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A Novel ISAC Transmission Framework based on Spatially-Spread Orthogonal Time Frequency Space Modulation
Shuangyang Li, Weijie Yuan, Chang Liu, Zhiqiang Wei, Jinhong Yuan, Baoming Bai, Derrick Wing Kwan Ng
TL;DR
The paper addresses the difficulty that communication channel fading strengths cannot be directly obtained from radar sensing. It proposes an SS-OTFS-based ISAC framework with radar and communication designs, and concludes that practical power allocation should lean toward radar sensing.
Problem
Communication channel fading coefficients, particularly their strengths, cannot be directly obtained from radar sensing.
Method
The paper proposes an SS-OTFS ISAC framework with beam tracking, angle estimation, radar power allocation, PEP-based communication design, and symbol-wise precoding using delay, Doppler, and angle estimates.
Results
The framework provides radar and communication designs, with equal power minimizing PEP when received signals from different paths are orthogonal.
Takeaways & Limitations
Radar sensing and communication require different power allocations, so practical ISAC power allocation should lean toward radar sensing.
Abstract
from arXiv · showhide
In this paper, we propose a novel integrated sensing and communication (ISAC) transmission framework based on the spatially-spread orthogonal time frequency space (SS-OTFS) modulation by considering the fact that communication channel strengths cannot be directly obtained from radar sensing. We first propose the concept of SS-OTFS modulation, where the key novelty is the angular domain discretization enabled by the spatial-spreading/de-spreading. This discretization gives rise to simple and insightful effective models for both radar sensing and communication, which result in simplified designs for the related estimation and detection problems. In particular, we design simple beam tracking, angle estimation, and power allocation schemes for radar sensing, by utilizing the special structure of the effective radar sensing matrix. Meanwhile, we provide a detailed analysis on the pair-wise error probability (PEP) for communication, which unveils the key conditions for both precoding and power allocation designs. Based on those conditions, we design a symbol-wise precoding scheme for communication based only on the delay, Doppler, and angle estimates from radar sensing, without the a priori knowledge of the communication channel fading coefficients, and also introduce the power allocation for communication. Furthermore, we notice that radar sensing and communication requires different power allocations. Therefore, we discuss the performances of both the radar sensing and communication with different power allocations and show that the power allocation should be designed leaning towards radar sensing in practical scenarios. The effectiveness of the proposed ISAC transmission framework is verified by our numerical results, which also agree with our analysis and discussions.
I. INTRODUCTION
The paper develops an SS-OTFS-based ISAC framework to address the mismatch between radar echo strengths and communication channel strengths. It uses angular-domain discretization to simplify sensing and communication designs without requiring communication fading coefficients a priori.
- Motivation: Radar sensing and communication share physical channels, but radar echoes and communication fading coefficients differ because they depend on different reflection and propagation factors.Radar echoes depend on antenna effective area and RCS, whereas communication fading depends on path loss and channel scatters.
- Motivation: Steering communication beams toward the strongest radar-indicated path may degrade communication performance because the strongest radar echo need not be the strongest communication path.This mismatch motivates communication designs that do not directly equate radar echo power with communication channel strength.
- SS-OTFS framework: The proposed SS-OTFS modulation introduces spatial spreading and de-spreading to discretize the angular domain and produce simple effective models for radar sensing and communication.The resulting structure supports system designs based on delay, Doppler, and angle estimates rather than prior communication fading coefficients.
- Radar sensing: For radar sensing, the paper derives effective models and develops beam tracking and angle-of-arrival estimation algorithms that exploit the effective sensing matrix structure.The proposed algorithms are designed around the special structure created by angular-domain discretization.
- Communication design: For communication, pair-wise error probability analysis identifies precoding and power-allocation conditions, motivating symbol-wise precoding based on radar-estimated delay, Doppler, and angle.The analysis links minimized PEP to a diagonal equivalent codeword-difference structure and equal power allocation across corresponding paths.
