Robust Beamforming and DOA Estimation

The traditional approach to the design of adaptive beamformers assumes that the desired signal components are not present in training data, and the robustness of beamformer is known to depend essentially on the availability of signal-free training data. However, in many important applications such as mobile communications, passive location, microphone array speech processing, medical imaging, and radio astronomy, the signal-free training data cells are unavailable. In such scenarios, the desired signal is always present in the training snapshots, and the adaptive beamforming methods become very sensitive to any violation of underlying assumptions on the environment, sources, or sensor array. In fact, the performances of the existing adaptive array algorithms are known to degrade substantially in the presence of even slight mismatches between the actual and presumed array responses to the desired signal [10]-[12]. Similar types of degradation can take place when the array response is known precisely but the training sample size is small, namely the mismatch between the actual and the estimated covariance matrix [13]-[15]. Therefore, robust approaches to adaptive beamforming appear to be of primary importance in these cases [16][17].


Robust beamforming overview
Beamforming is a ubiquitous task in array signal processing with applications, among others, in radar, sonar, acoustics, astronomy, seismology, communications, and medical imaging. Without loss of generality, we consider herein beamforming in array processing applications. The introduction to beamforming can be found in [1]- [9] and the references herein.
The traditional approach to the design of adaptive beamformers assumes that the desired signal components are not present in training data, and the robustness of beamformer is known to depend essentially on the availability of signal-free training data. However, in many important applications such as mobile communications, passive location, microphone array speech processing, medical imaging, and radio astronomy, the signal-free training data cells are unavailable. In such scenarios, the desired signal is always present in the training snapshots, and the adaptive beamforming methods become very sensitive to any violation of underlying assumptions on the environment, sources, or sensor array. In fact, the performances of the existing adaptive array algorithms are known to degrade substantially in the presence of even slight mismatches between the actual and presumed array responses to the desired signal [10]- [12]. Similar types of degradation can take place when the array response is known precisely but the training sample size is small, namely the mismatch between the actual and the estimated covariance matrix [13]- [15]. Therefore, robust approaches to adaptive beamforming appear to be of primary importance in these cases [16] [17].
Many approaches have been proposed to improve the robustness of the adaptive beamformer during the past three decades. Indeed, the literatures on the robust adaptive beamformer are quite extensive. We provide a brief review as fellows. For more recent detailed critical reviews, see [18] and [19]- [25].

Robust approaches for signal direction mismatch
For the specific case of the signal direction mismatch, there are several efficient methods have been developed. Representative examples of such techniques are the linearly constrained minimum variance (LCMV) beamformer [26], which is also denoted as the linearly constrained minimum power (LCMP) beamformer in other references [27] and this chapter, signal blocking-based algorithms [10] [28], and Bayesian beamformer [29]. Although method to compute the optimal weight vector, and the second-order cone (SOC) programming-based approach is used to solve the original problem. Ayman Elnashar et al make use of the diagonal loading technique to implement the robust beamformer [24], but the optimal value of diagonal loading level is not solved exactly, alternatively, the diagonal loading technique is integrated into the adaptive update schemes by means of optimum variable loading technique. R. G. Lorenz and S. P. Boyd also solve the similar beamformer by the Lagrange multiplier techniques [23], but they express the weight vector and the array manifold as the direct sum of the corresponding real and imaginary components. Almir Mutapcic et al show that worst-case robust beamforming with multiplicative uncertainty in the weights can be cast as a tractable convex optimization problem [37], but they can't give the solving method, In fact, the proposed robust beamformer with uncertain weights can be converted to that in [19] equivalently.
S. Shahbazpanahi et al consider the general-rank signal model, and the robust beamformer is proposed for the distributed sources [21], therein, an elegant closed-form solution is given, but its performance improvement depends on the constraint parameter severely, and is not up to optimal.

Weight norm constraint approaches for general mismatch
Jian Li et al propose a Capon beamforming approach with the norm inequality constraint (NIC) to improve the robustness against array steering vector errors and noise [22], although the exact solution is given, and optimal loading level can be computed via the proposed method, but by analysis and simulation, its efficiency is not as good as expectation. Since the constraint parameter determines its robustness, but how to select the constraint parameter is not discussed.
Quadratic inequality constraints (QIC) on the weight vector of LCMP beamformer can improve robustness to pointing errors and random perturbations in sensor parameter [27]. The weights that minimize the output power subject to linear constraints and an inequality constraint on the norm of the weight vector have the same form as that of the optimum LCMP beamformer with diagonal loading of the data covariance matrix. But the optimal loading level cannot be directly expressed as a function of the constraint in a closed form, and cannot be solved exactly. Hence, its application is restricted by the optimal weight vector finding. So that some numerically algorithms are proposed to implement the QICLCMP, such as Least Mean Squares (LMS) or Recursive Least Squares (RLS) [27], but the application effect isn't good as the expectation.
This chapter is organized as follows [38]. First, the norm inequality constraint Capon beamformer (NICCB) is introduced and analyzed particularly. Second, the choice of the norm constraint parameter and the selecting bound is discussed. Third, the norm equality constraint Capon beamformer (NECCB) is proposed and is solved effectively. Finally, the simulation analyses and the conclusion are given.

