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

On the Alignment Between Spectral Efficiency and Positioning Accuracy in Multi-LEO Clusters Konstantinos Kapoglis1 , Sharief Saleh2 , Henk Wymeersch2 , and Hazem Sallouha1 1 Department of Electrical Engineering (ESAT) - WaveCoRE, KU Leuven, 3000 Leuven, Belgium 2 Department of Electrical Engineering, Chalmers University of Technology, 412 58 Gothenburg, Sweden

arXiv:2610.13526v1 [cs.NI] 9 Oct 2026

E-mail: konstantinos.kapoglis@kuleuven.be Abstract—Low Earth Orbit (LEO) satellite constellations are expected to play a key role in future 6G Non-Terrestrial Networks (NTNs), where dense deployments enable cooperative multisatellite processing for communication and positioning. While existing works have addressed communication and localization objectives separately, the correlation and alignment between these objectives remain largely unexplored. In this paper, we investigate the correlation between Spectral Efficiency (SE) and positioning accuracy in cooperative multi-LEO satellite clusters. We consider an uplink direct-to-device scenario in which multiple satellites jointly process user signals and evaluate cluster performance using SE and Position Error Bound (PEB) metrics. We derive Timeof-Arrival (TOA)-based and Doppler-based PEBs, while communication performance is quantified through the achievable SE under joint signal processing. Using realistic LEO constellation simulations, we demonstrate that for clusters of seven satellites, SE exhibits a strong negative correlation with Doppler-based PEB (correlation coefficient of −0.72). In contrast, its correlation with TOA-based PEB is notably weaker, ranging between −0.55 and −0.37. These findings suggest that communication-oriented LEO clusters are largely compatible with Doppler-based positioning, but not necessarily with TOA-based positioning. Index Terms—LEO satellite constellations, NTN, ISAC, Direct to Device, Localization, Doppler, Time of Arrival

I. I NTRODUCTION The rapid emergence of Non-Terrestrial Networks (NTNs) has positioned Low Earth Orbit (LEO) satellite constellations as a key component of future 6G wireless systems [1]. Owing to their low orbital altitude, large constellation sizes, and global coverage capabilities, LEO satellites can complement terrestrial infrastructure by providing connectivity in underserved and remote regions while enabling truly ubiquitous Direct-to-Device (D2D) communication and positioning services [2,3]. The dense spatial diversity offered by modern mega-constellations, such as Starlink, Kuiper, and OneWeb, creates new opportunities for cooperative operation among multiple satellites where clusters of LEO satellites can exploit their spatial resources to improve communication performance [2] and localization accuracy [4]. From a communications perspective, existing research works explored the exploitation of large antenna arrays, which can be co-located or distributed [5] to overcome the severe link-budget limitations associated with D2D satellite communications. Inspired by the success of distributed multiple-input This work was supported by the SwarmNET project under KU Leuven C1 grant No. C14/24/093.

multiple-output (MIMO) and cell-free massive MIMO architectures in terrestrial networks [5], several recent works have proposed collaborative transmission and reception schemes in which multiple LEO satellites jointly process user signals [6]. Enabled by high-capacity inter-satellite links (ISLs), such cooperative satellite clusters effectively form a distributed antenna array in space, increasing array gain, improving spectral efficiency (SE), and extending coverage [7]. In [8], the authors proposed a distributed massive MIMO architecture using LEO satellites and formulated a resource optimization problem for maximizing the downlink throughput. In [6], the authors investigated a D2D uplink scenario where multiple satellites receive the transmitted signal from a handheld device and jointly process the signal in a central node. The impact of ISLs required to support distributed signal processing across multiple satellite clusters has been investigated in [7]. However, these existing works on multi-LEO clusters for D2D communication leave the question of whether communicationoptimal clusters are also favorable for positioning open. LEO constellations are increasingly being investigated as a platform for positioning, navigation, and timing (PNT) services [3,4]. Compared to Medium Earth Orbit (MEO) GNSS satellites, the significantly lower altitude and higher orbital velocity of LEO satellites provide stronger received signals and larger geometric diversity. Consequently, positioning techniques based on Time of Arrival (TOA), Doppler measurements, or a combination of both have received considerable attention. In particular, Doppler-based positioning has emerged as a promising approach due to the large Doppler shifts induced by the high relative velocity of LEO satellites [9]– [11], enabling accurate localization even when precise timing information is unavailable. In [12], the authors extract the theoretical Position Error Bounds (PEB) of differential Dopplerbased positioning using a parametric received signal model and compare them with the classical method. The convergence of communication and positioning functionalities has recently fueled interest in Integrated Sensing and Communication (ISAC) frameworks for terrestrial networks, but remains in its infancy for NTNs [13]. While existing studies have demonstrated the potential of LEO constellations to support high-capacity communications and accurate localization, communication and positioning objectives are often optimized using different criteria and may favor different satellite selections. In particular, to the best of our knowledge, the relationship between communication performance and positioning accuracy with

