0}\right)$$

16-QAM ($Q_m = 4$): nearest-neighbour Gray-coded approximation:

$$P_b^{\text{16QAM}} \approx \frac{3}{8}\,\text{erfc}\!\left(\sqrt{\frac{2E_b}{5N_0}}\right)$$

64-QAM ($Q_m = 6$):

$$P_b^{\text{64QAM}} \approx \frac{7}{24}\,\text{erfc}\!\left(\sqrt{\frac{E_b}{7N_0}}\right)$$

256-QAM ($Q_m = 8$):

$$P_b^{\text{256QAM}} \approx \frac{15}{64}\,\text{erfc}\!\left(\sqrt{\frac{E_b}{42N_0}}\right)$$

For reference, the erfc–Q function relationship: $Q(x) = \tfrac{1}{2}\text{erfc}\!\left(x/\sqrt{2}\right)$.

MCS Table (TS 38.214 Table 5.1.3.1-2) — All 29 Entries

MCS indices 0–27 are assigned; MCS 28–31 are reserved or used for re-transmission signalling. Spectral efficiency (SE) $= Q_m \times R$ bits/s/Hz.

NTN link budget note: satellite UEs with low EIRP typically operate at MCS 0–4 (QPSK, $R \leq 0.30$). High-throughput GEO broadband services with large dish antennas may reach MCS 22–24 (64QAM, $R \approx 0.65–0.75$).

Least-Squares (LS) Estimation at Pilot Positions

Let $Y(k)$ be the received OFDM subcarrier at pilot index $k \in \mathcal{P}$, and $X_{\text{DMRS}}(k)$ the known pilot symbol. The LS estimate is simply the element-wise division:

$$\hat{H}_{\text{LS}}(k) = \frac{Y(k)}{X_{\text{DMRS}}(k)}, \quad k \in \mathcal{P}$$

Estimation variance (normalising pilot power to unity):

$$\text{Var}\!\left[\hat{H}_{\text{LS}}(k)\right] = \frac{\sigma_n^2}{\left|X_{\text{DMRS}}(k)\right|^2} = \sigma_n^2$$

LS is unbiased but does not exploit channel correlation — its noise enhancement factor is $1/\text{SNR}$ per subcarrier. Interpolation between pilots (linear, spline, or DFT-based) is needed for data subcarriers.

MMSE Estimator

The MMSE estimator exploits the channel covariance matrix $\mathbf{R}_{HH} = E[\mathbf{H}\mathbf{H}^H]$ to suppress noise:

$$\hat{\mathbf{H}}_{\text{MMSE}} = \mathbf{R}_{HH}\!\left(\mathbf{R}_{HH} + \frac{\sigma_n^2}{\sigma_s^2}\mathbf{I}\right)^{-1}\hat{\mathbf{H}}_{\text{LS}}$$

where $\sigma_s^2$ is the average signal power. Writing $\text{SNR} = \sigma_s^2 / \sigma_n^2$:

$$\hat{\mathbf{H}}_{\text{MMSE}} = \mathbf{R}_{HH}\!\left(\mathbf{R}_{HH} + \frac{1}{\text{SNR}}\mathbf{I}\right)^{-1}\hat{\mathbf{H}}_{\text{LS}}$$

The minimum achievable MSE is:

$$\text{MSE}_{\text{MMSE}} = \text{tr}\!\left[\mathbf{R}_{HH} - \mathbf{R}_{HH}\!\left(\mathbf{R}_{HH} + \frac{1}{\text{SNR}}\mathbf{I}\right)^{-1}\!\mathbf{R}_{HH}\right]$$

At high SNR, MMSE approaches LS; at low SNR, the regularisation term $\mathbf{I}/\text{SNR}$ dominates and the estimate is shrunk toward zero, reducing noise at the cost of some bias — the ideal NTN operating mode where noise is the dominant impairment.

MMSE Equalizer on Data Subcarriers

Given the channel estimate $\hat{H}(k)$, the MMSE equalizer coefficient is:

$$w_{\text{MMSE}}(k) = \frac{\hat{H}^*(k)}{|\hat{H}(k)|^2 + 1/\text{SNR}}$$

Equalized symbol:

$$\hat{X}(k) = \frac{\hat{H}^*(k)}{|\hat{H}(k)|^2 + 1/\text{SNR}} \cdot Y(k)$$

Post-equalization SINR (assuming perfect channel knowledge):

$$\gamma(k) = \frac{|\hat{H}(k)|^2 \cdot \text{SNR}}{1} = |\hat{H}(k)|^2 \cdot \text{SNR}$$

The MMSE equalizer reduces to the matched filter (MRC) when $|\hat{H}|^2 \ll 1/\text{SNR}$ and to zero-forcing (ZF) when $|\hat{H}|^2 \gg 1/\text{SNR}$.

PTRS for Phase Noise Correction

Phase-Tracking Reference Signals (PTRS) track oscillator phase noise, which grows with carrier frequency. At Ka-band (26.5–40 GHz), LEO satellite links face severe phase noise from both UE TCXO and satellite oscillator.

The Common Phase Error (CPE) accumulated over one OFDM symbol $l$ is estimated by correlating PTRS pilots against the channel estimate:

$$\phi_{\text{CPE}}[l] = \angle\!\left\{ \sum_{k \in \mathcal{P}_{\text{PTRS}}} Y(k,l)\,\hat{H}^*(k,l) \right\}$$

The estimated CPE is then subtracted from all data subcarriers in symbol $l$:

$$\tilde{Y}(k,l) = Y(k,l) \cdot e^{-j\phi_{\text{CPE}}[l]}$$

PTRS density in time (symbols per PTRS symbol):

Frequency-domain PTRS density is typically sparse (1–2 RBs), since CPE is a constant phase rotation across the whole OFDM symbol (inter-carrier phase noise is a separate smaller impairment handled by pilot interpolation).

