基于压缩谱理论的高效超分辨波达方向估计算法研究
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摘要
随着多重信号分类(MUSIC)算法的提出，波达角估计迎来了超分辨发展阶段的崭新一页。时至今日，超分辨波达角估计技术在经历了三十余年的发展后已取得了异常辉煌的研究成果，然而随着应用的深入，人们发现理论上性能接近完美的超分辨算法在实际工程应用中依然存在稳健性差、计算量大、对阵列结构依赖性强等诸多缺陷。
为了推进超分辨波达角估计技术的工程化进度，本文以某工程项目为研究依托，以超分辨实施的高效性和算法原理适用的广泛性为研究向导，进行全面研究和深入创新，重点解决现存主流算法运算量大、对阵列结构和信号形式依赖性强的缺点，从而为超分辨技术向实际工程应用的转化提供参考。本文的主要研究内容和工作创新如下：
基于共轭镜像辐射源思想提出了空间谱对称压缩的全新概念，进而发展了一维对称压缩超分辨算法，即SC-MUSIC算法。大量仿真实验表明：对称压缩算法继承了标准MUSIC算法对于天线阵列结构的任意适应性，改善了原算法对靠近目标的分辨力，同时将原算法的计算量至少降低了一半。为将SC-MUSIC算法推广到任意平面阵，提出了变换域DOA概念并发展了2D-SC-MUSIC算法和TD-SC-MUSIC算法。
基于子空间扰动和泰勒级数近似原理分析了对称压缩类算法的“非渐进”理论性能，在此过程中，提出了两种用于求解对称压缩类DOA估计器噪声子空间的新算法，即求和分解算法和求积分解算法。基于这两种新算法，证明了对称压缩类DOA估计器的一致性和无偏性，给出了对称压缩类DOA估计器均方误差(MSE)的封闭表达式。
利用空间角度划分、子空间映射及变换域DOA等技术提出了高阶压缩谱概念，发展了HC-MUSIC和2D-HC-MUSIC两种高阶压缩算法。高阶压缩算法是对对称压缩算法的理论深化，其相比于对称压缩类算法进一步降低了计算量并改善了分辨力。在高阶压缩类算法关键技术的突破过程中，提出了两种求取高阶压缩谱噪声子空间的新算法，即连续求和分解算法和连续求积分解算法。相比于现存算法，这两种算法的计算量更小，应用范围更广。
依据连续求和定理，利用子空间扰动和泰勒级数近似原理对高阶压缩类算法的“非渐进”理论性能展开了研究，从理论上证明了高阶压缩类DOA估计器的一致性和无偏性，给出了新估计器的MSE表达式，同时揭示了新估计器MSE和经典MUSIC算法MSE的内在联系。
将实值运算和阵列结构的任意性折衷，提出了半实值超分辨DOA估计的概念，发展了两种基于半实值运算的超分辨算法，即SU-MUSIC算法和Capon-Like算法。新算法利用阵列接收数据协方差矩阵实部和虚部分裂处理获取信号DOA，不但获得了与同类实值算法相同的计算效率和估计性能，而且摆脱了算法对于阵列结构的依赖性。在推导新算法的过程中，以理论分析揭示了协方差阵整体子空间分解、求逆与实虚部分裂后子空间分解、求逆之间的内在数学关系。
The years since the publication of the Multiple Signal Classification (MUSIC)algorithm have witnessed a growing research interest in super-resolution techniquesfor estimating the direction of arrival (DOAs) of multiple signals. With more thanthirty years of great developments, brilliant achievements have been published forsuper-resolution DOA estimate. Unfortunately, growing examinations demonstratethat most of the state-of-the-art approaches in fact suffer from poor robustness,prohibitive computational complexity and fatal dependence on array configurationsin engineering, although they indeed perform almost perfectly in laboratories.
Aiming to make super-resolution estimators more implemental in practice, thispaper, which is supported by the national project, considers the problem of DOAestimation with the concerns of low computational cost as well as no dependence onarray geometries, showing numerical novel algorithms for super-resolution DOAestimate as well as offering a theoretical basis for real-world applications. The pri-mary contributions of this paper can be summarized as follows:
A new concept of symmetrical compressed (SC) spectral is proposed based onthe idea of mirror virtual sources, giving a SC-MUSIC estimator for finding1-DDOAs of multiple narrow-band signals. Like the conventional MUSIC, SC-MUSICis suitable for arbitrary linear array structures, but it involves a factor2on the red-uction of complexity. Numerical simulations illustrate that SC-MUSIC also showsimproved resolution probability for closely-spaced sources as compared to MUSIC.SC-MUSIC is extended to estimate2D DOAs with arbitrary plane array geometriesbased on another newly developed concept of transformed domain DOA (TD-DOA),leading to a corresponding2D-SC-MUSIC algorithm accordingly.