LEO clusters in currently deployed LEO constellations is an open research question. Understanding this relationship, which is essential for the design of future ISAC-enabled NTNs, motivates our work in this paper. In this paper, we investigate the relationship between SE and positioning accuracy in cooperative multi-LEO satellite clusters. We consider an uplink direct-to-device scenario in which multiple satellites jointly process user transmissions through distributed reception enabled by THz ISLs. To jointly evaluate communication and positioning performance, first, we formulate the achievable SE and derive theoretical PEBs for both Doppler- and TOA-based localization in cooperative satellite clusters. Subsequently, using a realistic Starlink constellation model, we evaluate all visible satellite clusters and analyze the correlation between communication- and positioningoriented performance metrics as a function of cluster size. To capture both linear and monotonic correlations, we analyze both Pearson and Spearman correlations. Our results reveal a strong alignment between SE and Doppler-based positioning performance, indicating that communication- and Dopplerpositioning objectives generally improve together. In contrast, TOA-based positioning exhibits a weaker and time-varying correlation coefficient, suggesting that communication-optimal clusters are not necessarily optimal for TOA localization. II. S YSTEM MODEL We consider an NTN consisting of a single user terminal (UT) with unknown position p = [𝑥, 𝑦, 𝑧] ⊤ , velocity v = [𝑣 𝑥 , 𝑣 𝑦 , 𝑣 𝑧 ] ⊤ and 𝐼 LEO satellites with known location p𝑖 = [𝑥𝑖 , 𝑦 𝑖 , 𝑧𝑖 ] ⊤ and velocity v𝑖 = [𝑣 𝑥𝑖 , 𝑣 𝑦𝑖 , 𝑣 𝑧𝑖 ] ⊤ , as illustrated in Figure 1. The UT is assumed to have a single omnidirectional antenna, while the LEO satellites use directional antennas. The UT sends its signal to all satellites simultaneously. We consider a scenario in which the UT sends 𝐿 orthogonal frequencydivision multiplexed (OFDM) symbols using a dedicated set of 𝐾 subcarriers with a subcarrier spacing Δ 𝑓 and a central frequency 𝑓𝑐 . The OFDM symbols duration is 𝑇sym = 𝑇 + 𝑇cp . 𝑇 = 1/Δ 𝑓 is the actual symbol duration while 𝑇cp is the cyclic prefix. We consider the UT transmitting a symbol 𝑠 to satellite 𝑖 over a Rician channel, yielding the received signal [7] √︃ √︃ 𝜅 1 ℎ𝑖𝑘,𝑙 𝑠 + 1+𝜅 ℎNL (1) 𝑦 𝑖 = 1+𝜅 𝑖 𝑠 + 𝜂𝑖 , where 𝑖 ∈ {1, . . . , 𝐼}, 𝑘 ∈ {0, . . . , 𝐾 −1} and 𝑙 ∈ {0, . . . , 𝐿 −1} denote the satellite, subcarrier, and OFDM symbol indices, respectively, ℎ𝑖𝑘,𝑙 and ℎNL denote the Line of Sight (LoS) and 𝑖 non Line of Sight (NLoS) multipath component, respectively, 𝜂𝑖 ∼ CN (0, 𝜎𝑛2up ) is the noise at the 𝑖th satellite receiver. 𝑖 We assume that all satellites have the same thermal noise distribution with a variance 𝜎𝑛2up . Moreover, in (1), 𝜅 denotes the Rician factor, which depends on the elevation angle √ [14], and 𝑃 is the transmit power of the UT so that 𝑠 = 𝑃. We define the LoS component as [12] ℎ𝑖𝑘,𝑙 = 𝛼𝑖 𝑒 − 𝑗2 𝜋 𝑘Δ 𝑓 ( 𝜏𝑖 −𝜈𝑖 𝑙𝑇sym ) 𝑒 𝑗2 𝜋 𝑓𝑐 𝜈𝑖 𝑙𝑇sym

(2)

