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Efficient Transmit Beamspace Design for Search-free Based DOA Estimation in MIMO Radar

Arash Khabbazibasmenj, Aboulnasr Hassanien, Sergiy A. Vorobyov, Matthew W. Morency

arXiv:1305.4979v1cs.ITmath.IT

TL;DR

Earlier transmit-beamspace methods had nonuniform transmit power and beam-magnitude variations associated with phase rotations. This paper designs beamspace matrices for search-free DOA estimation, and simulations report improved performance over traditional MIMO radar.

  • Problem

    Earlier methods had nonuniform transmit power and phase rotations accompanied by beam-magnitude variations that affect DOA estimation.

    Source: §I · p. 3

  • Method

    The paper structures the transmit beamspace matrix to establish rotational invariance and support search-free DOA estimation at the receiver.

    Source: §I · p. 3, §VI · p. 25

  • Results

    The proposed joint and SDD designs improve DOA estimation performance compared with traditional MIMO radar.

    Source: §I · p. 4, §VI · p. 28

  • Takeaways & Limitations

    The design supports search-free DOA estimation at the receiver while enforcing uniform transmit-array power.

    Source: §I · p. 3, §VI · p. 25

Abstract

from arXiv · show

In this paper, we address the problem of transmit beamspace design for multiple-input multiple-output (MIMO) radar with colocated antennas in application to direction-of-arrival (DOA) estimation. A new method for designing the transmit beamspace matrix that enables the use of search-free DOA estimation techniques at the receiver is introduced. The essence of the proposed method is to design the transmit beamspace matrix based on minimizing the difference between a desired transmit beampattern and the actual one under the constraint of uniform power distribution across the transmit array elements. The desired transmit beampattern can be of arbitrary shape and is allowed to consist of one or more spatial sectors. The number of transmit waveforms is even but otherwise arbitrary. To allow for simple search-free DOA estimation algorithms at the receive array, the rotational invariance property is established at the transmit array by imposing a specific structure on the beamspace matrix. Semi-definite relaxation is used to transform the proposed formulation into a convex problem that can be solved efficiently. We also propose a spatial-division based design (SDD) by dividing the spatial domain into several subsectors and assigning a subset of the transmit beams to each subsector. The transmit beams associated with each subsector are designed separately. Simulation results demonstrate the improvement in the DOA estimation performance offered by using the proposed joint and SDD transmit beamspace design methods as compared to the traditional MIMO radar technique.

I. INTRODUCTION

The paper proposes transmit beamspace designs for colocated MIMO radar that support search-free DOA estimation while enforcing uniform transmit power. Simulations report improved DOA estimation performance over traditional MIMO radar for both joint and spatial-division designs.

I. INTRODUCTION

  • Earlier methods had nonuniform transmit power and phase rotations accompanied by beam-magnitude variations that affect DOA estimation.The paper identifies these issues as shortcomings of the methods in.Source: §I · p. 3
  • The beamspace matrix is designed to match a desired beampattern while enforcing uniform power across transmit-array elements.The desired pattern may have an arbitrary shape and cover one or more spatial sectors.Source: §I · p. 3
  • The formulation supports an even, otherwise arbitrary number of waveforms and uses semidefinite-programming relaxation to obtain a convex problem.The convex problem can be solved efficiently using methods such as interior-point methods.Source: §I · pp. 3–4
  • A specific beamspace structure establishes transmit-array rotational invariance, enabling search-free DOA estimation at the receiver.The design pairs beams using conjugate-flipped weight vectors and requires rotational invariance within each pair.Source: §I · pp. 3–4
  • Unlike the earlier method, the proposed design gives paired beams identical magnitude responses in every direction and ensures uniform element power.It also uses accumulated phase rotations across all transmit beams and optimizes only half the beamspace-matrix entries, reducing computational load.Source: §I · p. 4
  • Spatial-division design divides the domain into subsectors and assigns a subset of transmit-beam pairs to each subsector.The beams are designed separately, allowing independent post-processing across subsectors with performance comparable to joint design.Source: §I · p. 4
  • The joint and SDD designs improve DOA estimation performance compared with traditional MIMO radar.The SDD design is reported to have performance comparable to the joint design.Source: §I · p. 4

II. SYSTEM MODEL AND MAIN IDEA

The paper designs the transmit beamspace matrix to shape the beampattern while enforcing rotational invariance, enabling search-free DOA estimation at the receiver. The system model describes orthogonal transmit waveforms and echoes from multiple targets.

