Ferkans — Interactive Telecom Tutor

What Is Different About the LEO Channel

Section 23.1 pinned down the orbital geometry. We now zoom in on the radio link itself. The LEO link has four salient features that make it unlike any terrestrial channel.

It is strongly line-of-sight. For any elevation above $\approx 10^\circ$ the terminal has a clear sky view of the satellite. Shadowing and multipath exist but are dominated by a single specular path.
The free-space path loss is enormous. At Ka band ( $28$ GHz) and a slant range of $1000$ km, the FSPL is $\approx 181$ dB. Closing the link at all requires high-gain terminal antennas and multi-Watt satellite EIRPs.
The channel is fast-varying through Doppler. The terminal sees $v_{\text{sat}}/c \approx 2.5 \times 10^{-5}$ velocity ratio, so at $28$ GHz the peak Doppler is $\approx 700$ kHz. This is two to three orders of magnitude larger than the peak Doppler on any terrestrial channel ( $\approx 1$ kHz at $1$ GHz / $200$ km/h).
Doppler is deterministic. Unlike terrestrial Doppler spectra (Jakes, classical), the LEO Doppler shift is known from the satellite ephemeris and the terminal position. Almost all of it can be pre-compensated. The uncompensable residual is much smaller but still hundreds of Hz.

The signal model throughout this section uses $\mathbf{H}$ for the narrowband channel vector from a single satellite to the terminal. At Ka band and LEO distances, $\mathbf{H}$ is essentially a steering- vector response scaled by a slow complex gain plus a residual Doppler-induced phase rotation.

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Definition:
LEO Link Budget and Received SNR

The narrowband received SNR on a clear-sky LEO link is

$\text{SNR} = \frac{P_t\, G_{\text{sat}}\, G_{\text{term}}\, \lambda^{2}} {(4\pi d_{\text{slant}})^2\, L_{\text{rain}}\, L_{\text{atm}}\, L_{\text{pol}} \, k_B T_{\text{sys}} W},$

where $P_t$ is the satellite (or terminal) transmit power, $G_{\text{sat}}$ and $G_{\text{term}}$ are the end antenna gains, $\lambda = c / f_0$ is the wavelength, $d_{\text{slant}}$ is the slant range, $L_{\text{rain}}$ is the rain-fade loss, $L_{\text{atm}}$ accounts for clear-air gaseous absorption, $L_{\text{pol}}$ is the polarization mismatch, $k_B$ is the Boltzmann constant, $T_{\text{sys}}$ is the receiver system noise temperature, and $W$ is the signal bandwidth. The ratio $G/T \equiv G_{\text{term}} - 10 \log_{10} T_{\text{sys}}$ (dB/K) is the terminal figure of merit used in every link-budget table. For a Starlink-class Ka-band terminal, $G/T \approx 10$ dB/K; for a phased-array UE on a car roof it is closer to $-5$ dB/K.

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Theorem: LEO Doppler Shift vs Elevation Angle

Consider a LEO satellite of circular-orbit velocity $v_{\text{sat}}$ passing through the zenith of a fixed terminal along a ground track aligned with the terminal meridian. At elevation angle $\theta_{\text{el}}$ during the approaching half of the pass, the instantaneous one-way Doppler shift at carrier frequency $f_0$ is

$\Delta f_D(\theta_{\text{el}}) = \frac{v_{\text{sat}}}{c} \cdot f_0 \cdot \frac{R_E \cos\theta_{\text{el}}} {\sqrt{R_E^2 \sin^2\theta_{\text{el}} + 2 R_E h + h^2}},$

and the peak value at the horizon ( $\theta_{\text{el}} = 0$ ) is

$f_D = \frac{v_{\text{sat}}}{c} f_0 \cdot \frac{R_E}{\sqrt{2 R_E h + h^2}} \approx \frac{v_{\text{sat}}}{c} f_0 \quad \text{for } h \ll R_E,$

so that at Ka band ( $f_0 = 28$ GHz, $v_{\text{sat}} = 7.56$ km/s) the peak is approximately $705$ kHz one-way, $1.41$ MHz two-way. At S band ( $f_0 = 2$ GHz) the peak drops to $\approx 50$ kHz one-way.

The instantaneous Doppler is the radial velocity divided by wavelength. At zenith the satellite's velocity is entirely tangential, so the radial component is zero and $\Delta f_D = 0$ (the Doppler sign flips here, from approaching-positive to receding-negative). At the horizon the radial component is nearly the full $v_{\text{sat}}$ , so the Doppler peaks. The closed-form expression uses the sine rule to express the radial projection in terms of $\theta_{\text{el}}$ , $R_E$ , and $h$ , and the small- $h$ -over- $R_E$ approximation gives the familiar $v_{\text{sat}} f_0 / c$ upper bound.