CN

UT

Fig. 1: An illustration of the system model with a cluster of LEO satellites communicating with and positioning a ground user terminal.

where 𝛼𝑖 is the complex channel gain with |𝛼𝑖 | = √ 1 , 𝛽up𝑖 = 𝛽upi 2  4 𝜋𝑑𝑖 𝑓c is the free space path loss (FSPL), 𝑑𝑖 is the distance 𝑐 between the 𝑖 th satellite and the UT and 𝑐 is the speed of light. The propagation delay as evaluated at the beginning of the 𝐿symbol observation period is 𝜏𝑖 = 𝑑𝑐𝑖 + 𝛿𝑡𝑖 and 𝛿𝑡𝑖 aggregates delays of clock bias and atmospheric effects. Furthermore, in (2), the Doppler shift factor for each link is expressed as 𝜈𝑖 = ⊤u 𝑖 − (v𝑖 −v) + 𝛿 𝑓𝑖 where u𝑖 = ∥pp𝑖𝑖 −p 𝑐 −p∥ is the unit LoS vector and 𝛿 𝑓𝑖 is the Carrier Frequency Offset (CFO) between the 𝑖 th satellite and the UT. We consider that the NLoS component is a Circularly Symmetric Complex Gaussian (CSCG) variable √︃ NL NL ℎ𝑖 = 𝑃𝑖 𝑔𝑖 where 𝑔𝑖 ∼ CN (0, 1) and 𝑃𝑖NL is its variance. We assume that the UT is static, i.e., v = 0, and that the ionospheric and tropospheric effects are not taken into account. Note that to isolate the effects of satellite geometry and cluster selection, we assume synchronized clocks and compensated carrier frequency offsets. The treatment of these impairments in LEO NTNs is discussed in [12,15]. III. C OMMUNICATION AND P OSITIONING WITH LEO C LUSTERS In this section, we formulate the communication and positioning models for LEO clusters. Subsequently, we present the respective performance metrics. A. Communication model We assume that the satellites to which the UT sends the signal are forming a cluster (cf. Figure 1). Satellites within a given cluster transmit the received signal to a central satellite called the Central Node (CN). The cluster is formed by employing THz ISLs. The CN satellite jointly estimates the received symbols, having knowledge of the estimation of the instantaneous Channel State Information (CSI) of each satellite and the received signal of each satellite. The CN satellite applies joint symbol detection, resulting in a received-signal estimate denoted by 𝑠ˆ = wy𝐶 𝑁 . In a given LEO cluster, the total uplink SE is [7] ! 𝑃 wC2 hh 𝐻 w 𝐻 𝑅 = log 1 + 2 . (3) 𝜎𝑛up wC2 w 𝐻 + w𝚺 𝑛isl w 𝐻 In

(3),

w

=

the estimation vector is given  by −1 √ 𝐻  2 𝐻 𝑃 ĥ C 𝑃C ĥĥ + 𝑃C2 𝚺h̃ + 𝚺 𝑛isl + 𝜎𝑛2up C2 ,

