DOA Estimation Based on Eigen Space and Toeplitz Processing

In order to efficiently and accurately estimate the direction of arrival of coherent signals, a Toeplitz estimation method based on eigen space multiple signal classification (MUSIC) algorithm is proposed. Firstly, the coherent signal is processed by Toeplitz, and then the MUSIC algorithm of the eigen space is used to estimate the DOA accurately. The computer simulation results show that the method can realize the accurate estimation of the angle under the condition of signal coherence, and the accuracy is improved compared with some other algorithms.


Introduction
Direction of Arrival (DOA) estimation is an important research direction of array signal processing, and it has been widely used in communication, navigation [1] , radar [2] , sonar and other fields.The traditional DOA estimation algorithm has Multiple Signal Classification (MUSIC [3] ) algorithm, Estimation of Signal Parameters by Rotational Invariance Technique (ESPRIT [4] ) algorithm, etc.These algorithms can achieve accurate angle estimation under ideal conditions, but in practical applications, the signal source may be coherent.At this time, the rank of the acceptance matrix will decrease, and the angle estimation accuracy of the algorithm will be greatly affected.In the case of signal coherence, it is necessary to perform decoherence processing on the signal.Usually, signal decoherence algorithms can be divided into two categories: One is dimension reduction processing; the other is non-dimension reduction processing.Common dimensionality reduction processing algorithms are based on space smoothing algorithm and matrix reconstruction algorithm.Among them, the algorithms based on spatial smoothing mainly include forward spatial smoothing algorithm [5,6] , two-way spatial smoothing algorithm [7] and so on.The spatial smoothing MUSIC algorithm restores the full-rank covariance matrix by finding the mean value of the sub-arrays of each covariance matrix.The forward spatial smoothing algorithm takes the sub-arrays from the front to the rear.The front and back sub-arrays are used to perform forward smoothing and backward smoothing respectively, and then the forward smoothing matrix and the backward smoothing matrix are averaged to obtain a two-way smoothing covariance matrix.Algorithms based on matrix reconstruction are mainly matrix decomposition algorithm [8,9] and vector singular value algorithm [10,11] .The matrix decomposition algorithm is to reconstruct the data covariance matrix through a specific method and then perform singular value decomposition, and use the results of singular value decomposition to estimate 2 DOA.The vector singular value algorithm is to pre-process the covariance matrix to obtain the data vector, and then perform singular value decomposition on the reconstructed matrix of the data vector, and use the result of the singular value decomposition to estimate DOA.Although the dimensionality reduction algorithm can accurately estimate the direction of arrival, it will lead to array loss.Algorithms for non-dimension reduction processing include modified MUSIC algorithm (MMUSIC [12] ), Toeplitz matrix reconstruction algorithm [13,14] , etc.The modified MUSIC algorithm is to reconstruct the covariance matrix, which is essentially a special case when the number of sub-arrays is 1 in the forwardbackward spatial smoothing algorithm, so it can only effectively de-coherently process two coherent signal sources.The Toeplitz matrix reconstruction algorithm is to perform Toeplitz processing on the data covariance matrix, and then perform DOA estimation.Toeplitz pre-processing methods include Toeplitz approximation method, Toeplitz correction method and phase Toeplitz method.Compared with the dimensionality reduction processing algorithm, the advantage of the Toeplitz algorithm is that there is no loss in the array aperture.Based on the traditional MUSIC algorithm, Zhang [15] proposed the eigen space MUSIC algorithm.The complexity of this algorithm is comparable to that of the MUSIC algorithm, but it can make full use of the characteristics of the signal subspace and noise subspace, and it can be used in coherent sources, small snapshots and low SNR The performance under the conditions is better than the traditional MUSIC algorithm.In this paper, the Toeplitz-ESMUSIC algorithm is proposed to estimate the DOA of coherent signals by combining the phase Toeplitz method with the eigen space MUSIC algorithm.The algorithm first performs Toeplitz processing on the covariance matrix of the signal, and then estimates the signal angle through the eigen space MUSIC algorithm.Simulation experiments show that the algorithm can estimate the signal accurately when the signal is coherent and the SNR is low, and the performance is better than the spatial smoothing algorithm when the SNR is low.Notations: In this paper, bold lowercase italic bold represents vector, and ordinary oblique bold represents scalar.( • )  , ( • )  , respectively, become transpose and conjugate transpose.