- Power allocation and evaluation: Radar sensing and communication require different power allocations, and the paper concludes that practical allocation should lean toward radar sensing.The framework’s effectiveness is verified through simulation results consistent with the analysis and discussions.
II. SYSTEM MODEL
The system uses an SS-OTFS-enabled ISAC transmitter in a MISO setting, transforming broadcast symbols across delay, Doppler, time, frequency, and spatial domains. Per-antenna symbol-wise precoding and power allocation precede spatial spreading for transmission.
- System assumptions: The considered system has one multi-antenna base station broadcasting a common message to K single-antenna UEs while sensing their echoes.The BS has NBS antennas, and each BS–UE link contains P independent resolvable paths.
- Domain transformations: OTFS transforms the broadcast message from the delay-Doppler domain through the time-frequency domain into the time-delay domain.The DD-domain symbol matrix X has dimensions M × N, with M subcarriers and N time slots.
- Precoding and spreading: The transmitter applies wideband precoding separately to each antenna’s signal to combat multipath interference before spatial spreading.Spatial spreading is applied after antenna-wise repetition and precoding to combat interference after spatial multiplexing.
- Precoding and spreading: Symbol-wise precoding restricts each precoding matrix to one nonzero element per row and column, with normalized Frobenius energy.Each W_nt is MN × MN and satisfies ∥W_nt∥F = MN.
- Power allocation: A diagonal power-allocation matrix assigns antenna powers α_nt under the total transmit-power budget α_total.The allocated powers scale the per-antenna transmitted symbol vectors before they are rearranged and spatially spread.
- Domain transformations: An NBS-point IFFT converts the antenna-indexed time-delay-angular signals into the time-delay-spatial domain for transmission.The resulting TDS-domain transmitted symbol matrix S has dimensions MN × NBS.
B. Communication Model
The communication model represents each UE’s multipath channel through delay, Doppler, and angle-dependent responses across the BS antennas. Under OTFS assumptions, the received signal is expressed in equivalent time-domain and delay-Doppler-domain forms.
- Channel representation: The antenna-dependent channel response combines angular phase progression with delay and Doppler shifts for every resolvable path.The model uses the far-field assumption and OTFS delay-Doppler channel characteristics.
- Channel representation: Each communication path is characterized by a fading coefficient, angle of departure, delay shift, and Doppler shift.The path parameters are denoted by h_i,p, ϕ_i,p, τ_i,p, and ν_i,p.
- Delay-Doppler structure: With rectangular filtering and a reduced cyclic prefix, the TDS-domain channel admits an equivalent matrix representation using delay permutations and Doppler diagonalization.The permutation matrix models delay influence, while the diagonal matrix Δ models Doppler influence.
- Delay-Doppler structure: The delay index lies in 0 ≤ l_i,p ≤ M −1, while the Doppler index lies in 0 ≤ k_i,p ≤ N −1.Fractional Doppler κ_i,p lies in −1/2 ≤ κ_i,p ≤ 1/2; fractional delays are neglected for typical wideband systems.
- Received signal: The received TD-domain vector equals the sum of path contributions over BS antennas and paths plus AWGN, and it can be rearranged into an angularly separated form.The UE has one antenna, so receiver-side spatial features disappear from the communication description.
- Received signal: The equivalent DD-domain received signal follows from the TD-to-DD OTFS connection, with transformed noise represented by (F_N ⊗ I_M)q_i.The DD-domain model is obtained after substituting the transmitted-signal representation into the received-signal equation.
C. Radar Model
The radar model extends the communication representation to transmit and receive antenna arrays, incorporating round-trip path parameters and radar noise. Spatial de-spreading produces a time-delay-angular radar representation.
- Radar channel: The radar channel includes transmit- and receive-array angular responses for each UE path, together with path-dependent radar reflection coefficients.The radar model accounts for wavelength and transmit/receive antenna gains.
