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An ESPRIT-Based Approach for 2-D Localization of Incoherently Distributed Sources in Massive MIMO Systems
Anzhong Hu, Tiejun Lv, Hui Gao, Zhang Zhang, Shaoshi Yang
TL;DR
The paper addresses limited ESPRIT-based capability for 2-D localization of incoherently distributed sources, especially the need to estimate coupled angular parameters without exhaustive search. It constructs signal-subspace and shifted-subarray relations to estimate nominal azimuth and elevation DOAs plus angular spreads. The resulting estimator improves with more base-station antennas and achieves much lower computational complexity while retaining comparable performance to traditional 2-D estimators.
Problem
Existing ID-source localization research is less adequate for 2-D settings, and prior ESPRIT or ML approaches respectively face angular coupling, search requirements, or high complexity.
Method
An ESPRIT-based signal-subspace method uses linear relations among a URA response matrix and three shifted subarrays to estimate 2-D nominal DOAs and angular spreads without searching.
Results
The proposed estimator has complexity less than 0.1% of existing methods in representative scenarios while providing comparable localization performance.
Takeaways & Limitations
Increasing the number of base-station antennas improves the proposed estimator’s performance, making the approach particularly suitable for massive MIMO systems.
Abstract
from arXiv · showhide
In this paper, an approach of estimating signal parameters via rotational invariance technique (ESPRIT) is proposed for two-dimensional (2-D) localization of incoherently distributed (ID) sources in large-scale/massive multiple-input multiple-output (MIMO) systems. The traditional ESPRIT-based methods are valid only for one-dimensional (1-D) localization of the ID sources. By contrast, in the proposed approach the signal subspace is constructed for estimating the nominal azimuth and elevation direction-of-arrivals and the angular spreads. The proposed estimator enjoys closed-form expressions and hence it bypasses the searching over the entire feasible field. Therefore, it imposes significantly lower computational complexity than the conventional 2-D estimation approaches. Our analysis shows that the estimation performance of the proposed approach improves when the large-scale/massive MIMO systems are employed. The approximate Cramér-Rao bound of the proposed estimator for the 2-D localization is also derived. Numerical results demonstrate that albeit the proposed estimation method is comparable with the traditional 2-D estimators in terms of performance, it benefits from a remarkably lower computational complexity.
I. INTRODUCTION
The paper targets 2-D localization of incoherently distributed sources in massive MIMO systems, where existing methods face search, dimensionality, or modeling limitations. It proposes an ESPRIT-based estimator using very large URAs to decouple angular parameters without searching.
- Motivation: Massive MIMO uses hundreds of base-station antennas to increase degrees of freedom, data rate, link reliability, and spectral efficiency.Its systems may require two-dimensional beamforming because practical array-aperture constraints favor multidimensional antenna arrays.
- Source model: Distributed sources emit over angular regions, modeling multipath transmission rather than single-DOA line-of-sight transmission.Incoherently distributed sources correspond to rapidly time-varying channels, which are typical in cellular mobile communications.
- Limitations of prior work: Prior 2-D ID-source estimators include exhaustive-search ML methods, while existing ESPRIT approaches cannot fully avoid coupling or search and may omit angular-spread estimation.Some existing approaches also constrain adjacent-antenna spacing or address only one-dimensional or single-source cases.
- Proposed contribution: The proposed ESPRIT approach estimates nominal azimuth and elevation DOAs and angular spreads from signal-subspace relations in very large URAs.The method constructs linear relations among the array response matrix and three shifted subarrays.
- Contributions: The approach removes the adjacent-antenna-distance constraint and decouples the 2-D angular parameters without searching.The paper also analyzes antenna-number effects, derives an approximate 2-D CRB, and compares computational complexity with existing methods.
III. THE ESPRIT-BASED APPROACH
The proposed method constructs a signal subspace from a first-order approximation of the array manifold and exploits shifted-subarray relations to estimate 2-D angular parameters. The same subspace framework also supports angular-spread estimation.
- III. THE ESPRIT-BASED APPROACH: Existing 2-D localization methods can require exhaustive multidimensional searches, whereas the proposed signal-subspace ESPRIT approach estimates angular parameters without searching.The method addresses mutual coupling of azimuth and elevation parameters in straightforward 1-D ESPRIT extensions.
- A. The Signal Subspace: A first-order Taylor expansion approximates each path response around the nominal azimuth and elevation DOAs.The approximation assumes the angular deviations, and therefore angular spreads, are sufficiently small.
- A. The Signal Subspace: The received signal becomes linearly related to the nominal array manifold and its partial derivatives, forming an array response matrix.The matrix is determined by the nominal DOAs, so those parameters can be obtained from it.