Capon beamformer under norm inequality constraint (NICCB)
The Capon beamformer can experience significant performance degradation when there is a mismatch between the presumed and actual characteristics of the source or array. The goal of NICCB is to impose an additional inequality constraint on the Euclidean norm of w for www.intechopen.com Fourier Transform Applications 90 the purpose of improving the robustness to pointing errors and random perturbations in sensor parameters, here w denotes the array weight vector. This requires incorporating a norm inequality constraint on w of the form: where ς is the norm constraint parameter. Consequently, the NICCB problem is formulated as follows: where R is the data covariance matrix, s is the presumed signal steering vector, and () H ⋅ denotes the conjugate transposition, ⋅ denotes the vector 2 l norm. For the convenience of analysis, and analyzing the choice of the norm constraint parameter, the solution to NICCB [22] is introduced as follows.

Solution to NICCB
Let S be the set defined by the constraints in above optimization problem, namely: Hence, ŵ belongs to the boundary of S . Therefore, ŵ is our sought solution to the NICCB optimization problem, which has the same form as the Capon beamformer with a diagonal loading term λI added to R , namely, NICCB also belongs to the class of diagonal loading approaches.
From the above analysis, we can see that if the Lagrange multiplier λ is obtained, the optimal weight vector for NICCB will be solved. In order to obtain the Lagrange multiplier λ , we must solve the following equation via Newton's method, and let: Hence, the key problem of NICCB is finding the optimal Lagrange multiplier by above equation (1.20). In this chapter, we will give the complete investigation on NICCB, and the existence of its solution is analyzed as follows.

Solution to the optimal Lagrange multiplier
In order to solve the equation (1.20), we perform an eigenvalue decomposition (EVD) of the sample covariance matrix as follows: In a word, we can conclude that when

Norm inequality constraint parameter selection
From above analysis, we can see that it is important to select the norm inequality constraint parameter ς for NICCB. If the norm inequality constraint parameter ς is large, it is inactive. On the contrary, if the norm inequality constraint parameter ς is small, there isn't a solution to satisfy NICCB.
We have analyzed that when , there is a unique solution ( If the norm inequality constraint parameter ς is out of the above bound, there is no solution to NICCB. Hence, the norm inequality constraint parameter ς should be chosen in the interval defined by the above inequalities.

Capon beamformer under norm equality constraint (NECCB)
From above analyses, we can see that the norm inequality constraint can enhance the robustness of NICCB. Since the inequality relationship has a wide range, the norm of the www.intechopen.com Fourier Transform Applications 96 weight vector will vary in the relevant wide range. If the fluctuation of weight vector norm is acutely, the performance improvement will be weakened greatly. Because the norm equality constraint (NEC) is stronger than the norm inequality constraint (QIC), NECCB will have more ascendant robust performance than NICCB. Hence, NECCB is proposed and is solved effectively in this chapter.
NECCB is to impose an additional equality constraint on the Euclidean norm of w . The NECCB problem is formulated as follows: Compare NECCB with NICCB, we can educe the conclusion as follows: (1) The solution to NICCB is obtained on the boundary of its constraint, similarly, for NECCB, the solution is also obtained on its constraint boundary. (2) The solving methods of the two beamformers (or the optimization problem) is different, such as the forenamed solution to NICCB, the Lagrange multiplier of NICCB is taken as positive real-value only, but for NECCB, the Lagrange multiplier is taken as arbitrary real-value, namely, it will be not only the positive real-value, but also the negative real-value. Hence, if we analyze from the point of view of the solving optimization problem, NECCB has two solutions to the optimal Lagrange multiplier, one is positive, and another is negative. Actually, the positive one is the solution to NICCB. For the sake of distinguishing the otherness, the negative solution is interested to NECCB. In order to solve NECCB, we must make use of the discussed results of NICCB, since the manipulation of some inequality, such as the inequality lessening and enlarging is only right for the positive real-value when we solve NICCB.
Similar to NICCB, the solution to NECCB can also be solved by the Lagrange multiplier methodology. And the optimal weight vector of NECCB has the same form as NICCB. The difference between NECCB and NICCB is only the Lagrange multiplier λ , for NICCB, 0 λ ≥ , here λ is arbitrary real-value.
Although the solution to NECCB has the same form as NICCB, but the bound of the Lagrange multiplier is different. In order to use the analyzed results of NICCB for NECCB, replace the Lagrange multiplier by its absolute value, namely the bound of the Lagrange multiplier λ for NECCB is given by:

Simulation analysis
In order to validate the correctness and the efficiency of the proposed algorithms, we analyze as follows. In our simulations, we assume a uniform linear array with N=10 omnidirectional sensors spaced half a wavelength apart. Through all examples, we assume that there is one desired source, namely, there is a signal from direction 0º, the Signal Noise Ratio (SNR) is -5dB. Therein, the presumed signal direction is equal to 5º (i.e., there is a 5º direction mismatch).
For the comparison, the benchmark standard Capon beamforming algorithm that corresponds to the ideal case when the covariance matrix is estimated by the maximun likelihood estimator (MLE) and the actual steering vector is used, this algorithm does not correspond to any real situation but is included in our simulations for the sake of comparison only, and is denoted by Ideal-SCB in the figure. The other algorithms include: standard Capon beamformer (SCB), NICCB, NECCB. For NICCB and NECCB, the constraint parameter is selected as the median of the allowable bound.

