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Transmit Energy Focusing for DOA Estimation in MIMO Radar with Colocated Antennas

Aboulnasr Hassanien, Sergiy A. Vorobyov

arXiv:1007.0436v1cs.IT

TL;DR

The paper addresses multiple-target DOA estimation in colocated MIMO radar when target directions lie within a spatial sector. It uses transmit beamspace beamformers to focus energy and create rotationally invariant data for MUSIC, ESPRIT, or PARAFAC, and reports lower CRB and superior simulation performance than existing techniques.

  • Problem

    Multiple-target DOA estimation in MIMO radar needs methods that improve estimation performance while avoiding exhaustive searches and reduced per-antenna SNR.

  • Method

    Transmit beamformers focus multiple orthogonal waveforms within a target sector and can produce rotationally invariant matched-filtered data for MUSIC, ESPRIT, or PARAFAC.

  • Results

    The proposed technique achieves lower CRB and superior simulation performance than existing techniques, while supporting search-free DOA estimation.

  • Takeaways & Limitations

    Transmit energy focusing combines improved SNR gain with search-free rotational-invariance-based direction finding for multiple targets.

Abstract

from arXiv · show

In this paper, we propose a transmit beamspace energy focusing technique for multiple-input multiple-output (MIMO) radar with application to direction finding for multiple targets. The general angular directions of the targets are assumed to be located within a certain spatial sector. We focus the energy of multiple (two or more) transmitted orthogonal waveforms within that spatial sector using transmit beamformers which are designed to improve the signal-to-noise ratio (SNR) gain at each receive antenna. The subspace decomposition-based techniques such as MUSIC can then be used for direction finding for multiple targets. Moreover, the transmit beamformers can be designed so that matched-filtering the received data to the waveforms yields multiple (two or more) data sets with rotational invariance property that allows applying search-free direction finding techniques such as ESPRIT for two data sets or parallel factor analysis (PARAFAC) for more than two data sets. Unlike previously reported MIMO radar ESPRIT/PARAFAC-based direction finding techniques, our method achieves the rotational invariance property in a different manner combined also with the transmit energy focusing. As a result, it achieves better estimation performance at lower computational cost. Particularly, the proposed technique leads to lower Cramer-Rao bound than the existing techniques due to the transmit energy focusing capability. Simulation results also show the superiority of the proposed technique over the existing techniques.

I. INTRODUCTION

MIMO radar DOA estimation benefits from virtual-array aperture and rotational-invariance methods, but existing approaches face search cost or reduced per-antenna SNR. The paper develops transmit beamspace designs that trade aperture for SNR and focus energy toward likely target sectors.

  • Existing DOA methods: MUSIC and ESPRIT provide high-resolution DOA estimation, while PARAFAC generalizes rotational-invariance processing to higher-dimensional data arrays.ESPRIT is described as a computationally efficient special case of PARAFAC-related decomposition.
  • Existing DOA methods: Existing MIMO radar methods either require fine-grid exhaustive searches or obtain rotational invariance by partitioning the extended virtual array into overlapped subarrays.Search-free ESPRIT/PARAFAC methods avoid exhaustive search but rely on the extended-array construction.
  • MIMO radar DOA estimation: MIMO radar forms an MN-element virtual array by matched-filtering N receive channels to M transmitted waveforms, extending aperture and improving angular resolution.The virtual-array advantage arises from jointly processing waveform-matched receive data.
  • Motivation: Full waveform diversity assigns energy E/M to each waveform omni-directionally, reducing SNR per virtual antenna for fixed total transmit energy E.The resulting lower SNR is important because subspace-method accuracy approaches the CRB only at relatively high SNR.
  • Proposed transmit beamspace approach: The proposed methods increase virtual-antenna SNR by transmitting fewer higher-energy waveforms and focusing energy within spatial sectors containing likely targets.Using fewer waveforms reduces virtual-array aperture from MN to KN, where K ≤ M, creating an SNR–aperture tradeoff.
  • Proposed transmit beamspace approach: Transmit beamformers are designed to focus sector energy, improve receive-antenna SNR and angular resolution, and enable search-free ESPRIT/PARAFAC-based DOA estimation.The paper also derives CRBs to examine dependence on waveform count, focusing, and effective virtual-array aperture.
  • Reported evaluation: The paper reports that its transmit beamspace DOA techniques outperform existing techniques in simulations.The paper organizes signal modeling, beamspace design, MUSIC/ESPRIT processing, CRB analysis, and simulations across Sections II–VII.