- A. The Signal Subspace: The received-signal covariance is decomposed by eigenvalue decomposition into signal and noise subspaces, with the largest 3K eigenvalues defining the signal subspace.The array response matrix is full rank under the paper’s massive-MIMO analysis, supporting this decomposition.
- A. The Signal Subspace: Angular spreads are obtained from the covariance of the coefficient vector, while the signal subspace remains linearly related to the array response matrix.This relation is the basis for estimating nominal DOAs from received snapshots.
B. The Proposed Estimator
The estimator uses three URA subarrays and signal-subspace linear relations to decouple and estimate 2-D nominal DOAs and angular spreads without constraining antenna spacing.
- Subarray construction: Three subarrays decouple the coupled elevation and azimuth nominal DOAs through constructed linear relations among their array response matrices.At least three subarrays are required for this decoupling.
- Model assumptions: The derivation retains the partial derivatives of the subarray functions, so the antenna-spacing restriction d much shorter than λ is unnecessary.Existing approaches approximate these derivatives as zero under the shorter-than-wavelength assumption.
- Signal-subspace relations: Each subarray array response matrix is linearly related to a selected signal subspace obtained from the received snapshots.This relation transfers the array-response structure into observable signal-subspace quantities.
- Nominal DOA estimation: The diagonal elements of two transform matrices encode functions of the 2-D nominal DOAs and are estimated from the selected subspaces using total least squares.Eigenvalue decompositions provide the transform-matrix eigenvalues used for DOA estimation.
- Eigenvalue matching: Eigenvalue products and quotients are used to match the two transform-matrix eigenvalue sets before nominal DOAs and angular spreads are recovered.The matching resolves different eigenvalue orderings across the decompositions.
3 TΦ2,1Φ3,1T−1T3,
This passage applies a substitution into an earlier relation and notes how diagonal elements are approximately formed before introducing a similar notation.
- The diagonal elements of Φ2,1Φ3,1 approximately formulate the diagonal elements of Λ3.
- A similar quantity is then denoted for the continuing derivation.
- Substituting (55) into (58) yields a subsequent expression.
3 TΦ2,1Φ−1 3,1T−1T3. (59)
The estimator matches transform eigenvalues, then uses the matched values to estimate nominal DOAs and angular spreads through closed-form processing.
- Eigenvalue matching: A third eigenvalue decomposition and a constructed quotient transform provide information for matching the eigenvalues of the first two transform matrices.Products are matched against Λ3, while quotients are matched against the constructed fourth transform.
- Eigenvalue matching: After matching and sorting, paired eigenvalues estimate the diagonal elements associated with the two subarray transformations.The pairs are indexed by source and subarray before DOA recovery.
- Parameter recovery: The nominal DOAs are estimated from the matched transform eigenvalues, after which the noise variance and array-response estimate support angular-spread estimation.The angular-spread estimates depend on the estimated nominal DOAs.
- Algorithm summary: The algorithm proceeds from the sample covariance and signal-subspace EVD through selected subspaces, TLS transforms, eigenvalue matching, and final parameter estimates.Its closed-form expressions avoid the complicated angular-parameter search used by many traditional approaches.
IV. ANALYSIS OF THE PROPOSED APPROACH
The analysis examines antenna-count effects, derives an approximate CRB, and compares computational complexity; performance improves as the number of BS antennas increases while complexity remains much lower.
- Performance analysis: Estimation performance improves as the number of BS antennas M increases.The analysis investigates the impact of M on both the rank of the array response matrix and estimator performance.
- Performance analysis: The paper derives an approximate CRB for estimation-error covariances and compares the proposed approach's computational complexity with existing approaches.The complexity comparison supports the stated lower-complexity characterization.
A. The Impact of the Number of the BS Antennas
With finite snapshots, the estimated signal subspace may deviate from the array-response subspace; as the number of BS antennas grows, this alignment and estimator accuracy improve.
- A. The Impact of the Number of the BS Antennas: Finite snapshots make the estimated signal subspace random and potentially misaligned with the array-response subspace.The sample covariance matrix and its eigenspaces fluctuate when the snapshot count is finite.
- A. The Impact of the Number of the BS Antennas: As M →∞, the array-response matrix approaches full rank and the estimated signal subspace almost surely converges to the same subspace.
- A. The Impact of the Number of the BS Antennas: Increasing the number of BS antennas therefore improves the proposed estimator’s accuracy under finite snapshots.
B. Approximate CRB of the Proposed Estimator
The paper derives an approximate CRB for the proposed estimator and compares its asymptotic computational complexity with alternative 2-D localization approaches.
- B. Approximate CRB of the Proposed Estimator: The approximate CRB measures the error spread of the estimated signal-parameter vector and serves as a reference for evaluating the estimator.
