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Optimum Design for Coexistence Between Matrix Completion Based MIMO Radars and a MIMO Communication System
Bo Li, Athina P. Petropulu, Wade Trappe
TL;DR
MIMO-MC radars reduce fusion-center communication through sparse sampling, but overlapping radar and communication bands create an interference-management problem. The paper designs communication transmit covariances, with and without radar sampling knowledge, and jointly designs covariance and sampling. These designs reduce effective interference power while retaining the radar’s matrix-completion setting and communication constraints.
Problem
Overlapping communication and radar frequency bands require spectrum sharing that limits interference to a MIMO-MC radar while supporting communication operation.
Method
The paper designs communication transmit covariance matrices using noncooperative and cooperative spectrum-sharing approaches, then jointly optimizes covariance and radar sampling.
Results
The cooperative approach reduces EIP more than the noncooperative approach, and joint design further reduces EIP for both Scheme I and Scheme II radars.
Takeaways & Limitations
Appropriately designed communication waveforms and radar sampling schemes can reduce interference affecting MIMO-MC radar receivers.
Abstract
from arXiv · showhide
Recently proposed multiple input multiple output radars based on matrix completion (MIMO-MC) employ sparse sampling to reduce the amount of data that need to be forwarded to the radar fusion center, and as such enable savings in communication power and bandwidth. This paper proposes designs that optimize the sharing of spectrum between a MIMO-MC radar and a communication system, so that the latter interferes minimally with the former. First, the communication system transmit covariance matrix is designed to minimize the effective interference power (EIP) to the radar receiver, while maintaining certain average capacity and transmit power for the communication system. Two approaches are proposed, namely a noncooperative and a cooperative approach, with the latter being applicable when the radar sampling scheme is known at the communication system. Second, a joint design of the communication transmit covariance matrix and the MIMO-MC radar sampling scheme is proposed, which achieves even further EIP reduction.
I. INTRODUCTION
MIMO-MC radars use sparse sampling and matrix completion to reduce forwarded data while preserving MIMO radar resolution. The paper designs communication and radar sampling strategies to reduce interference during spectrum sharing.
- Spectrum-sharing problem: The paper treats the MIMO-MC radar as the primary user and the MIMO communication system as the secondary user in overlapping spectrum.The communication system designs its transmit covariance matrix to minimize radar EIP while maintaining prescribed average capacity and transmit power.
- MIMO-MC radar operation: MIMO-MC radars forward sparse target-return or matched-filter samples, then reconstruct the full data matrix using matrix completion.The fusion center uses the completed matrix for standard array processing and target estimation.
- MIMO-MC radar benefits: MIMO-MC radars maintain high resolution while requiring significantly fewer communicated data, saving communication power and bandwidth.These savings are especially relevant for battery-operated receiver nodes and wireless links.
- MIMO-MC radar benefits: MIMO-MC radars reduce data while avoiding basis-mismatch issues inherent in compressive-sensing-based approaches.This provides a stated advantage over CS-based MIMO radars.
- Communication covariance design: Two covariance-design approaches are proposed: cooperative and noncooperative, with the cooperative approach applicable when the radar sampling scheme is known.When the sampling scheme is known, EIP can be greatly reduced, especially at low sub-sampling rates.
- Joint design: A joint design of communication covariance and radar sampling further reduces interference, using alternating optimization while preserving data-matrix completability.The sampling search uses row and column permutations of an initial binary sampling matrix with a large spectral gap.
III. SYSTEM MODEL
The system model considers spectrum sharing between a colocated MIMO communication system and a MIMO-MC radar operating at the same carrier frequency. The communication system adapts transmit covariance across time and space to reduce radar interference while maintaining average capacity.
- The MIMO communication and MIMO-MC radar systems share the same carrier frequency and may interfere with one another.
- The radar operates in two phases: receiving target-return measurements and forwarding selected samples to a fusion center.
- Only interference affecting the radar’s forwarded samples contributes to effective interference, rather than all received interference.
- The model assumes synchronized sampling times, constant channels over L symbols, and circularly symmetric complex Gaussian communication codewords.
- The communication system minimizes interference to the radar while maintaining average capacity over L symbol durations through time- and spatial-domain resource adaptation.
IV. SPECTRUM SHARING WITH SCHEME I RADARS
This section formulates spectrum sharing for Scheme I radars by designing communication transmit covariance matrices against effective, sampling-dependent interference. It considers noncooperative, cooperative, and joint approaches under communication power and capacity constraints.
- The communication waveform covariance matrices are designed to minimize interference at the Scheme I radar receiver.
- The effective interference power includes only communication interference corresponding to samples selected by the radar sampling pattern.
- The radar sampling scheme makes the effective interference channel vary across symbol durations, requiring potentially different covariance matrices R_xl.