Effectivity analyzing
In order to show the effectivity of the proposed algorithms, we first compare the pattern of the mentioned Capon beamforming algorithms. The Capon beamformer pattern is given in Fig. 1. Since the signal direction mismatch, the mainlobe of SCB departs from the signal direction. The performance of NICCB is slightly better than SCB, and NECCB is the best of all, the direction mismatch is overcame commendably and NECCB also has lower sidelobe level. Here, NICCB uses the positive optimal loading level, NECCB uses the negative optimal loading level. From the comparison, we can see that NECCB has the better performance than NICCB. The variation of the beamformer SNR versus samples number is given in Fig. 2. We can see that with the change of the samples number, the SNRs varies accordingly. The SNR of NICCB is almost closed to the SNR of SCB, and is lower than the SNR of Ideal-SCB, but NECCB is the best of all, especially for the small number, it has preferable performance. Hence, the norm constraint can improve the SNR, and NECCB has the highest SNR among the listed algorithms. The variation of the Capon beamformer output signal-to-noise-ratio (SNR) versus signal direction mismatch is given in Fig. 3. We can see that with the change of the signal direction mismatch, the SNR varies accordingly, when the angle error is in the range of [-7º, 7º], NECCB will has higher SNR than SCB, NICCB. The NECCB has the higher SNR can be explained by the Fig. 1 of the beam pattern comparison, NECCB not only has the good pointing performance, but also has the lower sidelobe level. Namely, for the same desired signal output, the output noise of NECCB is lower. The simulation results can also be explained as follows, for the used scene, the Signal Noise Ratio is -5dB, and for NECCB, the optimal Lagrange is negative, namely the optimal loading level is negative, but for others, the loading level is zero or positive. Therefore, for the NECCB beamformer, the output noise power is decreased, but for other beamformers, the output noise power is increased. Hence, the NECCB has higher output SNR than others. For the sake of saving space, the corresponding beam pattern comparison isn't given, but in the simulation, NECCB pattern also points to the actual signal direction exactly. Hence, NECCB has the better robustness in the signal direction mismatch case.
From above analysis, we can see that NECCB has the best robustness against the signal direction mismatch.

Correctness analyzing
Since NICCB and NECCB have the same form as that of SCB with diagonal loading. But their key problems are how to find their own optimal loading level or Lagrange multiplier, In order to show the impact of loading level on the Capon beamformer under norm constraint (NCCB) and attest the correctness of the proposed algorithms, the simulation results are given as follows.
The variation of the output SNR versus diagonal loading level is given in Fig. 4. We can see that with the change of the loading level in the bound of () () , the SNR of NCCB varies accordingly. When the loading level is positive, NCCB is NICCB, whereas, when the loading level is negative, NCCB is NECCB. By comparison, we can see that NECCB has higher SNR than NICCB, but for the optimal loading, namely when the loading level is equal to -6.09, NECCB has the best pointing performance, and its SNR is the highest one, where the optimal loading level -6.09 is calculated using the equation ( ) Hence, the loading level has a great impact on the SNR of the Capon beamformer, and determines the performance improvement.
The variation of the weight vector norm versus diagonal loading level is given in Fig. 5. We can see that with the change of the loading level in the bound of () () vector norm of NCCB varies accordingly. For most loading levels, the weight vector norm varies slightly, but when the loading level is small in negative domain, the weight vector norm is a little high, and the highest point is corresponding to the lowest point in Fig. 4. Therefore, the loading level has a great impact on the weight vector norm, especially for the negative loading.  From the above simulation results, we can see that the loading level has a great impact on the performance of the Capon beamformer, and NECCB has the best pointing performance, namely, the optimal negative loading is the best. This is also consistent to the theory analysis, for the robust beamformer with diagonal loading, the improvement is determined by the optimal loading level, when the loading level is optimal, the performance improvement will be the optimal, but for other values, the improvement will be little, or even worse.

Constraint parameter selection analyzing
For NCCB, there are two key problems, one is how to find the optimal loading level, and the other is how to select the norm constraint parameter. Although we have solved the two problems in theory, but there is another key problem, namely, how to select the optimal norm constraint parameter. Therefore, the impact of norm constraint parameter on NCCB is analyzed here particularly.
The variation of the output SNR versus norm constraint parameter is given in Fig. 6. We can see that with the change of the norm constraint parameter in the allowable bound of ( ) min max , ςς , the SNR of the Capon beamformer varies accordingly. NICCB has a little higher SNR than that of SCB, NECCB has the highest SNR. And with the norm constraint parameter increasing, the SNR of NECCB increases correspondingly, but the SNR of NICCB is inclined to the SNR of SCB. When the norm constraint parameter is equal to the maximum, the constraint is inactive, and the three SNRs tend to the same value. Hence, the SNR is determined by the choice of the norm constraint parameter, especially for NECCB.
Beamformer SNR (dB) The variation of the weight vector norm versus norm constraint parameter is given in Fig. 7.
When the norm constraint parameter is selected in the allowable bound of ( ) min max , ςς , the weight vector norms of NICCB and NECCB vary adaptively, and are equal to the square root of the constraint parameter approximatively, this is consistent with the theory, namely the solution is obtained on the constraint boundary. The slight difference is caused by the approximative computation. From above simulation results, we can see that if the norm constraint parameter is selected in the allowable bound, the norm constraint parameter has a great impact on the performance of NICCB and NECCB, especially for NECCB. But NECCB with the larger constraint parameter has the better pointing performance, namely, when the constraint parameter is selected as more larger in its allowable bound, the optimal negative loading has the optimal improvement.