Challenge 1 — Doppler-Induced Time Variation

For a LEO satellite at orbital velocity $v \approx 7.8$ km/s, the instantaneous Doppler shift at L-band (2 GHz) can reach $\pm 48$ kHz. Within one OFDM slot (14 symbols, 0.5 ms for 30 kHz SCS), the channel phase evolves as:

$$H(k, l) = H_0(k)\, e^{j 2\pi f_d l T_{\text{sym}}}$$

where $l$ is the OFDM symbol index and $T_{\text{sym}} = 1/\Delta f + T_{CP}$. Without compensation, this linear phase ramp causes inter-carrier interference (ICI) with power $\propto (f_d / \Delta f)^2$.

NTN pre-compensation: the UE applies frequency pre-compensation $\hat{f}_d$ based on GNSS/ephemeris. Residual Doppler is:

$$f_{d,\text{res}} = f_d - \hat{f}_d \lesssim 1 \text{ kHz (Rel-17 requirement)}$$

At $f_{d,\text{res}} = 1$ kHz and $\Delta f = 30$ kHz: $f_{d,\text{res}} / \Delta f = 0.033$ — ICI power penalty $\approx 0.02$ dB. Channel coherence bandwidth $\gg$ slot duration, so standard DMRS-based estimation (one or two DMRS symbols per slot) remains valid.

Challenge 2 — Ionospheric Faraday Rotation

The ionosphere rotates the polarization plane of electromagnetic waves by angle $\Omega_F$ (Faraday rotation):

$$\Omega_F = \frac{K_F}{\cos\theta} \cdot \frac{\text{TEC}}{f^2}$$

where $K_F \approx 2.365 \times 10^4$ rad·Hz²·m²/TECU, $\theta$ is the elevation angle, TEC is Total Electron Content (in TECU), and $f$ is frequency.

Mitigation: NTN L-band systems use circular polarization (RHCP/LHCP) to be immune to rotation; or dual-linear feeds with polarization-adaptive combining. Channel estimation must account for effective antenna gain reduction when Faraday rotation is not compensated.

Challenge 3 — Multipath & Delay Spread

The NTN path itself is LOS-dominated with negligible delay spread from the satellite geometry. However, near the ground terminal, scattering off buildings, terrain, and foliage introduces multipath with delay spread $\tau_{\text{rms}} \approx 0.1$–$3\,\mu$s (urban) — far smaller than the CP duration:

$$\tau_{\text{rms}} \ll T_{CP} = \frac{N_{CP}}{N_{FFT} \cdot \Delta f}$$

For SCS 15 kHz, extended CP: $T_{CP} = 16.67\,\mu$s. The frequency-domain channel is therefore nearly flat over the coherence bandwidth $B_c \approx 1/(5\tau_{\text{rms}}) \approx 0.07$–$2$ MHz, much wider than one OFDM subcarrier spacing. Standard subcarrier-level MMSE interpolation is sufficient.

The canvas below shows a 14 × 12 slot resource grid (14 OFDM symbols × 12 subcarriers, representing 1 PRB × 1 slot). Pilot positions follow 5G NR DL DMRS Type 1, mapping type A (DMRS symbols at $l = 2$ for normal CP, 14 symbols per slot) with one additional DMRS at $l = 11$. PTRS is shown at $l = 0, 7$ on subcarrier 0 (density = 2, single RB).

Legend: blue = data RE, orange = DMRS pilot, green = PTRS pilot, grey = DMRS CDM partner. Click canvas to toggle symbol label overlay.

The carrier-to-noise-density ratio is the fundamental NTN link quality metric. It collapses transmitter power, path loss, atmospheric absorption, and receiver sensitivity into a single dBHz figure from which spectral efficiency follows directly.

The full cascade in one expression:

$$\frac{C}{N_0} = \text{EIRP}_{tx} - \text{FSPL} - L_{atm} - L_{misc} + \left(\frac{G}{T}\right)_{rx} + 228.6 \quad [\text{dBHz}]$$

Term definitions:

• $\text{EIRP}_{tx} = P_{tx}[\text{dBW}] + G_{tx}[\text{dBi}] - L_{feeder}[\text{dB}]$ — effective isotropic radiated power of the transmitter
• $\text{FSPL} = 32.44 + 20\log_{10}(d[\text{km}]) + 20\log_{10}(f[\text{MHz}])$ — free-space path loss in dB
• $L_{atm} = L_{gas} + L_{rain} + L_{scint}$ — total atmospheric loss (gas absorption + rain fade + scintillation)
• $L_{misc}$ — polarisation mismatch, pointing loss, implementation margin (typically 1–2 dB)
• $G/T_{rx} = G_{rx}[\text{dBi}] - 10\log_{10}(T_{rx}[\text{K}])$ — receiver figure of merit in dB/K
• $228.6 = -10\log_{10}(k_B)$ — Boltzmann offset, where $k_B = 1.380649 \times 10^{-23}$ J/K

Reference geometry: LEO 550 km, elevation 45°

For a satellite at altitude $h = 550$ km and elevation angle $El = 45°$, the slant range is obtained from spherical geometry with Earth radius $R_E = 6371$ km:

$$d = \sqrt{R_E^2 \sin^2(El) + h^2 + 2hR_E} - R_E \sin(El)$$

Evaluating: $d \approx 750$ km. The free-space path loss at $f_c = 2.0$ GHz is:

$$\text{FSPL} = 32.44 + 20\log_{10}(750) + 20\log_{10}(2000) = 32.44 + 57.5 + 66.0 = 155.9 \approx 156.0 \text{ dB}$$

A parabolic reflector of diameter $D$ and aperture efficiency $\eta$ (typically 0.55–0.65) achieves gain:

$$G_{sat} = \eta \left(\frac{\pi D}{\lambda}\right)^2$$

In logarithmic form, substituting $\lambda = c / f_c$:

$$G_{sat}[\text{dBi}] = 10\log_{10}(\eta) + 20\log_{10}\!\left(\frac{\pi D f_c}{c}\right)$$

