A Fourth-Order Cumulants Orthonormal Propagator Rooting Method Based on Toeplitz Approximation

: A novel Toeplitz fourth-order cumulants (FOC) orthonormal propagator rooting method (TFOC-OPRM) to direction-of-arrival (DOA) estimation for uniform linear array (ULA) is addressed in this paper. Specifically, the modified (reduced-dimension) FOC(MFOC) matrix is achieved at first via removing the redundant information encompassed in the primary FOC matrix, and then the TFOC matrix which possesses Toepltiz structure can be recovered by utilizing the Toepltiz approximation method. To reduce computational complexity, we adopt an effective method which depends on the polynomial rooting technology. Finally, the DOAs of incident signals can be estimated by exploiting orthonormal propagator rooting method. The theoretical analysis coupled with simulation results show that the proposed resultant algorithm can reduce computational complexity significantly, as well as improve the estimation performance in both spatially-white noise and spatially-color noise environments.


Introduction
Direction-of-arrival estimation based on antenna array is one of the important directions of research hotspot in array signal processing, which has wide prospects of application in military and civil fields such as wireless communications, radar, passive sonar, biomedicine and seismic exploration [1][2][3][4].
Various high-resolution algorithms, such as multiple signal classification algorithm (MUSIC) [5] and estimating signal parameter via rotational invariance techniques (ESPRIT) [6] approaches, have been proposed to estimate direction-of-arrivals (DOAs) estimation of narrowband far-field signal sources.
However, these subspace-based DOA estimation algorithms described above are not only very sensitive to the noise, but also require the noise's characteristics of the sensors in advance. Furthermore, it is restricted that the total number of sources acting on the array must be less than or equal with that of sensors [7]. When the constrained condition cannot be met in practical environments, the estimation performance of those aforementioned algorithms may run into a stone wall. Fortunately, much more attentions has been paid to this issue, and much more efforts has been made to overcome the above drawbacks. Motivated by the truth that high-order cumulants-based (HOC) has been recognized as a promising technique for direction finding by adopting sensor array [8][9][10]. Besides, another key motivation of using HOC is the ability to resolve more number of sources than or equal to that of the array elements [11]. However, the process of eigenvalue decomposition (EVD) or singularvalue decomposition (SVD) requires large amount of calculation and time taken, which greatly affects the development of rapid source location. Marcos and co-workers [12][13] firstly proposed so-named propagator method (PM) to obtain the signal and noise subspaces by executing a linear-partition operation, which can decrease the computational complexity effectively. Specifically, the performance of the PM algorithm under the conditions of medium and high signal-to-noise ratio (SNR) can achieve the same as that of the traditional high-resolution algorithms but with higher calculation efficiency. Base on [12][13], numerous modifications of PM methods have been proposed, such as [14][15], to achieve low-complexity DOA estimation. In [16], an efficient HOPM algorithm is proposed by making full use of intrinsic multi-dimensional characteristics and affordable computability. A FOC-based and OPM-like(FOC-OPM) algorithm [17] is proposed to gain good location performance. However, the computational complexity of this method is high due to a great number of redundant information is still existed in the FOC matrix. To mitigate this shortcoming, the improved FOC algorithm [18] is proposed to low the computational complexity. However, the performance of the algorithm cannot be asymptotically optimal due to the estimation error of the FOC matrix. Zhang et al. [19] derives a root-MUSIC method using a co-prime linear array to improve the estimation accuracy with low complexity. In [20][21][22], a similar polynomial root-based method is chosen to realize lowcomplexity for DOAs estimation.
In this paper, a novel TFOC-OPRM algorithm is introduced. The contributions of this paper is twofold: Firstly, the reduced dimension matrix is obtained to reduce computational complexity by removing a large number of redundant elements from the original FOC matrix while maintaining the effective aperture of the virtual array in unchanged state. Secondly, the Toeplitz structure is recovered by the Toeplitz operation of the reduced dimension FOC matrix, and the DOA estimation of the recovered Toeplitz structure matrix is performed based on the polynomial root method.

Data Model
Consider M narrowband far-field sources ( ), ( 1, , ) impinging on a uniform linear array (ULA) with N equispaced omnidirectional sensors, where the distance between adjacent sensors is equal to half the wavelength. Assume that the incoming sources are stationary and mutually independent. The noise is the additive white/color Gaussian one, and statistically independent of the sources. Let the first sensor be the reference, and then the observed data received in time t at the th k sensor can be expressed ( ) exp( 2 ( ) sin ) where λ is the central wavelength, d is the spacing between two adjacent sensors. Therefore, the matrix form of (1) can be expressed as Assume that the source signals are zero-mean stationary random process, the FOC can be defined as [ ] ( , , , ) ( ( ) ( ) ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) , , , 1, , where B and S C represent the extended array manifold and the FOC matrix of incident source signals, It is obvious that ( ) θ b is a 2 1 N × vector, which means that the array aperture of ULA is extended. That is, the number of resolved source signals is no less than that of sensors.

