On the Alignment Between Spectral Efficiency and Positioning Accuracy in Multi-LEO Clusters

I. INTRODUCTION

• Dense LEO constellations support cooperative satellite clusters that can improve communication performance and localization accuracy.The introduction cites global coverage and spatial diversity as enabling opportunities for cooperative operation.Source: §I · p. 1
• Whether communication-optimal multi-LEO clusters also favor positioning remains open, motivating study of this relationship for future ISAC-enabled NTNs.Prior work explored cooperative multi-satellite communication and LEO positioning separately, while communication and positioning criteria may favor different satellite selections.Source: §I · pp. 1–2
• SE aligns more strongly with Doppler-based positioning than with TOA-based positioning, whose alignment is weaker and time-varying.The study investigates this relationship using an uplink D2D scenario with joint satellite processing and derives both TOA- and Doppler-based PEBs.Source: §I · p. 2

II. SYSTEM MODEL

• The model has a single UT transmit OFDM symbols over dedicated subcarriers to satellites with known locations and velocities, using Rician links.The UT has one omnidirectional antenna; satellites use directional antennas.Source: §II · p. 2
• The model assumes a static UT, synchronized clocks, compensated carrier frequency offsets, and no ionospheric or tropospheric effects.These assumptions isolate the effects of satellite geometry and cluster selection.Source: §II · p. 2

III. COMMUNICATION AND POSITIONING WITH LEO CLUSTERS

• The section formulates communication and positioning models for LEO clusters, then presents their respective performance metrics.Source: §III · p. 2

A. Communication model

• The central node uses each satellite's received signal and instantaneous CSI estimates for joint symbol detection.The satellites form a cluster using THz ISLs and forward their received signals to the central node.Source: A. Communication model · pp. 2–3
• The ISL model specifies transmit power, antenna gains, satellite separation, frequency, and receiver noise.Source: A. Communication model · p. 3

B. Localization Model

• The model derives PEBs for separate Doppler- and TOA-based localization methods from the parametric signal model's Fisher Information Matrix.Each satellite observes pilot symbols over the specified subcarriers.Source: B. Localization Model · p. 3
• The pilot-observation vector is CSCG, with independent observations for different satellite positions; its mean and covariance are specified.Source: B. Localization Model · p. 3
• The FIM includes UT coordinates and complex channel gains as nuisance parameters; its channel-derivative terms represent power, TOA, and Doppler information.Power-based location information is negligible in practice relative to TOA and Doppler; the model isolates either positioning method by zeroing the other method's term.Source: B. Localization Model · p. 3

• SE and PEB serve as the respective communication and positioning performance metrics for each satellite cluster.Source: C. Metrics for correlation analysis · p. 3
• The analysis calculates the Fisher information matrix for both TOA and Doppler measurements individually.Source: C. Metrics for correlation analysis · p. 3
• WPDOP and WVDOP are used as quality indicators to examine correlations with the two performance metrics.Source: C. Metrics for correlation analysis · p. 3

1) WPDOP:

• With receiver clock bias known, the position has three unknowns, and the geometry matrix collects visible satellites’ line-of-sight unit vectors.Source: 1) WPDOP · p. 3
• Each measurement is weighted by its link SNR, and the weight matrix is diagonal.The passage relates this choice to the inverse proportionality between time-delay estimation variance and SNR.Source: 1) WPDOP · p. 3
• A scaling matrix may be used to improve numerical stability, while the resulting PEB remains invariant to the transformation.Source: 1) WPDOP · p. 3

2) WVDOP:

• Each vector h_i represents the rate of change of satellite i’s line-of-sight unit vector.Source: 2) WVDOP:  · p. 4
• The Doppler-positioning geometry matrix HDop collects the h_i vectors as rows.Source: 2) WVDOP:  · p. 4
• The corresponding velocity dilution-of-precision measure is defined as WVDOP.Source: 2) WVDOP:  · p. 4

IV. SIMULATION SETUP AND RESULTS

• The section presents the simulation setup and analyzes communication-positioning correlations in LEO clusters.Source: §IV · p. 4

A. Orbit and signal setup

• The modeled first Starlink shell has 1,584 satellites at 530 km altitude and 53° inclination, with the user at 40°N, 74°W.Satellites above 15° elevation are considered; the first shell is described as the largest fully deployed Starlink shell.Source: A. Orbit and signal setup · p. 4
• The uplink uses 2 GHz S-band, while inter-satellite links use 1 THz; the OFDM signal has 60 MHz bandwidth and 64 subcarriers.Instantaneous CSI is assumed, and the channel is Rician.Source: A. Orbit and signal setup · p. 4
• Figure 2 tracks SE, TOA-PEB, Doppler-PEB, and mean elevation over time for clusters of 4, 5, 6, and 7 satellites.Table II lists the number of clusters per satellite count.Source: Figure 2 · p. 4, Table II · p. 4

B. Cluster forming and selection

• The analysis uses 15 satellites visible at every 10-second timestep, forms all combinations of N=4, 5, 6, and 7, and selects the nearest satellite as the central node.Figure 2 illustrates metric evolution for example clusters.Source: B. Cluster forming and selection · p. 4

C. Time evolution of correlation coefficient

• −0.05 to −0.53 for Pearson SE correlations with TOA-PEB and WPDOP, versus −0.26 to −0.76 with Doppler-PEB and WVDOP.The correlations are calculated over time for different cluster sizes.Source: Figure 3 · p. 5, C. Time evolution of correlation… · p. 5
• −0.4 to −0.82 for SE–Doppler-PEB versus −0.37 to −0.55 for SE–TOA-PEB; Spearman correlations vary less across cluster sizes.The stronger Doppler association is described as a robust monotonic relationship.Source: Figure 4 · p. 5, C. Time evolution of correlation… · p. 6
• Pearson coefficients measure linear association, while Spearman coefficients measure rank and monotonic association; the metrics are computed at each timestep and cluster size.The passage motivates timewise analysis by changes in satellite positions and velocities.Source: C. Time evolution of correlation… · p. 5
• For TOA, SE correlation strengthens with cluster size to −0.53; its correlation differs from WPDOP by at most 0.06 because both use LoS geometry.For four-satellite clusters, the best geometric diversity does not align with the clusters maximizing SE.Source: C. Time evolution of correlation… · p. 5
• Doppler-PEB depends on velocity diversity and perpendicular-to-LoS speed; its minimum absolute Pearson correlation rises from 0.26 at N=4 to 0.56 at N=7.For N>5, the reported linear negative relation between SE and Doppler PEB has |ρ| > 0.5.Source: C. Time evolution of correlation… · p. 5

D. Statistical comparison

• For SE–Doppler-PEB, mean |ρ| rises from 0.47 (N=4) to 0.72 (N=7), while its lead over SE–TOA-PEB grows from 0.08 to 0.26.The SE–Doppler-PEB correlation standard deviation remains 0.05 across cluster sizes; SE–TOA-PEB rises from 0.03 (N=4) to 0.05 (N=7).Source: D. Statistical comparison · p. 6, Figure 5 · p. 6

V. CONCLUSION

• The study evaluated SE alongside TOA- and Doppler-based PEBs in uplink D2D clusters through realistic LEO constellation simulations.Source: §V · p. 6
• In the Starlink-like scenario, SE-maximizing clusters tended to align with Doppler-PEB-minimizing clusters, whereas TOA alignment was weaker and more geometry-dependent.Source: §V · p. 6