II. SYSTEM MODEL AND MAIN IDEA

  • The M × K beamspace matrix assigns a transmit beamforming weight vector to each orthogonal waveform.Matched filtering the received signal against each waveform yields the virtual data vectors used in subsequent processing.Source: §II · pp. 5–7
  • Designing the beamspace matrix enforces the rotational invariance property (RIP), which supports computationally efficient, search-free DOA estimation such as ESPRIT.The approach also allows practical design requirements such as equal power across transmit elements.Source: §II · p. 6
  • The system model treats received echoes as reflections from L narrowband far-field targets in noise, with target reflection coefficients following the Swerling II model.The model assumes each coefficient remains constant during one pulse and changes from pulse to pulse.Source: §II · p. 6
  • For each target, paired virtual data components have the same magnitude and differ by a phase rotation that can be used to estimate its DOA.The receiver obtains these components by matched filtering against the orthogonal waveforms.Source: §II · pp. 6–7

III. PROBLEM FORMULATION

The design minimizes the difference between a desired and actual beampattern while imposing equal power across transmit elements and rotational invariance. The resulting formulation is non-convex, motivating a structured beamspace matrix and SDP relaxation.

III. PROBLEM FORMULATION

  • The design minimizes the difference between the desired and actual beampattern while enforcing equal transmit power across array elements.The desired pattern is defined over the spatial domain, and each antenna is constrained to transmit Pt/M power.Source: §III · p. 8
  • Uniform element power is practically motivated by transmit antennas using amplifiers with the same dynamic power range.Allowing large power variation across elements can degrade system performance because of amplifier nonlinearities.Source: §III · p. 8
  • The beamspace matrix is also constrained to enforce RIP between paired transmit beams, enabling search-free DOA estimation.The constraints must hold across directions, which motivates imposing a specific structure on W.Source: §III · pp. 8–9
  • The resulting optimization is non-convex and difficult to solve efficiently, so the paper proposes a structured matrix and SDP relaxation.The equal-power and RIP constraints are non-convex, and enforcing the latter across all directions makes the formulation difficult.Source: §III · p. 9

A. Two Transmit Waveforms

For two transmit waveforms, the paper imposes a conjugate-flipped beam structure to guarantee RIP. It then uses SDP relaxation to obtain a convex approximation to the beampattern-design problem.

A. Two Transmit Waveforms

  • With two orthonormal waveforms, a beam and its conjugate-flipped counterpart have equal-magnitude responses, differing only by a phase rotation and thereby satisfying RIP.The structure makes the RIP constraint implicit in the two-waveform optimization.Source: A. Two Transmit Waveforms · pp. 9–10
  • The design approximates the spatial domain with a finite set of uniformly or nonuniformly chosen directions for optimization.The chosen directions approximate the full angular interval [−π/2, π/2].Source: A. Two Transmit Waveforms · p. 10
  • The two-waveform design is a non-convex QCQP, which the paper relaxes to a convex SDP by dropping the rank-one constraint.If the relaxed solution has rank greater than one, randomization produces candidate vectors that are mapped to feasible points and ranked by the original objective.Source: A. Two Transmit Waveforms · pp. 11–12

B. Even Number of Transmit Waveforms

For an even number of transmit waveforms, the paper groups the virtual data into two sets and imposes paired beam structure to obtain RIP. This generalization offers more degrees of freedom and uses SDP relaxation for the resulting design.