Conclusion
From the above analysis, we can conclude as follows. (I) The proposed algorithm is correct and effective. (II) The norm constraint can improve the robustness of the Capon beamformer. Especially, the equality constraint has the preferable improvement to overcome the steering vector mismatch, and also has good robustness for the samples number. (III) When the norm constraint parameter is selected in the allowable bound, NECCB has the best performance, namely the optimal negative loading has the optimal improvement, this is because that the norm equality constraint is stronger than the norm inequality constraint.

Improved pattern synthesis method with linearly constraint minimum variance criterion
Antenna pattern synthesis becomes the fundamental research contents with the wide application of the array antenna in communication, radar and other areas, and catches the attentions widely. The array antenna pattern synthesis is the task which solves the weight values of the every element to force the antenna pattern inclining to the anticipant shape. Dolph has first given the method of getting the weight function for uniform linear array to achieve the Chebychev pattern [39], therefore the optimal solution can be achieved in the sense of giving the mainlobe width and the maximum lowest sidelobe level. However, how to implement the pattern synthesis for the arbitrary array antenna efficiently is a challenging research task in array signal processing society.
Currently, the methods of pattern synthesis can be classified as the two types, one is the traditional vector weight methods [40][41][42], the other one is the matrix weight methods [43], therein, the intelligent computer methods are used to improve the calculating efficiency of the optimal weight vector, such as the genetic evolution algorithms [44] and the particle swarm optimization algorithms [45]. However, for any pattern synthesis method, the iterative operation can't be avoided, and the iterative number determines the operation load directly, the operation load, or titled as the compute efficiency is the key metric to evaluate the validity of the pattern synthesis.
Guo Q et al propose the pattern synthesis method for the arbitrary array antenna with the linearly constraint minimun variance criterion (LCMV-PS) [45], compared with the traditional vector weight methods, this algorithm has the small iterative number and the preferable convergence. However, by analysis and simulation, it is found that the iterative coefficient determines its performance, namely, the iterative coefficient not only determines the pattern shape, but also determines the iterative number, or titled as the compute load. Therefore, how to select the iterative formula and its iterative coefficient is the key problem to reduce the compute load and enhanced the applicability.
In this chapter, for the LCMV-PS method proposed in [45], by analyzing its implementation and jammer power iterative formula, the improved fast robust LCMV-PS method is proposed [46] . This algorithm takes into account the effect of the relative amplitude between synthesis pattern and its reference upon the pattern synthesis adequately, via adding a proportion constant to the iterative formula, the effect of their relative amplitude upon the changing ratio of the jammer power is strengthened, not only the iterative efficiency of the jammer power is improved, namely the iterative number is reduced, and the pattern synthesis efficiency is improved, but also the selecting bound of the iterative coefficient is extended, namely the effect of the iterative coefficient upon the pattern synthesis is weakened, and the application area and applicability of the pattern synthesis method is enhanced greatly. The last simulation attests its correctness and effectiveness.

Pattern synthesis method with LCMV criterion
The problem of array pattern synthesis can be simplified as follows, namely for the given element number M and element position {} 1 , solving the complex weight vector w, and force the array pattern ( ) P θ with the definite width and maximum value in the desired direction, at the same time, make the sidelobe level according to the requirement.
The target of the pattern synthesis method for arbitrary arrays based on LCMV criterion (LCMV-PS) is making all the sidelobe peak level equal to the minimum that the array can achieve as possible. Furthermore, this method constructs many illusive jammers in the sidelobe region, and the jammer power will be justed by the synthesis pattern amplitude in its relative direction, namely, if the synthesis pattern amplitude is high in this direction, the illusive jammer power will be increased. Therefore, the LCMV-PS method can be simple described as follows: 1. Specify the mainlobe region [ ] 12 , Set the initial value of jammer power where N is the number of the uniformly distributed jammers with one degree spacing, namely, 12  If k≥1, there is the iterative formula as follows: is the array manifold matrix. σ is a given small quantity, and I is the identity matrix, σ I is added to prevent the covariance matrix from being ill-conditioned. 4. Calculate the weight vector w according with the following LCMV beamforming algorithm, then synthesize the pattern. If it is satisfactory, stop; otherwise, go to step (2) and continue. Therein, w is solved by the below LCMV optimization problem, namely: where C is the M×m constraint matrix, and f is the m×1 constraint value vector. Its optimal solution is: In the constraint condition of the optimization problem, the constraint of the mainlobe can be imposed, the constraint of the sidelobe can also be added, in other words, the constraint condition and parameter can be selected according to the pattern synthesis requirement.