- Radar channel: Radar paths use round-trip delay and Doppler shifts satisfying ˜τ_i,p = 2τ_i,p and ˜ν_i,p = 2ν_i,p.The radar reflection coefficient is associated with the path’s propagation and reflection characteristics.
- Radar channel: With co-located transmit and receive antennas, each path has equal AoD and AoA values.The radar receive and transmit steering responses therefore use the same angle parameter.
- Radar observation: Under rectangular receive filtering and a reduced CP, the TDS-domain radar response is represented by an equivalent sensing matrix.The radar noise model includes AWGN and interference power remaining after interference cancellation.
- Radar observation: The received TDS-domain radar vectors are stacked across receive antennas, transmit antennas, UEs, and paths to form the aggregate radar observation.The stacked representation includes an equivalent TDS-domain radar-noise vector.
- Radar observation: Spatial de-spreading transforms the TDS-domain radar received vector into the TDA domain, where the equivalent radar noise has one-sided PSD ˜N0.This representation is used for subsequent radar sensing in the time-delay-angular domain.
D. Model Simplifications with Spatial Spreading and De-spreading
Spatial spreading and de-spreading discretize angular features and yield sparse, approximately orthogonal effective channel structures. This simplifies communication and radar models by associating paths with angular indices.
- Angular discretization: The angular resolution is 2/NBS, and transmit and receive angular indices represent discretized angle locations.The analysis identifies angular-domain values that concentrate around specific indices when the angle satisfies the grid condition.
- Model simplification: With sufficiently many BS antennas, spatial spreading and de-spreading approximately eliminate multipath and multiuser interference in communication and radar channels.This follows from the asymptotical orthogonality of the angular-domain responses.
- Angular sparsity: The angular-domain channel becomes sparse; for ϕ_i,p = π/4 and NBS = 128, nonzero values concentrate near transmit index 84 and receive index 46.The example is presented as consistent with the angular-domain analysis.
- Angular discretization: The framework assumes sufficiently large NBS so angular indices are integer-valued and all paths are separable by their angular features.The path-separation condition is ai,p ≠ ai′,p′ for distinct UE or path pairs.
- Effective models: The effective radar sensing matrix is a block matrix whose (˜a_i,p, a_i,p)-th sub-block contains the path response and whose remaining sub-blocks are zero.This structure follows after the communication and radar models are simplified in the angular domain.
- Effective models: Angular-domain discretization creates direct interactions between a signal transmitted on one antenna and the distortion associated with one path.The resulting input-output relationships are simplified for both communication and radar, making the representation suitable for ISAC transmission.
III. RADAR SENSING DESIGNS BASED ON SS-OTFS MODULATION
SS-OTFS enables angular-domain discretization and a sparse effective radar sensing matrix, supporting simple beam tracking, AoA estimation, and power allocation designs.
- Radar sensing structure: SS-OTFS sparsifies the effective radar sensing matrix, discretizing the angular domain for direction-selective transmission by assigning power to corresponding antennas.This avoids sophisticated precoding designs for transmitting toward desired directions.
- Beam tracking: The proposed beam-tracking design uses wider beams directed toward AoAs estimated at the previous time instant.
- Radar power allocation: Radar power is allocated equally across antennas associated with each path, assuming sufficiently separated AoAs and disjoint antenna sets.This guarantees that the ISAC signal is reflected or received by the corresponding UE.
- AoA estimation: The AoA estimator identifies indices associated with the largest diagonal-subblock traces and maps them to receive-antenna angular indices.The sparse block structure leaves only sub-blocks associated with estimated angular indices nonzero.
- Radar power allocation: The radar-oriented allocation maximizes the minimum effective radar SNR by assigning more power toward directions with weaker radar echoes.This differs from communication water-filling, which favors directions with larger channel gains.
- Radar–communication mismatch: Communication fading coefficients cannot be obtained directly from radar sensing, complicating power allocation that balances radar and communication performance.The paper therefore adapts the allocation using the statistical distribution of communication fading coefficients.
