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Low-Complexity and High-Resolution DOA Estimation for Hybrid Analog and Digital Massive MIMO Receive Array
Feng Shu, Yaolu Qin, Tingting Liu, Linqing Gui, Yijin Zhang, Jun Li, Zhu Han
TL;DR
Large fully digital receive arrays offer high-resolution DOA estimation but impose high RF-chain cost. This paper develops structured HAD phase-alignment and Root-MUSIC methods with limited candidate searching. Simulations show that HDAPA and Root-MUSIC-HDAPA achieve the hybrid CRLB at substantially lower complexity than pure linear-search methods.
Problem
Large-scale digital receive arrays provide high-resolution DOA estimation but have excessive beamforming computation, circuit complexity, and cost; using HAD structures for spatial-spectrum DOA estimation remains challenging.
Method
The paper proposes HADPA and HDAPA phase-alignment estimators plus Root-MUSIC-HDAPA, which combines an approximately analytical solution with limited searching over feasible directions.
Results
Root-MUSIC-HDAPA and HDAPA achieve the hybrid CRLB with substantially lower complexity than pure linear-search methods such as APA.
Takeaways & Limitations
The proposed Root-MUSIC-HDAPA and HDAPA methods balance DOA accuracy, computational complexity, and number of time blocks for HAD receive arrays.
Abstract
from arXiv · showhide
A large-scale fully-digital receive antenna array can provide very high-resolution direction of arrival (DOA) estimation, but resulting in a significantly high RF-chain circuit cost. Thus, a hybrid analog and digital (HAD) structure is preferred. Two phase alignment (PA) methods, HAD PA (HADPA) and hybrid digital and analog PA (HDAPA), are proposed to estimate DOA based on the parametric method. Compared to analog phase alignment (APA), they can significantly reduce the complexity in the PA phases. Subsequently, a fast root multiple signal classification HDAPA (Root-MUSIC-HDAPA) method is proposed specially for this hybrid structure to implement an approximately analytical solution. Due to the HAD structure, there exists the effect of direction-finding ambiguity. A smart strategy of maximizing the average receive power is adopted to delete those spurious solutions and preserve the true optimal solution by linear searching over a set of limited finite candidate directions. This results in a significant reduction in computational complexity. Eventually, the Cramer-Rao lower bound (CRLB) of finding emitter direction using the HAD structure is derived. Simulation results show that our proposed methods, Root-MUSIC-HDAPA and HDAPA, can achieve the hybrid CRLB with their complexities being significantly lower than those of pure linear searching-based methods, such as APA.
I. INTRODUCTION
Large receive arrays improve DOA resolution but make digital beamforming costly, motivating HAD architectures. The paper models subarray outputs as a virtual digital array and proposes lower-complexity DOA estimators.
- Large-scale digital receive arrays provide high-resolution DOA estimation but incur excessive beamforming computation, circuit complexity, and cost.
- HAD beamforming balances beamforming computation, hardware cost, and implementation complexity for large receive arrays.
- Using each subarray output as a virtual antenna models the HAD array as a large digital virtual array for spatial-spectrum DOA estimation.
- HADPA and HDAPA exploit the subarray structure to estimate DOA with lower complexity than conventional APA.
- Root-MUSIC-HDAPA supplies an approximately analytical solution, while limited candidate-direction searching resolves HAD direction-finding ambiguity.
III. PROPOSED LOW-COMPLEXITY
The conventional APA estimator maximizes receive power through exhaustive analog phase searching. Its accuracy improves with finer stepsizes, but approaching the CRLB requires high computational cost.
- A. Conventional APA: APA computes N phase values per search step for the N RF-chain phase shifters.
- A. Conventional APA: APA aligns the analog optimizing vector with the array manifold so antenna signals coherently combine and maximize output receive power.
- A. Conventional APA: APA searches direction angles from −π/2 to π/2 using Q subintervals with stepsize ∆θ.
- A. Conventional APA: APA requires L(Q+1)KM floating-point operations because new analog phases and receive data are needed at every search point.
- A. Conventional APA: To approach the CRLB, APA needs a sufficiently small ∆θ, which increases Q and computational amount.
