Research of direction-of-arrival estimation in fewer snapshots based on niche artificial bee colony algorithm

Abstract

This paper focuses on the issue of the large calculation when searching for the optimal solution of weighted subspace fitting algorithm (WSF) in the Direction of Arrival (DOA) estimation and introduces the artificial bee colony algorithm (ABC) to optimize solving process. Taking advantage of Niche technique and coordinating with improved strategy and control exclusion method, the proposed algorithm can overcome the weakness that bee colony algorithm itself can only get the global solutions, and hence get the local optimization solution for the multimodal problem. The experiment results demonstrate the proposed algorithm has small computation load, less controlling parameters, and high accuracy.

Keywords

DOA weighted subspace fitting algorithm artificial bee colony algorithm niche

1 Introduction

DOA estimation is one of the most important research directions of array signal processing [1]. In addition to the application in the field of military and radar, it has been more widely used in the aspects of civil communication, wireless internet and earthquake early warning. Over the last several decades, there appear lots of algorithms, from traditional beam forming algorithm (BF) [2] to MUSIC and ESPRIT [3, 4] of high resolution and so on. The research directions of DOA algorithm are various. Some scholars focus on improving array structure, from traditional linear array, circular array to arbitrary array manifold [5, 6], and from scalar sensor array to electromagnetic vector sensor [7]. Some pay attention to doing researches about application environment, from common Gaussian noise expanding to colored noise and even impulsive noise environment [8, 9]. Some focus on the signal sources, from independent signals to the coherent signals, and even the circumstances of both independent signals and coherent signals. Some may pay attention to researching into the array aperture extension ability to improve utilization ratio of the proposed array. Some work on improving operation speed and reduce algorithm complexity. Overall, as a hot point of signal processing, DOA estimation is being favored by more and more scholars.

Among many direction-finding algorithms of high resolution, subspace algorithm is a popular part. MUSIC is the most typical subspace decomposition algorithm and can get accurate DOA estimation of incidence signals on the condition of large number of samples. However, as for short-time small snap signals, there is no obvious differentiation between the large eigenvalue and the small eigenvalue from covariance matrix of data sampling and hence part of the power from signal subspace will divulge into the noise subspace, which causes the estimation performance of subspace decomposition algorithm decreases severely. But subspace fitting method (SF) [10] can get good estimation performance even during low signal-to-noise ratio and small snapshots [7]. The algorithm performance is better than decomposition algorithm. Of all the spatial spectrum estimation algorithm, subspace fitting method has a simple principle and high estimation accuracy. However, the direction angle function is nonlinear during the DOA estimation, and it needs multi-dimensional searching when getting the optimal solution, whose process is very complicated. Especially when the number of signal sources increases, the computational load of subspace fitting method grows exponentially, which cannot be real-time applied in practice as a result of its complex calculation [11, 12].

The paper mainly focuses on the disadvantage of operation speed and great computation load for the traditional direction-finding methods, and takes advantage of swarm intelligence algorithm to improve operation speed and reduce computation complexity [12, 13]. The paper tries to combine artificial bee colony algorithm and high-resolution technique, and proposes a fast method based on improved niche artificial bee colony algorithm and weighted subspace fitting method (Niche-ABC-WSF) [14] on the condition of small snapshots. The proposed method has four advantages as follows. First, it is fit for short-time small snapshots. Second, it has high estimation accuracy on the condition of low SNR, which is almost equal to WSF. Third, it has good convergence performance and lower computational load. Forth, it can be easily applied in the real practice and do not need extra parameter setting or search range [15].

From the ordinary scalar sensor arrays to the special vector sensor arrays, all kinds of combination forms are applied in the DOA estimation. Different sensor arrays can provide practical basis for the above algorithms. In recent years, algorithms based on different sensor arrays have been proposed in turn. For example, the linear array [16], the planar array [17], the uniform circular array [18, 19] and so on.