For $D = 0.5$ m, $f_c = 3.5$ GHz, $\eta = 0.6$, $c = 3 \times 10^8$ m/s:

$$G_{sat} = 0.6 \times \left(\frac{\pi \times 0.5 \times 3.5 \times 10^9}{3 \times 10^8}\right)^2 = 0.6 \times (18.33)^2 = 0.6 \times 336.0 = 201.6$$ $$G_{sat}[\text{dBi}] = 10\log_{10}(201.6) \approx 23.0 \text{ dBi}$$

Half-power beamwidth (approximate for a uniformly illuminated circular aperture):

$$\theta_{3\text{dB}} \approx \frac{70\lambda}{D} \text{ (degrees)}$$

At $f_c = 3.5$ GHz ($\lambda = 8.57$ cm), $D = 0.5$ m: $\theta_{3\text{dB}} \approx 70 \times 0.0857 / 0.5 \approx 12.0°$. A tighter beam increases gain but demands precise attitude control and beam-hopping scheduling.

DL Table 1 — LEO 550 km · S-band 2.0 GHz · Elevation 45°

Scenario: Low-power direct-to-device (D2D) link. Satellite carries a modest phased-array providing 10 dBi. UE is a smartphone-class handheld with isotropic receive antenna.

DL Table 2 — LEO 550 km · Ka-band 26.5 GHz · Elevation 45°

Scenario: Fixed terminal with 30 cm dish (CPE-class). Higher gain on both ends, but significantly larger FSPL and rain attenuation at Ka-band.

Required C/N₀ for a given MCS is derived from the minimum $E_b/N_0$ and the information bit rate $R_b$:

$$\left(\frac{C}{N_0}\right)_{req} = \frac{E_b}{N_0}\bigg|_{min} + 10\log_{10}(R_b)$$

For 1 PRB at numerology $\mu = 1$ ($\Delta f = 30$ kHz), 11 data symbols per slot, DDDSUUDDDD TDD pattern (2000 slots/s per 0.5 ms slot duration at μ=1):

$$R_b = 12 \; \text{SCs} \times 11 \; \text{sym} \times Q_m \times R_{code} \times 2000 \; \text{slots/s} \quad [\text{bits/s}]$$

The table below shows MCS 0–28 (Table 5.1.3.1-1, TS 38.214), computed $R_b$, required C/N₀, and link margin against the S-band scenario C/N₀ = 64.3 dBHz. Green rows close the link; red rows do not.

At C/N₀ = 64.3 dBHz (S-band, handheld), MCS up to approximately MCS 16 (16QAM, $R_{code} \approx 0.642$) closes with positive margin. Higher MCS orders require the Ka-band fixed-terminal scenario (C/N₀ ≈ 79.7 dBHz) or reduced satellite–UE range.

The 3GPP NR peak DL throughput formula sums across component carriers $j$:

$$R_{peak} = \sum_j \nu_j \cdot Q_{m,j} \cdot f_j \cdot R_{max,j} \cdot N_{PRB,j} \cdot 12 \cdot (14 - N_{OH}) \cdot 2^\mu \cdot 10^3 \cdot \rho_{TDD} \quad [\text{bps}]$$

Where:

• $\nu_j$ — number of MIMO layers (spatial streams)
• $Q_{m,j}$ — modulation order (2 QPSK, 4 16QAM, 6 64QAM, 8 256QAM)
• $f_j$ — scaling factor (set to 1 for peak calculation)
• $R_{max,j}$ — maximum code rate = 948/1024 ≈ 0.9258
• $N_{PRB,j}$ — number of active PRBs in bandwidth
• $(14 - N_{OH})$ — data OFDM symbols per slot, $N_{OH}=2$ for FR1 PDSCH DMRS type A
• $2^\mu$ — slots per subframe; $\mu=0$: 1, $\mu=1$: 2, $\mu=2$: 4
• $\rho_{TDD}$ — DL slot fraction in TDD pattern (e.g. 0.7 for 7:3 DL:UL)

Worked example — Handheld LEO, 20 MHz, μ=1

$N_{PRB} = 51$, $Q_m = 4$, $R_{code} = 0.642$, $\nu = 1$, $N_{OH} = 2$, $2^\mu = 2$ slots/subframe, $\rho_{TDD} = 0.7$:

$$R_{peak} = 1 \times 4 \times 1 \times 0.642 \times 51 \times 12 \times 12 \times 2 \times 10^3 \times 0.7$$ $$= 4 \times 0.642 \times 51 \times 144 \times 1400 \approx 67.6 \text{ Mbps}$$

On the uplink the UE is power-limited. Maximum transmit power per 3GPP is $P_{max} = 23$ dBm for Class 3 handhelds, up to 33 dBm for fixed CPE (Class 1).

$$\text{EIRP}_{UE} = P_{tx}[\text{dBm}] - 30 + G_{UE}[\text{dBi}] - L_{body}[\text{dB}] \quad [\text{dBW}]$$

Body loss $L_{body} \approx 3$ dB for handheld. The satellite acts as receiver; its G/T depends on the aperture array size.

UL C/N₀ formula: $\text{EIRP}_{UE} - \text{FSPL} - L_{gas} - L_{misc} + (G/T)_{sat} + 228.6$. FSPL = 156.0 dB, $L_{gas}$ = 0.3 dB, $L_{misc}$ = 1.0 dB constant for all rows.

Interactive NTN Link Budget Calculator

Adjust the sliders below. C/N₀ and recommended MCS update in real time. The canvas shows the link budget waterfall.

9.1 5G NR HARQ Architecture

NR uses 16 parallel HARQ processes per UE in the DL and 16 in the UL (TS 38.321 §5.4.2). Each process is an independent stop-and-wait channel: the transmitter sends a Transport Block (TB), then either awaits an ACK and advances to the next TB, or awaits a NACK and retransmits. Multiple processes run concurrently to keep the channel utilised during the propagation + processing latency.

