Virtual Extension Of Co-arrays Exploiting Coprime Arrays For Underdetermined DOA Estimation

Finding more number of sources than sensors which is known as underdetermined DOA estimation is very crucial in case of limited resources like limited number of sen-sors and shortage of spaces for sensors' distribution.The primary focal point of this thesis is to contemplate undetermined observation models in array processing where the proportion of unknown sources obtained concurrently by the array can be substantially greater than the number of physical sensors.Clearly,there exists a higher degree of re-dundancy in Uniform Linear Array(ULA)which should be resolved towards minimum redundancy.A recurring question in array design for both signal reception and spatial spectrum estimation is that of how to beneficially deploy the elements of a sparse array in order to optimally sample the spatial frequency spectrum.So,the challenge exists in finding the different sets of locations with consecutive integers and minimum repetitions ensuring optimal spatial sample spacings.The fundamental concern of the research is to deploy the array elements more suitably to detect more sources than imposed by the theoretical limit which directly depends on the number of array elements.The array configurations are exploited to get the advantages of detecting increased number of sig-nals with highest possible resolution.To make array configurations cost effective,for a given cost,it is pre-requisite to maximize the array aperture dimensions while plac-ing minimum sensors simultaneously achieving maximum higher resolution precisely diminishing the redundancy.The main contribution of this dissertation is to mitigate the challenge of increas-ing the array aperture lengths from a limited number of array elements to achieve the highest possible resolution in such a way that the array correlation matrix will contain minimum possible repeatable entries by reducing the number of redundant spacings present in the array.There is a representative example of class of arrays called the Min-imum Redundancy Array(MRA)which achieves minimum redundancy but this class of arrays needs an exhaustive searching program to determine the locations of the array elements.Moreover,there is no systematic approach or specified formula for design-ing an MRA with unknown sensors' distributions.Co-array equivalence plays a pivotal role in the layout of distinct sparse structures for undetermined estimation of DOA.Re-cently,coprime array has been proposed which employs this concept to boost the DOA estimation capability beyond the theoretical limit of the number of sources that can be detected with a certain number of antenna elements.Moreover,Signal autocorrelation can be projected at a much more sparse spacing than the physically dense spacing of the sample,accordingly noise sinusoids can be predicted more appropriately.Coprime sampling utilizes a pair of coprime factors to undersample the signal resulting a longer virtual linear array by deriving the difference co-array of a coprime array.Our research approach is based on the idea of estimating the DO As from the outputs of a "virtual array" which is obtained from the co-array equivalence of novel array geometry based on coprime array.In the first work,an optimal coprime structure is designed to form a larger size con-tinuous virtual array using only the basic configuration of coprime array by exploiting the difference and sum co-array without the need of additional array elements or fre-quencies at low-cost.In our work,coprime geometries of non-uniform linear arrays have been proposed to increase the degrees of freedom of the array by exploiting the covariance matrix of the received signals.Thus,we can achieve larger array aperture by exploiting both difference and sum co-array concurrently.Specifically,MN+2M+1 non-negative consecutive lags could be attained using only M N-1 physical sen-sors by the concurrent use of difference and sum co-array to form a continuous virtual array.In this work,using the method of Virtual Extension of Coprime Array imbibing Difference and Sum(VECADS),the achievable number of non-negative consecutive lags is larger than the conventional coprime array which is MN+M obtained by us-ing 2M+N-1 number of sensors.It gives us facilities to be applied in systems where the available space and power are limited.As in coprime arrays there exist some holes,which could stave off the full co-array to be implemented with Multiple Signal Classification(MUSIC)algorithm for DOA estimation as MUSIC does not outfit in the co-array encompassing holes.Till now consecutive lags have been extracted from coprime array to obtain ULA like virtual array which prevents to be used full co-array length.Full array length can be ensured as ULA by fulfilling the holes of co-array using suitable method.Therefore,new investigations on recovering the holes are significant to use MUSIC algorithm.Consequently,a novel technique to mitigate the problem is proposed in this thesis.In the second work,a novel array structure is proposed which is capable of provid-ing a dramatic increase in the degrees of freedom(DOF)than the basic conventional coprime array.Henceforth,more sources than the actual number of physical sensors can be resolved significantly.A direction finding technique is developed which uses the outputs of a continuous virtual array,computed from the novel array geometry based on difference sets of conventional coprime array using an interpolation procedure to-wards higher resolution DOA estimation.An optimal new augmentation technique with Translocated and Axis Rotated Compressed Subarrays(CATARCS)is designed on the coprime array concept to enhance the resolution of DOA estimation as well as the DOF.These outputs are obtained by a rank-incremented iterative power factorization inter-polation technique which is first time used in our work for DOA estimation.Given measurements of the difference array outputs of covariance matrix,the interpolator can be used in principle to compute the outputs of the missing elements of the covariance matrix for full length virtual array.In our research work,the covariance matrix of the virtual array is computed directly from the covariance matrix of the real array with miss-ing elements.Hence,the proposed technique takes full advantage of the longer virtual array.Successively,our work extends to get longer virtual arrays using fewer sensors by proposing another novel array geometry.In the third contribution,a novel array structure exploiting coprime arrays is pro-posed which can be very proficient to determine the number of consecutive lags in proportion with the number of array elements.The proposed method comprises novel array structure by configuring three subarrays positioned in alignment with some pre-scribed values.By increasing array elements in third subarray while keeping other sub-arrays fixed,explicit number of consecutive lags could be obtained proportionately.The proposed method offers maximization of consecutive lags in remarkable number by calculating the fourth order difference co-array unifying interpolation.Forth order difference co-array is achieved by exploiting the second order difference co-array twice.The consideration of third subarray in addition with two coprime subarrays leads to a novel array structure which can significantly enhance DOF.An effective interpolation technique Nuclear Norm Minimization(NNM)is considered to fill the holes subsisting in the virtual co-array in order to exploit full virtual co-array length.This interpolation method uses convex framework which is trackable and very simple to implement yield-ing a freedom of fixing any predefined extra tuning parameter.The proposed method in this work is named as VEFODCI which stands for virtual extension of coprime arrays by exploiting fourth order difference co-array with interpolation.Sparse Bayesian Learn-ing(SBL)is used for DOA estimation exploiting the proposed novel array structure by imposing interpolation to fill the holes.The array geometry of sensor distributions proves that the proposed method used in this thesis is less susceptible from mutual cou-pling effect.In the fourth contribution,a novel configuration of array structure with three lin-ear coprime subarrays is designed to propose the ability of achieving the maximum degrees of freedom than-the-state-of-the-art which could be very prone for high reso-lution DO A estimation.Afterwards,the previous structure was modified by appending one more subarray with exploitation of fourth order difference co-array to ascertain the minimum number of required consecutive lags necessitating for estimation of sources spaced in lower angular separation by assuring a cost-effective approach for array ge-ometry while keeping the number of sensors minimum as possible.An enumerated quarterback is implemented unifying displacement and densification on these three co-prime linear subarrays to perpetrate covetable number of consecutive lags for high res-olution DOA estimation.One more noteworthy crying predominance of the proposed technique over our previously proposed technique is that it can retain similar or larger number of consecutive lags by adding less number of sensors in subarray 4 than others while ensuring shorter aperture size as the number of sensors requisiting in subarray 4 is lesser than others.This can be very advantageous in the space of subarray 4 is shorter and if the cost of designing subarray 4 is higher as it is placed in far distance from the first three coprime linear subarrays.