ĥ = [ ℎˆ 1 , . . . , ℎˆ 𝑀 ] ⊤ is the estimated channel vector and h̃ = [ ℎ˜ 1 , . . . , ℎ˜ 𝑀 ] ⊤ is the channel estimation error, i.e., h = ĥ − h̃. 𝜎𝑛2up is the uplink noise power. The matrix C  √ 1 1 is defined as C = diag 𝑃isl ( 𝜷up ) 2 ◦ ( 𝜷isl ) − 2 where 𝑃isl is the transmit power in the ISL, 𝐺 Tisl is the THz transmit antenna gain, 𝐺 Risl is the THz receive antenna gain. Finally, ⊤ ⊤ 𝜷up = 𝛽up1 , . . . , 𝛽upI ∈ R𝐼 , 𝜷isl = 𝛽isl1 , . . . , 𝛽isl𝐼 ∈ R𝐼 ,  ⊤ 𝐼 nisl = 𝑛isl1 , . . . , 𝑛islI ∈ C represent the uplink FSPL, ISL FSPL and ISL noise, respectively. 𝛽isl𝑛𝑛˜ is the FSPL between 2  4𝜋 𝑑 isl𝑛 𝑛˜ 𝑓isl where satellites 𝑛 and 𝑛, ˜ calculated as 𝛽isl𝑛𝑛˜ = 𝑐 𝑑isl𝑛𝑛˜ is the distance between satellites 𝑚 and 𝑛 in meters, and 𝑓isl is the ISL frequency in Hz, 𝑛isl𝑛 is the noise signal  2 at the 𝑛th satellite, modeled as 𝑛isl𝑛 ∼ CN 0, 𝜎nisl . B. Localization Model This section considers UT localization using LEO clusters combining observations from individual satellites. We consider two separate localization methods, namely Doppler shift and TOA. To calculate the PEBs of the proposed positioning system, we construct the Fisher Information Matrix (FIM) from the parametric signal model in (2), capturing the relationship between UT location and the channel parameters, namely, Doppler shift and signal delay. We assume that each of the 𝐼 satellites observes 𝐿 pilot symbols transmitted by the UT over 𝐾 subcarriers. h We define the i ⊤ vector of LoS channel 𝑘,𝑙 𝑘,𝑙 𝑘,𝑙 components h ≜ ℎ1 · · · ℎ 𝐼 , the vector of variances  NL ⊤ NL of the NLoS component p ≜ 𝑃1 · · · 𝑃NL and the 𝐼 vector of received pilot observations y ≜ [𝑦 1 , · · · , 𝑦 𝐼 ] ⊤ . Since 𝑔𝑖 ∼ CN (0, 1) and 𝜂𝑖 ∼ CN (0, 𝜎𝑛2up ) are independent across p𝑖 , it follows that [16]  √︃ 𝑃𝑖NL 𝑃 2 𝜅 𝑃 𝑘,𝑙 ℎ , 𝜎 + , (4) 𝑦 𝑖 ∼ CN 𝑛up 1+𝜅 𝑖 1+𝜅 Thus, the observation vector y is a CSCG random vector. Note that in (4) 𝑦 𝑖 ’s are independent for different p𝑖 . The mean vector and the covariance matrix of y are, respectively √︂

𝜅𝑃 𝑘,𝑙 diag{pNL } 𝑃 h , Ry = 𝑁 0 I 𝐼 + , (5) 1+𝜅 1+𝜅 We consider a position-domain vector denoted by 𝝃 = [p⊤ , ℜ{𝛼1 }, ℑ{𝛼1 }, . . . , ℜ{𝛼 𝑁 }, ℑ{𝛼 𝑁 }] ⊤ ∈ R3+2𝑁 , where ℜ{𝛼𝑖 } and ℑ{𝛼𝑖 } are nuisance parameters representing, respectively, the real and imaginary parts of the channel gain between UT and the 𝑖th satellite. The (𝑛, 𝑚)th element of the FIM, denoted by Jp ∈ R (3+2𝑁 ) × (3+2𝑁 ) , is given by " ( )  𝐿−1 −1 ∑︁ 𝐾∑︁ 𝜕my 𝐻 −1 𝜕my Ry [Jp ] 𝑛,𝑚 = 2R 𝜕 [𝝃] 𝑛 𝜕 [𝝃] 𝑚 𝑙=0 𝑘=0 (6)  # −1 𝜕Ry −1 𝜕Ry + tr Ry R , 𝜕 [𝝃] 𝑛 y 𝜕 [𝝃] 𝑚 my =

where [𝝃] 1:3 represent the UT coordinates 𝑥, 𝑦, 𝑧, respectively, and tr{.} denotes the trace of a matrix. The derivative of the LoS channel with respect to position is given by

" 𝜕ℎ𝑖𝑘,𝑙 2𝜋𝑘Δ 𝑓 𝑘,𝑙 p𝑖 − p = ℎ𝑖 +𝑗 (p𝑖 − p) 𝜕p 𝑐 𝑑𝑖 𝑑𝑖2 | {z } | {z } Term 2

Term 1

#
2𝜋𝑙𝑇sym (𝑘Δ 𝑓 + 𝑓𝑐 ) +𝑗 v𝑖 − (v⊤ . 𝑖 u𝑖 )u𝑖 𝑐 𝑑𝑖 | {z }

(7)