HARQ Process State Machine

The NDI (New Data Indicator) bit in DCI format 1_0 / 1_1 is toggled to signal a new TB vs. a retransmission for the same process. The RV (Redundancy Version) field (2 bits, values 0–3) selects which portion of the LDPC rate-matched output to transmit. Convention: initial transmission always uses RV = 0; retransmissions cycle 0 → 2 → 3 → 1.

LLR Combining for IR-HARQ

The receiver maintains a soft buffer per HARQ process. On each retransmission carrying RV $r$, the corresponding coded-bit positions are updated. The combined log-likelihood ratio at bit position $n$ after $N_{\mathrm{tx}}$ transmissions:

$$L_{\mathrm{total}}[n] = \sum_{i=1}^{N_{\mathrm{tx}}} L_i[n], \qquad L_i[n] = 0 \text{ if bit } n \text{ not included in RV}_i$$

The effective code rate after $N_{\mathrm{tx}}$ transmissions (each carrying $E$ coded bits for a $K$-bit information block):

$$R_{\mathrm{eff}} = \frac{K}{N_{\mathrm{tx}} \times E}$$

Chase Combining (CC-HARQ) vs. Incremental Redundancy

Chase Combining retransmits the identical RV = 0 each time, giving pure MRC gain. SNR accumulates linearly:

$$\mathrm{SNR}_{\mathrm{CC}}^{(N)} = N \cdot \mathrm{SNR}^{(1)} \quad \Longrightarrow \quad 10\log_{10}(N) \text{ dB gain per doubling}$$

Incremental Redundancy (IR-HARQ) transmits different RVs, expanding the effective codeword. After two IR transmissions (RV0 + RV2) the code rate halves, giving ~3 dB additional coding gain over CC. Shannon bound for combined SNR:

$$R_{\max} = \log_2\!\left(1 + \mathrm{SNR}_{\mathrm{combined}}\right) \text{ bits/s/Hz}$$

NR uses IR-HARQ by default. CC is only applicable when the soft buffer is already full (buffer-limited combining).

9.2 NTN HARQ Timing Problem — Full Analysis

Bent-Pipe (Transparent) Satellite RTT

For a transparent satellite the HARQ feedback loop traverses four link hops — UE to satellite, satellite to gateway, gateway to satellite, satellite back to UE — plus baseband processing at both ends:

$$RTT_{\mathrm{bent\text{-}pipe}} = 2\,\frac{d_{\mathrm{slant}}}{c} + 2\,\frac{d_{\mathrm{feeder}}}{c} + T_{\mathrm{proc,\,gNB}} + T_{\mathrm{proc,\,UE}}$$

where $c = 3 \times 10^5$ km/s. For LEO at 550 km altitude, minimum-elevation geometry gives $d_{\mathrm{slant}} \approx 750$ km (10° elevation), so the service-link one-way delay $= 750 / (3\times10^5) \approx 2.5$ ms and the round-trip service-link delay = 5 ms. With a co-located feeder link gateway ($d_{\mathrm{feeder}} \approx 550$ km, one-way 1.83 ms), total RTT:

$$RTT_{\mathrm{LEO,bent\text{-}pipe}} \approx 2 \times 2.5 + 2 \times 1.83 + 1 + 5 \approx 14.7 \text{ ms}$$

For GEO at 35 786 km: service-link one-way = 119 ms, feeder one-way ≈ 119 ms. Total RTT ≈ $4 \times 119 + \text{processing} \approx 480\text{–}600$ ms.

K1 and K2 Extended Values — TS 38.213 Rel-17

K1 is the PDSCH-to-HARQ-ACK feedback timing offset in slots (UE sends PUCCH carrying HARQ-ACK exactly K1 slots after the last PDSCH symbol). In terrestrial NR: K1 $\in \{1, \ldots, 15\}$ slots. For NTN, Rel-17 extended K1 to up to 32 slots.

K2 is the UL scheduling offset (DCI → PUSCH start) — extended from 0–7 to 0–32 slots.

For NTN $\mu = 1$ (0.5 ms slots), K1 = 50 would cover 25 ms one-way (LEO). The Rel-17 cap of K1 = 32 covers 16 ms one-way, sufficient for LEO but not GEO. GEO requires HARQ disabling (Solution 1).

9.3 Four Solutions Standardised in Rel-17 (TS 38.821)

Solution 1: HARQ Disabling

DCI format 1_1 carries a harq-FeedbackEnablingDisabling bit. When set to disabled, the UE transmits no HARQ-ACK after receiving a PDSCH. The gNB transmits continuously without waiting for feedback.

Solution 2: K1/K2 Extension

K1 extended from 1–15 to 1–32 slots. For LEO bent-pipe at $\mu = 1$ (0.5 ms/slot), RTT ≈ 12.8 ms requires K1 = 26 slots — within the new limit. For GEO, RTT ≈ 600 ms would require K1 ≈ 1200 slots — far beyond feasibility. Scope: LEO only.

Solution 3: Regenerative (On-Board Processing) Architecture

When the gNB is hosted on the satellite, the HARQ loop closes within the service link only. The feeder link carries N2/N3 backhaul, not HARQ.

$$RTT_{\mathrm{regen}} \approx 2 \times \frac{d_{\mathrm{slant}}}{c} + T_{\mathrm{proc}} \approx 5 + 5 = 10 \text{ ms (LEO 550 km)}$$

K1 = 21 slots at $\mu = 1$ covers this RTT. Architecture requires baseband processing hardware on the satellite payload — higher cost but dramatically reduced HARQ burden.

Solution 4: HARQ Process Count Extension (Rel-17/18)

Rel-17 extends the maximum HARQ process count from 16 to 32 for DL and UL independently. Rel-18 studies further extension to 64 or 256 processes to support MEO and partial GEO coverage with HARQ enabled. The UE capability field maxNumberHARQ-ProcessesForPDSCH reports the supported maximum.