II. MIMO RADAR SIGNAL MODEL

The model describes a colocated MIMO radar transmitting orthogonal waveforms and receiving noisy returns from multiple targets. Matched filtering separates waveform components, which are stacked into virtual-array data and used to form a sample covariance matrix.

  • The radar has M colocated transmit antennas and N colocated receive antennas, with L targets present.
  • Each transmit antenna emits a unit-energy baseband waveform, and waveforms from different antennas are orthogonal.The total energy within one pulse is E.
  • The received vector contains target returns, receive-array steering information, and zero-mean white Gaussian noise.Target reflection coefficients follow a Swerling II model: constant during a pulse and independently varying between pulses.
  • Matched filtering exploits waveform orthogonality to extract the received component associated with each transmitted waveform.
  • The extracted components are stacked into an MN × 1 virtual data vector, whose sample covariance uses Q snapshots.

III. TRANSMIT BEAMSPACE BASED MIMO RADAR SIGNAL MODEL

The transmit beamspace model focuses orthogonal waveform energy into a target sector using K beams, where K can be no larger than the number of transmit antennas. This focusing can increase virtual-element SNR and improve DOA performance under fixed total energy.

  • Transmit beamspace processing forms K directional beams over a sector Θ, transmitting an independent waveform over each beam.The beamspace dimension satisfies K ≤ M, and the total transmit energy remains fixed at E.
  • The transmit beamspace matrix C transforms the M × 1 transmit-array manifold into a K × 1 beamspace manifold.Its columns are unit-norm beamforming weight vectors.
  • Matched filtering the beamspace received signal to each waveform produces K received signal components that can be stacked into a KN × 1 virtual data vector.
  • Transmit beamspace design can focus energy in one sector or divide it among several disjoint sectors by shaping the transmit beampattern through C.
  • Increasing signal power at each virtual array element is attributed to transmit beamforming gain and the E/K energy factor per beam.The resulting SNR increase is associated with lower CRB and improved DOA estimation.

IV. TRANSMIT BEAMSPACE DESIGN

The transmit beamspace is designed to concentrate fixed transmit energy within a sector while controlling its spatial distribution and out-of-sector radiation. Two design approaches target uniform in-sector energy, reduced leakage, and improved estimation performance.

  • The section presents two methods for designing the transmit beamspace weight matrix C.
  • Unlike passive beamspace dimension reduction, transmit beamspace processing focuses energy into the desired sector instead of spreading it omni-directionally.Energy that would be radiated out of sector can be redirected into the desired region.
  • The resulting increase in signal strength within the desired sector can potentially lower the CRB and improve best achievable estimation performance.
  • The design requirements are approximately uniform energy within Θ and minimized energy transmitted outside Θ.
  • B. Spheroidal Sequences Based Transmit Beamspace Design: The spheroidal-sequence design maximizes the ratio of beamspace energy inside Θ to total radiated energy.With orthogonal beamspace vectors, the solution is formed from the K principal eigenvectors of matrix A.
  • B. Spheroidal Sequences Based Transmit Beamspace Design: For θ ∈ Θ, the combined transmit power distribution is approximately constant, although power across individual waveforms need not be uniform.The matrix can be rotated to achieve uniform per-waveform power, even if orthogonality is lost.

C. Convex Optimization Based Transmit Beamspace Design

The convex-optimization design targets transmit beamformers that provide rotationally invariant virtual data for search-free DOA estimation while limiting out-of-sector radiation. Its constraint parameter γ specifies the acceptable worst-case transmit power outside the sector.

  • The convex optimization method is proposed as a general transmit beamspace design applicable to arbitrary transmit-array geometry.
  • ESPRIT requires two received-data sets related by phase rotation, while PARAFAC uses rotational invariance among multiple received-data sets.
  • The transmit beamspace matrix C can be designed so that the virtual data vectors possess the rotational invariance required by these methods.
  • The optimization minimizes the mismatch between C^H a(θ) and a phase-shift vector while constraining out-of-sector transmit power.
  • The out-of-sector constraint imposes ∥C^H a(θ_j)∥ ≤ γ, where γ is a user-chosen bound on worst-case radiation outside Θ.γ is analogous to a stop-band attenuation parameter in bandpass filter design.