However, most of the DOA algorithms need to know the number of sources before estimation, if the number is unknown or deviated, the algorithms will fail. So the key step of the DOA is the estimation of sources number. The typical methods of source number detection algorithm are hypothesis testing [20], Akaike information criterion (AIC) [21], the minimum description length criterion (MDL) [22], Gerschgorin circle method (GDE) [23], Canonical correlation technique (CCT) [24], etc. These methods have the advantages and disadvantages, the method based on hypothesis testing requires some prior knowledge generally; AIC method apply to Gauss white noise background, and the AIC algorithm is not a consistent estimation; GDE algorithm has a higher probability of error at low SNR; the CCT algorithm is a kind of source number detection algorithm in the case of anisotropic noise, and the algorithm is very good in high SNR, it is also significantly better than the information theory and other algorithms in low SNR, but it will lose array aperture. In recent years, the Nature Inspired Computation methods such as the artificial neural network [8], genetic algorithm [9], ant colony algorithm [10], particle swarm optimization algorithm [11], bee colony algorithm [25] and membrane computing (MC) [26] were employed in array signalprocessing.

The paper is organized as follows. The second part will introduce the signal model and traditional algorithm. And the third section mainly introduces niche bee colony algorithm and proposes the improved algorithm. Other intelligence swarm algorithms and comparing simulation experiments with the proposed algorithm are introduced in the forth section. In the final part, we analyze the whole method and discuss the strength and weakness, and then draw the conclusion.

2 Signal model

2.1 Mathematic model of uniform linear array (ULA)

As illustrated in Fig. 1, supposing there are N far-field, narrow-band signals impinging on the antennas in the spatial uniform linear array, the array is made up of M array elements, where the array spacing is d = λ/2, and the direction angles of N signals are respectively, θ₁, θ₂, …, θ_N.

Fig.1

Uniform linear array.

Supposing the received data of M array elements at a certain time is denoted as $X (t) = [x_{1} (t), x_{2} (t), \dots x_{M} (t)]$ (1) the received data of the array element of lth ideal situation can be denoted as $x_{l} (t) = \sum_{l = 1}^{N} s_{i} (t - τ_{li}) + n_{l} (t)$ (2)

where s_i (t) is the incident signal, τ_li is the time delay in the array elements of the ith signal source and n_i (t) is the noise and interference at the moment t.

We can put (1) into the form of vector matrix, and we get $X (t) = AS (t) + N (t)$ (3) where X (t) is a M × 1 snapshot vector, and N (t) is a M × 1 noise data vector, and S (t) is N × 1 incident signal vector, and A is a M × N array. We can have $\begin{matrix} N (t) & = & [n_{1} (t), n_{1} (t), \dots n_{M} (t)]^{T} \\ S (t) & = & [s_{1} (t), s_{2} (t), \dots s_{M} (t)]^{T} \\ A (ω) & = & [a_{1} (ω), a_{2} (ω), \dots a_{N} (ω)] . \end{matrix}$ where A (ω) is the steering vector, and a (ω) is the incidence direction angle of signal source.

Considering about the covariance matrix of array snapshot data, $R_{X} = E [X (t) X^{H} (t)] = R_{S} + R_{N} = {AR}_{S} A^{H} + σ^{2} I$ (4) where R_x is the covariance matrix of signal and R_S is the covariance matrix of signal, and R_N is the covariance matrix of noise.

Then we make eigen decomposition of R_X: $R_{X} = U_{S} Σ_{S} U_{S}^{H} + U_{N} Σ_{N} U_{N}^{H}$ (5)

Under the ideal condition, the eigenvalue from the eigen decomposition of covariance matrix satisfies λ₁ > λ₂ ⋯ > λ_N > λ_N+1 = λ_N+2 ⋯ λ_M, and the eigenvector is mutually orthogonal and linearly orthogonal.

As shown in Fig. 2, the array antenna array element spacing is assumed to, far-field signal source and antenna array in the same plane, the incident angle of the i-th signal source is θ_i (i = 1, 2, ⋯ , N). Then the steering vector can be identified as:

Fig.2

The model of uniform line array.

$\begin{matrix} a (θ_{i}) & = & [1, exp (- j 2 π \frac{d}{λ} sin θ_{i}) \dots, \\ {exp (- j 2 π (M - 1) \frac{d}{λ} sin θ_{i})]}^{T} \end{matrix}$ (6)

The λ represents the wavelength of the signals.

2.1.1 The model of plane array

As shown in Fig. 3, assuming a plane array composed by of plansensors, the lower left corner of the array as a reference point. There are equal interval distribution along the X and the Y axis. The azimuth and pitch angle of incident signal are and. This time delay of the received signal compared with the time delay of the reference lattice element. $τ_{i} = \frac{1}{c} (x_{i} cos φ_{i} cos θ_{i} + y_{i} sin φ_{i} cos θ_{i})$ (7)c is the light-speed.

