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Beam Squint-Aware Integrated Sensing and Communications for Hybrid Massive MIMO LEO Satellite Systems

Li You, Xiaoyu Qiang, Christos G. Tsinos, Fan Liu, Wenjin Wang, Xiqi Gao, Björn Ottersten

arXiv:2203.00235v1cs.ITeess.SPmath.IT

TL;DR

Terrestrial ISAC designs cannot be directly adopted in LEO satellite systems because wave propagation differs. This paper proposes beam-squint-aware hybrid precoding using statistical channel information; simulations show communications and sensing can both be performed well in a partially connected system at ζ = 0.7.

  • Problem

    Terrestrial ISAC designs cannot be directly adopted in LEO satellite systems because their wave-propagation properties differ.

    Source: §I · p. 1

  • Method

    The paper characterizes propagation with beam squint and proposes an sCSI-based hybrid analog/digital transmitter that weights communication and sensing objectives.

    Source: §I · p. 2

  • Results

    Simulations show that communications and sensing can both be well performed with a partially connected structure at ζ = 0.7.

    Source: §V · p. 12

  • Takeaways & Limitations

    Different weighting factors let satellite ISAC systems adjust to differing communication and target-sensing requirements.

    Source: §V · p. 12

Abstract

from arXiv · show

The space-air-ground-sea integrated network (SAGSIN) plays an important role in offering global coverage. To improve the efficient utilization of spectral and hardware resources in the SAGSIN, integrated sensing and communications (ISAC) has drawn extensive attention. Most existing ISAC works focus on terrestrial networks and can not be straightforwardly applied in satellite systems due to the significantly different electromagnetic wave propagation properties. In this work, we investigate the application of ISAC in massive multiple-input multiple-output (MIMO) low earth orbit (LEO) satellite systems. We first characterize the statistical wave propagation properties by considering beam squint effects. Based on this analysis, we propose a beam squint-aware ISAC technique for hybrid analog/digital massive MIMO LEO satellite systems exploiting statistical channel state information. Simulation results demonstrate that the proposed scheme can operate both the wireless communications and the target sensing simultaneously with satisfactory performance, and the beam-squint effects can be efficiently mitigated with the proposed method in typical LEO satellite systems.

I. INTRODUCTION … II. SYSTEM MODEL AND PROBLEM FORMULATION

The paper develops beam-squint-aware ISAC for massive-MIMO LEO satellites using statistical CSI, since satellite propagation conditions and rapidly varying wideband channels make terrestrial ISAC designs and instantaneous CSI impractical. It motivates this approach within SAGSIN and formulates a system supporting simultaneous communication and target sensing.

I. INTRODUCTION

  • The paper proposes a first satellite ISAC study using statistical CSI because long propagation delays, large Doppler shifts, and rapidly varying wideband channels hinder instantaneous CSI estimation.The authors describe the approach as the first investigation of ISAC adoption in satellite systems.Source: §I · pp. 1–2
  • The work characterizes beam-squint-aware statistical propagation and relates beam squint to system bandwidth, carrier frequency, and antenna-array aperture.Source: §I · p. 2
  • The proposed hybrid analog/digital transmitter jointly supports communications and sensing, while a weighting coefficient trades communication energy efficiency against sensing-beampattern performance.An sCSI-based hybrid-precoding algorithm is developed to mitigate beam squint and improve communication energy efficiency and sensing-beampattern matching.Source: §I · p. 2

A. Related Works

  • LEO SATCOM studies have examined sCSI-based transmission and hybrid precoding for rate or energy efficiency, while sub-arrayed MIMO radar balances phased-array and fully independent-transmit architectures.The cited radar work describes subarrays as a compromise between one RF chain with high array gain and one RF chain per antenna.Source: A. Related Works · p. 2
  • Terrestrial ISAC research explores hybrid beamforming, OFDM, multibeam operation, and massive MIMO; wideband arrays can cause beam squint, and existing mitigation designs often rely on transmitter iCSI.Source: A. Related Works · p. 2

B. Organization

  • The paper proceeds from the communication-and-sensing system model and trade-off formulation to equivalent fully digital optimization, hybrid-precoder algorithms, simulations, and conclusions.Source: B. Organization · p. 2

C. Notations

  • The notation section defines matrix and vector conventions, complex and probability notation, matrix operators, norms, and the empty-set symbol.Source: C. Notations · p. 3

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The modeled LEO satellite system serves K single-antenna users and detects targets using a massive-MIMO UPA; it employs OFDM with M subcarriers across bandwidth Bw.The text also defines carrier frequency and wavelength, signal duration Ts = 1/Bw, and equal antenna spacing along both array axes.Source: §II · p. 3

A. Channel Model with Beam Squint Effects in the Communication Module

The section models the satellite downlink as a multipath channel with a frequency-dependent array response that accounts for beam squint. It identifies the array-delay-to-signal-duration ratio as the condition governing when beam squint becomes significant.

