ICA算法及其在阵列信号处理中的应用研究
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摘要
独立分量分析(Independent Component Analysis, ICA)是解决盲源分离(Blind Source Separation, BSS)问题的主要方法之一。该方法可以在信源信号和信道参数均未知的条件下,仅利用信源信号之间相互统计独立的特性,就能从接收到的混合信号中辨识出信道参数,估计出信源信号。ICA在通信系统、语音信号处理、图像信号处理、生物医学信号、环境和金融数据的分析等领域,具有很大的应用价值。
ICA按照所处理信号的不同可以分为实数ICA和复数ICA。实数ICA是目前研究最广泛的ICA方法,主要应用于实数信号处理领域。复数ICA是近几年来部分学者为了处理复数信号而提出来的ICA方法。虽然目前有了一定的发展,但是相对于实数ICA,复数ICA的研究还不是很成熟,其理论和其应用方面的研究都有待于进一步深化和完善。
为此,本文重点研究和分析了复数ICA的基本理论,从算法的适用范围、收敛速度和实际应用角度出发,对现有的复数ICA算法进行了改进。同时本文也对实数ICA的基本理论进行了研究和分析,并从硬件实现角度对典型的实数快速ICA算法进行了改进。研究的主要内容和创新点如下：
首先,由于实数快速ICA算法硬件实现比较困难,基于Huber M估计函数的实数快速ICA算法虽然硬件实现容易,但稳健性不够好,针对这一问题,本论文提出了一种基于Tukey双权函数的硬件实现容易而且稳健的实数快速ICA算法。该算法采用稳健性较好的Tukey双权函数作为实数快速ICA算法代价函数中的非线性函数,该函数的实现只涉及到加法和乘法,因此相对于原算法,该算法硬件实现容易；相对基于HuberM估计函数的实数快速ICA算法,该算法的稳健性更好。
其次,针对强不相关变换算法仅适用于谱系数不相同的非圆信源信号的问题,提出了两种适用范围更广的复数ICA算法。这两种算法以非圆信号二阶统计量不为零的特点构造代价函数,然后分别采用不同的优化方法对代价函数进行优化。它们不仅适用于原算法所适用的谱系数不相同的非圆信源信号,而且适用于原算法所不适用的谱系数相同的非圆信源信号,扩大了算法的适用范围。
再次,针对复数快速ICA算法只适用于非高斯圆信源信号的问题,提出了一种适用范围更广的复数快速ICA算法。该算法通过修正复数快速ICA的代价函数,得到新的代价函数,并采用近似的复数牛顿迭代方法对代价函数优化。该算法不但适用于原算法所适用的非高斯圆信源信号,而且适用于原算法所不适用的非高斯非圆信源信号。另外,针对复数快速ICA算法具有二阶收敛速度,收敛速度不够快的问题,采用具有三阶收敛速度的牛顿方法对算法的代价函数进行优化,推导出了收敛速度更快的复数快速ICA算法,并将其应用到波达方向估计中。该算法不但收敛速度快,而且可以直接计算出信号的波达方向,相对传统高分辨率波达方向估计方法性能更好,分辨率更高。
最后,在没有任何信号先验信息的条件下,复数非高斯最大化算法很难选择合适的学习速率,针对这一问题,本论文提出了一种不需要设置学习速率的复数非高斯最大化算法。该算法将分离矩阵满足归一化的条件,以惩罚函数的形式引入到代价函数中,得到新的代价函数,在复数域中直接对代价函数优化。另外,针对峭度最大化盲波束形成算法也存在设置学习速率的问题,采用近似的复数牛顿方法对原代价函数进行优化,推导出了一种不需要设置学习速率的固定点峭度最大化盲波束形成算法。改进后的复数非高斯最大化算法和峭度最大化盲波束形成算法都是固定点迭代算法,都不需要设置学习速率,因此更适合在盲条件下应用。
综上所述,本文研究了实数ICA算法和复数ICA算法以及盲波束形成算法,并针对算法中存在的不足,进行了相应的改进。仿真实验证实,本文提出的改进算法,均能够获得很好的效果。
Independent component analysis is one of the most primary methods to solve the problem of blind source separation. By using mutual statistical independence property of source signals, it can separate source signals from mixed signals without any known paraemters of source signals and channel. ICA has large application value in communication system, speech signal processing, image processing, biomedicine, environment analysis, financial data analysis and other fields.