Fig.3

The model of plane array.

2.1.3 The model of uniform circular array

As shown in Fig. 4, assuming uniform circular array with M sensors. The circle radius is r, the center of the circle as the origin of coordinate, the azimuth is and the pitch angle is, so the steering vector can be determined as: $a (φ_{i}, θ_{i}) = {[e^{- j η \cos (φ_{i} - γ_{0})}, \dots, e^{- j η \cos (φ_{i} - γ_{M})}]}^{T}$ (8)

Fig.4

The model of uniform circle array.

Where, $\begin{matrix} η = \frac{2 π r}{λ} \cos θ_{i} \\ γ_{n + 1} = \frac{2 π m}{λ} \\ m = 0, \dots, M - 1 \end{matrix}$

2.2 Weighted subspace fitting algorithm (WSF)

When the data is long infinitely, the covariance matrix of signal is of full rank, and there is a diagonal matrix Σ_S which is composed by big eigenvalue, corresponding to eigen vecor λ₁, λ₂ ⋯ , λ_N. The space spanned by US is called signal subspace, and is same as the space spanned by array manifold A (ω), which means span {U_S} = span {A (ω)}.

At the meantime, there exists a full-rank matrix T, which enables U_S = A (ω) T. When there exists noise, span {U_S} ≠ span {A (ω)}. Therefore, we consider a weighted fitting relation to evaluate the approximating level of T to array manifold. First, we find the matrix which makes (8) valid and makes the best fitting under the criterion of the least square, which means. $θ, \overset{\land}{T} = min {∥ U_{S} - A \overset{\land}{T} ∥}_{F}^{2}$ (9)

Then, the least square solution of the fixed $\hat{T}$ is obtained by A. $\hat{T} = (A^{H} A)^{- 1} A^{H} U_{S} = A^{+} U_{S}$ (10)

Finally, the formula (10) into the Equation (9),

$\begin{matrix} θ & = & min {∥ U_{S} - {AA}^{+} U_{S} ∥}_{F}^{2} \\ = & min tr {P_{A}^{⊥} U_{S} U_{S}^{H}} \\ = & max tr {P_{A} U_{S} U_{S}^{H}} \end{matrix}$ (11) where $P_{A}^{⊥}$ is the orthogonal matrix of the projection matrix of the array manifold.

If we can further extend the fitting algorithm of weighted subspace: $θ, \hat{T} = min {∥ U_{S} W^{1 / 2} - A \overset{\land}{T} ∥}_{F}^{2}$ (12)

Then: $θ = min tr {P_{A}^{⊥} U_{S} {WU}_{S}^{H}} = max tr {P_{A} U_{S} {WU}_{S}^{H}}$ (13)

Optimization solution of WSF method is the essence of the visible value, to solve the function of the maximum or minimum value, according to the characteristics of DOA estimation, we can find many extreme points in the search space with formula corresponding to a plurality of incident angles.

Figure 5 is an example of the WSF spectrum of two signal sources impinge on a 10 sensor ULA with SNR from 0 dB to –15 dB. The DOAs are 20° and 30°, and 100 snapshots are applied. It is obvious to notice in Fig. 5 that as the SNR decreases, not only the global optima move apart from the true DOAs, but also the number of local maximums increases. This characteristic complicates the optimization of the likelihood function, especially for lowerSNR.

Fig.5

The spectrum of likelihood function. (a) SNR = 0 dB; (b) SNR = –15 dB.

2.3 Artificial bee colony algorithm

Artificial bee colony algorithm is an important branch of research of swarm intelligence algorithm. It is a new type of swarm intelligence optimization algorithm based on the mechanisms of honey bees which is proposed by Karaboga D in 2005.It was introduced into China in 2008. The algorithm simulates the intelligent behavior of the natural organisms in the natural world, and extracts the model from the behavior of their groups to establish some rules for these behaviors, so as to construct a stochastic optimization algorithm. The test of Literature [26] based on several standard functions shows that its performance is better than the current mainstream optimization algorithms, such as GA algorithm, PSO algorithm, DE algorithm and so on. Because the ABC algorithm is simple in operation and has few parameters and excellent performance, it has been paid more and more attention by scholars from various countries since the day of the day. It is widely used in the fields of communication, chemical industry, economy, image and signal processing.