Term 3

Three terms can be identified in the derivative’s expression. The first term reflects the information acquired from power measurements. The second term reflects the information acquired from TOA measurements. The third term reflects the information acquired from the Doppler measurements. The term v𝑖 − (v⊤ 𝑖 u𝑖 )u𝑖 is the component of the velocity perpendicular to the LoS. It is worth noting that in practice, the power-based measurement in term 1 provides negligible location information compared to terms 2 and 3. Therefore, for example, to compute the FIM for only TOA-based positioning, we may set term 3 to zero, and similarly, to compute the FIM for only Doppler-based positioning, we set term 2 to zero. Here, we focus on LoS-based positioning and therefore only consider location information extracted from the LoS component. Under this assumption, the NLoS parameters are treated 𝜕R as nuisance parameters, i.e., 𝜕 𝜉𝑛y = 0 and hence, the trace term √︃  in (6) is zero. We then compute1 the PEB = tr [Jp −1 ] 1:3,1:3 . C. Metrics for correlation analysis To investigate the correlation between communication and positioning performance across satellite clusters, we use spectral efficiency (SE) and the Position Error Bound (PEB) as the respective performance metrics. For a cluster of LEO satellites, we calculate the SE denoted by 𝑅 in 3 and the FIM (6) for both cases, TOA and Doppler measurements individually. Having these two parameters as metrics to calculate the communication and localization efficiency of each cluster, we use some variables as quality indicators and examine their correlation with our two metrics. More specifically, we adopt Weighted Position Dilution of Precision (WPDOP) and Weighted Velocity Dilution of Precision (WVDOP). 1) WPDOP: Assuming the receiver clock bias is known, there are three unknowns (𝑥, 𝑦, 𝑧). The geometry matrix collects  the LoS⊤ unit vectors  across all 𝐼 visible satellites ⊤ ⊤ . The received signal-to-noise H = − u⊤ ; −u ; · · · ; −u 𝐼 1 2 (SNR) ratio for the 𝑖-th link is SNR𝑖 = 𝛽 𝑃𝜎 2 . Since the up𝑖

𝑛up

ranging variance follows 𝜎𝑖2 ∝ 1/SNR𝑖 from the Cramér– Rao bound on time-delay estimation, the measurement weight is taken as 𝑤 𝑖 = SNR𝑖 and the weight matrix is W = diag(𝑤 1 , 𝑤 2 , . . . , 𝑤 𝑛 ). The weighted cofactor matrix and the corresponding weighted position dilution of precision are then √︁
−1 (8) Q𝑊 = H⊤ WH , WPDOP = tr(Q𝑊 ). 1 To ensure numerical stability, a scaling matrix D may be introduced such that J̃p = D⊤ Jp D. Matrix inversion is then performed on J̃p . The resulting PEB is invariant to this transformation.

TABLE I: Simulation Parameters

TABLE II: Number of Clusters per Satellite Count

Value

Parameter

Value

#Satellites #Sat./orbital plane Altitude Inclination Phasing Min. elevation angle Carrier freq. 𝑓 𝑐 Subcarr. spacing Δ 𝑓 # Subcarriers 𝐾 # OFDM symbols 𝐿 Bandwidth 𝐵 = 𝐾 Δ 𝑓

1584 72 530 km 53◦ 39 15◦ 2 GHz 937.5 KHz 64 252 60 MHz

UT Tx Power UT Gain UT Rx Gain ISL freq ISL Tx power ISL Tx Gain ISL Rx gain ISL BW ISL Noise T. Up. Noise T. UT location

-6 dBW 5 dBi 35 dBi 1 THz 5 dBW 60 dBi 60 dBi 20 GHz 7000 K 290 K 41◦ N, 74◦ W

Number of satellites

Total number of clusters

4 5 6 7

1365 3003 5005 6435

4 SE [bps/Hz]

Parameter

3.8 3.6 3.4

𝑁 =4 𝑁 =6

0.03

0.02 500 400 300 200 60

(9)

IV. S IMULATION SETUP AND RESULTS In this section, we present the simulation setup and analyze communication-positioning correlations in LEO clusters. A. Orbit and signal setup To quantify the communication and positioning performance of cooperative LEO clusters, we simulate a realistic satellite scenario using the MATLAB Satellite Communications Toolbox. Specifically, we consider the first shell of the Starlink constellation, which consists of 1584 satellites deployed at an altitude of 530 km with an inclination of 53◦ . This shell is selected because it is the largest fully deployed Starlink shell and has been widely adopted as a representative LEO constellation in the literature [7]. The UT’s position, 40◦ N, 74◦ W, lies within the high-density coverage of the selected Starlink shell. We consider an uplink frequency in the S-band at 2 GHz. The other parameters for the UT to satellite and ISLs follow [7] and are detailed in Table I. We assume that the UT sends the uplink signal to all satellites with an elevation angle above 15 degrees with respect to the UT. These satellites can be divided into clusters with 𝑁 satellites per cluster (cf. Table II). In addition, we consider an ISL frequency of 1 THz and assume that instantaneous CSIs are available. We consider a Rician channel with 𝜅 of a suburban environment [14]. We considered an OFDM signal with a bandwidth of 60 MHz and a total number of 64 subcarriers with a spacing of 937.5 kHz. The total symbol duration time is 𝑇sym = 1.067𝜇𝑠. A summary of orbit and signal parameters is presented in Table I.