IV. SENSING-ASSISTED COMMUNICATION DESIGN
The sensing-assisted communication design uses radar-estimated delay, Doppler, and angle parameters while accounting for the absence of direct communication fading information. Its PEP analysis links reliability to the codeword-difference matrix, its rank, determinant, precoding, and power allocation.
- Design basis: Radar estimates determine communication delay, Doppler, and transmit angular indices, but have no direct relationship with communication fading coefficients.The design therefore minimizes communication PEP using radar-derived parameter estimates without requiring fading coefficients a priori.
- PEP analysis: The conditional PEP analysis uses equivalent and weighted codeword-difference matrices, whose positive-semidefinite Hermitian structure enables eigenvalue-based derivation.The effective channel coefficients are unknown to the BS, so their distributions are considered.
- PEP analysis: The PEP decreases exponentially with order r_i as the noise PSD is reduced, where r_i is the matrix rank and represents transmission diversity gain.
- Practical design: The proposed communication design develops practical precoding matrices and power allocation schemes to approach the lowest PEP upper bound across possible delay and Doppler shifts.
- Design criteria: For full-rank codeword-difference matrices, precoding should maximize the determinant, while power allocation should maximize the associated power product.The determinant depends on delay shifts and precoding matrices, allowing their effects to be analyzed separately from power allocation.
- Design criteria: A diagonal codeword-difference matrix is sufficient for attaining the determinant upper bound, corresponding to orthogonal received signals from different paths.The bound is independent of delay and Doppler shifts.
B. Precoding Design
The precoding design uses radar-derived angular, delay, and Doppler estimates to improve path orthogonality, relaxing an unattainable strict-diagonal condition to diagonal dominance. Simulations show larger determinant gains, especially with more resolvable paths and rich scattering.
- Proposed precoding: The proposed precoding assigns different virtual delay and Doppler indices to different paths using radar-estimated angle, delay, and Doppler parameters.Its rationale is to improve OTFS-induced orthogonality when paths share delay or Doppler values.
- Practical trade-off: The precoding design inevitably reduces communication SNR through antenna shaping and equal power assignment, although the reduction is small when the beam is suitably configured.
- Design constraint: The determinant dilemma shows that precoding matrices cannot simultaneously satisfy all required orthogonality conditions for distinct paths.The paper therefore relaxes the target from a strict diagonal codeword-difference matrix to a diagonally-dominant one.
- Diagonal dominance: When paths share the same delay or Doppler indices, the proposed precoding makes the codeword-difference matrix more likely to be diagonally dominant.
- Numerical evaluation: The proposed precoding increases the determinant relative to no precoding, with values aligning more closely to the upper bound for small error-sequence distances.
- Numerical evaluation: The determinant improvement becomes more obvious with more resolvable paths, indicating greater communication benefit in rich-scattering scenarios.
- Fractional Doppler: Fractional Doppler produces a dense, complex effective channel matrix and high detection complexity, while the proposed precoding can reduce UE-side detection complexity.
C. Power Allocation for Communication
Communication and radar sensing favor different power allocations: equal power minimizes the communication PEP under the analyzed fading model, whereas practical ISAC allocation should prioritize radar sensing.
- Communication allocation: Under a total communication-power constraint, equal power allocation minimizes the PEP because communication fading coefficients share the same distribution.
- Communication allocation: Communication performance degradation from unsuitable power allocation becomes more severe as the number of resolvable paths increases.
- ISAC trade-off: Radar sensing benefits more directly from tailored power allocation because radar reflection coefficients are assumed known at the BS, unlike communication fading coefficients.
- ISAC trade-off: In practical ISAC scenarios, power allocation should prioritize radar sensing, while the communication allocation can be adapted to statistical fading information.
- Scope boundary: A precise radar–communication power-allocation relationship requires statistical models for both coefficient types and depends on system settings such as frame size.The paper leaves this analysis for future work.