Mathematical model of array signal
Assuming that the -element uniform linear array model satisfies the far-field condition, the incident angles of the  narrowband signals are   ( = 1,2, … , ), and  > , select the first array element as the reference array, and the array element spacing is , then the corresponding vector of the array can be obtained by the following formula: Then the output of the m-th array element is: where   () is the kth signal received by the antenna array;  is the signal wavelength; where  is the number of signal sources, and   () is Gaussian white noise with a variance of  2 on the array element.
Then the  signal source DOA direction matrix can be expressed as: ) And remember: (5) where in: () is  × 1 dimensional noise vector, then the data received by the array can be expressed by the following formula: () = ()() + () (6) According to formula (6), the covariance matrix of array snapshot data can be expressed as:  = [()  ()] =     +  2   (7)  where   = [()  ()] is the signal covariance matrix; it represents the M-order identity matrix.

Toeplitz processing algorithm
First perform Toeplitz pre-processing on the covariance matrix, and record the covariance matrix after restoring the Toeplitz structure as  ̃.

Toeplitz Approximation Method
The Toeplitz approximation method (TOP) is actually to average the elements on the oblique diagonal and the elements on each oblique line parallel to the diagonal.Since the covariance matrix is a Hermitian matrix, only the covariance matrix The upper triangular elements in are processed.Toeplitz preprocessing process boils down to: ̃() =  ̃ * (−) (9) where  is the number of array elements; ( • ) * means complex conjugate;  ̅  (−) is the nth oblique line parallel to the main diagonal and above the main diagonal in  ̃.The element value above,  ̃ represents the i-th row and j-th column element, it can be seen that the element in  ̃ is  ̃ =  ̃( − ).

Modified Toeplitz method
The modified Toeplitz method (MTOP) averages the magnitudes of the elements on the diagonal of the covariance matrix, while the phases of the elements are unchanged.Its processing can be summarized as: where (−) is the phase of  ̂(−).

Phase Toeplitz method
The phase Toeplitz method (PTOP) is to average the phase of the elements on the diagonal of the covariance matrix, while the magnitude of the elements is unchanged.Its processing can be summarized as: where (−) is the phase magnitude of  ̂(−).
The three methods were analyzed and verified experimentally, and it was found that the phase Toeplitz method only changed the phase of the elements on the diagonal of the covariance matrix, but did not change the magnitude of the elements of the covariance matrix, so the effect is the best, so this paper adopts the phase Toeplitz method for decoherence processing.

Eigen Space MUSIC Algorithm
The traditional MUSIC algorithm only utilizes the noise subspace.Next, we introduce a new spatial spectrum estimation method, which makes full use of the signal subspace and the noise subspace.Define the matrix   :  (15)  where, (⋅) + represents the pseudo-inverse operation,   = [0, … ,1,0, … ,0]  is a vector of  × 1, the ith element is 1, and the others are 0,   is the power of the i-th source.Define an exchange matrix  of order : where, the matrix  is an  ×  matrix, and its sub-diagonal elements are all 1, and the rest of the elements are all 0.
Restructure the covariance matrix: where  is the original covariance matrix,   is the reconstructed covariance matrix, and (⋅) * means conjugate.Next,   is subjected to eigen decomposition, and the covariance matrix   is divided into signal subspace and noise subspace according to the university order of eigenvalues, and the following expression is obtained:

Simulation experiment and result analysis
This section analyzes and compares the performance of the original algorithm and the algorithm in this paper through simulation experiments.
The conditions of the simulation experiment are set as follows: the signal-to-noise ratio (SNR) is defined as the ratio of the total power of the array received signal to the total power of the noise, namely: In addition, the DOA estimation accuracy is measured by the root mean square error RMSE, without loss of generality, assuming that the number of independent experiments is n, the number of signal sources is  , the real DOA is ( 1 , … ,   ) , and the DOA of the i-th experiment is estimated as ( ̂1(), … ,  ̂()), RMSE is defined as:

