Applications Of Large Random Matrix Theory To Array Signal Parameters Estimation

The high performance parameter estimation method is always a front research task in the field of array signal processing,of which source enumeration and direction of arrival(DOA)estimation serve as core technologies in diverse applications,such as radar,sonar,telemetry and remote sensing.At present,most of parameter estimation methods for array signals are based on the classical statistical signal processing,which considers that the number of samples of the observed data grows to infinite while the dimension is fixed.These methods are suitable for a small sensor array which can easily meet the requirement that the sample size of the observed data is much larger than the dimension.However,a large number of sensors are more and more popular in the modern system,such as the phased array radar,the multiple input multiple output(MIMO)system and the distributed sensor network,etc.For a large array,the number of samples of the observations may be on the same order of magnitude as the number of sensors due with the limit of storage capacity and the real-time signal processing.When the methods established on the classical asymptotic system are used to process the large sensor array signals,they almost can not obtain consistent estimations of the signal parameters.Large arrays raise new challenges to the technology of signal parameters estimation.The large random matrix theory researches the asymptotic behaviors and statistical distributions of sample covariance matrices and their eigenvalues as well as eigenvectors in the general asymptotic system.The general asymptotic system assumes the sample size and the dimension of a random matrix both tend to infinite with a fixed ratio.Compared with the classical asymptotic system,the general asymptotic system is more suitable to describe the relationship between the number of samples and the number of sensors.The large random matrix theory brings new thoughts and theories to investigate the signal parameters estimation methods for a large sensor array.Recently,the spectral analysis of sample covariance matrices and the Stieltjes transformation have been used to study consistent estimations of signal parameters.Although many source enumeration and DOA estimation methods are proposed and obtain unbiased parameter estimation results when applied in processing large array signals.Compared with the maturing status of signal parameters estimation methods of the classical asymptotic system,the methods of the general asymptotic system need to be further researched and perfected.Based on the large random matrix theory,this dissertation analyzes the reasons causing performance degradation when the methods of the classical asymptotic system used in a large sensor array.This dissertation focuses on the source enumeration and DOA estimation in the situations where the number of samples is on the same order magnitude as the number of sensors,the number of samples is less than the number of samples and the non-Gaussian distributed observations.To develop and perfect the signal parameters estimation methods of the general asymptotic system,this dissertation investigates the way of realizing source enumeration and DOA estimation based on statistical distribution properties of sample covariance matrices,sample eigenvalues and sample eigenvectors.The major contributions of this dissertation are as follows.(1)A source enumeration method is proposed based on corrected Rao's score test(CRST).This method solves the problem that information theory criterion(ITC)like methods are always not suitable when the number of samples is less than the number of sensors.Different from the traditional likelihood function in the ITC like methods,the proposed method uses a statistic of the corrected Rao's score test to build a new likelihood function and avoid the negative effect from zero samples eigenvalues.The covariance matrix of the noise components of the observations is proportional to an identity matrix and can be test by the CRST statistic.The CRST statistic is formed by the presumptive noise sample eigenvalues,and it will be of norm distribution when there is no signal sample eigenvalue in the presumptive noise part.The number of signals is inferred via the generalized Bayesian information criterion(GBIC)with the CRST statistic.Simulation results show that the proposed method has a high detection probability than the compared methods when the number of samples is less than the number of sensors.(2)A source enumeration method is proposed based on the sphericity test under the Gaussian and non-Gaussian noise.Instead of joint probability of observations or sample eigenvalues distribution,this method introduces a statistic used to testing the sphericity of a positive definite covariance matrix,to overcome the limitation of the Gaussian observations assumption.Under the white noise assumption,the identity structure of the noise subspace covariance matrix can be tested by a sphericity test statistic.The observations are decomposed into signal and noise subspace components under a presumptive number of sources.When the presumptive noise subspace components do not contain signals,the corresponding sphericity test statistic will be of a certain Gaussian distribution,and the number of sources can be estimated via the GBIC.Simulation results show that the proposed method not only provides high detection probability in both the Gaussian and the non-Gaussian noise,and performs better than the compared methods when the number of sample is less than the number of sensors.(3)A covariance matrix shrinkage method is proposed to make an improvement of the DOA estimation under a large uniform linear array.This method provides a shrinkage target with Toeplitz structure and deduces a closed-form expression of the shrinkage coefficient.The shrinkage coefficient is calculated based on the unbiased estimations of the moments of a covariance matrix with Wishart distribution.The statistical property of the calculated shrinkage coefficient is discussed through theoretical analysis and simulations.The calculated shrinkage coefficient can ensure that the proposed covariance matrix estimation is a good compromise between the sample covariance matrix and the shrinkage target.Simulation results show that the performance of DOA estimation can be improved in the case of low signal-to-noise ratio(SNR)with small sample size when the proposed covariance matrix estimation used in the multiple signal classification(MUSIC).(4)A source enumeration method is proposed based on the likelihood ratio test(LRT)of a spike covariance matrix.And the MUSIC method is modified via compensating the phase transformation of the signal sample eigenvectors.The unbiased estimations of signal eigenvalues and the noise power are realized based on the phase transformation theory of the spike covariance matrix.Then the LRT statistic can be calculated using the presumptive noise sample eigenvalues.If the presumptive number of sources is equal to the truth,the LRT statistic will be of the normal distribution.The number of sources can be inferred via the GBIC with the LRT statistic.The phase transformation in the DOA estimation function of MUSIC method can be calculated using the unbiased estimations of signal eigenvalues and the noise power.The simulation results show that the proposed source enumeration method realizes a higher detection probability than the compared methods when the number of samples is on the same order magnitude as the number of sensors.And the modified MUSIC method improves the resolution of DOA estimation when used to estimate two space adjacent signals.