Mean Angle [°]

√︄    −1  ⊤ WVDOP = tr HDop WHDop

PEB Doppler [m]

2) WVDOP: : The rate of change of the unit vector in LoS direction of the satellite 𝑖 is given by h𝑖 =  1 1 ⊤u ) u v = v − (v where h𝑖 is the 𝑖-th row 𝑖 𝑖 𝑖 𝑖 𝑑𝑖 ⊥,𝑖 𝑑𝑖 of the Doppler-positioning geometry matrix. The matrix  − that collects all vectors is defined as HDop = ⊤ 1 ⊤ 1 ⊤ 1 ⊤ − 𝑑2 v⊥,2 ; · · · ; − 𝑑𝑛 v⊥,𝑛 . The corresponding 𝑑1 v⊥,1 ; Weighted Velocity Dilution of Precision is defined as

PEB TOA [m]

3.2

𝑁 =5 𝑁 =7

50 40 30

0

20

40

60

80

100

120

140

Time [sec] Fig. 2: Evolution of (top to bottom) SE, TOA-PEB, Doppler-PEB, and mean elevation angle over time for clusters with 𝑁 = 4, 5, 6 and 7.

B. Cluster forming and selection To ensure a consistent analysis over time, the visible satellites to the UT are stored at 10-second timesteps, and we choose to observe only the subset that are visible to the UT at all timesteps. Accordingly, we have a subset of 15 satellites that are always visible to the UT throughout our simulation time frame. For each timestep, the Earth-Centered, Earth-Fixed (ECEF) position and velocity of the satellite are also stored in order to calculate the respective Doppler shift and TOA. For each timestep, we form all combinations of the 15 satellites that contain 𝑁 = 4, 5, 6, and 7 satellites. The total number of combinations is shown in Table II. The satellite that is closer to the user is chosen as the CN satellite. For each cluster, we compute the WPDOP and WVDOP defined in Section III-C along with the SE and PEBs. Figure 2 shows, for example clusters with different sizes the evolution over time of SE, TOA-based PEB, Doppler-based PEB, and the average elevation angle between the satellites in the corresponding cluster and the UT. While the Doppler-based PEB exhibits a negative correlation with SE, the TOA-based PEB exhibits a different behavior, which does not follow the trend of

Correlation coefficient (𝜌)

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

𝑁 =4

0

50

100

150

0

Time [sec] (a)

50

100

𝑁 =5

150

0

Time [sec] (b)

𝑁 =6

50

𝑁 =7

100

150

0

Time [sec] (c)

50

100

150

Time [sec] (d)

Correlation coefficient (𝜌)

Fig. 3: Pearson correlation coefficient 𝜌 between SE and (a) WPDOP, (b) TOA-PEB, (c) Doppler-PEB, and (d) WVDOP over time for four cluster sizes. 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

𝑁 =4

0

50

100

150

0

Time [sec] (a)

50

100

Time [sec] (b)

𝑁 =5

150

0

𝑁 =6

50

𝑁 =7

100

Time [sec] (c)

150

0

50

100

150

Time [sec] (d)

Fig. 4: Spearman correlation coefficient 𝜌 between SE and (a) WPDOP, (b) TOA-PEB, (c) Doppler-PEB, and (d) WVDOP over time for four cluster sizes.

the SE nor the Doppler PEB curves. To further investigate this behavior, we present Pearson and Spearman correlation analyses in the following subsections. C. Time evolution of correlation coefficient To examine the relation between the SE and PEBs, we compute Pearson and Spearman correlation coefficients for each timestep and cluster size, separately. The Pearson correlation coefficient shows the strength of a linear relation, while the Spearman correlation coefficient shows the rank and monotonic relation. The strength of the monotonic relation reveals whether one of these variables can serve as an indicator of the other. Analyzing such correlations over time is crucial, as the satellites’ positions and velocities change at each timestep; thus, the considered metrics change, as they are position- and velocity-dependent. Finding a strong, stable correlation across different timesteps indicates a robust relation. In Figure 3a and Figure 3b, we observe that the Pearson correlation coefficient of SE with TOA-based PEB and WPDOP varies between −0.05 and −0.53 for different timestamps, while the same coefficient of SE with Doppler-based PEB, in Figure 3c, and with WVDOP, Figure 3d, varies between −0.26 and −0.76 for different numbers of satellites. This observation suggests that TOA-based PEB is influenced by the geometric diversity of the LoS vectors. In the case of Figure 3a, Figure 3b, for 𝑁 = 4 satellites, the clusters that provide the best geometric diversity do not align with the clusters that maximize the SE, which are the ones closest to the user because of the lower FSPL and better multipath conditions. As