- The cooperative and joint approaches assume that the communication system knows the radar sampling scheme; higher cooperation is expected to improve performance but increases security and coordination costs.
- Noncooperative Spectrum Sharing: The noncooperative problem minimizes total interference without knowledge of the radar sampling scheme, subject to total transmit power and average capacity constraints.
- The optimization problem is convex, and its constraints can also be applied separately to each symbol duration while preserving convexity.
B. Cooperative Spectrum Sharing
The cooperative approach uses the radar’s known sampling scheme to optimize communication covariance matrices against time-varying effective interference channels. Its minimum EIP is never greater than the noncooperative approach and can be substantially lower in favorable channel geometries.
- The cooperative approach shares the radar sampling scheme Ω_I with the communication system and solves the resulting covariance-design problem under the same communication constraints.
- The achieved average capacity equals the required capacity C at the optimum because reducing covariance until the constraint is active also reduces the objective.
- The cooperative solution allocates more power toward communication-channel directions with stronger received signal and toward directions causing smaller radar interference.
- For any transmit power and capacity constraints, cooperative EIP is less than or equal to noncooperative EIP.
- When the radar-induced null-space intersection is available but the corresponding noncooperative intersection is empty, cooperation can achieve zero EIP while satisfying communication constraints.
- Radar subsampling can reduce the dimension of the interference-channel row space, increasing communication waveform design flexibility and enabling smaller interference.
C. Joint Communication and Radar System Design for Spectrum Sharing
The joint design alternates communication covariance optimization with row and column permutations of a radar sampling matrix, while preserving matrix-completion suitability. The resulting iterations reduce effective interference power.
- The joint design optimizes communication transmit covariance matrices together with the MIMO-MC radar sampling scheme.
- Candidate sampling matrices must permit data-matrix completion through uniform randomness or a large spectral gap.
- The sampling search restricts candidates to row and column permutations of an initial matrix Ω0, preserving its singular values and spectral gap.
- Brute-force search has complexity Θ(Mr,R!L!), so alternating row and column assignment problems provide a lower-complexity search.
- Alternating optimization decreases EIPI, and the joint-design strategy is expected to further reduce EIP at the Scheme I radar receiver.
V. SPECTRUM SHARING WITH SCHEME II MIMO-MC RADARS
For Scheme II MIMO-MC radars, the paper formulates effective interference through the random matched-filter model and optimizes communication spatial spectra against that interference.
- The Scheme II radar uses a random matched-filter signal model to characterize effective interference power.
- The effective interference expression sums interference contributions across radar receive antennas.
- Communication spatial spectra are optimized to minimize the resulting effective interference power.
- The interference expression is weighted by coefficients derived from the radar waveforms and sampling selections.
- The matched-filter formulation includes all matched filters and does not use matrix completion in that comparison.
A. Noncooperative Spectrum Sharing
The noncooperative strategy designs the communication spectrum using only the interference channel, minimizing radar interference while respecting communication-system constraints. Cooperative variants exploit additional radar information and can be compared through effective-interference objectives.
- Noncooperative Spectrum Sharing: Without radar knowledge beyond interference channel G2, the transmitter minimizes interference power at the radar receivers using (P0).
- Cooperative Spectrum Sharing: In the fully cooperative case, the radar shares diagonal matrices Δ_lξ with the communication system.
- Scheme II Objective: The effective interference power EIPII is structured like TIP and IPFMFB but is reweighted by diagonal matrices Δ_lξ.
- Theorem 2: For any Pt and C, (P4) achieves Scheme II effective interference no larger than (P0) and (P3) with none or partial information sharing.
- Joint Design: Joint design couples communication covariance matrices and the radar sampling scheme through an equivalent EIPII formulation.
- Joint Design: The joint optimization alternates convex covariance updates with sampling updates solved through row and column assignment problems.
VI. SPECTRUM SHARING BETWEEN MISMATCHED SYSTEMS
The proposed spectrum-sharing techniques extend to mismatched radar and communication symbol durations by reweighting interference terms according to sampling-time relationships. The same algorithmic framework remains applicable, subject to the stated model assumptions.
- Mismatched Systems: The mismatched-system analysis considers radar and communication systems with different waveform and symbol rates.
- Down-sampling: Communication symbols not sampled by the radar receiver introduce zero interference power to the radar receiver.
- Over-sampling: When communication symbols are oversampled, one symbol contributes interference across multiple radar sampling instances.
- Interference Weighting: In mismatched cases, effective-interference expressions retain the matched-case form but use modified diagonal weighting matrices.
- Information Requirement: The communication system needs only the radar sampling times to calculate the corresponding diagonal matrices.
- Scope: The proposed algorithms from Sections IV and V remain applicable to mismatched cases, while further investigation is left for future work.