Proof

Radial velocity from geometry

In the Earth-centered inertial frame, the satellite moves at speed $v_{\text{sat}}$ tangent to its circular orbit. The line from the terminal to the satellite makes an angle $\eta$ with the orbit tangent. By the sine rule in the triangle formed by the Earth's center, the terminal, and the satellite, $R_E / \sin(\eta) = d_{\text{slant}} / \sin(90^\circ + \theta_{\text{el}})$ , giving $\sin\eta = R_E \cos\theta_{\text{el}} / d_{\text{slant}}$ . The radial velocity is $v_{\text{rad}} = v_{\text{sat}} \sin\eta$ .

Convert to frequency shift

The one-way narrowband Doppler is $\Delta f_D = (v_{\text{rad}} / c) f_0$ (approaching-positive convention). Substituting $v_{\text{rad}} = v_{\text{sat}} R_E \cos\theta_{\text{el}} / d_{\text{slant}}$ and $d_{\text{slant}} = \sqrt{R_E^2 \sin^2\theta_{\text{el}} + 2 R_E h + h^2} - R_E \sin\theta_{\text{el}}$ (keeping only the leading $\sqrt{\cdot}$ term for the small-angle expansion) gives the stated formula.

Horizon limit

At $\theta_{\text{el}} \to 0$ , $\sin\theta_{\text{el}} \to 0$ and $\cos\theta_{\text{el}} \to 1$ , so $d_{\text{slant}} \to \sqrt{2 R_E h + h^2}$ and $\Delta f_D \to v_{\text{sat}} f_0 R_E / (c \sqrt{2 R_E h + h^2})$ . For $h = 600$ km this evaluates to $v_{\text{sat}} f_0 / c \cdot R_E / 2800 \approx 0.947 v_{\text{sat}} f_0 / c$ — essentially the plain $v_{\text{sat}} f_0 / c$ bound. $\blacksquare$

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LEO Doppler vs Elevation Angle at S, Ku, and Ka Bands

The one-way Doppler shift of a LEO satellite swings from $+f_D$ at horizon-approach to $-f_D$ at horizon-departure, passing through zero at zenith. The amplitude is linear in the carrier frequency $f_0$ , which is why S band ( $2$ GHz) is an order of magnitude friendlier than Ka band ( $28$ GHz) on this axis alone. We plot the full elevation sweep for a single overhead pass and mark the minimum-elevation cutoff where operational service actually starts.

Parameters

h

(km, altitude)600

f_0

(GHz)28

overlay S/Ku/Ka bands

Ka Band

The Ka-band radio spectrum, nominally $26.5$ – $40$ GHz, with $17.7$ – $20.2$ GHz typically used for satellite downlinks and $27.5$ – $30$ GHz for uplinks. The main attraction of Ka band for NTN is the $2$ – $3$ GHz of contiguous bandwidth; the penalty is rain fade (dominant excess loss) and the largest Doppler shifts of any current satcom band. Starlink, OneWeb, and Kuiper all operate on Ka band as their primary broadband spectrum.

Doppler Shift

The frequency offset $\Delta f_D = (v_{\text{rad}}/c) f_0$ induced by relative radial motion between transmitter and receiver. For a LEO satellite at $v_{\text{sat}} \approx 7.5$ km/s and $f_0 = 28$ GHz, the peak one-way Doppler is approximately $700$ kHz — two orders of magnitude larger than any terrestrial case. Unlike terrestrial Doppler, LEO Doppler is deterministic: it can be computed from satellite ephemeris and pre-compensated open-loop.

Rain Fade

Atmospheric attenuation due to precipitation along an Earth-space path, dominant above about $10$ GHz. Modeled statistically by ITU-R Recommendation P.618. At Ka band, typical rain-fade exceedance levels are $\approx 3$ dB for $0.1\%$ of year and $\approx 10$ – $15$ dB for $0.01\%$ of year. Mitigations include static margin, adaptive coding and modulation, gateway site diversity, and — the topic of this chapter — macro-diversity across multiple simultaneously serving satellites.

Definition:
Narrowband Satellite Channel Vector

Consider a single LEO satellite with an on-board digital array of $N_t$ elements illuminating a single-antenna user at angular position $(\phi, \theta)$ (relative to the satellite's bore-sight). The narrowband downlink channel at a given time instant $t$ is

$\mathbf{H}(t) = \beta\, e^{j 2 \pi \Delta f_D\, t}\, \mathbf{a}(\phi, \theta) + \tilde{\mathbf{H}}(t),$

where $\beta$ is the slow complex gain (incorporating path loss, rain fade, and polarization), $\Delta f_D$ is the instantaneous Doppler shift of the dominant path, $\mathbf{a}(\phi, \theta)$ is the satellite array's steering vector, and $\tilde{\mathbf{H}}(t)$ collects diffuse scatter contributions with total variance $\sigma_{\text{NLOS}}^2 \ll |\beta|^2$ . The Rician K-factor $K = |\beta|^2 / \sigma_{\text{NLOS}}^2$ is typically $10$ – $20$ dB for Ka-band LEO with the terminal outdoors.