A. Channel Model with Beam Squint Effects in the Communication Module

  • The downlink is multipath, with paths assumed to share an AoD; Doppler combines satellite and UT mobility, and satellite Doppler is nearly the same across paths.The common-AoD assumption follows from the satellites’ much higher altitude than surrounding UT-side scatterers.Source: A. Channel Model with Beam Squint… · p. 3
  • The channel parameters are treated as invariant over the time intervals of interest and should be updated when UT or satellite positions change significantly.Source: A. Channel Model with Beam Squint… · p. 4
  • At each OFDM subcarrier, the channel response is the product of a frequency-dependent UPA response and a Rician-distributed channel gain.The gain has Rician factor κ_k and power γ_k.Source: A. Channel Model with Beam Squint… · p. 4
  • In wideband massive MIMO with a large UPA, the array response varies across OFDM subcarriers and depends on AoD and the ratio of element delay to signal duration.Beam squint appears when signal duration is comparable to or much smaller than the array element delay.Source: A. Channel Model with Beam Squint… · p. 4

B. Downlink Transmission Signal Model in the Communication Module

The communication module models downlink transmission through a hybrid satellite precoder and designs it using statistical channel state information. It defines per-user communication metrics and accounts for the hybrid precoder’s power consumption.

B. Downlink Transmission Signal Model in the Communication Module

  • The final transmit signal at each subcarrier is x[m] = B[m]s[m], with K user data streams and a hybrid precoder formed by analog and baseband precoders.The data vector has zero mean and identity autocorrelation; the analog precoder is frequency-dependent and unit-modulus.Source: B. Downlink Transmission Signal Model… · p. 4
  • Each user’s precoding vector is the RF analog precoder multiplied by that user’s baseband precoding vector.For user k, b_k[m] = W_RF[m]w_BB,k[m].Source: B. Downlink Transmission Signal Model… · p. 4
  • The model defines each user’s received signal and SINR per subcarrier, then uses SINR to define ergodic rate and communication-module EE.Source: B. Downlink Transmission Signal Model… · pp. 4–5
  • Because delay and satellite and user mobility make downlink channels difficult to estimate, precoding is designed from statistical channel information.The statistical information comprises channel gain and the UPA response vector.Source: B. Downlink Transmission Signal Model… · p. 4
  • Total transmission power includes power-amplifier consumption and static hybrid-precoder power, with static power accounting for RF chains, the local oscillator, the baseband precoder, and the RF analog precoder.The power-amplifier efficiency is denoted by 1/ξ; RF analog-precoder power is mainly related to antenna count and transmission bandwidth.Source: B. Downlink Transmission Signal Model… · p. 5

C. Sensing Beampattern Considering Beam Squint Effects … E. Problem Formulation

The section develops a beam-squint-aware sensing model and detection procedure, then formulates hybrid precoding to maximize communications energy efficiency subject to sensing performance.

C. Sensing Beampattern Considering Beam Squint Effects

  • The sensing beampattern uses subcarrier steering responses and covariance, so beam squint makes it depend on OFDM frequency as well as target AoD.The section also gives an optimal sensing precoder for Pr ≤ K targets.Source: C. Sensing Beampattern Considering… · p. 5

D. Probability of Muti-Target Detection

0. Besides, xr[m] is the probing signal, satisfying

  • The reflected signal combines target array responses and reflection coefficients with the probing signal, plus additive white Gaussian noise.Source: §0 · p. 5, D. Probability of Muti-Target Detection · p. 5
  • Detection probability is calculated using central and noncentral chi-squared distributions with 2Pr degrees of freedom, with PFA denoting false-alarm probability.Source: §0 · p. 5