B. Proposed Low-Complexity HADPA DOA Estimator
HADPA lowers phase-search complexity by decomposing each phase into element-specific and subarray-common components, applying APA followed by DPA.
- B. Proposed Low-Complexity HADPA DOA Estimator: APA exhaustively searches the full angle range and computes N phase values simultaneously, causing high complexity.
- B. Proposed Low-Complexity HADPA DOA Estimator: HADPA decomposes αk,m into an element-specific phase and a common subarray phase.
- B. Proposed Low-Complexity HADPA DOA Estimator: The decomposition implements phase alignment in two steps: APA first and DPA second.
- B. Proposed Low-Complexity HADPA DOA Estimator: The second DPA step removes the common factor associated with each subarray before forming the receive output.
- B. Proposed Low-Complexity HADPA DOA Estimator: HADPA obtains the maximum receive power after phase alignment and comparison of candidate outputs.
C. Proposed Low-Complexity HDAPA DOA Estimator
HDAPA reverses the alignment order, using DPA before APA to generate a finite set of candidate directions and select the maximum-power solution.
- C. Proposed Low-Complexity HDAPA DOA Estimator: HDAPA first performs exhaustive DPA using one data block, then applies APA across M candidate directions using the next M blocks.
- C. Proposed Low-Complexity HDAPA DOA Estimator: Digital beamforming permits arbitrarily small ∆θ during the DPA search.
- C. Proposed Low-Complexity HDAPA DOA Estimator: The periodic virtual-array pattern produces M feasible estimated directions, including pseudo-solutions.
- C. Proposed Low-Complexity HDAPA DOA Estimator: HDAPA evaluates the candidate directions and selects the angle yielding the maximum receive power.
- C. Proposed Low-Complexity HDAPA DOA Estimator: Because digitally processed signals can be stored, DPA requires one block for a full angle-range search instead of APA’s Q+1 blocks.
IV. PROPOSED LOW-COMPLEXITY HYBRID ROOT-MUSIC-HDAPA ESTIMATOR AND HYBRID CRLB
The proposed Root-MUSIC-HDAPA estimator combines virtual-array modeling, MUSIC spatial-spectrum estimation, polynomial rooting, and finite candidate selection to estimate DOA with low complexity despite hybrid-array ambiguity.
- Root-MUSIC-HDAPA estimator: Root-MUSIC-HDAPA treats each subarray output as a virtual antenna and models the HAD array as a large digital virtual array after digital phase alignment.The virtual array has manifold a_M(θ_0), while g(θ_0) is a common factor from summing elements within each subarray.
- Root-MUSIC-HDAPA estimator: The method forms a MUSIC pseudo-spectrum from the virtual-array covariance matrix and its signal- and noise-subspace decomposition.The noise subspace is formed from the K−1 smallest singular-value vectors of R_yy.
- Root-MUSIC-HDAPA estimator: Instead of pure linear search, Root-MUSIC-HDAPA converts the near-zero MUSIC denominator into a polynomial whose degree is 2K −2 and solves for its roots.Reciprocal-conjugate root symmetry is used when identifying candidate roots and associated emitter directions.
- Root-MUSIC-HDAPA estimator: The digital beamformer removes pseudo-solutions, while periodicity creates M feasible direction candidates that must be disambiguated.The method evaluates receive-power values for the M candidates and selects the direction associated with the largest value.
- Root-MUSIC-HDAPA estimator: The resulting estimated direction is the candidate associated with the largest element of the computed receive-power set.This finite candidate search completes the Root-MUSIC-HDAPA estimation process.
B. Hybrid CRLB
The paper derives a hybrid CRLB for DOA estimation in the HAD receive structure with a single emission source and a ULA.
- Hybrid CRLB: The hybrid CRLB is derived to evaluate the proposed HDAPA and Root-MUSIC-HDAPA estimators.The derivation is based on equation (43) and is provided in Appendix A.
- Hybrid CRLB: For a HAD beamforming structure with a single emission source and ULA, the variance of an unbiased DOA estimator is lower bounded by the stated hybrid CRLB.The result is presented as Theorem 1.
C. Complexity Analysis and Comparison
The complexity comparison emphasizes that Root-MUSIC-HDAPA and HDAPA use fewer time-domain blocks than HADPA and APA, reducing overall computational burden when M is much smaller than Q.