The key structural feature — and the one Section 23.3 exploits — is that the dominant LOS direction is deterministic from the satellite ephemeris. The array response $\mathbf{a}(\phi,\theta)$ can be computed offline, and the Doppler phasor $e^{j 2\pi \Delta f_D t}$ is pre-compensated before OFDM processing. What remains is a slowly varying amplitude $\beta$ and the residual diffuse term.

This is a drastically simpler model than the terrestrial multipath channels of Chapters 2 and 10. The simplification is real: LEO channels genuinely look like this, with Rician $K$ factors well above $10$ dB for typical Ka-band deployments.

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Example: Doppler Numbers at S, Ku, and Ka Bands

A LEO satellite at $h = 600$ km orbits with $v_{\text{sat}} = 7.56$ km/s. Compute the peak one-way Doppler shift and the peak Doppler rate of change $|d f_D / dt|$ at zenith for carriers $f_0 \in \{2, 12, 28\}$ GHz (S, Ku, Ka bands). Compare with a typical OFDM subcarrier spacing of $\Delta f = 120$ kHz (5G NR FR2 numerology $\mu = 3$ ).

Solution

Peak Doppler

$f_D \approx v_{\text{sat}} f_0/c$ .

S band: $(7560/3 \cdot 10^8) \cdot 2 \cdot 10^9 \approx 50$ kHz.
Ku band: $(7560/3 \cdot 10^8) \cdot 12 \cdot 10^9 \approx 302$ kHz.
Ka band: $(7560/3 \cdot 10^8) \cdot 28 \cdot 10^9 \approx 705$ kHz.

Doppler rate at zenith

At zenith the Doppler crosses zero but its derivative is maximal: $|df_D / dt|_{\text{zenith}} \approx v_{\text{sat}}^2 f_0 / (c h)$ . Plug numbers (using $h = 600$ km):

S band: $\approx 630$ Hz/s.
Ku band: $\approx 3.8$ kHz/s.
Ka band: $\approx 8.8$ kHz/s.

Compare to OFDM subcarrier spacing

With $\Delta f = 120$ kHz, the uncompensated Ka-band peak Doppler of $705$ kHz is $\approx 5.9 \Delta f$ — far beyond the tolerable $f_D/\Delta f < 0.01$ bound for OFDM (Section 23.4). Pre-compensation via ephemeris therefore is mandatory. After pre-compensation the residual uncertainty on $\Delta f_D$ is $O(100)$ Hz, comfortably below $\Delta f$ . $\blacksquare$

⚠️Engineering Note

Rain Fade at Ka Band

Rain fade is the single largest source of variable excess loss on a Ka-band satellite link and is the reason operators run careful link-margin calculations. The ITU-R P.618 model predicts rain attenuation from local rainfall rate statistics. Representative numbers at $f_0 = 28$ GHz, $30^\circ$ elevation, temperate climate:

0.01% of year exceedance (heavy rain): $\approx 10$ – $15$ dB.
0.1% exceedance (moderate rain): $\approx 3$ – $5$ dB.
1% exceedance (light rain): $\approx 0.5$ – $1$ dB.
Clear sky: $< 0.3$ dB (atmospheric absorption only).

A $10$ dB rain fade reduces the SNR to a tenth of its clear-sky value and can kill the link entirely if the terminal is operating near its sensitivity threshold. Mitigation strategies:

Static link margin — design the link for a worst-case rain event. Wastes spectral efficiency during the $99\%$ of time the sky is clear.
Adaptive modulation and coding (ACM) — drop to a lower-rate MCS when rain attenuation is detected. Standard in modern DVB-S2X.
Site diversity — route traffic through a gateway at a different geographic location. Gateways are cheap relative to satellites, so large operators deploy several per region.
Macro-diversity across satellites. If a terminal can pick from $M$ visible satellites at different elevation angles, the rain path is different for each, and the effective rain margin improves by a factor roughly proportional to $\sqrt{M}$ . This is one of the payoffs of the cell-free architecture of Section 23.3.

Practical Constraints

•
Ka-band link margin must budget $\geq 5$ dB for rain at $99.9\%$ availability
•
Terminal $G/T$ drops during rain due to increased sky noise temperature
•
Rain fade affects both uplink and downlink independently

📋 Ref: ITU-R P.618-13 (rain attenuation), 3GPP TR 38.811 §6.6.6

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Doppler Shift Profile During a LEO Pass

The one-way Doppler shift traced over a single LEO pass. The curve rises from zero at acquisition, peaks at approximately

f_D = v_{\text{sat}} f_0 / c

as the satellite crosses the terminal's horizon, passes through zero at the zenith moment (sign reversal), and settles back to zero at the far horizon. At Ka band the range is more than

1.4

MHz peak-to-peak, and the rate of change at zenith is

\approx 9

kHz/s.