ICA can be classified as real valued ICA and complex valued ICA according to the differences of the processed signals. Real valued ICA enjoys the most extensive researching, which is mainly applied in real valued signal processing. Complex valued ICA is proposed by some researchers to process complex valued signal in recent years, which is mainly applied in the fields of array signal processing, frequency domain signal and functional magnetic resonance imaging processing. Although complex valued ICA has some development, comparied with real valued ICA, its research is not mature. Its research on theory and application need to be deeper.
In this paper, we mainly research on and analyze the basic theory of complex valued ICA and improve the complex valued ICA algorithms from their scope of application, convergence rate and practical application. At the same time, we also research on and analyze the basic theory of real valued ICA and propose an improved ICA algorithm, which can be realized easily by hardware. The content and innovation of this paper can be summarized as follows:
Firstly, to overcome the problem that real valued fast ICA algorithm is difficult to be realized by hardware, while the real valued fast ICA algorithm based on Huber M-estimator is easy, but its robustness is not good enough. We propose a new real valued fast ICA algorithm wich is easy to be realized by hardware based on tukey biweight function. It uses tukey biweight function which has better robustness as nonlinear function of real valued fast ICA algorithm and only involves addition and multiplication operation, so it can be realized easily by hardware. Compared with the FastICA algorithm based on Huber M-estimtor, the new algorithm has better robustness.
Secondly, to overcome the problem that strong-uncorrelating transform algorithm is only applicable to the non-circular source signals which have different spectrum coefficients, we propose two complex valued ICA algorithms with wider range of application. They use the property that second-order statistics are not zero to contruct cost function and optimize it by different optimization methods. The new algorithms are not only applicable to non-circular source signals with different spectrum coefficients, but also any statical independent complex valued source signals that contain non-circular signals. They extend the range of application of original algorithm.
Thirdly, to overcome the problem that complex valued FastICA algorithm is only applicable to circular source signals, we propose an improved complex valued FastICA algorithm which has wider range of application. It constructs new cost function by modifying the cost function of complex valued fast ICA algorithm, and uses approximate Newton method to optimize the new cost function. The new algorithm is not only applicable to circular signals, but also to any non-Gaussian complex valued source signals. Besides, to overcome the problem that the quadratic convergence rate of complex valued FastICA algorithm is not fast enough, we use complex Newton method that has third order convergence rate to optimize the cost function, and deduce a new complex valued FastiCA algorithm with faster convergence rate and apply it in estimating direction of the arrival signals. It not only has faster convergence rate compared with the original algorithm, but also can directly compute the direction of arrival signals, compared with traditional estimation method of the arrival signal direction with high resolution. It also has better perfeormantce and resolution.
Finally, to overcome the problem that complex valued non-Gaussian maximization algorithm need the setting of learning rate, but it is difficult to choose suitable learing rate without any system information. We propose a fixed-point complex valued non-Gaussian maximization algorithm. It constructs cost function by introducing penalty function, which is a separated matrix satisfying normalization condition to original cost function, and optimizes the cost function directly in complex valued field. Besides, kurtosis maximization blind beamforming algorithm also has the same problem of setting learning rate. To overcome the problem, we use complex valued Newton-Like method to ptimize cost function and deduce a fixed point kurtosis maximization blind beamforming algorithm without settig learning rate. The improved complex valued non-Gaussian maximization algorithm and kurtosis maximization blind beamforming algorithm are both fixed point algorithm without setting any learning rate, so they are more suitable for practical application.
