⚡ electrical engineering

Performance Analysis of LEO-Terrestrial Systems in Presence of Doppler Effect

This paper presents a novel stochastic geometry-based framework to analytically evaluate the impact of residual Doppler shift on the coverage probability of multiuser LEO-terrestrial OFDMA systems, demonstrating through S-band and Ka-band simulations that even compensated Doppler effects significantly degrade subcarrier orthogonality and must be accounted for in future network designs.

Original authors: Islam M. Tanash, Nuria Gonzalez-Prelcic, Risto Wichman

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Islam M. Tanash, Nuria Gonzalez-Prelcic, Risto Wichman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Fast-Moving Train" Problem

Imagine you are standing on a train platform, and a high-speed train (the LEO Satellite) is zooming past you. You are trying to have a conversation with someone on the train using a walkie-talkie.

Because the train is moving so fast, the sound of their voice changes pitch as they approach and then recede. This is the Doppler Effect (like the change in pitch of a siren as an ambulance drives by). In satellite terms, this "pitch change" messes up the radio signal, making it hard to understand.

To fix this, the train station (the Ground Network) has a smart computer. It calculates exactly how fast the train is moving and tells the train to "pre-tune" its radio so that, for the person standing right in the middle of the platform, the voice sounds normal.

The Problem: The platform is huge. While the person in the middle hears the voice perfectly, the person standing at the far left edge and the person at the far right edge are still hearing a slightly different pitch. The "pre-tuning" worked for the center, but not for the edges. This leftover mess is called Residual Doppler.

What This Paper Does

The authors of this paper are like engineers trying to figure out exactly how bad that "leftover mess" is for a whole crowd of people on the platform. They want to know: If we have a giant crowd of users spread out under the satellite's beam, how much of the signal will get garbled because of this leftover pitch shift?

They focus on a specific type of communication called OFDMA. Think of this as a choir where every singer (user) is assigned a specific note (frequency) to sing. If everyone sings their note perfectly, the music is beautiful. But if the "pitch" of the whole room is wobbling (due to the Doppler effect), the singers start hitting the wrong notes. They start bleeding into each other's notes, creating a muddy noise called Inter-Carrier Interference (ICI).

The "Math Magic" They Used

Instead of just guessing or running millions of computer simulations, the authors used a branch of math called Stochastic Geometry.

The Analogy: Imagine throwing darts at a circular board (the satellite's coverage area). The authors didn't track every single dart. Instead, they used probability to predict exactly how many darts would land near the center versus the edge, and how that distribution affects the overall "muddiness" of the signal.
The Result: They created a formula (a "recipe") that predicts the Coverage Probability. In plain English: "What are the odds that a random user on the ground will get a clear signal?"

Key Findings (The "So What?")

The paper ran simulations to test their formula against real-world scenarios (using both S-band and Ka-band frequencies). Here is what they found:

The "Center" Fix Isn't Enough: Even if you perfectly compensate for the Doppler shift in the center of the beam, the people on the edges still suffer. It's like tuning a piano for the middle keys; the high and low keys will still sound slightly out of tune. This leftover error significantly hurts performance.
Higher Frequencies = Bigger Problems: If you try to use higher frequencies (like the Ka-band, used for high-speed internet), the Doppler effect is much more aggressive. It's like trying to balance a pencil on its tip; a tiny wobble causes a huge fall. The paper shows that for high-speed internet from space, you need to be very careful about this residual error.
Wider Spacing Helps: If you space the "notes" in the choir further apart (increasing subcarrier spacing), the singers are less likely to step on each other's toes. The paper shows that making the frequency gaps wider makes the system more robust against the Doppler wobble.
Bigger Cells = More Chaos: If the satellite's "spotlight" (beam) is very wide (covering a huge city), the difference in speed between the center and the edge is massive. This creates more interference. Smaller, tighter beams actually perform better because everyone is closer to the "center" of the tune.
Distance Matters: Surprisingly, when the satellite is very far away (high altitude), the angle at which it is seen from the ground changes. Sometimes, being further away actually reduces the "wobble" effect, even though the signal is weaker because of the distance. It's a trade-off between signal strength and pitch stability.

The Takeaway

This paper tells us that building the next generation of satellite internet (like Starlink) isn't just about pointing a satellite at the ground. We have to account for the fact that the satellite is moving so fast that it creates a "ripple effect" across the entire coverage area.

Even with smart computers trying to fix the signal, residual errors will always exist. If we ignore these errors, our internet connections will drop or slow down. The authors have given engineers a precise mathematical tool to predict these drops and design better systems that can handle the "wobble" of a fast-moving satellite.