B. Even Number of Transmit Waveforms

  • The multi-waveform formulation provides more degrees of freedom for obtaining better performance than the two-waveform case.The paper constructs two summed data vectors whose target components satisfy the phase relation used for DOA estimation.Source: B. Even Number of Transmit Waveforms · pp. 12–13
  • For even K, pairing beams as conjugate-flipped weight vectors guarantees RIP between the two groups of virtual data vectors.The groups contain the first K/2 and last K/2 data vectors; target components have equal magnitude and differ only in phase.Source: B. Even Number of Transmit Waveforms · pp. 13–14
  • The multi-waveform design is approximated by dropping rank-one constraints to form a convex problem, then using randomization to recover weight vectors.The relaxed matrices are solved separately, and candidate rank-one solutions are mapped to feasible points before selecting the best one.Source: B. Even Number of Transmit Waveforms · p. 15

C. Optimal Rotation of the Transmit Beamspace Matrix

The optimal beamspace solution can be rotated without changing its transmit beampattern, and the paper uses this freedom to improve energy preservation in the desired sectors. The rotation is selected to increase the squared magnitude of the signal-component sums used for DOA estimation.

C. Optimal Rotation of the Transmit Beamspace Matrix

  • Because problem (38) has nonunique solutions, the authors choose an optimal solution with better energy preservation.After an approximate solution is obtained, they still seek the rotation that gives the best energy preservation.Source: C. Optimal Rotation of the Transmit… · p. 15
  • The signal-component sums matter because DOA is estimated from phase differences between the target's signal components in the newly defined vectors.The paper illustrates the rotation optimization for K=4.Source: C. Optimal Rotation of the Transmit… · pp. 15–16
  • A unitary rotation of the beamspace columns preserves the transmit beampattern and antenna-element power distribution while changing the beams.The method applies this rotation after designing the beamspace matrix with the desired beampattern and rotational invariance property.Source: C. Optimal Rotation of the Transmit… · p. 16
  • The rotation maximizes the integrated squared magnitude of the signal-component sums over the desired sectors, since small sums degrade DOA estimation.The targets' directions are unknown beforehand, so the design maximizes this quantity over the sector rather than at known target locations.Source: C. Optimal Rotation of the Transmit… · pp. 15–16

U tr(UEUHD) (46)

The rotation optimization is posed over a Grassmannian manifold, and a steepest-descent-based algorithm can be used to address it.

U tr(UEUHD) (46)

  • Optimization problem (46) is an optimization over the Grassmannian manifold.The paper points to existing steepest-descent-based algorithms for addressing the problem.Source: U tr(UEUHD) (46) · p. 17
  • An existing steepest-descent-based algorithm can be used to address the optimization problem.The cited algorithm is described as steepest-descent-based.Source: U tr(UEUHD) (46) · p. 17

D. Spatial-Division Based Design (SDD)

The spatial-division based design (SDD) avoids incoherent summation by assigning separate beam pairs to different parts of the desired sector. It designs those pairs separately with nonoverlapping beampatterns and equal energy across covered parts.

D. Spatial-Division Based Design (SDD)

  • SDD designs different transmit-beam pairs separately to avoid incoherent summation of the signal terms.The alternative is presented as an easier way to design the beamspace matrix than designing all transmit beams jointly.Source: D. Spatial-Division Based Design (SDD) · pp. 17–18
  • The beamforming vectors' beampatterns do not overlap and cover different parts of the desired sector with equal energy.Each pair consists of a beamforming vector and its paired conjugate vector.Source: D. Spatial-Division Based Design (SDD) · p. 18

V. SIMULATION RESULTS

In Example 1, nonuniform transmit power across waveforms severely degrades DOA estimation even when the transmit beampattern is unchanged. The simulations also compare designs across target sectors of varying widths under fixed transmit power and repeated Monte-Carlo trials.