V. NUMERICAL RESULTS
The numerical results evaluate SS-OTFS radar sensing under varied beam widths and power allocation settings, alongside the proposed transmission setup. The results report accurate AoA estimation across beam widths and indicate that the proposed radar-oriented allocation is suitable for the considered sensing scenarios.
- Simulation Setup: The simulations use BPSK signals and define communication and radar SNRs through average symbol power and total radar power, respectively.The setup also assumes uniformly distributed complex Gaussian reflection coefficients.
- AoA Estimation: Increasing beam width reduces the received echo power in the considered AoA-estimation experiment.The reported average received power is associated with the beam-width setting.
- Radar Power Allocation: Nrange = 4 yields average received power equal to 1/5 of the Nrange = 0 case because power is evenly assigned across Nrange + 1 antennas.The observation supports the suitability of the proposed power allocation for the considered radar sensing issues.
B. Precoding Performance
The section evaluates the proposed precoding and contrasting power allocations for communication and radar sensing. Precoding improves coded FER performance, radar-oriented allocation improves radar sensing, and the resulting communication degradation is comparatively small, motivating radar-prioritized allocation.
- Precoding Performance: With the same channel coding and power allocation, precoding provides roughly 1.7 dB average bit-SNR gain over transmission without precoding at the reported FER.The comparison uses coded BPSK, a terminated (7, 5) convolutional code, and equal power allocation.
- Precoding Performance: Precoded transmission has a steeper FER slope than transmission without precoding, indicating improved diversity gain.The paper attributes this to greater likelihood that the codeword difference matrix has full rank when virtual delay and Doppler indices differ.
- Power Allocation: Suitable radar-oriented power allocation significantly improves radar-sensing performance, especially when the number of targets is large.The comparison considers K = 4, P = 2 and K = 2, P = 1 cases using miss-detection probability.
- Power Allocation: Radar-oriented power allocation worsens communication performance, but the degradation is relatively small compared with the radar-sensing improvement.The authors therefore recommend designing power allocation with priority given to radar sensing.
- Power Allocation: Equal power allocation provides better communication FER performance, with larger improvement as the number of paths increases.The proposed precoding scheme is applied in this comparison.
- Framework Summary: The SS-OTFS framework derives simplified radar and communication models, enabling radar algorithms and communication precoding based on sensed parameters.The communication precoding uses delay, Doppler, and angle estimates without prior fading-coefficient knowledge.
APPENDIX A PROOF OF THEOREM 1
The proof analyzes the Gram determinant of the codeword difference matrix through orthogonal projections and path-dependent cross terms. It establishes how orthogonality and shared delay or Doppler indices affect the determinant-related bounds.
- Gram-Determinant Analysis: The codeword difference matrix is treated through its Gram determinant, which equals the squared volume of the parallelotope formed by the associated path vectors.This geometric interpretation supports the subsequent determinant recursion.
- Gram-Determinant Analysis: The Gram determinant is recursively expressed as the previous determinant multiplied by the squared norm of an orthogonal projection.The projection is taken onto the complement of the span of preceding vectors.
- Orthogonality Condition: Equality in the determinant recursion occurs when each new path vector is orthogonal to all preceding path vectors.The proof connects this condition to the corresponding codeword-difference terms.
- Orthogonality Condition: The proof uses orthogonality relations across distinct paths to derive identity-matrix products and complete the associated lemma.A zero-determinant contradiction is used in the argument.
- Path-Dependent Terms: For sufficiently large N, TDA-domain OTFS symbols are modeled as approximately i.i.d. Gaussian because each is a phase-rotated superposition of N delay-Doppler symbols.This assumption supports the analysis of non-diagonal codeword-difference terms.
- Path-Dependent Terms: When paths share a delay or Doppler index, the proof compares the resulting cross terms and finds that one case is more likely to produce larger absolute values under the i.i.d. assumption.The comparison distinguishes zero-mean products from strictly non-negative squared terms.