Artificial bee colony algorithm is to simulate the foraging behavior of workers. It consists of three basic components, including the food source, employed bees and unemployed bees, and it is defined into two kinds of behavior, including the recruitment of bees to honey and honey. The recruited bees and the behavior of give up the food source of the artificial bee colony algorithm as shown in Fig. 6:

Fig.6

Principle of bee colony algorithm.

In the initial time, the bee acts as the scout bees to search the environment. The search made the decision by the system’s priori knowledge, and it can also be completely random. After an investigation, if the bees find the food source, the bees use the storage capacity to record position information of its own and began to gather honey. At this time, the bees will become the “employees” of honeybees in the food source. After unloading and then returning to the hive honey will have the following options:

It can give up the food source and become a non-employed bee.

It can dance swing dance for the corresponding food source to recruit more bees, and then back to the food source of honey.

It can continue to collect honey in the same food source and not to recruit.

There are options for non-employed bees:

It can become a scout bee to hive and search near the food source. The search can be determined by prior knowledge, which can also be completely random.

It can be observed in the end after being hired as swing dance follow bee, and begin to search for the corresponding neighborhood and food source of honey

Artificial bee colony algorithm is to simulate the actual bee optimization function processing mechanism of honey. It is divided into three categories: employed bees, onlooker bees and scout bees. Both onlookers and scouts are also called unemployed bees. The basic idea of the algorithm is following the initial population generated from a random start. The algorithm searches the fitness value of the better half of individuals using a competitive strategy for survival in a preferred to retain individual, and the operation is known as employed bees searching. Then the roulette wheel selection method is to choose the optimal individual, and the greedy search is executed around. So the other half have to be produced. This process is called the following bee searching. A new population is formed by the individuals of theunemployed bees to avoid the loss of population diversity, and the iterative population the algorithm of is established. Through continuous iterations, the algorithm retains the excellent individuals and eliminates the inferior individuals to approximate the global optimal solution. The algorithm is chosen to the process as follows:

Step 1. Initialization parameters, including the number of population NP, control parameters “limit” and the number of iterations “G”;

Step 2. Generating the initial population X^t in the solution space by the random optimization function in the initial iteration t = 0 in the solution space; $X_{i}^{t} = X_{l}^{t} + rand (0, 1) (X_{u}^{t} - X_{l}^{t})$ (14)

Where X^t represents the first individual in the T generation population, t ∈ {1, 2 ⋯ D} are D dimensional solution vector. $X_{l}^{t}, X_{u}^{t}$ is the upper and lower bounds of the T generation population respectively.

Step 3. calculating the fitness value of the individual in the population;

Step 4. determining the size of the initial employed bees is NP/2 according to the fitness value, the other half of the population is initialized as onlooker bees population;

Step 5. Searching for nectar and generating new individuals for the employed bee population according to the formula (15), and the best individual to retain the formation of a new employed bee population can be denoted as; $V_{i}^{t} = X_{t}^{t} + φ_{i}^{t} (X_{i}^{t} - X_{k}^{t})$ (15)

$φ_{i}^{t}$ is a random number between [–1,1]. k ∈ {1, 2 ⋯ NP/2}, k ≠ i, and the range of values V is the optimization function of the feasible space.

Step 6. Using the greedy operator comparison V and X population in order to retain a better fitness population, and using the $X_{i}^{t + 1}$ to updated the employed bees population. $X_{i}^{t + 1} = {\begin{matrix} V, f (V) \geq f (X_{i}^{t}) \\ X_{i}^{t}, f (V) < f (X_{i}^{t}) \end{matrix}$ (16)

Step 7. Following the bee in accordance with the probability formula (16) and using the roulette selection method to lead the bee population to select the superior individual, and forming the following bee population; $P_{i} = \frac{f_{i}}{\sum_{i = 1}^{NP / 2} f_{i}}$ (17)

Step 8. Forming the corresponding iterative population and recording the population fitness function value according to the individual of Step 6 and 7; and determining whether a new scout bee exists according to the limits; and terminating the algorithm according to the maximum number of iterations or according to the conditions set by the optimal solution output, otherwise go to Step 3.