E. Problem Formulation

  • The formulation maximizes communications EE subject to sensing guarantees, using a unitary rotation for dimensional matching and an ε tolerance on hybrid-versus-sensing precoders.The rotation does not affect the sensing beampattern.Source: E. Problem Formulation · pp. 5–6
  • The RF analog precoder supports fully connected or partially connected structures, with nonzero elements selected from the corresponding structure set [39],.In the partially connected structure, antennas are divided into Mt groups, each with Ng = Nt/Mt elements connected to the same RF chain.Source: E. Problem Formulation · p. 6
  • The fractional objective and coupled nonconvex constraints make the problem difficult; the proposed approach uses an equivalent digital precoder, balances communication and sensing, and applies alternating optimization,.Source: E. Problem Formulation · p. 6

III. BEAM SQUINT-AWARE EQUIVALENT FULLY DIGITAL PRECODING · A. Upper Bound of the Ergodic Rate · B. Fractional Programming

The section develops an equivalent fully digital precoder by replacing costly Monte Carlo estimation of the ergodic rate with a tight upper bound, then solving the resulting fractional problem through iterative transforms and constrained optimization.

III. BEAM SQUINT-AWARE EQUIVALENT FULLY DIGITAL PRECODING

  • The equivalent fully digital procedure converts the design into subproblems with Dinkelbach’s algorithm, then uses the Lagrangian dual and quadratic transforms to obtain convex problems handled by a Lagrange multiplier method.The algorithm iteratively updates auxiliary variables and the precoder until the objective converges.Source: §III · p. 6, B. Fractional Programming · p. 8

A. Upper Bound of the Ergodic Rate

  • Jensen’s inequality gives a tight upper bound on each ergodic rate, using the concavity of log(1 + SINRk[m]) with respect to channel gain gk, and avoids costly Monte Carlo estimation, [42, Lemma 2].Monte Carlo estimation is described as having high computational complexity.Source: A. Upper Bound of the Ergodic Rate · p. 6

B. Fractional Programming

  • The fully digital design applies Dinkelbach’s algorithm to form auxiliary subproblems whose solutions converge to the original problem’s globally optimal point,.The objective’s numerator and denominator are continuous and positive-valued.Source: B. Fractional Programming · p. 6
  • The framework alternates between updating the Dinkelbach variable η(i) for fixed B(i) and optimizing B(i) for fixed η(i),.This alternating optimization handles the auxiliary subproblems.Source: B. Fractional Programming · p. 6
  • The Lagrangian dual transform introduces auxiliary variables λ to handle fractional expressions inside logarithms, yielding an equivalent problem with the same optimal objective value [45].For fixed B, the transformed problem is concave in λ, whose optimal updates follow by setting the derivative to zero.Source: B. Fractional Programming · pp. 6–7
  • The quadratic transform introduces auxiliary variables ρ to convert the sum-of-ratios term into an equivalent problem, with ρ updated by setting its derivative to zero.The resulting problem is convex in the precoding vectors bk,m when η, λ, and ρ are fixed.Source: B. Fractional Programming · p. 7
  • A Lagrange multiplier for the power constraint yields the precoding-vector update; its value is chosen to satisfy complementary slackness and can be found by bisection.The multiplier update uses the power constraint’s complementary-slackness condition.Source: B. Fractional Programming · pp. 7–8

C. Convergence and Computational Complexity

Algorithm 1 converges to a stationary point through finite-termination Dinkelbach iterations and monotonically nondecreasing subproblem objectives. Its complexity is analyzed in terms of the outer and inner iteration counts, with the precoding-vector pseudoinverse as the dominant stated operation.

Convergence

  • Algorithm 1 converges to a stationary point because Dinkelbach’s algorithm terminates finitely and each subproblem objective is monotonically nondecreasing.The objective numerator and denominator are continuous and positive-valued functions of B; each subproblem is handled iteratively using the Lagrangian dual transform and quadratic transform.Source: C. Convergence and Computational… · p. 8
  • Dinkelbach’s algorithm converts the fractional EE problem into subproblems solved iteratively using the Lagrangian dual transform and quadratic transform.The resulting iterative scheme underlies the convergence analysis of the proposed equivalent fully digital precoder.Source: C. Convergence and Computational… · p. 8

Computational Complexity

  • The precoding-vector update requires O(N_t^3) complexity for its pseudo-inverse, while updates of λ_k,m and ρ_k,m each require O(N_t).The bisection search for b_k,m has only a small number of iterations and is ignored; the analysis assumes I Dinkelbach iterations and average J iterations for each subproblem.Source: C. Convergence and Computational… · p. 8

IV. BEAM SQUINT-AWARE HYBRID PRECODING

The section formulates hybrid precoding as a weighted approximation to communications and sensing precoders. It then adopts alternating optimization to address the non-convex design problem and unit-modulus RF constraints.