the number of satellites increases, the correlation coefficient increases as well, because the distance of the satellites from the UT influences the linear relationship, which reaches a maximum negative value of −0.53. The maximum difference at the same time and cluster size between SE–PEB-TOA and SE–WPDOP correlations is 0.06, which is attributed to PEBTOA being based on unit LoS vectors, similar to the WPDOP. The Doppler-based PEB is affected by the diversity of the velocity vectors and the satellites’ velocity component perpendicular to the LoS direction, which is maximized when the satellite has the maximum elevation angle from the user’s perspective. It can be seen that the difference of the Pearson correlation coefficient in Figure 3c and Figure 3d increases with the number of satellites, with a maximum difference of 0.19 between 𝑁 = 4 and 𝑁 = 5 satellites and approximately 0.1 between 𝑁 = 6 and 𝑁 = 7. satellites. For 𝑁 = 4 satellites, the minimum absolute value of 𝜌 is 0.26, while for 𝑁 = 7 satellites, it is 0.56. With fewer satellites, the velocity diversity affects the correlation. However, as the cluster size increases, the distance of the satellites from the user has a more pronounced impact on the correlation. The maximum difference for the same timestamps, with 𝑁 = 4, in the correlation between SE and PEB-Doppler and between SE and SE-WVDOP is 0.15, which is explained by the fact that both Doppler-based PEB and WVDOP are constructed from the velocity component perpendicular to the LoS. Accordingly, it can be seen that there is a strong linear negative relation (|𝜌| > 0.5) between SE and PEB in the case of joint communication and Doppler-based positioning for 𝑁 > 5

D. Statistical comparison To assess temporal stability, we compute the mean and standard deviation of the correlation coefficient over time, where the mean captures average strength and the standard deviation measures variability. We focus on the Spearman correlation, as the previous section indicates a stronger monotonic than linear relationship between the considered metrics. Figure 5 shows the SE Spearman correlation coefficient statistics, namely mean and standard deviation, across all timestamps for different numbers of satellites per cluster. The figure shows that the mean Spearman correlation coefficient for SE-Doppler-PEB is higher, and its variance is comparable, when compared to that of SE-TOA-PEB. The difference in the mean in the case of SE-Doppler-PEB compared to SE-TOA-PEB is 0.08 in the case of 𝑁 = 4 satellites, while it reaches 0.26 in the case of 𝑁 = 7, indicating an increasing monotonic relationship with 𝑁. Furthermore, for SE-WPDOP and SE-TOA-PEB correlations, the total number of satellites in the cluster has little impact, since the mean correlation coefficient doesn’t vary more than 0.01 between different numbers of satellites, whereas for SE-WVDOP and SE-Doppler-PEB, the correlation becomes stronger, as the mean value of |𝜌| increases from 0.47 to 0.72 when 𝑁 increases from 4 to 7. The standard deviation of the SE–Doppler-PEB Spearman correlation remains low and nearly constant at 0.05 across all cluster sizes. Similarly, the standard deviation for the SE–TOA-PEB correlation is small, increasing slightly from 0.03 for (𝑁 = 4) to 0.05 for (𝑁 = 7). These results indicate a robust monotonic relationship, particularly for Doppler-PEB. V. C ONCLUSION This paper investigated the relationship between SE and positioning accuracy in cooperative multi-LEO satellite clusters. Considering an uplink D2D scenario. We evaluated cluster performance using SE together with TOA- and Doppler-based PEBs. Through simulations of a realistic LEO constellation,

1 Correlation coefficient (𝜌)

satellites, whereas this correlation is weaker (|𝜌| < 0.5) in the case of joint communication and TOA-based positioning. This indicates that the correlation is more pronounced in joint communication with Doppler-based positioning than with TOA-based positioning. As shown in Figure 4a and Figure 4b, the Spearman correlation coefficient between SE and TOAPEB and between SE and WPDOP varies between −0.37 and −0.55 while they show relatively small differences of less than 0.1, for different numbers of satellites for the same time indexes. In Figure 4c and Figure 4d, the same coefficient between SE and Doppler-PEB and between SE and WVDOP ranges between −0.4 and −0.82. The absolute minimum value is higher by 0.14 compared to the absolute minimum value of the corresponding Pearson coefficient and by 0.04 from the absolute maximum value, which indicates that there is a robust monotonic relationship. Furthermore, the Spearman correlation shows smaller variation across different numbers of satellites, indicating that cluster size has less impact on the monotonic relationship than on the linear one.