Experiment 1: Performance Comparison of Several Toeplitz Algorithms
In the experiment, a uniform line array with an array element spacing of half a wavelength and an array element number of M=22 was used.Assume that the number of coherent signal sources K=2, the DOAs are 0 ° and 2 ° respectively, and the number of snapshots is 250.The signal-to-noise ratio is changed from -10dB to 10dB, and 100 Monte Carlo experiments are performed at about 2dB intervals to compare the RMSE of the three Toeplitz methods.The RMSE of several algorithms are shown in Figure 1.It can be seen from Figure 1 that the RMSE curve of the phase Toeplitz method(PTOP) decreases gently with the increase of SNR, and the effect is better than that of the modified Toeplitz method(MTOP), which is equivalent to the Toeplitz approximation method(TOP).

Experiment 2: The spatial spectrum comparison between the Toeplitz-ESMUSIC method and the MMUSIC method.
In the experiment, a uniform line array with an array element spacing of half a wavelength and an array element number of M=10 was used.Assuming that the number of signal sources K=4, the DOA are −30 °, 10 °, 30 °, 60 °, respectively, and the angles corresponding to −30 ° and 10 ° are coherent signals; the number of snapshots is 300, The signal-to-noise ratio is -10dB.The spatial spectra using the Toeplitz-ESMUSIC method and the MMUSIC method are shown in Figure 1.

Experiment 3: Estimation performance comparison of Toeplitz-ESMUSIC method and MMUSIC method.
In the experiment, a uniform line array with an array element spacing of half a wavelength and an array element number of M=10 was used.Assuming that the number of signal sources K=4, the DOAs are −30 °, 10 °, 30 °, and 60 ° respectively, where the angles corresponding to −30 ° and 10 ° are coherent signals, and the number of snapshots is 200.The signal-to-noise ratio is changed from -10dB to 10dB, and Monte Carlo experiments are performed about 300 times at intervals of 2dB, and the RMSE of the Toeplitz-ESMUSIC algorithm and the MMUSIC algorithm are compared.The spatial spectra using the Toeplitz-ESMUSIC method and the MMUSIC method are shown in Figure 3.  3, it can be seen that as the SNR increases, the RMSE of the above two methods decreases, and the estimation accuracy increases.However, the RMSE of the MMUSIC method is obviously large, indicating that the MMUSIC algorithm cannot distinguish the coherent signals among the four signal sources, that is, the MMUSIC algorithm cannot accurately estimate the signal direction when there are coherent signals in the four signal sources.When the SNR of Toeplitz-ESMUSIC algorithm is greater than -6dB, its RMSE is close to 0, and the curve declines gently, which shows that Toeplitz-ESMUSIC algorithm has superior estimation performance and strong stability.

Experiment 4: Estimation performance comparison of the Toeplitz-ESMUSIC method with other decoherence methods.
In the experiment, a uniform line array with an array element spacing of half a wavelength and an array element number of M=8 was used.Assume that the number of coherent signal sources K=2, the DOAs are −10 ° and 10 ° respectively, and the number of snapshots is 200.For the DOA estimation of the spatial smoothing algorithm, set the number of array sub-arrays to 3. The signal-to-noise ratio is changed from -10dB to 10dB, and Monte Carlo experiments are performed about 300 times at intervals of 2dB.Compare the RMSE of Toeplitz ESMUSIC algorithm, SS MUSIC algorithm and ESVD MUSIC algorithm.The RMSE of several methods is shown in Figure 4.It can be seen from Figure 4 that when the signal-to-noise ratio of two coherent signal sources is lower than 0, the RMSE of Toeplitz ESMUSIC algorithm is significantly lower than that of traditional spatial smoothing algorithm and ESVD-MUSC algorithm.When the signal-to-noise ratio is greater than 0, the RMSE of the three algorithms shows that Toeplitz ESMUSIC algorithm has a stronger estimation effect than the general spatial smoothing algorithm and ESVD-MUSC algorithm when the signal is coherent.In the experiment, a uniform line array with an array element spacing of half a wavelength and an array element number of M=8 was used.Assume that the number of coherent signal sources K=2, the DOAs are −10 ° and 10 ° respectively.For the DOA estimation of spatial smoothing algorithm, set the number of array subarrays to 3. The experimental equipment is Dell G15 notebook computer, which includes i7-11800H processor and RTX3060 graphics card.The Toeplitz-ESMUSIC algorithm, MMUSC algorithm, SS-MUSC algorithm and ESVD-MUSC algorithm are simulated under the conditions of 100, 200 and 500 snapshots respectively.Carry out several experiments and record the average running time of each algorithm under different snapshot numbers, as shown in Table 1.It can be seen from Table 1 that the running time of MMUSIC algorithm is significantly less than that of the other three algorithms due to its simple structure.The running time of Toeplitz-ESMUSIC algorithm, SS-MUSIC algorithm and ESVD-MUSIC algorithm are basically the same.Under different snapshot numbers, the running times of several algorithms are basically the same, which shows that the number of snapshots has little impact on the running time of the algorithm.