In conclusion, the real valued ICA, complex valued ICA and kurtosis maximization blind beamforming algorithm are researched in this paper, and improved algorithms are proposed to overcome the problems existing in the algorithms. Experimental results indicate that all the improved algorithms could attain good results.
ICA按照所处理信号的不同可以分为实数ICA和复数ICA。实数ICA是目前研究最广泛的ICA方法,主要应用于实数信号处理领域。复数ICA是近几年来部分学者为了处理复数信号而提出来的ICA方法。虽然目前有了一定的发展,但是相对于实数ICA,复数ICA的研究还不是很成熟,其理论和其应用方面的研究都有待于进一步深化和完善。
为此,本文重点研究和分析了复数ICA的基本理论,从算法的适用范围、收敛速度和实际应用角度出发,对现有的复数ICA算法进行了改进。同时本文也对实数ICA的基本理论进行了研究和分析,并从硬件实现角度对典型的实数快速ICA算法进行了改进。研究的主要内容和创新点如下：
首先,由于实数快速ICA算法硬件实现比较困难,基于Huber M估计函数的实数快速ICA算法虽然硬件实现容易,但稳健性不够好,针对这一问题,本论文提出了一种基于Tukey双权函数的硬件实现容易而且稳健的实数快速ICA算法。该算法采用稳健性较好的Tukey双权函数作为实数快速ICA算法代价函数中的非线性函数,该函数的实现只涉及到加法和乘法,因此相对于原算法,该算法硬件实现容易；相对基于HuberM估计函数的实数快速ICA算法,该算法的稳健性更好。
其次,针对强不相关变换算法仅适用于谱系数不相同的非圆信源信号的问题,提出了两种适用范围更广的复数ICA算法。这两种算法以非圆信号二阶统计量不为零的特点构造代价函数,然后分别采用不同的优化方法对代价函数进行优化。它们不仅适用于原算法所适用的谱系数不相同的非圆信源信号,而且适用于原算法所不适用的谱系数相同的非圆信源信号,扩大了算法的适用范围。
再次,针对复数快速ICA算法只适用于非高斯圆信源信号的问题,提出了一种适用范围更广的复数快速ICA算法。该算法通过修正复数快速ICA的代价函数,得到新的代价函数,并采用近似的复数牛顿迭代方法对代价函数优化。该算法不但适用于原算法所适用的非高斯圆信源信号,而且适用于原算法所不适用的非高斯非圆信源信号。另外,针对复数快速ICA算法具有二阶收敛速度,收敛速度不够快的问题,采用具有三阶收敛速度的牛顿方法对算法的代价函数进行优化,推导出了收敛速度更快的复数快速ICA算法,并将其应用到波达方向估计中。该算法不但收敛速度快,而且可以直接计算出信号的波达方向,相对传统高分辨率波达方向估计方法性能更好,分辨率更高。
最后,在没有任何信号先验信息的条件下,复数非高斯最大化算法很难选择合适的学习速率,针对这一问题,本论文提出了一种不需要设置学习速率的复数非高斯最大化算法。该算法将分离矩阵满足归一化的条件,以惩罚函数的形式引入到代价函数中,得到新的代价函数,在复数域中直接对代价函数优化。另外,针对峭度最大化盲波束形成算法也存在设置学习速率的问题,采用近似的复数牛顿方法对原代价函数进行优化,推导出了一种不需要设置学习速率的固定点峭度最大化盲波束形成算法。改进后的复数非高斯最大化算法和峭度最大化盲波束形成算法都是固定点迭代算法,都不需要设置学习速率,因此更适合在盲条件下应用。
综上所述,本文研究了实数ICA算法和复数ICA算法以及盲波束形成算法,并针对算法中存在的不足,进行了相应的改进。仿真实验证实,本文提出的改进算法,均能够获得很好的效果。
Independent component analysis is one of the most primary methods to solve the problem of blind source separation. By using mutual statistical independence property of source signals, it can separate source signals from mixed signals without any known paraemters of source signals and channel. ICA has large application value in communication system, speech signal processing, image processing, biomedicine, environment analysis, financial data analysis and other fields.