V. TRANSMIT BEAMSPACE BASED DOA ESTIMATION

The section develops MUSIC- and ESPRIT-based DOA estimators for transmit beamspace MIMO radar. Carefully designed beamspace weights provide rotational invariance for ESPRIT and can generate multiple data sets for PARAFAC.

  • Transmit beamspace processing supports subspace-based DOA estimation techniques including MUSIC and ESPRIT.
  • Transmit Beamspace Based MUSIC: The transmit beamspace covariance matrix is estimated from matched-filtered data and decomposed into signal and noise subspaces.The signal-subspace eigenvectors correspond to the largest eigenvalues, while the noise-subspace eigenvectors correspond to the smallest.
  • Transmit Beamspace Based ESPRIT: Choosing beamspace weights through (22)–(23) makes the relevant matrix relationship satisfy the rotational invariance property.
  • Transmit Beamspace Based ESPRIT: With two transmit beams, the resulting data vectors obey rotational invariance, enabling ESPRIT-based estimation of the parameter matrix Ψ.
  • Transmit Beamspace Based ESPRIT: For K > 2, proper beamspace-weight design produces more than two rotationally invariant data sets, allowing PARAFAC instead of ESPRIT.
  • Transmit Beamspace Based ESPRIT: Earlier ESPRIT methods rely on a ULA transmit array and divide total transmit energy across M waveforms, reducing SNR per virtual antenna.The cited method may also suffer performance degradation with array perturbation errors.

VI. PERFORMANCE ANALYSIS

The section analyzes transmit beamspace MIMO radar against alternative array-partitioning techniques. Comparisons address coherent gain, aperture, per-element SNR, computational complexity, and CRB.

  • Performance comparisons consider transmit subapertures, transmit array partitioning, and transmit beamspace processing.
  • Transmit array partitioning provides coherent transmit gain, whereas the transmit-subaperture technique does not.
  • The comparison evaluates effective virtual-array aperture, SNR gain per virtual element, and eigendecomposition-based DOA-estimation complexity.
  • Array Model: For a ULA transmitter and receiver with half-wavelength spacing, the transmit and receive apertures are (M −1)λ/2 and (N −1)λ/2, respectively.
  • Array Model: The resulting virtual-array model is analyzed through alternative choices of geometry and transmit-energy distribution.

1) Traditional MIMO radar:

The traditional and partitioned MIMO radar cases trade virtual-array aperture, per-element SNR, and computational cost. Partitioning can increase SNR or control aperture, but may reduce distinct virtual-array elements.

  • Traditional MIMO radar: Traditional MIMO radar has effective virtual-array aperture (M + N −2)λ/2 but only M + N −1 distinct elements.
  • Traditional MIMO radar: E/M SNR gain per virtual element can lead to poor DOA estimation performance at low SNR.
  • Traditional MIMO radar: O(M3N3) is the eigendecomposition-based DOA-estimation complexity for the traditional MIMO radar case.
  • Transmit subaperturing: TS-based MIMO radar provides M/2 times the SNR gain of traditional MIMO radar, with effective aperture controlled by the subaperture separation ζ.
  • Transmit array partitioning: The extreme two-subaperture comparison may require power amplifiers with much higher amplifying gain than using M transmit antennas.
  • Transmit subaperturing: Selecting ζ = Nλ/2 produces a 2N-element ULA with effective aperture (2N −1)λ/2 and complexity O(23N3).
  • Transmit array partitioning: TAP improves SNR per virtual element through coherent gain, but its virtual array contains N + 1 distinct elements and has aperture Nλ/2.

4) Transmit beamspace based MIMO radar:

The proposed transmit beamspace design uses two focused beams to combine broad virtual-array aperture with enhanced SNR and low eigendecomposition complexity. Its CRB framework also covers the compared techniques as special cases.