- Complexity comparison: Root-MUSIC-HDAPA requires computational amount characterized by the expression given after equation (37).The supplied passage introduces the corresponding FLOP analysis without exposing the complete expression.
- Complexity comparison: M + 1 time blocks are required by Root-MUSIC-HDAPA and HDAPA, whereas HADPA and APA require Q + 1 blocks.The first pair is independent of stepsize, while Q depends on stepsize and is generally larger than M.
- Complexity comparison: Computational complexity is a linear function of the corresponding number of time blocks for each method.This relationship is summarized in Table I's complexity comparison.
V. SIMULATION RESULTS
The simulations evaluate RMSE, CRLB attainment, antenna and snapshot effects, and computational complexity for the proposed hybrid DOA estimators. Root-MUSIC-HDAPA achieves the lowest complexity while the proposed methods attain the hybrid-structure CRLB under the reported settings.
- RMSE and CRLB: As stepsize decreases, the three linear-search methods improve RMSE; APA and HADPA approach the fully-digital CRLB, whereas HDAPA and Root-MUSIC-HDAPA converge to the hybrid CRLB.Smaller stepsizes increase complexity.
- RMSE and CRLB: When M = 8 and N = 32, Root-MUSIC-HDAPA reaches the CRLB curve when SNR is larger than 5dB.As M increases, RMSE degrades gradually and the corresponding CRLB value increases.
- Parameter effects: As the total number of antennas N increases, the accuracy of Root-MUSIC-HDAPA improves accordingly for fixed L = 32 and M = 4.The antenna-count experiment varies N from 32 to 128.
- Parameter effects: As the number L of snapshots increases, RMSE improves and reaches the corresponding CRLB across SNR values of 0dB, 5dB, and 10dB.The reported trend holds for the tested SNR and snapshot settings.
- Complexity: Root-MUSIC-HDAPA has the lowest complexity, with near-an-order-of-magnitude lower complexity than HDAPA and lower complexity than APA and HADPA.The complexity advantage remains as antenna count increases; its search directions are fixed independently of resolution requirement.
- Overall findings: Root-MUSIC-HDAPA and HDAPA achieve the hybrid-structure CRLB with dramatically lower complexity than HADPA and APA, while HADPA reaches the fully-digital CRLB at higher complexity.The proposed Root-MUSIC-HDAPA combines an approximately closed-form step with limited finite-direction linear searching.
APPENDIX A DERIVATION OF CRLB FOR HYBRID STRUCTURE
The appendix derives the Fisher information matrix for the hybrid analog-and-digital receive array by expanding its subarray and Kronecker-product structure. It reduces the required matrix terms using the array manifold and its derivative before assembling the CRLB.
- Fisher information matrix: The hybrid-structure Fisher information matrix is derived from the receive-array model and its constituent terms F1, F2, and F3.The derivation computes these terms through expressions involving the covariance matrix, array manifold, and derivative matrices.
- Manifold expansion: The array manifold a(θ) and its derivative with respect to θ are used to expand products such as aHBDa and related Fisher-information terms.The derivation abbreviates a(θ) and its derivative as a and ȧ for convenience.
- Hybrid-array structure: The block-diagonal matrix B consists of K M ×M all-ones matrices, enabling the hybrid array structure to be represented with Kronecker products.The appendix introduces EM as the M ×M all-ones matrix and uses B = IK ⊗ EM in subsequent expansions.
- Term reduction: The appendix progressively reduces aHBDa, F1, F2, and F3 through matrix identities and substitutions into the Fisher information matrix.The final substitution combines the separately derived terms into the expression for F.
A DAEMaA) + Tr(aH
The final appendix steps simplify the Fisher information matrix and state the resulting Cramer-Rao lower bound for the hybrid receive-array structure.
- Fisher information matrix: The derived expressions for F1, F2, and F3 are substituted into the Fisher information matrix to obtain its simplified form.This completes the algebraic reduction of the hybrid-structure Fisher information matrix.
- CRLB: The Cramer-Rao lower bound is then given for direction estimation with the hybrid analog-and-digital receive array.The appendix identifies this expression as the completion of the hybrid-structure CRLB derivation.