V. SIMULATION RESULTS

  • Across 500 independent Monte-Carlo runs, simulations use a uniform linear transmit array with M=10 antennas, a non-uniform linear receive array with N=10 elements, zero-mean white Gaussian noise, and constant total transmit power Pt=M.Source: §V · pp. 18–19
  • Examples 1–4 use, respectively, one centrally located 20° sector, two separated 20° sectors, one centrally located 10° sector, and one centrally located 30° sector; each uses two optimal waveforms.Source: §V · p. 18
  • The comparisons include traditional MIMO radar with uniform transmit power density and the proposed jointly optimal transmit beamspace design.The paper also identifies SDD as an alternative tested design, while omitting it in examples where two waveforms suffice or adjacent divided sectors risk sidelobe-related energy loss and degradation.Source: §V · pp. 18–19

A. Example 1 : Effect of the Uniform Power Distribution Across the Waveforms

  • Nonuniform transmit power across waveforms severely degrades DOA estimation, even when the transmit beampattern is unchanged.Example 1 compares the proposed method using W0 and W0U2×2; the RMSE curves are shown in Fig. 1.Source: A. Example 1 : Effect of the Uniform… · p. 19

B. Example 2 : Two Separated Sectors of Width 20◦Degrees Each

For two separated sectors, joint and spatial-division designs outperform traditional MIMO in reported RMSE and target resolution. In the single-sector example, the proposed design also improves beampattern gain and moderate- and high-SNR RMSE.

B. Example 2 : Two Separated Sectors of Width 20◦Degrees Each

  • The optimal transmit beamspace method provides the most even power concentration in desired sectors, while SDD concentrates power beyond traditional MIMO but distributes energy unevenly, with one sector peaking at 15 dB.Source: B. Example 2 : Two Separated Sectors… · p. 20
  • Both joint and SDD designs have lower RMSE than traditional MIMO, with SDD performing very close to the jointly optimal design.This comparison is reported for two separated sectors.Source: B. Example 2 : Two Separated Sectors… · p. 21
  • As SNR increases, target resolution transitions from near-zero failure to near-one success at the lowest SNR for jointly optimal transmit beamspace design, second-lowest for SDD, and highest for traditional MIMO.Source: B. Example 2 : Two Separated Sectors… · p. 21

C. Example 3 : Single and Centrally Located Sector of Width 10◦Degrees

  • In the single-sector example, the proposed design has 5 dB higher transmit power gain than traditional MIMO and lower RMSE at moderate and high SNR.The tested sector is [−10°, 0°]; only two waveforms are used, so the comparison includes the proposed design and traditional MIMO.Source: C. Example 3 : Single and Centrally… · p. 21
  • For the single-sector test, the beamspace method saturates at 3° because each target is 3° from the sector edge; closely spaced-target results agree with Example 2.The resolution test places targets at −3° and −1°.Source: C. Example 3 : Single and Centrally… · p. 22

D. Example 4 : Single and Centrally Located Sector of Width 30◦Degrees

In Example 4, the jointly optimal transmit beamspace design has lower RMSE than traditional MIMO radar. The conclusion describes a paired-beam design and an SDD alternative that improve DOA estimation performance over traditional MIMO radar.

D. Example 4 : Single and Centrally Located Sector of Width 30◦Degrees

VI. CONCLUSION

  • The beamspace matrix minimizes the difference between desired and actual beampatterns, with an even but otherwise arbitrary number of waveforms designed jointly in pairs.Source: §VI · pp. 25–26
  • Each beam pair has rotational invariance, with phase rotation and exactly matching magnitude responses across spatial directions.Source: §VI · p. 26
  • SDD assigns transmit beamspace pairs to spatial subsectors, enabling independent postprocessing with DOA performance comparable to joint design.Source: §VI · p. 27
  • The proposed joint and SDD transmit beamspace designs improve DOA estimation performance over traditional MIMO radar.Source: §VI · p. 28
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