Combining K1 extension with process-count extension enables LEO bent-pipe operation with HARQ active:

$$N_{\mathrm{proc}} = \left\lceil \frac{RTT}{T_{\mathrm{slot}}} \right\rceil = \left\lceil \frac{14.7}{0.5} \right\rceil = 30 \text{ processes at } \mu=1$$

The Rel-17 cap of 32 processes covers LEO with margin. For GEO, Solution 1 (disabling) remains the only standardised option.

9.4 RLC-Level Retransmission for NTN

3GPP TS 38.322 defines RLC Acknowledged Mode (AM) ARQ, which operates independently of HARQ. When HARQ is disabled (GEO, MEO), RLC AM is the primary retransmission mechanism.

Effective frame-error-rate after combined HARQ + RLC retransmission, assuming independence:

$$P_{\mathrm{FER,\,RLC}} = P_{\mathrm{FER,\,HARQ}}^{N_{\mathrm{HARQ,tx}}} \cdot P_{\mathrm{RLC}}^{N_{\mathrm{RLC,tx}}}$$

For GEO with HARQ disabled ($N_{\mathrm{HARQ,tx}} = 1$), a single RLC retransmission reduces FER from $\sim10^{-2}$ to $\sim10^{-4}$ at the cost of an added RTT delay (~600 ms for GEO) — acceptable for non-real-time traffic.

9.5 Quality of Service (QoS) Impact

5G QoS flows are classified by 5QI (5G QoS Identifier, TS 23.501 Table 5.7.4-1). NTN propagation delay adds directly to E2E latency, which must be compared against per-5QI packet-delay budgets:

$$L_{\mathrm{E2E}} = RTT_{\mathrm{NTN}} + T_{\mathrm{backhaul}} + T_{\mathrm{core}} + T_{\mathrm{application}}$$

PDCP ROHC (Robust Header Compression, TS 38.323): reduces per-packet IP/UDP/RTP overhead from 40+ bytes to 1–4 bytes, improving efficiency on power- and bandwidth-limited NTN S-band links where every bit of overhead matters.

GBR (Guaranteed Bit Rate) flows — voice, real-time video — are practically limited to LEO NTN. GEO NTN is suited to eMBB data, IoT, and delay-tolerant applications only. 3GPP TS 38.821 §6.3 reflects this in the service model definitions for NTN.

10.1 gNB Estimation and Measurement Procedures

SRS-Based UL Channel Estimation

SRS (Sounding Reference Signal, TS 38.211 §6.4.1.4) is a UE-transmitted pilot mapped on configurable comb patterns (comb-2, comb-4) in the frequency domain. Up to 64 SRS ports support full-dimension MIMO sounding. LS estimate at subcarrier $k$:

$$\hat{H}_{UL}(k) = \frac{Y_{\mathrm{SRS}}(k)}{X_{\mathrm{SRS}}(k)}$$

where $Y_{\mathrm{SRS}}(k)$ is the received frequency-domain sample and $X_{\mathrm{SRS}}(k)$ is the known SRS sequence. The estimate is used for:

• TDD UL-DL reciprocity → DL beamforming weight computation
• Link adaptation: SINR estimate → CQI prediction for DL scheduling
• UE location estimation (time-of-arrival, angle-of-arrival) for TA calibration

In NTN, SRS periodicity is typically relaxed (long-period SRS) since the satellite beam is largely fixed relative to the ground in beam-Earth-fixed mode, reducing the rate of channel variation compared to high-mobility terrestrial UEs.

PMI Feedback and Codebook

Type I codebook (TS 38.214 §5.2.2.2.1): two-stage structure $\mathbf{W} = \mathbf{W}_1 \times \mathbf{W}_2$.

• $\mathbf{W}_1$: wideband beam selection — one of $N_1 \times N_2$ beams on the 2D DFT codebook grid indexed by oversampling factors $O_1, O_2$
• $\mathbf{W}_2$: subband co-phasing / co-polarisation combination within the selected beam group

Type II codebook: linear combination of $K$ beams with amplitude and phase coefficients — higher angular resolution, higher feedback overhead.

NTN impact: due to propagation latency, PMI fed back by the UE is $RTT/2$ stale. For LEO this is ~7 ms (14 subframes at $\mu=1$) — acceptable for slowly varying channel geometry. For GEO (~300 ms stale PMI), real-time CSI feedback is impractical; the gNB instead uses ephemeris-based beam prediction: beam steering angles are computed from orbital mechanics without UE feedback.

CQI Measurement and Link Adaptation

UE measures SINR per subband on CSI-RS:

$$\mathrm{SINR}_{\mathrm{meas}}(k) = \frac{|H(k)|^2 P_s}{\sigma_n^2}$$

Maps to CQI index 0–15 per TS 38.214 Table 5.2.2.1-3 (4-bit CQI). CQI 0 = out of range; CQI 15 → 64QAM 948/1024 code rate → ~5.5 bps/Hz.

In NTN, the CQI report arrives $RTT/2$ stale at the gNB. The scheduler must apply a link adaptation margin $\Delta_{\mathrm{LA}}$ to account for channel variation during the reporting delay:

$$\Delta_{\mathrm{LA}} = 10\log_{10}\!\left(1 + \frac{\sigma_{\mathrm{channel}}^2}{\bar{H}^2}\right) \text{ dB}$$

For LEO: channel variation over ~7 ms is low (geometric — beam footprint moves ~52 m in 7 ms at 7.5 km/s orbital velocity); $\Delta_{\mathrm{LA}} \approx 1$–3 dB. For GEO: channel is essentially static; $\Delta_{\mathrm{LA}} \approx 0$ dB for the geometry, though rain fade events can cause sudden drops.