Conclusion
This paper proposes a Toeplitz-ESMUSIC method to solve the coherence of array signals.First, the coherent source signal is pre-processed using the phase Toeplitz technique.Then, ES-MUSIC algorithm is used to estimate the DOA accurately, so that it can make full use of the information of signal subspace and noise subspace.The method in this paper does not affect the estimation of DOA when incoherent signals exist, and can estimate the power of incoherent or coherent signal sources, making it more suitable for DOA estimation under low SNR conditions.Compared with the MMUSIC algorithm, although the Toeplitz-ESMUSIC method has no prior information that can reflect the signal, it can still estimate the signal direction in the case of multiple signal sources.Compared with algorithms such as spatial smoothing, the Toeplitz-ESMUSIC method does not reduce the degree of freedom, the aperture of the array can be fully utilized, and it still has good estimation performance under the condition of low signal-to-noise ratio.

Figure 1 .
Figure 1.RMSE images of three Toeplitz methods estimate

Figure 2 .
Figure 2. Spatial spectrum images of Toeplitz-ESMUSIC and MMUSIC methodsFrom the comparison of the curves in Figure2, it can be seen that when four signal sources and two signal sources are coherent signals, the angle can be accurately estimated by using the Toeplitz-

Figure 3 .
Figure 3. RMSE images of the two algorithms in the case of signal coherenceFrom the comparison of the curves shown in Figure3, it can be seen that as the SNR increases, the RMSE of the above two methods decreases, and the estimation accuracy increases.However, the RMSE of the MMUSIC method is obviously large, indicating that the MMUSIC algorithm cannot distinguish the coherent signals among the four signal sources, that is, the MMUSIC algorithm cannot accurately estimate the signal direction when there are coherent signals in the four signal sources.When the SNR of Toeplitz-ESMUSIC algorithm is greater than -6dB, its RMSE is close to 0, and the curve declines gently, which shows that Toeplitz-ESMUSIC algorithm has superior estimation performance and strong stability.

Figure 4 .
Figure 4. RMSE images of different decoherence algorithms in the case of signal coherence 4.5.Experiment 5: Time comparison between Toeplitz ESMUSIC method and other decoherent methods.In the experiment, a uniform line array with an array element spacing of half a wavelength and an array element number of M=8 was used.Assume that the number of coherent signal sources K=2, the DOAs are −10 ° and 10 ° respectively.For the DOA estimation of spatial smoothing algorithm, set the number of array subarrays to 3. The experimental equipment is Dell G15 notebook computer, which includes i7-11800H processor and RTX3060 graphics card.The Toeplitz-ESMUSIC algorithm, MMUSC algorithm, SS-MUSC algorithm and ESVD-MUSC algorithm are simulated under the conditions of 100, 200 and 500 snapshots respectively.Carry out several experiments and record the average running time of each algorithm under different snapshot numbers, as shown in Table1.It can be seen from Table1that the running time of MMUSIC algorithm is significantly less than that of the other three algorithms due to its simple structure.The running time of Toeplitz-ESMUSIC algorithm, SS-MUSIC algorithm and ESVD-MUSIC algorithm are basically the same.Under different snapshot numbers, the running times of several algorithms are basically the same, which shows that the number of snapshots has little impact on the running time of the algorithm.

Table 1 .
Running schedule of various algorithms