Improvement of the jammer power iteration formula
From the step of the LCMV-PS method, we can see that the key is the jammer power iteration in step (2), since it not only determines the synthesis pattern shape, but also determines the final iterative number.
By the particular analysis of the LCMV-PS implementing steps, it is not difficulty to find that although the relative difference of the synthesis pattern and the reference pattern This expression indicates that the jammer powers between the adjacent iterations are different by a proportional factor, when the iterative coefficient K is given, the jammer power ratio of the adjacent iterations is determined by the relative amplitude of the synthesis pattern and the reference pattern From the pattern synthesis process of the LCMV-PS method, when the synthesis pattern is higher than the reference pattern, the jammer powers should increase, and is in direct proportion to the difference of the two patterns. When the synthesis pattern is more higher than the reference pattern, the jammer powers should increase more larger. But when the synthesis pattern is close to the reference pattern, the change of the jammer power should be small, namely the adjustment should be precise at this time. Although the original iterative formula is consistent to the analyzing idea, and the change ratio of the iterative jammer powers is θ Pr −− . Therefore, for the original method, K is the main parameter to determine the iterative effect and efficiency, and by the simulation, it is found that the synthesis pattern will be good when the parameter K is selected in a small region, such as the reference value K=0.1 in [42]. Actually, for the difference element number or array parameter, the optimal value of K will vary correspondingly. Hence, for the original method, how to select the optimal parameter K is the key matter, it not only determines the effect of the synthesis pattern, but also determines the efficiency of the jammer power iteration, namely the iterative number.
Since in the iterative process, it is the factor determining the change quantity and direction of the jammer powers iteration, for the given K, the second item is constant, but the first item is the linear function of ( ) k1 k1 P θ Pr − − , and its proportional coefficient is K, namely the slope K determines the change quantity of the jammer power with With the slope of the linear function increasing, namely for the given parameter K p >1, the change ratio of ( ) will be larger, namely the efficiency of the jammer power iteration will be improved. At the same time, for the given parameter K p , when the effect of the jammer power iteration is better, the selection of K will be loosened, namely K can be selected in a wider region. Therefore, if the constant factor K p (K p >1) can be added as this method, the efficiency of the jammer power iteration will be improved, and the bound for selecting the iterative coefficient K will be enlarged, namely the selection of K will be simplified greatly.
Hence, in order to improve the iterative efficiency of the LCMV-PS method and simplify the selection of the iterative coefficient K, the iterative formula of the jammer power can be improved as follows, namely where K p is used to adjust the effect of the relative amplitude ( ) k1 k1 P θ Pr − − of the synthesis pattern and reference pattern upon the change ratio of the jammer power, namely is used to adjust the iterative efficiency of the pattern synthesis method, and other parameters have the same sense as forenamed. If K p >1, the iterative efficiency will be advanced, whereas the iterative efficiency will be reduced. It is important that the effect of the iterative coefficient K upon the pattern synthesis is reduced greatly by adding the parameter K p , therefore, the selection of parameter K will be simplified greatly.
Compared with the LCMV-PS method proposed in [42], the iterative formula of the jammer power is added by a constant K p to adjust the iterative efficiency in this chapter, if K p >>1, the efficiency of the proposed method will be improved greatly, therefore, the iterative number will be reduced, so that the operation load will be reduced greatly by the proposed method. At the same time, the bound for selecting K will also be enlarged greatly, and the application area and applicability of the pattern synthesis method is enhanced.

Simulations
Since the proposed method has the higher iterative efficiency and stronger applicability as compared with the LCMV-PS method proposed in [42], the simulation keystone is to compare the iterative efficiency of the two methods, and the effect of the iterative coefficient upon the two methods. The simulations are as follows.

Efficiency analyzing
In order to validate the efficiency of the proposed method, the single beam and multi-beam pattern synthesis examples of the uniform and non-uniform linear array are given respectively, and the single beam pattern synthesis examples of uniform and non-uniform planar array are also given respectively. For the convenience of comparison, the proposed improved LCMV-PS method is denoted as I-LCMV-PS, the LCMV-PS in [42] is denoted as LCMV-PS, the reference pattern is denoted as Reference.
The single beam synthesis pattern of the uniform linear array is given in Fig. 8. Therein, the element number is 32, the elements inter-space is half wavelength (λ/2), the mainlobe is point to 0º, the mainlobe width is 22º, the iterative parameter of jammer power K=0.5, and K p =100. When the optimal weight vector is solved under the LCMV criterion, the mainlobe direction constraint is used only. Therein the iterative number of I-LCMV-PS is 5, but the iterative number of LCMV-PS is 20. The single beam synthesis pattern of the non-uniform linear array is given in Fig. 9 30.825], the mainlobe is also point to 0º, the mainlobe width is also 22º, and K=0.5, K p =100. When the optimal weight vector is solved under the LCMV criterion, the mainlobe direction constraint is used only. The iterative number of the two methods are 5 and 20, respectively.  In order to attest the applicability of the improved algorithm to the planar array, and its effectivity of multi-beam synthesis, namely the arbitrary array and arbitrary pattern synthesis, the particular simulation examples are given as Fig. 11.~Fig. 16.

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The multi-beam synthesis patterns of the uniform and non-uniform linear array are given in Fig. 11. and Fig. 12. The parameters are same as Fig. 8. and Fig. 9., the two beams point to 45º and -45º respectively. Therein the iterative number of I-LCMV-PS is 6, but the iterative number of LCMV-PS is 25. The single beam synthesis patterns of the uniform and non-uniform planar array with I-LCMV-PS are given in Fig. 13.~ Fig. 16. Therein, they have the same element number 36, the uniform planar array is phalanx, and element space is half wavelength, but the element position of the non-uniform planar array is given as Fig. 10.  (1) From the above simulations, we can see that the two methods have the preferable synthesis pattern, but I-LCMV-PS has small iterative number, namely it has the higher pattern synthesis efficiency.