Assume that we have an ULA with 10 sensors and 4 narrow sources. The space distance of adjacent sensor element is half signal wavelength. The number of snapshots is 50 and the SNR is 10 dB. The number of initial population is shown in Fig. 7. We run the bootstrap method 200 times and the ABC method to get the Fig. 8. The Fig. 9 is the simulation curve of spectral line for 50 times.

Fig.7

The diagram of initial population.

Fig.8

The estimation of source number.

Fig.9

Simulation diagram of power spectrum.

The Figs. 7 and 9 indicate that the performance of method based on ABC is effective. The number of sources is accurately estimated. In each basic membrane, the initial population outputs the new population after selection, crossover, mutation and finally determines by the fitness function. The communication protocol of the membrane structure dominates the final results in the output membrane. The intuitive results of spectrum are shown in Fig. 9.

2.4 Niche principle

Niche originated from biology, which is proposed by Grinnell in 1917. In biology, the niche refers to a kind of organization environment. In nature, some characteristics of similar shapes are in a specific environment and species clusters in the same mate between offspring, where offspring in general is limited to the environment, and gradually forms species.

In the field of evolutionary computation, niche intelligent optimization algorithm for solving multimodal function optimization problems. The goal is a multimodal optimization problem with multiple solutions of the problem, including the global optimization and local optimization. The niche technology is in fact: first it can make the swarm intelligence optimization algorithm to form species, and converge to different locations to multimodal optimization problems; second it has the ability to find multiple optimal solutions. In many domains, each value can be viewed as a niche, and the species is a subgroup of similar individuals living in a niche in the composition. The formation of niche and maintaining of it are two important problems to realize niche optimization. The traditional artificial bee colony algorithm only converges to a global optimal solution, which can be combined with niche for multimodal function optimization. The niche for artificial bee colony, first solves the niche identification, and selection of improvement strategies.

3 Improved ABC algorithm based on niche

DOA estimation algorithm generally estimates the information of multiple angles of incidence at the same time. According to WSF algorithm principle, DOA estimation can be used as a multi-peak optimization problem. In the context of different noises, the literature [12, 13] directly uses the ABC algorithm to solve the DOA problem. The computational complexity is large and it is easy to fall into local extremum. In [14], the search space is partitioned into different spaces according to the number of sources, and the ABC algorithm runs separately for each space to get the optimal solution. This method is less efficient, requiring the number and size of artificial grid division and having to set the threshold to determine the performance of the estimated merits, which is of a great relationship with the establishment.

The main changes are as follows:

Step 1. Initialize the parameters, set the number of sub-populations according to the number of sources, define the population size, set the dynamic radius and boundary parameters according to the niche identification technique, and define the limiting times and the maximum iterations;

Step 2. Randomly initialize the subpopulation in the WSF solution space, calculate the shared fitness function according to the shared fitness value of the individual by using the formula (18), and record the optimal solution and corresponding position in the initial population; $f_{i}^{'} = \frac{f_{i}}{\sum_{j = 1}^{N} F (d_{i j})}$ (18)

In the formula, $F (d_{ij}) = {\begin{matrix} 1 - (d_{ij} / r)^{α}, \begin{matrix} if & d_{ij} \end{matrix} < r \\ \begin{matrix} 0, & else \end{matrix} \end{matrix}$ , where r is the shared radius, d_ij is the Euclidean distance between the two individuals, and α is the control parameter.

Step 3. In each niche, each individual following bee selects the closest fitness value according to the Euclidean distance from the leading bee, and the individual with the better fitness value is retained and the iterated population is kept. Proceed until the required number of populations is reached;

Step 4. According to the roulette selection strategy, following bees select the leading bees, finding new sources of nectar, and choosing the optimal alternative solution; According to the search rules, decide whether to change the owing bee into a scouting bee, in order to research the solution space;

Step 5. Update the optimal solution and corresponding position in the niche according to the rule; and the exclusion strategy to keep the niche search space independent.

Step 6. According to the maximum number of iterations, determine whether to end the cycle, and output the final results.

In order to further optimize the computational complexity of the algorithm to improve the search efficiency, according to the characteristics of DOA direction finding and WSF algorithm, the maximum value of WSF function is defined as the basic incidence angle. Search radius is at least 5 degrees.