The non-asymptotic performance of SC-MUSIC is theoretically analyzed byusing the theory of subspace perturbation and Taylor’s expansion series, for whichtwo approaches convenient for computing the noise subspace of SC-Musicale alsopresented. These two methods are referred to as sum-decomposition algorithm andproduct-decomposition algorithm, respectively. With the above results, the unbiase-dness and consistency of SC-MUSIC are theoretically proved. Furthermore, a clos-ed-form expression is derived to predict the Mean Square Error (MSE) of DOA es- timate by SC-MUSIC, which agrees closely with simulations.
A novel concept of High-order compressed (HC) spectral is proposed based onseveral key techniques including spatial dividing, subspace mapping and TD-DOA,leading to two promising estimators named HC-MUSIC and2D-HC-MUSIC, res-pectively. Although HC-MUSIC can be taken as a further developed version of SC-MUSIC, the former shows a much better efficiency to the latter, and the foundationsof the two algorithms are significantly different. During the derivations of HC-MU-SIC, two new methods for calculating the noise subspace of HC-MUSIC are pres-ented, which are named as continuous sum-decomposition algorithm and continuousproduct-decomposition algorithm, respectively. Compared with existent techniques,the proposed approaches involve much lower computational load, and can be alsoused with more scenarios.
The unbiasedness, consistency as well as the MSE for DOA estimation by HC-MUSIC is asymptotically analyzed with a finite number of snapshots, based on con-tinuous sum-decomposition algorithm. Furthermore, the relationship between MSEfor DOA estimate by conventional MUSIC and that by the newly proposed HC-MU-SIC estimator is theoretically discussed.
A new concept of DOA estimate with semi-real-valued operations is proposedby combining real-valued computations with arbitrary array structures, leading totwo novel efficient estimators named SU-MUSIC algorithm and Capon-Like tech-nique, respectively. Both the two methods exploit results behind only either the realpart-or the imaginary part-of the array output covariance matrix (AOCM), givingequivalent reductions on complexity as well as much more applicability on arrayconfigurations as compared to their real-valued versions. Besides, numerical resultsproduced during the derivations of SU-MUSIC and Capon-Like estimators firstlyreveals the mathematical relationships among the subspace decomposition-and theinverse-on the entire AOCM as well as those on only the real part-or the imaginarypart-of AOCM.
为了推进超分辨波达角估计技术的工程化进度，本文以某工程项目为研究依托，以超分辨实施的高效性和算法原理适用的广泛性为研究向导，进行全面研究和深入创新，重点解决现存主流算法运算量大、对阵列结构和信号形式依赖性强的缺点，从而为超分辨技术向实际工程应用的转化提供参考。本文的主要研究内容和工作创新如下：
基于共轭镜像辐射源思想提出了空间谱对称压缩的全新概念，进而发展了一维对称压缩超分辨算法，即SC-MUSIC算法。大量仿真实验表明：对称压缩算法继承了标准MUSIC算法对于天线阵列结构的任意适应性，改善了原算法对靠近目标的分辨力，同时将原算法的计算量至少降低了一半。为将SC-MUSIC算法推广到任意平面阵，提出了变换域DOA概念并发展了2D-SC-MUSIC算法和TD-SC-MUSIC算法。
基于子空间扰动和泰勒级数近似原理分析了对称压缩类算法的“非渐进”理论性能，在此过程中，提出了两种用于求解对称压缩类DOA估计器噪声子空间的新算法，即求和分解算法和求积分解算法。基于这两种新算法，证明了对称压缩类DOA估计器的一致性和无偏性，给出了对称压缩类DOA估计器均方误差(MSE)的封闭表达式。
利用空间角度划分、子空间映射及变换域DOA等技术提出了高阶压缩谱概念，发展了HC-MUSIC和2D-HC-MUSIC两种高阶压缩算法。高阶压缩算法是对对称压缩算法的理论深化，其相比于对称压缩类算法进一步降低了计算量并改善了分辨力。在高阶压缩类算法关键技术的突破过程中，提出了两种求取高阶压缩谱噪声子空间的新算法，即连续求和分解算法和连续求积分解算法。相比于现存算法，这两种算法的计算量更小，应用范围更广。
依据连续求和定理，利用子空间扰动和泰勒级数近似原理对高阶压缩类算法的“非渐进”理论性能展开了研究，从理论上证明了高阶压缩类DOA估计器的一致性和无偏性，给出了新估计器的MSE表达式，同时揭示了新估计器MSE和经典MUSIC算法MSE的内在联系。
将实值运算和阵列结构的任意性折衷，提出了半实值超分辨DOA估计的概念，发展了两种基于半实值运算的超分辨算法，即SU-MUSIC算法和Capon-Like算法。新算法利用阵列接收数据协方差矩阵实部和虚部分裂处理获取信号DOA，不但获得了与同类实值算法相同的计算效率和估计性能，而且摆脱了算法对于阵列结构的依赖性。在推导新算法的过程中，以理论分析揭示了协方差阵整体子空间分解、求逆与实虚部分裂后子空间分解、求逆之间的内在数学关系。
The years since the publication of the Multiple Signal Classification (MUSIC)algorithm have witnessed a growing research interest in super-resolution techniquesfor estimating the direction of arrival (DOAs) of multiple signals. With more thanthirty years of great developments, brilliant achievements have been published forsuper-resolution DOA estimate. Unfortunately, growing examinations demonstratethat most of the state-of-the-art approaches in fact suffer from poor robustness,prohibitive computational complexity and fatal dependence on array configurationsin engineering, although they indeed perform almost perfectly in laboratories.