Convergence analyzing
In order to compare the convergence characteristic, limit by the chapter length,   Fig. 11. give the convergence of the synthesis pattern for the uniform linear array with I-LCMV-PS and LCMV-PS respectively. Therein, the parameters are as same as 2.3.1. From Fig. 10., we can see that when the iterative number is larger than 4, I-LCMV-PS can achieve the preferable pattern synthesis effect, but for LCMV-PS, the iterative number must be larger than 20, because I-LCMV-PS has the higher efficiency of the jammer power iteration.  In order to attest the convergence and synthesis precision of the improved algorithm, Fig.  19. gives the pattern synthesis error versus the iterative number. Therein, the pattern synthesis error is calculated as follows: where ( ) Pr θ is the reference pattern, ( ) k P θ is the synthesis pattern after k-th iteration. From the error curve, we can see that the improved algorithm has the fast convergence performance, and fall rapidly from beginning, at last, the two curves converge at the same value, namely the convergence is consistent with the Fig. 17. and Fig. 18. Therefore, the proposed algorithm has the good convergence and synthesis precision.
In order to analyze the effect of the iterative coefficient upon the pattern synthesis, Fig. 20. and Fig. 21. give the synthesis pattern versus iterative coefficient for the uniform linear array with I-LCMV-PS and LCMV-PS respectively. Therein, the parameters are as same as 2.3.1. From Fig. 20., we can see that when K p =100, the selection of iterative coefficient K has very little effect upon the pattern synthesis, in simulation, when 0.005<K<2000, the preferable performance can be achieved, the 2000 is the upper bound in simulation, if the parameter is larger than this value, the good performance can also be achieved. Actually, if K p larger, the selecting bound for K wll be wider, it is consistent with the theory analysis. But from Fig.  21., we can also see that the efficiency of the pattern synthesis is determined by K, in the simulation, we find that when 0.1<K<1.6, the preferable effect is achieved. By the comparison, we can see that the improved LCMV-PS method has the lower dependence upon the iterative coefficient, and enables the selecting bound for the iterative coefficient K enlarged greatly, and enhanced its application area and applicability.

Conclusion
From the above analysis and simulation, we can conclude as follows: (I) The proposed jammer power iterative formula is correct and effective. (II) By the improvement for the iterative formula of the original method, the iterative efficiency is increased greatly, and the iterative number is reduced greatly, therefore the operation load of the pattern synthesis is reduced efficiently. (III)The improved jammer power iterative formula enlarges the selecting bound for the iterative coefficient, and reduces the effect of the parameter upon the pattern synthesis, and enhances the application area and applicability of the proposed pattern synthesis method.

Unitary Root-MUSIC
The problem of estimating the direction-of-arrival (DOA) of narrowband sources from sensor array data has received considerable attention. The eigen-based methods for DOA estimation represent a class techniques that offer a much better resolution performance than that of conventional beamformers. In eigen-based methods, signal and noise subspaces are identified first via a M M × generalized EVD (GEVD) of the array data/noise correlation matrix pencil, where M equals the number of array elements. A search is then conducted over a null spectrum associated with the noise subspace, to locate the minima, from which the source DOA's can be determined. In the case where a uniform linear array (ULA) is employed, the null-spectrum searching can be converted into a polynomial rooting problem. Two well known examples are the Root-MUSIC [47] and Root-Minimum-Norm [48] method. They belong to the so-called weighted root-form eigen-based methods. Compared to their spectrum-searching or spectral-form counterparts, root-form methods exhibit a higher resolution capability in dealing with closely spaced sources.Rao and Hari [49] argue that a zero of the null spectrum, having a large radial error, will cause the corresponding spectral minima to be less defined,but does not affect the DOA estimates. As for the mean-squared errors of the DOA estimates,Li and Vaccaro [50] show that both spectral and root-form methods yield the same expression. It should be borne in mind, however, that the result holds only when each of the sources has a minimum corresponding to it in the null spectrum.
A major issue regarding eigen-based methods is the heavy computational load associated with the GEVD. This is more significant when M is large. To remedy this the concept of beamspace transformation was proposed [51] as a means of reducing the dimension of the array data. Ta.S.L proposed a novel iterative implementation of beamspace root-form methods without the need for large-order polynomial rooting [52] . Marius.P, Alex.B.G and Martin.H proposed the unitary root MUSIC algorithm reduces the computational complexity because it exploits the eigendecomposition of a real-valued covariance matrix [53] .
In this chapter [54], combining the algorithms of Root-MUSIC and Unitary-root MUSIC, the Root-MUSIC algorithm with real-valued eigendecomposition is given.

Array signal model
Assume a uniform linear array (ULA) is composed of M sensors, and let it receive q ( qM < ) narrowband signals impinging with unknown directions of arrival ((DOA) available. The tth measured snapshot of the array is generally modeled as: It is generally assumed that signals are uncorrelated with the noise ( ) t n . The sensor noise is assumed to be a zero-mean spatially and temporally uncorrelated Gaussian process with the uunknown diagonal covariance matrix given by is the qq × source waveform covariance matrix.