  • Transmit beamspace based MIMO radar: With K = 2 and beamspace weights designed by (22)–(23), the proposed method forms a virtual array with 2N distinct elements.
  • Transmit beamspace based MIMO radar: The proposed method has effective aperture (2N −1)λ/2 and SNR gain Gbeam · E/M.
  • Transmit beamspace based MIMO radar: O(23N3) is the eigendecomposition-based DOA-estimation complexity for the proposed transmit beamspace method.
  • Transmit beamspace based MIMO radar: The comparison table summarizes transmit beamspace MIMO radar alongside the other considered techniques.
  • Cramer–Rao Bound: The CRB expression derived for transmit beamspace processing can also compute CRBs for the other techniques summarized in Table 1.
  • Cramer–Rao Bound: Under specified statistical assumptions, the virtual data model has the form used in conventional array processing for stochastic CRB computation.
  • Special Cases: Selecting C = I_M recovers the traditional MIMO radar model, while other C choices recover TS and TAP models.

VII. SIMULATION RESULTS

Simulations compare the proposed transmit beamspace MIMO radar with traditional, TS-based, and TAP-based methods under fixed transmit energy. The proposed method combines focused energy, high per-waveform power, and large effective aperture, producing the lowest CRB and strongest estimation performance.

  • Transmit designs: The proposed spheroidal-sequence design makes individual waveform power uniform within the desired sector after vector rotation.Without rotation, total power is uniform but individual waveforms have different distributions; rotation equalizes individual waveform power.
  • CRB comparison: The proposed transmit beamspace radar achieves the lowest CRB among all methods.Its advantage combines energy focusing, high power per waveform, and the large effective aperture of the corresponding virtual array; the same conclusion applies to deterministic CRB.

B. Example 2: MUSIC-based DOA Estimation

MUSIC-based DOA estimation is evaluated through source-resolution probability and RMSE as SNR varies. The proposed transmit beamspace radar has the lowest SNR threshold and outperforms the comparison methods in RMSE.

  • Source resolution: All methods achieve 100% correct source resolution at high SNR, but resolution probability drops as SNR decreases past each method’s threshold.The SNR threshold is the level at which this transition occurs.
  • Source resolution: The proposed transmit beamspace radar has the lowest SNR threshold and the best source-resolution probability performance.TS with ζ = λ/2 has the highest threshold, while traditional MIMO and TAP have the second- and third-highest thresholds.
  • RMSE: TAP outperforms traditional MIMO at low SNR, whereas traditional MIMO is better at high SNR.The results attribute this crossover to the relative importance of per-virtual-antenna SNR gain at low SNR and effective aperture at high SNR.
  • RMSE: The proposed transmit beamspace radar outperforms all comparison methods in MUSIC-based DOA estimation RMSE.TS with ζ = λ/2 has the poorest RMSE; TS with ζ = Nλ/2 outperforms traditional MIMO and TAP.
  • Evaluation boundary: RMSE comparisons above 10° are not meaningful because the desired sector width is 10°.The paper specifically excludes the portions of curves exceeding the sector width from meaningful method comparisons.

C. Example 3: ESPRIT-based DOA Estimation

ESPRIT-based DOA estimation is tested using virtual-array subarrays and transmit-beamspace designs that preserve rotational invariance. The proposed method achieves the best source-resolution and RMSE performance across the comparisons.

  • ESPRIT setup: ESPRIT partitions the virtual array into overlapping or non-overlapping subarrays depending on the method.The traditional MIMO radar uses two overlapping subarrays of size (M − 1)N, while other methods use two non-overlapping N-element subarrays in the 2N virtual array.
  • RMSE interpretation: Large effective aperture is more important at high SNR, while high SNR gain per virtual antenna is more important at low SNR.This trade-off explains the crossover between TAP and traditional MIMO in the ESPRIT RMSE results.
  • RMSE: The proposed ESPRIT estimator outperforms all aforementioned estimators in RMSE.TAP beats traditional MIMO at low SNR but loses at high SNR, while TS with ζ = Nλ/2 outperforms both.
  • Transmit beamspace design: The proposed transmit beamspace design focuses multiple orthogonal waveforms while maintaining rotational invariance for search-free DOA estimation.Spheroidal sequences maximize SNR gain at each receive antenna; convex optimization controls out-of-sector dissipation and preserves rotational invariance.
  • Complexity: The proposed method’s computational complexity can be controlled by selecting the transmit beamspace dimension.The dimension corresponds to the number of transmit beams.
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