RSRP / RSRQ / SINR / RSSI (TS 38.215)

$$\mathrm{RSRP} = \frac{1}{N_{\mathrm{ref}}} \sum_{k \in \mathrm{CSI\text{-}RS}} |H(k)|^2 \cdot \frac{P_{TX}}{N_{SC}}$$ $$\mathrm{SINR} = \frac{P_{\mathrm{signal}}}{P_{\mathrm{interference}} + P_{\mathrm{noise}}}$$ $$\mathrm{RSRQ} = N_{RB} \cdot \frac{\mathrm{RSRP}}{\mathrm{RSSI}}$$

Typical NTN values for LEO S-band handheld UE: RSRP = −90 to −120 dBm, SINR = 0–20 dB. Link budget depends on: satellite EIRP, UE G/T, slant range, atmospheric losses (ITU-R P.676, P.618), and adjacent-beam interference.

10.2 NTN-Specific MAC/RRC Procedures

SIB19 — NTN System Information Block (TS 38.331 §6.3.1)

SIB19 is the NTN-specific SIB introduced in Rel-17. It carries satellite ephemeris and NTN configuration parameters required by the UE to perform autonomous TA pre-compensation and Doppler pre-compensation before initial access.

Using SIB19 + GNSS fix, the UE computes its TA independently:

$$TA_{\mathrm{UE}} = \frac{2 \cdot d_{\mathrm{slant}}(t) + d_{\mathrm{feeder}}}{c}$$

The gNB confirms or adjusts via the standard TA command (MAC CE). TA range extended to 541 ms in Rel-17 (vs. 3.8 ms terrestrial).

RRC Configuration Additions for NTN (TS 38.331)

• ta-Report: UE reports measured TA back to gNB for cross-validation against ephemeris-computed TA
• nr-NTN-Config in ServingCellConfig: enables NTN-specific timing procedures and grants access to extended K1/K2 tables
• Rel-18 additions: multiple beam configurations, inter-satellite routing capability, enhanced mobility for LEO handover (beam-sweeping trigger thresholds)

Doppler Pre-Compensation

UE applies a frequency offset $\Delta f_{\mathrm{UE}}$ equal and opposite to the estimated Doppler before UL transmission, using satellite velocity from SIB19 ephemeris and UE GNSS position. Residual Doppler after pre-compensation must be $< 0.1 \times \Delta f_{\mathrm{SCS}}$ (i.e., <1.5 kHz for 15 kHz SCS).

$$\Delta f_{\mathrm{Doppler}} = -\frac{v_{\mathrm{radial}}}{c} \cdot f_c$$

At LEO 550 km, max radial velocity ~7.5 km/s, $f_c$ = 2 GHz: max Doppler ≈ 50 kHz. Pre-compensation brings residual below 2 kHz (using GNSS ± ~30 m/s velocity error).

10.3 Transparent vs. Regenerative Architecture

10.4 Multi-Cell NTN: Interference and Frequency Reuse

Beam Footprint and Frequency Reuse

A LEO NTN satellite typically illuminates the Earth with a multi-beam antenna producing dozens to hundreds of spot beams (e.g., Starlink: ~1500+ beams per satellite). Each beam subtends a footprint of ~50–200 km diameter depending on altitude and antenna aperture.

Frequency reuse factor for LEO: adjacent beams can reuse the full frequency band (reuse-1), similar to dense small-cell terrestrial deployments, because:

• Satellite beam gain isolation provides ~20–30 dB first adjacent-beam suppression
• Short satellite dwell time (~5–10 min for LEO 550 km) limits interference duration
• Doppler profile differs between adjacent beams (different off-nadir angles) allowing frequency-domain discrimination

ICI Between Beams

Inter-Cell Interference (ICI) from a neighbouring satellite beam:

$$I_{\mathrm{ICI}} = P_{tx,\mathrm{neighbor}} \times \frac{G_{\mathrm{beam,\,sidelobe}}}{G_{\mathrm{beam,\,main}}} \times \mathrm{FSPL}_{\mathrm{neighbor}}^{-1}$$

where $G_{\mathrm{beam,\,sidelobe}}/G_{\mathrm{beam,\,main}}$ is the beam isolation ratio (typically −20 to −30 dB for a well-designed phased array at first sidelobe). Full-frequency reuse requires this ratio to keep SINR above the minimum usable threshold (~0 dB for QPSK 1/5, up to ~20 dB for 64QAM 9/10).

3GPP NTN Spectrum Sharing with TN

NTN systems operating on S-band frequencies (2 GHz band) must coexist with terrestrial MNO deployments. 3GPP TS 38.821 §6.6 defines:

• Guard zones: exclusion radius around gateways where NTN UL is suppressed to protect adjacent TN UL
• Time-domain coordination: satellite beam scheduling aligned to TN network subframe structure to limit interference windows
• Power control: NTN UE reduces UL transmit power when within guard zone proximity to TN base stations

Walker-Delta Constellation Coverage

LEO constellations use Walker-Delta or Walker-Star orbital geometries to ensure continuous global coverage. A Walker-Delta constellation is specified as $T/P/F$ where $T$ = total satellites, $P$ = orbital planes, $F$ = phase offset factor. The phasing ensures adjacent planes' satellites are offset by $F \times 360°/(T)$, maintaining uniform angular spacing between planes at any latitude.

Handover rate at $\mu_{\mathrm{lat}}$ latitude for altitude $h$:

$$f_{\mathrm{HO}} \approx \frac{v_{\mathrm{orbit}}}{\pi \cdot R_{\mathrm{footprint}}} = \frac{7.5 \text{ km/s}}{\pi \times 100 \text{ km}} \approx 0.024 \text{ Hz} \approx 1 \text{ HO per 43 s}$$

This is orders of magnitude faster than terrestrial handover rates for static UEs. 3GPP TS 38.821 §7 defines beam-sweeping handover and conditional handover (CHO) procedures for LEO NTN to handle this.

Interactive HARQ RTT Calculator

Use the controls below to compute RTT and required HARQ process count for any orbit altitude and numerology, and compare against 3GPP Rel-17/18 process limits.