The effective array aperture extended
As proven in [23], an array of N arbitrary identical omnidirectional sensors can be extended to at most of 2 1 N N − + . Especially, the number of virtual elements is 2 1 N − for ULA according to [23]. In order to discuss the effective aperture of ULA , four real elements ( 4) N = are considered, and ( ) θ b can be expressed in detail as follows C , but also keeps the extended array aperture unchanged.

The TFOC-OPRM Method
When the incident targets is a statistically independent signal sources, the ideal 4 R is with a Toeplitz structure. However, in practical applications, for example, due to finite sampling snapshots and the low SNR , the matrix 4 R obtained at this time does not meet the Toeplitz structure any more, instead, it becomes a diagonally dominant matrix. The happening of such condition will have a negative impact on the performance of the final DOA estimation. In order to improve the DOA estimation accuracy of the antenna array, the first task is to recover the Toeplitz structure of matrix 4 where T S represents Toeplitz matrices, and the entries of the Toeplitz matrix 4T R can be written as where the element (  (10) where Toep stands for the Toeplitization operator.
Although conventional algorithms, such as MUSIC and ESPRIT , can be applied to estimate DOAs based on the 4T R , the computational burden is much heavier due to the EVD and SVD involved. Therefore, we apply OPM for estimating the DOAs to reduce the complex computations effectively.
The presented propagator method is based on the following partition where the dimensions of 4T1 The relation (13) shows that the 4T R is orthogonal to the columns of H Q , and the propagator matrix P can be obtained by minimizing the cost function ( ) where F • indicates the Frobenius norm, and the optimal solution P is given by In order to introduce the orthonormalization, the orthonormalized matrix 0 Q is obtained as follows H 1/2 0 ( ) − = Q Q Q Q (16) Therefore, the following spectral function ( ) p θ can be formed to estimate the DOAs of source signals It can be seen from function (17) that the M DOAs of the incoming signals can be obtained by means of one-dimensional (1 D) − spectrum-peak search over θ . However, to further reduce the computational burden, we can improve the function (17) in order to derive a computationally more efficient search-free modification estimator based on polynomial rooting [24]. In order to further reduce the computational complexity of the algorithm, the method based on polynomial roots is used to improve the spatial spectrum estimation function, so as to obtain more efficient estimators in the calculation, with the specific description of the algorithm given as follows.
Then the denominator of the estimator (17) can be re-expressed with the following polynomial format So far as it is concerned, the specific operational steps of the proposed Toeplitz fourth-order cumulant orthogonal propagation method based on polynomial roots under limited sampling snapshots can be summarized as follows: Step 1 Estimate 4 C from the received data by (5).
Step 2 Obtain the dimension reduction matrix 4 R by removing the redundant items from the expanded matrix 4 C .
Step 4 Estimate the linear operator P according to (14) and (15), then calculate the standard orthonormalized matrix 0 Q based on (16).
Step 5 Obtain the polynomial Function (19), and further to it, obtain the M roots closest to the unit circle, that is, the roots of the ( ) f z .
Step 6 Obtain the direction estimation of the incoming wave of the incident target signal source from (20).

Complexity analysis
As for the analysis of computational complexity, the main parts of computation are considered, that is, the construction of the cumulant matrix, the linear operation, the spectral peak search operation, the (9(2 1) 2(2 1) 1 (2(2 1) (2 1)) 2(2 1) 1 From the above analysis, it can be obviously seen that the computational complexity of TFOC-OPRM algorithm proposed is significantly lower than that of both FOC-OPM algorithm and MFOC-OPM algorithm. The main reason is that the polynomial roots method has been involved to reduce the computational complexity further.

Results and Discussion
In this section, the proposed TFOC-OPRM algorithm, as well as FOC-OPM [17] and MFOC-OPM [18]  in the formula equals 0.8 and1.5 for experiment two and three, respectively.

Experiment 1: The spatial spectrum estimation
In the first experiment, the input SNR and the number of snapshots are set to be10dB and 500 , respectively. Shown in the Fig. 1 is the spatial spectrum of the proposed TFOC-OPRM , FOC-OPM and MFOC-OPM algorithms in both spatially-white noise and spatially-color noise situations. It can be observed from the curves in the chart that all of the three algorithms have successfully located the peak corresponding to the incident angle. Further analysis indicates that the angular resolution of the proposed TFOC-OPRM algorithm is much higher than that of both MFOC-OPM and FOC-OPM algorithms. The reason is that the proposed TFOC-OPRM algorithm recovers the Toeplitz structure of 4 R , making the Toeplitz matrix 4T R closer to the real situation.