IV. BEAM SQUINT-AWARE HYBRID PRECODING

  • The design minimizes weighted Euclidean distances between the hybrid precoder product and the equivalent fully digital communications and sensing precoders,.The formulation is defined for each subcarrier and includes the communications-only equivalent fully digital precoder.Source: §IV · p. 8
  • The weighting coefficient ζ ∈[0, 1] can be set according to different system requirements, with sensing and communications terms defined in (26).The formulation uses S ∈{SFC, SPC}.Source: §IV · p. 8
  • Alternating optimization iteratively designs the hybrid precoders and unitary matrices to handle objective and constraint non-convexity and unit-modulus RF analog precoder elements.Methods are developed for both fully and partially connected structures and apply to arbitrary subcarriers.Source: §IV · p. 8

A. Fully Connected Structure

For the fully connected structure, the RF analog precoder satisfies WRF ∈SFC, and the method alternates updates of WRF, WBB, and U with the other two variables fixed. The updates combine an SVD-based unitary-matrix solution, a relaxed least-squares digital precoder, and an MM-based RF analog precoder.

A. Fully Connected Structure

  • The fully connected design alternately updates WRF, WBB, and U, keeping the other two fixed at each step.Algorithm 2 summarizes the hybrid precoder procedure.Source: A. Fully Connected Structure · pp. 8–9

Update of the Unitary Matrix U

  • Given WRF and WBB, the unitary-matrix subproblem becomes an orthogonal Procrustes problem with an analytical SVD solution.The update uses factors obtained from the SVD of the specified matrix.Source: A. Fully Connected Structure · pp. 8–9

Update of the Digital Precoder WBB

  • With WRF and U fixed, the digital-precoder subproblem is relaxed by ignoring its constraint, yielding a least-squares problem with a closed-form solution.The resulting WBB is normalized so the Euclidean distance in (53a) can be sufficiently small,.Source: A. Fully Connected Structure · p. 9

Update of the RF Analog Precoder WRF

  • With the other variables fixed, the RF analog-precoder subproblem is tackled using a majorization-minimization method.The method defines Y = GGH and its maximum eigenvalue λmax(Y) before presenting the RF update.Source: A. Fully Connected Structure · p. 9

Algorithm 2: Hybrid Precoding with the Fully Connected Structure

  • Algorithm 2 initializes the precoders, evaluates the objective, and repeatedly updates U, WBB, and WRF using Eqs. (51), (54), and (58).The algorithm also specifies a threshold ǫ2 and maximum iteration count cmax.Source: A. Fully Connected Structure · p. 9

B. Partially Connected Structure · 1) Update of the Unitary Matrix U: · 2) Update of the Digital Precoder WBB:

The partially connected architecture constrains the RF analog precoder to a block-diagonal structure with row-dependent element-column indexing. Its unitary-matrix update reuses the fully connected algorithm, while its digital-precoder update uses the same A and C definitions and a closed-form projection for Q7.

B. Partially Connected Structure

  • The partially connected RF analog precoder is constrained to SPC and uses phase-valued entries whose column index depends on the row index.The passage gives the entry form as [WRF]i,j = exp {φi,j} with j = ⌈i/Ng⌉.Source: B. Partially Connected Structure · p. 9
  • Because WRF is block diagonal, the equality constraint in (48c) can be simplified.The supplied passage truncates the expression after ||WRFWBB||2.Source: B. Partially Connected Structure · p. 9

1) Update of the Unitary Matrix U:

2) Update of the Digital Precoder WBB:

3) Update of the RF Analog Precoder WRF: · C. Convergence and Computational Complexity · V. SIMULATIONS

The section describes the hybrid-precoder update and its convergence and complexity, then evaluates communication and sensing performance in simulations. The results show beam-squint-aware precoding benefits and a flexible communication–sensing tradeoff across architectures and weighting factors.