1. Problem Statement

Low Earth Orbit (LEO) satellite systems offer significant advantages over Medium Earth Orbit (MEO) and Geostationary (GEO) systems, including lower latency and reduced power consumption. However, the high mobility of LEO satellites induces significant Doppler shifts, which pose a major challenge for Orthogonal Frequency-Division Multiple Access (OFDMA) systems.

While common Doppler compensation techniques exist (compensating at a reference point within the cell), they cannot eliminate the residual (differential) Doppler shift experienced by users at different locations within the same beam due to varying elevation angles and distances. This residual shift causes a loss of subcarrier orthogonality, leading to Inter-Carrier Interference (ICI). The paper addresses the lack of system-level analyses that quantify how this geometry-induced residual Doppler affects fundamental performance metrics, specifically coverage probability, in LEO-terrestrial networks.

2. Methodology

The authors propose a novel stochastic geometry-based approach to model and analyze the system performance.

System Model:
- A single-beam LEO satellite serves a circular cell on the Earth's surface (approximated as flat for analytical tractability).
- Ground terminals are uniformly distributed within the cell.
- Doppler Modeling: The Doppler shift is derived based on the relative geometry between the satellite and the user. The total shift is decomposed into a common component (compensated at the cell center) and a residual component ( $\Delta \delta_t$ ).
- Channel Model: Assumes a dominant Line-of-Sight (LoS) link, neglecting multipath effects, which is valid for suburban/rural scenarios with minimum elevation angles ( $\alpha_{min} = 10^\circ$ ).
Signal Analysis:
- The paper adapts an existing OFDM model to treat residual Doppler as a user-dependent frequency mismatch.
- It derives expressions for useful signal power ( $\Omega$ ) and ICI ( $I$ ) based on the sinc-squared function of the residual Doppler shift and OFDM symbol duration.
- The Signal-to-Interference-plus-Noise Ratio (SINR) is formulated by incorporating the ICI caused by the residual Doppler.
Coverage Probability Derivation:
- The authors derive a closed-form expression for the coverage probability ( $P_{cov}$ ).
- This derivation explicitly accounts for the spatial distribution of users, the residual Doppler shift, and the resulting ICI.
- The analysis utilizes the Cumulative Distribution Function (CDF) of the horizontal distance between the user and the satellite's projection.

3. Key Contributions

Analytical Framework: The paper provides the first system-level analytical formulation of coverage probability for LEO-terrestrial OFDMA systems that explicitly incorporates geometry-induced residual Doppler and its resulting ICI.
Closed-Form Expression: A tractable, closed-form expression for coverage probability is derived, allowing for rapid evaluation without exhaustive simulations.
Doppler Characterization: The study accurately models the residual Doppler shift after common compensation, distinguishing it from the total Doppler shift and quantifying its impact on subcarrier orthogonality.
Validation: The theoretical results are rigorously validated against extensive Monte-Carlo simulations for both S-band and Ka-band scenarios.

4. Key Results

The numerical results and simulations reveal several critical insights:

Impact of Residual Doppler: Even after compensating for the common Doppler component, the residual Doppler shift significantly degrades coverage probability compared to an ideal, fully compensated scenario. This degradation is more severe in higher frequency bands (e.g., Ka-band) due to larger absolute frequency shifts.
Subcarrier Spacing: Increasing the OFDM subcarrier spacing reduces the impact of residual Doppler. A wider spacing makes the system more robust to frequency variations, thereby reducing ICI and improving coverage.
Cell Size (Beamwidth): As the cell radius (determined by the antenna's Half-Power Beamwidth) increases, the variation in residual Doppler across users becomes more pronounced, leading to a decrease in coverage probability.
Satellite Altitude and Geometry:
- Altitude: Increasing satellite altitude reduces the velocity component along the Line-of-Sight (reducing Doppler) but increases path loss. The path loss eventually dominates, causing a decline in coverage at very high altitudes.
- Horizontal Distance: Counter-intuitively, increasing the horizontal distance between the cell center and the satellite's sub-point (at a fixed altitude) can improve coverage. This is because the reduction in residual Doppler distortion (and thus ICI) outweighs the penalty of increased path loss.

5. Significance

This work is significant for the design of future 5G/6G Non-Terrestrial Networks (NTN) integrating LEO satellites. It demonstrates that residual Doppler is a critical impairment that cannot be ignored, even with standard compensation techniques.

The findings suggest that network designers must:

Carefully select subcarrier spacing to balance spectral efficiency against Doppler robustness.
Optimize antenna beamwidths to limit the spatial variation of Doppler shifts within a cell.
Consider satellite altitude and orbital geometry not just for path loss, but for their impact on Doppler-induced interference.

By providing an accurate analytical tool, this paper enables more efficient resource allocation and link budget planning for LEO satellite-terrestrial networks.