11.1 IMT-2030 NTN Standards Timeline

11.2 6G NTN Key Requirements vs. Rel-17 Baseline

11.3 Interactive Satellite D2D Link Budget

Adjust the parameters below to compute a simplified NTN link budget and determine whether the link closes (received SNR ≥ required SNR). Uses free-space path loss plus receiver noise figure; ignores rain fade, pointing loss, and atmospheric absorption for clarity.

The D2D NTN carrier-to-noise density is given by: $$C/N_0 = \mathrm{EIRP}_{UE1} - L_{\mathrm{FSPL}} + G/T_{UE2} + 228.6 \;\text{dB-Hz}$$ where $\mathrm{EIRP}_{UE1} = P_{t,UE1}\,\text{[dBW]} + G_{t,UE1}\,\text{[dBi]}$ is the transmitting UE’s effective radiated power, $L_{\mathrm{FSPL}}$ is the free-space path loss to the receiving UE’s position (via satellite relay), $G/T_{UE2}$ is the receive UE’s gain-to-noise-temperature ratio (dB/K), and $+228.6$ converts Boltzmann’s constant to dB (i.e. $-10\log_{10}(k_B) = 228.6$ dB).

11.4 6G NTN Architecture Evolution

Beyond Rel-17's two-architecture model, 6G introduces a richer set of deployment options enabled by miniaturised on-board processing, laser ISLs, and AI-native network management.

NTN ISAC signal model. In the IMT-2030 ISAC scenario the received signal vector simultaneously carries communication payload and radar echoes: $$\mathbf{y} = \mathbf{H}_{\mathrm{comm}}\mathbf{x} + \mathbf{H}_{\mathrm{radar}}\mathbf{x} + \mathbf{n}$$ where $\mathbf{x}$ is the dual-function waveform (e.g. AFDM chirp or CP-OFDM), $\mathbf{H}_{\mathrm{comm}}$ is the communication channel matrix, $\mathbf{H}_{\mathrm{radar}}$ is the target-return channel (delay-Doppler spread), and $\mathbf{n}$ is AWGN. AFDM chirp parameter: $c_1 = p_1/(2N)$, with Discrete Affine Fourier Transform (DAFT): $$X[k] = \sum_{n=0}^{N-1} x[n]\, e^{-j2\pi(c_1 n^2 + kn/N + c_2 k^2)}$$ The quadratic phases $c_1 n^2$ and $c_2 k^2$ spread energy across the delay-Doppler plane, providing full diversity in doubly-dispersive NTN channels.

12.1 Beam Management in NTN (3GPP Rel-17/18)

Terrestrial beam management (P1/P2/P3 procedures, TS 38.214 §5.2.1) assumes quasi-static beams. NTN introduces earth-fixed vs satellite-fixed beam modes:

Mode	Beam footprint	Doppler to UE	Beam switching rate	3GPP support
Earth-fixed	Stationary on ground	Varies across footprint	High (steered electronically)	Rel-17 (TS 38.821)
Satellite-fixed	Moves with satellite	Uniform across beam	Low (fixed grid)	Rel-17

Earth-fixed beams require active digital beamforming (phased array); satellite-fixed allow simpler bent-pipe repeaters. Rel-18 adds AI/ML-assisted beam prediction using ephemeris data — gNB can predict beam-UE geometry up to 1–2 orbital periods ahead.

Beam failure recovery (BFR): Standard T_BFR timer (Rel-15 TS 38.321 §5.17) is too short for NTN. Rel-17 extends BFR timeout; ephemeris-aware BFR predicts beam failure before it occurs.

12.2 Handover in NTN

Three handover types specific to NTN (TS 38.821 §6.3):

• Intra-satellite: Between beams of same satellite — sub-100 ms, similar to terrestrial HO
• Inter-satellite: LEO passes out of range → new satellite; interrupt window 50–500 ms; requires prior resource allocation on target
• NTN→TN (or TN→NTN): Service continuity during coverage gaps; UE maintains dual registration

Key challenge: HO preparation latency. For LEO μ=1: RTT ≈ 25 ms → measurement report → HO command round-trip ≈ 50 ms. LEO visible window ≈ 5–12 min → ≫ HO latency, so proactive HO scheduling (ephemeris-driven) is used instead of reactive RSRP triggering.

Rel-17 solution: Conditional handover (CHO, TS 38.331 §5.4.4.5) pre-configures HO commands; UE executes when trigger condition met — no additional round-trip needed.

Handover trigger condition (A3 event, TS 38.331 §5.5.4.4): $$RSRP_{\mathrm{target}}(t) > RSRP_{\mathrm{serving}}(t) + \text{hysteresis} + A3\text{-offset}$$ In standard terrestrial NR this condition is evaluated over a time-to-trigger (TTT) window of 0–5120 ms. For LEO NTN, the proactive CHO variant uses ephemeris-predicted geometry: $$\text{execute HO if } RSRP_{\mathrm{target}}(t) > T_{\mathrm{CHO}}$$ where $T_{\mathrm{CHO}}$ is a pre-configured absolute RSRP threshold (dBm) set at RRC reconfiguration time. This eliminates the A3 measurement-report round-trip (saves ~2× RTT ≈ 50 ms for LEO at $\mu=1$).

12.3 NTN-Aware Scheduling

Standard 5G NR scheduling (TS 38.214) assumes tight feedback loops. NTN RTT blows these assumptions:

Function	Terrestrial	LEO NTN	GEO NTN
HARQ round-trip	~8 ms (μ=1)	~25 ms one-way → ≥50 ms	~270 ms one-way → ≥560 ms
Link adaptation	CQI report delay ~5 ms	~50 ms stale	~560 ms stale
Scheduling request	Immediate	Pre-grant (TS 38.321)	Pre-grant mandatory
Buffer status report	On-demand	Periodic (avoid RTT overhead)	Periodic mandatory
Power control	TPC closed-loop	Open-loop (Rel-17 TS 38.101-5)	Open-loop mandatory

Pre-grant scheduling: gNB sends scheduled UL grants before UE requests them, based on traffic prediction (AI/ML). Eliminates the SR→grant→data round-trip for latency-sensitive traffic. Standardized in 3GPP Rel-17 TS 38.321 §5.4.4.