3) Update of the RF Analog Precoder WRF:

  • With WBB and U fixed, the RF analog precoder WRF is updated by solving its subproblem; Algorithm 3 sketches hybrid precoder design for the partially connected structure.The iterative procedure calculates the objective and updates U, WBB, and WRF using Eqs. (51), (60), and (63).Source: 3) Update of the RF Analog Precoder WRF · p. 10

C. Convergence and Computational Complexity

  • Algorithms 2 and 3 are block coordinate descent methods guaranteed to converge; their dominant costs involve the fully connected case’s baseband-precoder pseudoinverse and the partially connected case’s SVD and equation calculations.The supplied complexity passages give operation types but their asymptotic expressions are truncated.Source: C. Convergence and Computational… · p. 10

V. SIMULATIONS

  • Algorithm 1 converges rapidly across power budgets, while Algorithms 2 and 3 generally converge within a few iterations for different weighting factors.Figure 2 reports average convergence performance; the objective functions are defined in Eqs. (29a) and (48a).Source: Figure 2 · p. 10, §V · p. 10
  • Sensing range resolution improves with more antennas, while communication interference on target sensing decreases; weighting factors let the architectures balance sensing and communication requirements.For the fully connected structure, ζ = 0.4 enables target detection with a marked EE rise; for the partially connected structure, ζ = 0.7 supports both functions well.Source: §V · pp. 11–12, Figure 4 · p. 12, Figure 5 · p. 12
  • The beam squint-aware scheme outperforms its unaware counterpart in communication EE, with the advantage increasing as antenna count grows; hybrid architecture also improves EE at large array sizes.The comparison attributes the growing gap to beam-squint effects and the hybrid architecture’s lower RF-chain power consumption.Source: Figure 6 · p. 12, §V · pp. 12–13
  • The beam squint-aware hybrid scheme significantly mitigates beam squint in both horizontal and vertical sensing directions, with a small beampattern-gain loss at some subcarrier frequencies.The illustrated case uses a target at AoD (−0.3, 0.7).Source: Figure 8 · p. 13, §V · p. 13
  • With PFA = 10^-7, ζ = 0.9 uses more power than ζ = 0.4 to reach 90% detection probability, and the proposed scheme improves detection probability over the beam squint-unaware scheme.The passage notes that a proper weighting coefficient can enhance both communication EE and target detection performance.Source: §V · pp. 13–14, Figure 9 · p. 14

VI. CONCLUSION

The paper proposes a beam squint-aware hybrid precoding scheme using statistical CSI for simultaneous communications and target sensing in massive MIMO LEO satellite systems. Numerical results show gains over a method that does not account for beam squint, with satisfactory performance for both functions.

VI. CONCLUSION

  • Numerical results show performance gains over the method without considering beam squint, while supporting communications and sensing simultaneously with satisfactory performance.The conclusion does not report numerical values.Source: §VI · p. 14
  • The scheme converts the coupled nonconvex problem into equivalent fully digital precoder design, handled through fractional programming.It then minimizes Euclidean distances between the hybrid precoder product and the communication and sensing precoders.Source: §VI · p. 14
  • A weighting coefficient trades off communications and detection, and alternating optimization designs the hybrid precoders using statistical CSI.The proposed beam squint-aware hybrid precoding scheme targets simultaneous communications and target sensing in massive MIMO LEO satellite systems.Source: §VI · p. 14

APPENDIX A PROOF FOR THE UPPER BOUND IN (28)

The appendix proves the upper bound in (28) by showing that the logarithmic rate expression is concave in x = |gk[m]|^2, then applying Jensen’s inequality. The resulting bound uses the expression evaluated at the expectation of x.

APPENDIX A PROOF FOR THE UPPER BOUND IN (28)

  • The logarithmic expression inside the ergodic-rate expectation is represented using x = |gk[m]|^2.Source: Appendix A · p. 14
  • The second derivative of f(x) is nonpositive, establishing concavity with respect to x = |gk[m]|^2.The appendix checks the second derivative to verify this property.Source: Appendix A · p. 14
  • Jensen’s inequality, E{f(X)} ≤ f(E{X}), yields the upper bound on the ergodic rate.Source: Appendix A · p. 14

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