Root-MUSIC
Root-MUSIC is the polynomial rooting form of MUSIC, namely, the spectrum peak searching is replaced by polynomial rooting in MUSIC implementation.
In Root-MUSIC, the polynomial should be defined as follows where, i u is the eigenvector corresponding to the M-q minimum eigen-value of the data covariance matrix, and Therefore, the source DOA information can be obtained when the above roots are solved. At the same time, we found the item of z * in the polynomial, and the solving process will become complex and difficult. In order to simplify problem, the above polynomial can be modified as Here, the polynomial order is 2(M-1), and has (M-1) pairs roots, the every pair roots have the mutual conjugate relationship. In the (M-1) pairs roots, there are q roots 1 ,, q zz are distributed on the unit circle, and For the ULA, the corresponding DOA of signal can be calculated as where λ is the signal wavelength, d is the array space.
A simple alternative method is proposed in Ref. [55]. From above analysis, we can see that the signal space is orthogonal to the noise space, therefore Now that c has been determined, the polynomial ( ) From the well known conventional Root-MUSIC polynomial of Eq.(3.7), we can conclude that it is a function of z, namely Here the orthogonal property of the signal and noise subspace is used.

Root-MUSIC with real-valued eigendecomposition
The matrix R is the centro-Hermitian if * = RJ R J (3.19) where J is the exchange matrix with ones on its antidiagonal and zeros elsewhere, and   Step 1. Compute R and FB R with the use of (3.4) and (3.6).and the estimate is given by Step 2. Compute C , and the Q is dependent on the number of array sensors. The estimate of the real-valued covariance matrix is given by Step 5. Compute the coefficient of the polynomial by (3.53).
Step 6. Find the root of the polynomial (3.51), and select the q roots that are nearest to the unit circle as being the roots corresponding to the DOA estimates.
Step 7. DOA estimate, using: where k z represents one of the q roots selected for DOA estimation.
From the above analysis, we can conclude that the RVED-Root-MUSIC has a lower computational complexity than the conventional root-MUSIC technique thanks to the eigendecomposition of the real-valued matrix instead of that of the complex matrices, and the asymptotic performance of it is better than of conventional root-MUSIC due to the FB averaging effect..

Simulations
In this section, we present some simulation results to illustrate the performance of RVED-Root-MUSIC. We consider a ULA with M=8 elements and the inter-element space is equal to a half of wavelength. There are three signals with SNRs of 30 dB impinges on the array   figure 24. and 25., the x-axis denotes the SNR, and y-axis denotes the departure of signal DOA .
From the detecting results and the comparison between RVED-Root-MUSIC and Root-MUSIC, we can conclude that RVED-Root-MUSIC can detect DOA of signal quickly and effectively. At the same time, the results validate the correctness and effective of this algorithm.

Conclusion
An improved version of the Root-MUSIC algorithm, called Root-MUSIC with real-valued eigendecomposition (RVED-Root-MUSIC), has been presented in this chapter. The computational complexity is reduced significantly by exploiting the one-to-one correspondence between centro-Hermitian and real matrices, allowing a transformation to real matrices, which can be maintained for all steps of the algorithm. Due to the inherent forward-backward averaging effect, RVED-Root-MUSIC can separate two completely coherent sources and provide improved estimates for correlated signals.

Real-value space ESPRIT algorithm and its implement
The recovery of signal parameters from noisy observations is a fundamental problem in (real-time) array signal processing. Due to their simplicity and high-resolution capability,the subspace estimation schemes have been attracting considerable attention. Among them the most representative are MUSIC and ESPRIT methods. MUSIC utilizes the orthogonal characteristic of noisy subspace of data covariance matrix,but ESPRIT exploits the rotational invariance structure of the signal subspace [57,58] . T h e v i r t u e o f E S P R I T i s t h e l o w computational burden,and not requiring spectrum peak searching by contrast with MUSIC. Comparing with Root-MUSIC, ESPRIT obtains the information of signal direction of arriving (DOA) via exploiting the rotational invariance of every subarray (every subarray 's signal subspace), but Root-MUSIC estimates the signal DOA by solving the polynomial, which is constructed by using the orthogonal between the steering vector and noise subspace.
Unitary ESPRIT achieves even more accurate results than previous ESPRIT techniques by taking advantage of the unit magnitude property of the phase factors that represent the phase delays between the two subarrays [59] . It has been shown in [63] that constraining the phase factors to the unit circle can also give some improvement for correlated sources. For centro-symmetric sensor arrays with a translational invariance structure, Unitary ESPRIT provides a very simple and efficient solution to this task.
Although Unitary ESPRIT effectively doubles the number of data samples, the computational complexity is reduced by transforming the required rank-revealing factorizations of complex matrices into decompositions of real-valued matrices of the same size. Thus, we obtain increased estimation accuracy with a reduced computational load. This reduction can be achieved by constructing invertible transformations that map centro-Hermitian matrices to real matrices.
The real-value ESPRIT algorithm is proposed by [62] and [63], which is on the foundation of the Unitary ESPRIT, by constructing a transformation matrix, transforms the complex data of original array into real-value data. Thus lowered the computational burden. Moreover this algorithm is also applicable to centro-symmetric sensor arrays.
This chapter bases on the foundation of the algorithm that above references proposes and reference [64], analyzes the rotational invariance principle of RVS-ESPRIT algorithm, and the relationship of RVS-ESPRIT and complex space ESPRIT(CS-ESPRIT), definitely give: 1. The rotational invariance relationship of the real-value space array steering, 2. The rotational invariance relationship of the real-value space signal subspace, 3. The rotational invariance relationship between the array steering and the signal subspace of the real-value space, 4. The rotational invariance relationship between the real-value space signal subspace and the complex value space signal space, 5. The rotational invariance relationship between the real-value space array steering and the complex value space array steering.
And give the implementing algorithm of REV-ESPRIT. At last compares its performance with other algorithm by simulation.
This chapter is organized as follows [65] . It starts with a review of the signal model and the rotational invariance subspace principle. Next the RVS-ESPRIT algorithm is analyzed, among which includes the transformation from the complex space to real-value space, the rotational invariance principle of real-value space, and its implementing algorithm. Finally, the computer simulations with the comparison the performance of RVS-ESPRIT and the well-known LS-ESPRIT algorithm are given.