A. Spectrum Sharing between a Scheme I radar and a MIMO Communication System
For Scheme I, cooperative and joint-design spectrum sharing reduce radar interference while preserving matrix-completion recovery and communication-capacity requirements. The joint-design method generally provides the strongest overall reduction, with useful performance concentrated at moderate sub-sampling rates.
- Performance under different sub-sampling rates: The cooperative spectrum-sharing method outperforms the noncooperative method in both EIP and matrix-completion relative recovery error.This advantage is especially significant when the sub-sampling rate is low.
- Performance under different sub-sampling rates: The joint-design method achieves smaller EIP and relative recovery errors than the other three methods in the larger-array scenario.It optimizes the radar sampling matrix starting from the same initial sampling matrix used by the other methods.
- Performance under different sub-sampling rates: The optimal sub-sampling range is p = [0.4, 0.6], where joint-design sharing reduces EIP by at least 20% versus selfish communication.Lower p can cause matrix-completion failure, while p > 0.6 requires more samples with little or no recovery-error improvement.
- Conclusion: Scheme I sub-sampling reduces both effective interference from communication and data forwarded to the fusion center.The paper’s simulations report this benefit alongside the spectrum-sharing results.
- Performance under different capacity constraints: The communication average-capacity constraint holds with equality in both scenarios, while the proposed sharing designs allocate power toward high-rate, low-EIP directions.The larger-array case shows only marginal cooperative improvement over noncooperative sharing, but joint design further reduces both metrics.
- Performance under different capacity constraints: Under all tested capacity values, cooperative and joint-design sharing achieve smaller EIP and relative recovery errors than selfish and noncooperative methods.For selfish and noncooperative sharing, both errors increase as communication capacity increases.
3) Performance under different number of targets:
Additional simulations examine target count, radar transmit power, interference-channel strength, and Scheme II operation. Joint-design sharing generally performs best, but recovery degrades with many targets and Scheme II offers less cooperative benefit at small sub-sampling rates.
- Performance under different number of targets: As the number of targets increases, relative recovery error increases because retained samples become insufficient for reliable matrix completion under noise.The EIPs remain constant because communication-waveform design is unaffected by target number.
- Performance under different number of targets: The joint-design method works effectively for Scheme I when a moderate number of targets are present.All methods show large recovery error for large target counts.
- Performance under different levels of radar TX power: With increasing radar transmit power, EIP increases more slowly, while relative recovery errors improve.Across the tested methods, joint design performs best, followed by cooperative and then noncooperative sharing.
- Performance under different channel variance: When the interference channel strengthens, communication transmit power increases to satisfy the capacity constraint, increasing EIP and relative recovery error.The joint-design method remains best, followed by cooperative and noncooperative sharing.
- Scheme II radar: For Scheme II, all four proposed sharing methods significantly reduce interference relative to selfish communication.The reported Scheme II simulations vary the sub-sampling rate from 0.2 to 1 under fixed capacity and power constraints.
- Scheme II radar: At small p, fully cooperative sharing improves less for Scheme II because the effective reference channel remains full rank, preventing zero-EIP directions.The paper contrasts this behavior with the stronger small-p improvement observed for Scheme I.
- Scheme II radar: The joint-design method achieves much smaller EIP and relative recovery errors than the other four methods for Scheme II.This supports its effectiveness for the Scheme II radar in the reported simulations.
2) Performance under different capacity constraints:
Across capacity, target-count, radar-power, and channel-variance settings, the proposed spectrum-sharing methods reduce interference and recovery error. Joint design is consistently strongest, while cooperative gains depend on system dimensions and sampling information.
- The selfish communication method is inferior to all proposed spectrum-sharing methods in the evaluated scenarios.
- The joint-design spectrum-sharing method achieves considerably smaller EIP and relative recovery errors than the partially and noncooperative methods.
- The joint-design method remains effective with a moderate number of targets when sufficient matrix-completion samples are used.
- As radar TX power increases, the performance gap between joint design and the other methods becomes larger.
- Cooperative spectrum sharing significantly reduces EIP by allocating communication power to directions in the effective interference channel's null space.
- For both radar schemes, joint design achieves much smaller EIP and relative recovery errors when radar TX and RX antenna counts are moderately large.
APPENDIX A
The appendix derives the effective interference power at Scheme II radar receive antennas from antenna-level interference terms and the sampling matrix. It expresses the result through trace and diagonal-matrix operations.
- The interference power at each radar receive antenna is represented using trace terms involving communication receive covariance matrices.
- Substituting the antenna-level quantity β_m into the preceding expression yields the effective interference power for the Scheme II radar.
- For Scheme II, the sampling matrix enters through diagonal matrices formed from rows of Ω_II and elementwise products of sampling vectors.