Aiming to make super-resolution estimators more implemental in practice, thispaper, which is supported by the national project, considers the problem of DOAestimation with the concerns of low computational cost as well as no dependence onarray geometries, showing numerical novel algorithms for super-resolution DOAestimate as well as offering a theoretical basis for real-world applications. The pri-mary contributions of this paper can be summarized as follows:
A new concept of symmetrical compressed (SC) spectral is proposed based onthe idea of mirror virtual sources, giving a SC-MUSIC estimator for finding1-DDOAs of multiple narrow-band signals. Like the conventional MUSIC, SC-MUSICis suitable for arbitrary linear array structures, but it involves a factor2on the red-uction of complexity. Numerical simulations illustrate that SC-MUSIC also showsimproved resolution probability for closely-spaced sources as compared to MUSIC.SC-MUSIC is extended to estimate2D DOAs with arbitrary plane array geometriesbased on another newly developed concept of transformed domain DOA (TD-DOA),leading to a corresponding2D-SC-MUSIC algorithm accordingly.
The non-asymptotic performance of SC-MUSIC is theoretically analyzed byusing the theory of subspace perturbation and Taylor’s expansion series, for whichtwo approaches convenient for computing the noise subspace of SC-Musicale alsopresented. These two methods are referred to as sum-decomposition algorithm andproduct-decomposition algorithm, respectively. With the above results, the unbiase-dness and consistency of SC-MUSIC are theoretically proved. Furthermore, a clos-ed-form expression is derived to predict the Mean Square Error (MSE) of DOA es- timate by SC-MUSIC, which agrees closely with simulations.
A novel concept of High-order compressed (HC) spectral is proposed based onseveral key techniques including spatial dividing, subspace mapping and TD-DOA,leading to two promising estimators named HC-MUSIC and2D-HC-MUSIC, res-pectively. Although HC-MUSIC can be taken as a further developed version of SC-MUSIC, the former shows a much better efficiency to the latter, and the foundationsof the two algorithms are significantly different. During the derivations of HC-MU-SIC, two new methods for calculating the noise subspace of HC-MUSIC are pres-ented, which are named as continuous sum-decomposition algorithm and continuousproduct-decomposition algorithm, respectively. Compared with existent techniques,the proposed approaches involve much lower computational load, and can be alsoused with more scenarios.
The unbiasedness, consistency as well as the MSE for DOA estimation by HC-MUSIC is asymptotically analyzed with a finite number of snapshots, based on con-tinuous sum-decomposition algorithm. Furthermore, the relationship between MSEfor DOA estimate by conventional MUSIC and that by the newly proposed HC-MU-SIC estimator is theoretically discussed.
A new concept of DOA estimate with semi-real-valued operations is proposedby combining real-valued computations with arbitrary array structures, leading totwo novel efficient estimators named SU-MUSIC algorithm and Capon-Like tech-nique, respectively. Both the two methods exploit results behind only either the realpart-or the imaginary part-of the array output covariance matrix (AOCM), givingequivalent reductions on complexity as well as much more applicability on arrayconfigurations as compared to their real-valued versions. Besides, numerical resultsproduced during the derivations of SU-MUSIC and Capon-Like estimators firstlyreveals the mathematical relationships among the subspace decomposition-and theinverse-on the entire AOCM as well as those on only the real part-or the imaginarypart-of AOCM.
引文
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