Under a Poisson traffic model with arrival rate $\lambda$ packets/s, the probability that a UE has data ready at slot $n$ (slot duration $T_{\mathrm{slot}} = 2^{-\mu}$ ms) is: $$P_{\mathrm{data}}(n) = 1 - e^{-\lambda T_{\mathrm{slot}}}$$ For bursty IoT-NTN traffic ($\lambda = 0.1$ pkt/s, $T_{\mathrm{slot}} = 0.5$ ms at $\mu=1$): $P_{\mathrm{data}} \approx 5 \times 10^{-5}$ — pre-grant is wasteful for sparse IoT but essential for voice/video ($\lambda \geq 50$ pkt/s) where $P_{\mathrm{data}} \approx 1$.

Rel-19/20 outlook: Joint TN-NTN scheduling, unified RAN controller across orbital planes, AI-driven ephemeris-based resource pre-allocation.

12.4 6G AI/ML Beam Prediction for NTN

Rel-18 introduced the AI/ML framework for NR air-interface (TR 38.843). For NTN, the key application is predictive beam management: because satellite trajectories are deterministic, a neural network can forecast the optimal beam weight vector well ahead of the actual beam-failure event.

The general ML-based beam predictor takes the form: $$\hat{\mathbf{w}}(t+\Delta t) = f_{\mathrm{ML}}\!\left(\mathbf{H}(t),\; \mathbf{v}_{\mathrm{sat}}(t),\; \mathbf{r}_{UE}(t)\right)$$ where $\mathbf{H}(t) \in \mathbb{C}^{N_r \times N_t}$ is the measured channel matrix at time $t$, $\mathbf{v}_{\mathrm{sat}}(t)$ is the satellite velocity vector (from ephemeris), $\mathbf{r}_{UE}(t)$ is the UE position estimate (GPS or GNSS), and $\Delta t$ is the prediction horizon (up to one orbital pass, ~6–10 min for LEO 600 km).

The output $\hat{\mathbf{w}} \in \mathbb{C}^{N_t}$ is the predicted beamforming weight vector; it is pre-loaded into the on-board BFN so the satellite switches beams without any uplink measurement report during the handover window. Prediction accuracy metric (3GPP TR 38.843 §6.3.3): beam top-$K$ accuracy $P(\hat{b} \in \mathcal{B}_K^*) \geq 0.9$ where $\mathcal{B}_K^*$ is the set of $K$ best beams from exhaustive sweep.

ML Architecture	Input features	Prediction horizon	Top-1 accuracy (reported)
LSTM / GRU	CQI sequence + position	1–5 s	~85–90%
Transformer (attention)	CSI + ephemeris	10–60 s	~92–96%
Graph Neural Network	Multi-satellite topology	30–120 s	~88–93%

12.5 NTN Spectral Efficiency: Residual Doppler & SINR Floor

The achievable spectral efficiency of an NTN link is: $$\eta_{\mathrm{NTN}} = \log_2\!\left(1 + SINR_{\mathrm{eff}}\right) \quad \text{[bits/s/Hz]}$$ where the effective SINR after Doppler pre-compensation is: $$SINR_{\mathrm{eff}} = \frac{P_r / (N_0 B)}{1 + \gamma_{\mathrm{ICI}} + \gamma_{\mathrm{IBI}} + \gamma_{\mathrm{inter-beam}}}$$

The three interference terms in the denominator are:

• Residual Doppler ICI floor $\gamma_{\mathrm{ICI}}$: After UE pre-compensation, residual Doppler $\Delta f_D$ causes inter-carrier interference. For CP-OFDM with $N$ subcarriers and SCS $\Delta f$: $$\gamma_{\mathrm{ICI}} \approx \frac{\pi^2}{3}\left(\frac{\Delta f_D}{\Delta f}\right)^2$$ At $\mu=1$ (30 kHz SCS) with residual $\Delta f_D = 1$ kHz: $\gamma_{\mathrm{ICI}} \approx -30$ dB (negligible). At $\Delta f_D = 5$ kHz: $\gamma_{\mathrm{ICI}} \approx -16$ dB (limits SNR to ~16 dB — bottleneck for high MCS).
• Inter-beam interference $\gamma_{\mathrm{inter-beam}}$: adjacent satellite spot beams reuse the same frequency; isolation depends on beam roll-off and satellite altitude. Typical isolation for Ka-band HTS: 18–25 dB. In 6G, digital BFN with null-steering targets $\gamma_{\mathrm{inter-beam}} < -30$ dB.
• HARQ-off ARQ penalty: For GEO where HARQ is disabled, uncorrected PHY errors must be recovered by RLC AM ARQ. The effective FER budget requires initial BLER $\leq 10^{-1}$ for RLC retransmission to converge within 3 ARQ rounds.

Combining with the IMT-2030 spectral efficiency targets ($\geq 10$ bits/s/Hz DL, $\geq 5$ bits/s/Hz UL), the required $SINR_{\mathrm{eff}}$ thresholds are: $$SINR_{\mathrm{eff}} \geq 2^{10} - 1 \approx 1023 \approx 30.1\;\text{dB (DL)}$$ $$SINR_{\mathrm{eff}} \geq 2^{5} - 1 = 31 \approx 14.9\;\text{dB (UL)}$$ These are per-subcarrier targets achievable only with massive MIMO beamforming gain $\geq 20$ dBi at the satellite and Ka/Q-band link budgets; confirming why sub-THz feeder links and digital BFN are core 6G NTN requirements.

A.1 Orbital Mechanics

A.2 Doppler Shift & Pre-Compensation

A.3 Link Budget Formulas

A.4 HARQ & Timing Formulas

A.5 NTN System Parameters & Constants

A.6 NTN Acronyms & Glossary