Experiment 2: RMSEs and NPS versus SNR
The main objective of this experiment is to evaluate the performance of TFOC-OPRM algorithm, FOC-OPM algorithm and MFOC-OPM algorithm in terms of RMSEs and NPS with the change of input SNR . The number of sampling snapshots is 2000 L = , the input SNR changes from 8dB to 24dB , with the step being 2dB . In both Fig. 2 and Fig. 3 are the performance curves of RMSEs and NPS of the proposed algorithm and the comparison algorithms as the input SNR changes, respectively. It can be seen from Fig. 2 that the RMSEs of the three algorithms decrease monotonically with the increase of the input SNR . Further analysis shows that in the environment of spatial-white noise, with the increase of input SNR , the RMSEs performance curve of TFOC-OPRM algorithm is better than that of FOC-OPM algorithm and that of MFOC-OPM algorithm; in the environment of spatial-color noise, the RMSEs performance curve of TFOC-OPRM algorithm is better than that of MFOC-OPM algorithm.
In addition, when the input SNR changes between 8dB and 14dB , the proposed TFOC-OPRM algorithm manages to achieve almost the same RMSEs performance as the FOC-OPM algorithm. But when the input SNR is higher than14dB , the performance of TFOC-OPRM becomes better than that of FOC-OPM . Moreover, the performance of the improved TFOC-OPRM algorithm is identical to that of the MFOC-OPM algorithm no matter whether in spatially-white noise situation or spatially-color noise situation. From Fig. 3, it can be concluded that the NPS performance of the proposed TFOC-OPRM algorithm is better than that of the FOC-OPM algorithm and MFOC-OPM algorithm in the case of low input SNR . With the increase of input SNR , the NPS of all of the three algorithms ultimately is 1. In addition to that, the proposed algorithm not only removes a lot of redundant data in the original FOC , but also restores the Toeplitz structure of the reduced dimensional FOC . Moreover, it adopts the method of finding roots of polynomials. Therefore, the proposed TFOC-OPRM algorithm not only reduces the computational complexity, but also improves the accuracy of DOA estimation.

Experiment 3: RMSEs and NPS versus snapshots
The main objective of this experiment is to verify the performance of the RMSEs and the NPS of The reason is that the number of sampling snapshots is relatively small, resulting in too less data acquired. In other words, the estimated matrix 4 R deviates greatly from the ideal matrix 4 R . With the increasing number of sampling snapshots, we can see that the performance curve tends to be stable gradually. At the same time, it can be observed that the TFOC-OPRM algorithm proposed in this paper achieves more satisfactory estimation performance than RMSEs algorithm and NPS algorithm, either in the condition of spatial-white noise or in the condition of spatial-color noise. Note that the computational complexity of proposed algorithm is significantly lower than that of the FOC-OPM algorithm due to the fact that the redundant information of the original cumulants matrix is removed.
Moreover, the Toeplitz approximate method is performed on the reduced-rank 4 R to improve estimation performance. Meanwhile, compared to MFOC-OPM method, the TFOC-OPRM algorithm has lower computational burden, which exploits polynomial rooting instead of spectral search.

Experiment 4: The calculation complexity versus snapshots
In this simulation experiment, we further verify the advantages of TFOC-OPRM algorithm in terms of computational complexity, also by comparing the algorithms with FOC-OPM and MFOC-OPM .  ). Viewing from the simulation results in Fig. 6, with the increasing number of sampling snapshots, the computational complexity of the proposed TFOC-OPRM algorithm is far lower than that of the FOC-OPM algorithm and the MFOC-OPM algorithm, and this advantage will be more obvious with the further increase of the number of sampling snapshots. The reason is that the proposed TFOC-OPRM algorithm not only eliminates a large number of redundant data in the original FOC , but also adopts the polynomial root method. This is consistent with the theoretical analysis given in section 3.3.

Conclusions
In this paper, a novel computationally efficient TFOC-OPRM localization algorithm have been proposed. Specifically, the extended effective array aperture can resolve the number of sources more than or equal to that of the array elements. Moreover, resorting to Toeplitz approximate method, the Toeplitz structure of the reduced-dimension 4 R matrix is recovered to provide a more satisfactory estimation performance than the compared algorithms. In addition, compared to MFOC-OPM algorithm, the proposed TFOC-OPRM algorithm can obtain good estimation performance, as well as has lower computational burden because of the advantage of the polynomial root method in computational complexity. Simulation results validate the effectiveness of the proposed algorithm both in spatially-white noise and spatially-color noise situations.