Satellite Coverage Footprint vs Altitude

The slant range at a given minimum elevation sets the radius of the coverage footprint on the ground. We plot how this footprint radius grows with altitude for three minimum-elevation thresholds. Lower elevation means a larger footprint per satellite (fewer satellites needed for global coverage) but higher path loss and more rain margin. Operators pick the threshold depending on whether they want few expensive satellites (GEO-like) or many cheap satellites (LEO- like). This trade-off is the reason the Starlink shell sits at $\approx 550$ km rather than further out.

Parameters

min elevation (deg)10

Common Mistake: Jakes Spectrum Does Not Apply to LEO

Mistake:

A common mistake is to apply the Jakes / classical Doppler spectrum (Chapter 2, terrestrial fading) to the LEO channel, assuming a U-shaped distribution of Doppler around a peak $f_D$ . Under this assumption the channel is modelled as rich scatter with rapid decorrelation and the system designer reaches for OFDM subcarrier spacings in the tens of kilohertz.

Correction:

Jakes assumes isotropic scatter around a moving terminal in a dense multipath environment. The LEO channel has almost no scatter — $10$ – $20$ dB Rician K — and a single dominant LOS component with a deterministic Doppler shift set by ephemeris. The right picture is not a Doppler spectrum but a Doppler line whose frequency varies predictably over the pass. After pre-compensation, the effective spectrum is a narrow peak near zero with $O(100)$ Hz residual. Design OFDM for the residual, not for the raw Doppler, and budget for open-loop pre-compensation as a first-class processing step.

Quick Check

A LEO constellation at $h = 550$ km operates at $f_0 = 20$ GHz (downlink Ka). Using the small- $h$ -over- $R_E$ approximation, what is the peak one-way Doppler shift at low elevation? (Take $v_{\text{sat}} = 7.58$ km/s.)

$\approx 50$ kHz

$\approx 250$ kHz

$\approx 505$ kHz

$\approx 1.4$ MHz

Correction:

\approx 505

kHz

$f_D \approx v_{\text{sat}} f_0 / c = (7580 / 3 \times 10^8) \cdot 2 \times 10^{10} \approx 505$ kHz. A more careful calculation with the exact formula at $10^\circ$ elevation yields $\approx 495$ kHz, confirming the simple bound.

Why This Matters: Why Ka Band for LEO Despite the Doppler Penalty

Ka band ( $17.7$ – $20.2$ GHz downlink, $27.5$ – $30$ GHz uplink) is the main broadband spectrum for Starlink, OneWeb, and Kuiper despite incurring a $14\times$ larger Doppler than S band. The reason is bandwidth: the Ka band offers contiguous $2$ – $3$ GHz chunks, compared to the $10$ – $20$ MHz typical of L/S-band satcom. Bandwidth beats Doppler because Doppler is a per-symbol effect (a few hundred Hz residual after pre-compensation) while bandwidth determines the system throughput directly. The design challenge — which this chapter addresses — is to handle the Doppler with open-loop pre-compensation and robust waveform choices so that the Ka-band bandwidth translates into actual delivered data rate.

Key Takeaway

The LEO channel is LOS-dominated with a large but deterministic Doppler shift and a manageable residual. Ephemeris-driven pre-compensation removes almost all of the raw Doppler, leaving $O(100)$ Hz of residual that OFDM can tolerate. The remaining engineering effort goes into (a) closing the Ka-band link budget (rain fade and path loss), and (b) exploiting macro-diversity across multiple visible satellites, which is the subject of Section 23.3.

The LEO Channel Model: Path Loss, Doppler, Time-Variance

What Is Different About the LEO Channel

Definition: LEO Link Budget and Received SNR

Theorem: LEO Doppler Shift vs Elevation Angle

Radial velocity from geometry

Convert to frequency shift

Horizon limit

LEO Doppler vs Elevation Angle at S, Ku, and Ka Bands

Parameters

Ka Band

Doppler Shift

Rain Fade

Definition: Narrowband Satellite Channel Vector

Example: Doppler Numbers at S, Ku, and Ka Bands

Peak Doppler

Doppler rate at zenith

Compare to OFDM subcarrier spacing

Rain Fade at Ka Band

Doppler Shift Profile During a LEO Pass

Satellite Coverage Footprint vs Altitude

Parameters

Common Mistake: Jakes Spectrum Does Not Apply to LEO

Quick Check

Why This Matters: Why Ka Band for LEO Despite the Doppler Penalty

Key Takeaway

Definition:
LEO Link Budget and Received SNR

Definition:
Narrowband Satellite Channel Vector