𝑁 =4 𝑁 =5 𝑁 =6 𝑁 =7

0.5 0 −0.5 −1

P

SE-

DO WP

PEB SE-

TOA

pler

PEB SE-

Dop

P

DO WV

SE-

Fig. 5: Statistics of Spearman correlation with SE (mean ± std over time) for different cluster sizes.

we analyzed the extent to which communication-oriented and positioning-oriented cluster selection objectives are correlated. In the considered Starlink-like scenario, SE-maximizing clusters tended to align with Doppler-based PEB minimizing clusters, while TOA-based PEB exhibited weaker and more geometry-dependent alignment. R EFERENCES [1] M. M. Azari et al., “Evolution of Non-Terrestrial Networks From 5G to 6G: A Survey,” IEEE Com. Surveys & Tutorials, vol. 24, no. 4, 2022. [2] Z. M. Bakhsh et al., “Multi-Satellite MIMO Systems for Direct Satelliteto-Device Communications: A Survey,” IEEE Communications Surveys & Tutorials, vol. 27, no. 3, 2025. [3] S. Saleh et al., “Integrated 6G TN and NTN Localization: Challenges, Opportunities, and Advancements,” IEEE Communications Standards Magazine, vol. 9, no. 2, 2025. [4] H. Sallouha et al., “On the Ground and in the Sky: A Tutorial on Radio Localization in Ground-Air-Space Networks,” IEEE Communications Surveys & Tutorials, vol. 27, no. 1, 2025. [5] S. Kim et al., “Cell-Free Massive Non-Terrestrial Networks,” IEEE Journal on Selected Areas in Communications, vol. 43, no. 1, 2025. [6] Y. Omid, Z. M. Bakhsh, F. Kayhan, Y. Ma, and R. Tafazolli, “Space MIMO: Direct Unmodified Handheld to Multi-Satellite Communication,” in GLOBECOM 2023 - 2023 IEEE Global Communications Conference, 2023. [7] Z. M. Bakhsh et al., “Direct Uplink Connectivity in Space MIMO Systems With THz and FSO Intersatellite Links,” IEEE Systems Journal, vol. 19, no. 4, 2025. [8] M. Y. Abdelsadek et al., “Distributed massive mimo for leo satellite networks,” IEEE Open Journal of the Com. Society, vol. 3, 2022. [9] S. Saleh et al., “6G RIS-Aided Single-LEO Localization with Slow and Fast Doppler Effects,” in 2024 IEEE Globecom Workshops, 2024. [10] M. L. Psiaki, “Navigation using carrier Doppler shift from a LEO constellation: Transit on steroids,” NAVIGATION, vol. 68, no. 3, 2021. [11] M. A. Hasan et al., “A double-difference doppler shift-based positioning framework with ephemeris error correction of leo satellites,” IEEE Systems Journal, vol. 18, no. 4, 2024. [12] Q. Bader, S. Saleh, G. Seco-Granados, and A. Noureldin, “Theoretical Limits of Differential Doppler Positioning Using LEO Satellite Signals,” in 2025 IEEE Global Communications Conference, 2025. [13] L. You et al., “Ubiquitous Integrated Sensing and Communications for Massive MIMO LEO Satellite Systems,” IEEE Internet of Things Magazine, vol. 7, no. 4, 2024. [14] “3GGP Specification # 38.811 - Study on New Radio (NR) to support Non-terrestrial Networks.” [15] Y. Ettefagh et al., “Integrated Cellular and LEO-based Positioning and Synchronization under User Mobility,” IEEE Transactions on Vehicular Technology - (early access), 2026. [16] J. Lin, Y. Jing, and X. Yu, “Cramér-Rao Lower Bound Analysis of Positioning With Planar Large Intelligent Surfaces Under Rician Channel,” IEEE Trans. on Wireless Com., vol. 23, no. 7, 2024.

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