Signal model
Assume that there are two completely same subarray, their space Δ is already known, and every subarray consists of m elements. Consider N ( ) Nm < narrowband plane waves from far-field of the array, these plane waves are assumed to be impinging on the array from directions 12 ,,, N θ θθ , among them, i θ , is angle between the array normal and the direction of the ith signal of the N narrowband planes waves imping. Because the structure of two arrays is completely same, therefore, for a signal, the difference of the two subarray outputs is only one phase difference i ϕ , . Suppose the first subarray receives the data for 1 X , the second receives the data for 2 X , then: Where, the direction matrix of subarray 1 is , the direction matrix of subarray 2 is 2 = AA Φ , S is the space signal vector, 1 N and 2 N are the noise vectors of the subarray 1 and 2, respectively,and are assumed to be white Gaussian, and among the formula:

The rotational invariance subspace principle
From the above mathematics model, we can know that the signal direction information is included in A and Φ , because Φ is a diagonal matrix, so that we can obtain the DOA of signal through solving Φ , that is: where λ is the center wave-length of Arriving the wave. So if we obtain the rotational invariance relationship Φ of the two subarray, we can get the signal DOA information. First uniting the two subarray models, namely: Under the ideal condition, the covariance matrix is estimated as fellows: Let the eigendecompositions of the covariance matrix, there is: Very obviously, the eigenvalue that gets from the top have the relationship as follows: where S U is signal subspace that spanned by eigenvectors which are corresponding to large eigenvalues, N U is noise subspace that spanned by eigenvectors which are corresponding to small eigenvalues.
We know that the signal subspace is spanned by large eigenvector is equal to that is spanned by array direction matrix in the above eigendecomposition, that is: At this time, existing a nonsingular matrix T , which can make: Obviously above-mentioned structure is coming into existence to the two subarrays, so have: Moreover, from the relationship of the two subarrays with regard to signal direction matrix, we can know: Again from (4.10),we can know: where 1 − =⋅ ⋅ Ψ T Φ T . (4.12) reflects the rotational invariance characteristic of the signal direction matrix of the two subarrays, but (4.14) reflects the rotational invariance characteristic of the received signal data subspace of the two subarrays.
If the signal direction matrix A is full rank, we can obtain form (4.14) as fellows: So that,the diagonal matrix which is consisted of the eigenvalues of Ψ certainly be equal to Φ , but the every column of T is the eigenvectors of Ψ . Therefore, once we get the rotational invariance matrix Ψ , we can obtain the signal DOA from (4.4) directly.

The transformation from complex space into realvalue space
We know that the uniform linear array is centro-symmetric, and its signal direction matrix satisfy the nether formula: where, M J is the M M × exchange matrix with ones on its antidiagonal and zeros elsewhere, and the signal direction matrix makes reference to the first element of the array, the diagonal matrix  So that, (4.50) reflects the rotational invariance relationship of the real-value space array steering, but (4.51) reflects the rotational invariance relationship between the real-value space array steering and the complex value space array steering.
Resembling the derivation of (4. 50 So that, (4.57) reflects the rotational invariance relationship of the real-value space signal subspace, but (4.58) reflects the rotational invariance relationship between the real-value space signal subspace and the complex value space signal space.
Utilizing the character that the space spanned by array direction matrix is equal to which is spanned by the signal subspace, so a nonsingular matrix T T exists, and satisfying Comparing with (4.57), we can obtain that: This formula reflects the rotational invariance relationship between the array steering and the signal subspace of the real-value space.

The real-value space ESPRIT algorithm
The observational data of M elements are given as: is the observational data vector which is consists of M elements observational signals.
Step 2. Get the estimating formula of R by ( )  Step 4. Solve the rotational invariance of (4.57) by least square method (or total least square method), and gain T ∧ Ψ .
Step 5. Compute the eigendecompositions of ( ) 2a r c t a n

Simulations
In order to validating the correctness and the effective of the proposed algorithm, we present some simulation results to illustrate the performance of RVS-ESPRIT. We consider a ULA with M=8 element and the interelement space is equal to a half of wavelength. There are three signals impinge on the array from 1 80 θ = − , 2 20 θ = − , 3

Conclusion
This chapter carrys on the detailed theories analysis of RVS-ESPRIT based on the theory of CS-ESPRIT, and gives the concrete implementing algorithm. Because the eigendecompositions of RVS-ESPRIT is in real domain, so the calculation speed is raised consumedly, then the speed of DOA estimating is improved largely also. Due to the inherent forward-backward averaging effect, RVS-ESPRIT can separate two completely coherent sources and provides improved estimates for correlated signals.