The time complexity analysis of the algorithm: Suppose that the number of the source is n, both the number of scouting bees and the number of following bees are (n-1). therefore:

The time required to calculate the fitness function is (n - 1) × t: The computational complexity isO (n - 1);

The time to calculate Euclidean distance between individuals is: (n - 1) (n - 2) × t/2 the computational complexity is O ((n - 1) ²);

The time to calculate the fitness function of each pair of individuals is (n - 1) (n - 2) × t/2, the computational complexity is: O ((n - 1) ²)

The time to update the nectar source is 2 (n - 1) t, the computational complexity is O (n - 1);

So the total time complexity required for the process is O ((n - 1) ²).

4 Experimental simulation

In this part, we mainly perform a series of experiments to verify the performance of the proposed algorithm. The common parameters are 2, the exclusion factor is 20, the number of iterations is 100, the search limit is 20 times, and the population number is NP = 40. Each test function runs 100 times independently, taking its performance average.

First of all, a simple simulation of a variety array forms were shown in Figs. 10 to 13. The parameters of each simulation are shown in the top of the figure. The experimental results show that the direction finding resolution increased with the increase of the number of sensors and the SNR. But the ULA is superior to other forms under the same conditions. ULA requires less number of sensors and SNR, which reduces the complexity of the system, and does not require complex protocol control. So the basic array of senor in experiment 1 to 6 is ULA.

Fig.10

The music method for uniform linear array direction finding.

Fig.11

The music method for uniform circular array direction finding.

Fig.12

The music method for L shape array direction finding.

Fig.13

The music method for uniform plane array direction finding.

Experiment 1: Assuming a total of 2 incident signals, the incident direction angle is 20°, and the 40°, and SNR is 10 dB, and the array number of the uniform linear array is 8, and the array element spacing is d = λ/2, and the snapshot number are 20. Test the convergence of the proposed algorithm.

It can be seen from Fig. 14 that the convergence rate of the proposed algorithm is good, and the estimated value of the two angles is close to the true value within about 20 times. The estimated value of the incident angle of 20° is 19.9844° at the average of 17 iterations and the estimated value of the incident angle of 40° is 39.9800° at the average of 12 iterations. It can be seen that the proposed algorithm can correctly estimate the incident angle at a SNR of 10 dB and an average of about 15 iterations at a shortsnapshot.

Fig.14

Convergence test.

Experiment 2: Basic condition as shown in Experiment 1, test the RMSE of the proposed algorithm, WSF algorithm and reference [13] algorithm under the same conditions. $RSME = \sqrt{\frac{1}{{NN}_{k}} \sum_{i = 1}^{N_{k}} {\sum_{i = 1}^{N} [\overset{\land}{θ_{i} (k)} - θ_{i}]}^{2}}$

It can be seen from Fig. 15 that the proposed algorithm has a large error at a signal-to-noise ratio less than –5, which indicates that the proposed algorithm has poor performance at low signal-to-noise ratio, but when the SNR is greater than zero, the performance is significantly better than the other two algorithms. Overall, the proposed algorithm has good estimation accuracy.

Fig.15

Comparison of mean square error of three methods.

Experiment 3: Assuming that one narrow source impinging on ULA with four sensors. The space of adjacent sensors is half of the wave length of signals received. DOA parameters of the sources are 40°. The number of snapshots is L. And re-sample times is B = 200. Each group of experiments is carried for 30 times to compare the two algorithms under the conditions of different snapshot number. It is shown in Fig. 16 that the distribution of solutions is convergent with the increasing of snapshot number.

Fig.16

The success rate of different method with different number of snapshots.

5 Conclusion

In order to optimize the computational complexity of DOA estimation, a novel niche algorithm is proposed to solve the problem of multi - modal function optimization by using the optimization ability of artificial bee colony algorithm. Based on the deep research and analysis of the existing multi-modal optimization algorithms, various improvement measures are proposed. Each individual in the algorithm can move closer to the nearest peak, and the iterative population can be determined by the squeezing mechanism which converges to a single optimal peak in order to ensure the algorithm to find all the peak points as much as possible. Using the niche technique of dynamic boundary recognition, we can ensure that the following bees are mined around each peak to improve the searching accuracy. In addition, the algorithm itself does not need additional parameters, which is easy to implement. The experimental results show that the proposed algorithm is an effective DOA estimation method for DOA multimodal function optimization problem with low computational complexity, good diversity and high accuracy.

Footnotes

Acknowledgments

The content of this research was supported by the National Natural Science Foundation of China (61571149) and the Central University Basic Research Foundation (heucf160808).

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