- C. Complexity Analysis: The proposed algorithm has complexity O(M^3+M^2T+MK^2), which approaches O(M^3) as M →∞ when M is much larger than K.Its three main steps have complexities O(M^2T), O(M^3), and O(MK^2).
- C. Complexity Analysis: The COMET-based 2-D approach has complexity O(D1M^3+M^2T) → O(D1M^3), where D1 is its search dimension over all users’ parameters.
- C. Complexity Analysis: The modified subspace approach has complexity O(D2M^3+M^2T) → O(D2M^3), with D1 = D2^K^2 and therefore D1 ≫ D2.
- C. Complexity Analysis: The complexity comparison concludes that the proposed approach is significantly lower than the other localization approaches.
V. NUMERICAL RESULTS
Numerical tests compare the proposed estimator with existing approaches across complexity, antenna count, SNR, angular spread, user-terminal count, and scatterer count. The proposed method achieves substantially lower complexity, improves with more antennas or SNR, performs best for modest angular spreads, and is largely insensitive to scatterer count.
- Complexity: The proposed approach has complexity O(6.0×10^6) at M = 100, below 0.1% of COMET's O(1.4641×10^14) and DISPARE's O(1.21×10^10).The comparison includes the sample covariance calculation, with complexity O(5.0×10^6).
- Antenna count: At average received SNR 10 dB, the proposed approach's RMSEs decrease rapidly as the BS antenna count M increases.The corresponding RMSEs for the subspace-based approach and DISPARE are almost invariant because they reach their best performance at smaller M.
- Antenna count: When M = 144, the proposed approach has azimuth and elevation DOA RMSEs close to and, while its angular-spread RMSEs are superior.The paper attributes the improvement with M to convergence of the estimated signal subspace toward the array response matrix A.
- SNR: At M = 100, the proposed approach's RMSEs decrease rapidly with increasing SNR, whereas other approaches decrease slowly.The results indicate a trade-off between increasing SNR and increasing the number of BS antennas for improving performance.
- Scatterers: Across the scatterer-count test, the RMSEs of the proposed approach and the comparison methods remain almost invariant.Because path gains are temporally independent and multipaths are indistinguishable, increasing multipaths does not increase the number of independent snapshots.
VI. CONCLUSIONS
The paper proposes an ESPRIT-based method for 2-D localization of multiple incoherently distributed sources in massive MIMO systems. It decouples angular parameters without search, improves with more BS antennas, and achieves comparable performance at much lower complexity.
- The proposed approach performs 2-D localization of multiple incoherently distributed sources in massive MIMO systems.
- The method does not constrain adjacent-antenna spacing and decouples the 2-D angular parameters.
- The estimation performance improves as the number of BS antennas increases.
- In representative scenarios, the proposed estimator’s complexity is less than 0.1% of that of existing methods.
- Simulations show performance comparable to other approaches in massive MIMO systems.
APPENDIX A PROOF OF PROPOSITION 1
The appendix analyzes the large-array behavior of the constructed matrix A and shows that its signal-subspace estimate converges almost surely to A's column space as M grows.
- Matrix structure: The matrix A contains three types of columns: steering vectors and their azimuth- and elevation-derivative vectors.
- Matrix scaling: The norms of the last 2K columns of A become proportional to M as M → ∞.
- Matrix structure: As M → ∞, normalized inner products between columns associated with different sources tend to zero, while each source's three vectors remain linearly independent.
- Eigenvalue argument: The factorization A = ˜A T_A uses orthonormal columns in ˜A and an upper-triangular T_A whose condition number remains bounded as M grows.
- Eigenvalue argument: A ˆR_c A^H has at most 3K nonzero eigenvalues, with corresponding eigenvectors in the column space of A.
- Conclusion: As M → ∞, the largest 3K eigenvalues of ˆR_x correspond to eigenvectors whose columns converge almost surely to the same subspace as A.
APPENDIX B DERIVATION OF THE APPROXIMATE CRB
The appendix derives an approximate Cramér–Rao bound by approximating the array manifold and covariance, defining the parameter vector and Fisher information matrix, and applying block-matrix inversion.
- Approximation: The array manifold for the varying azimuth and elevation directions is approximated before deriving the covariance representation.
- Approximation: The covariance model is expressed using D_k and B_k matrices, with D_k defined from the nominal steering vector.
- Fisher information matrix: The received signal is modeled as a zero-mean circularly symmetric complex-valued Gaussian vector.
- Fisher information matrix: Because the received signal is approximated, the appendix derives an approximate Fisher information matrix and approximate Cramér–Rao bound.
- Fisher information matrix: The parameter vector includes nominal azimuths, nominal elevations, angular spreads, and a noise-related parameter.
- Cramér–Rao bound: Applying the block matrix inversion lemma yields the approximate covariance matrix of the angular-parameter estimation error.