ICA can be classified as real valued ICA and complex valued ICA according to the differences of the processed signals. Real valued ICA enjoys the most extensive researching, which is mainly applied in real valued signal processing. Complex valued ICA is proposed by some researchers to process complex valued signal in recent years, which is mainly applied in the fields of array signal processing, frequency domain signal and functional magnetic resonance imaging processing. Although complex valued ICA has some development, comparied with real valued ICA, its research is not mature. Its research on theory and application need to be deeper.
In this paper, we mainly research on and analyze the basic theory of complex valued ICA and improve the complex valued ICA algorithms from their scope of application, convergence rate and practical application. At the same time, we also research on and analyze the basic theory of real valued ICA and propose an improved ICA algorithm, which can be realized easily by hardware. The content and innovation of this paper can be summarized as follows:
Firstly, to overcome the problem that real valued fast ICA algorithm is difficult to be realized by hardware, while the real valued fast ICA algorithm based on Huber M-estimator is easy, but its robustness is not good enough. We propose a new real valued fast ICA algorithm wich is easy to be realized by hardware based on tukey biweight function. It uses tukey biweight function which has better robustness as nonlinear function of real valued fast ICA algorithm and only involves addition and multiplication operation, so it can be realized easily by hardware. Compared with the FastICA algorithm based on Huber M-estimtor, the new algorithm has better robustness.
Secondly, to overcome the problem that strong-uncorrelating transform algorithm is only applicable to the non-circular source signals which have different spectrum coefficients, we propose two complex valued ICA algorithms with wider range of application. They use the property that second-order statistics are not zero to contruct cost function and optimize it by different optimization methods. The new algorithms are not only applicable to non-circular source signals with different spectrum coefficients, but also any statical independent complex valued source signals that contain non-circular signals. They extend the range of application of original algorithm.
Thirdly, to overcome the problem that complex valued FastICA algorithm is only applicable to circular source signals, we propose an improved complex valued FastICA algorithm which has wider range of application. It constructs new cost function by modifying the cost function of complex valued fast ICA algorithm, and uses approximate Newton method to optimize the new cost function. The new algorithm is not only applicable to circular signals, but also to any non-Gaussian complex valued source signals. Besides, to overcome the problem that the quadratic convergence rate of complex valued FastICA algorithm is not fast enough, we use complex Newton method that has third order convergence rate to optimize the cost function, and deduce a new complex valued FastiCA algorithm with faster convergence rate and apply it in estimating direction of the arrival signals. It not only has faster convergence rate compared with the original algorithm, but also can directly compute the direction of arrival signals, compared with traditional estimation method of the arrival signal direction with high resolution. It also has better perfeormantce and resolution.
Finally, to overcome the problem that complex valued non-Gaussian maximization algorithm need the setting of learning rate, but it is difficult to choose suitable learing rate without any system information. We propose a fixed-point complex valued non-Gaussian maximization algorithm. It constructs cost function by introducing penalty function, which is a separated matrix satisfying normalization condition to original cost function, and optimizes the cost function directly in complex valued field. Besides, kurtosis maximization blind beamforming algorithm also has the same problem of setting learning rate. To overcome the problem, we use complex valued Newton-Like method to ptimize cost function and deduce a fixed point kurtosis maximization blind beamforming algorithm without settig learning rate. The improved complex valued non-Gaussian maximization algorithm and kurtosis maximization blind beamforming algorithm are both fixed point algorithm without setting any learning rate, so they are more suitable for practical application.
In conclusion, the real valued ICA, complex valued ICA and kurtosis maximization blind beamforming algorithm are researched in this paper, and improved algorithms are proposed to overcome the problems existing in the algorithms. Experimental results indicate that all the improved algorithms could attain good results.
引文
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