Convolution Neural Networks for Localization of Near-Field Sources via Symmetric Double-Nested Array

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Introduction
Neural networks are developing rapidly with deep learning, which have important applications in the field of array signal processing. By treating the assignment of angle-of-arrival (AOA) label as a separate binary classification, a convolutional neural network (CNN) method in [1] is presented for multispeaker AOA estimation. By employing an end-to-end regression rather than a classification, the complex-valued residual network (ResNet) in [2] is proposed to improve the training performance for AOA estimation in shortrange multiple-input multiple-output (MIMO) communications. In [3], some small deep feedforward networks are trained for AOA estimation under uniform circular array, which can reduce computational complexity and have similar satisfactory performance. In the case of the rectilinear or strictly noncircular sources, a symmetric uniform linear array of cocentered orthogonal loop and dipole (COLD) antennas is exploited in [4] to achieve the localization of near-field sources, where the DOA, range, and polarization parameters are separated by employing the rank reduction (RARE) principle. Moreover, by using the noncircular information for a symmetric uniform linear array, a novel method is proposed in [5] to realize the localization and classification of the mixed sources. Considering that the model-driven approaches depend on the preformulated models [6,7], the data-driven approach of deep neural network (DNN) [8] implements autoencoders and classifiers to estimate the AOA, which can adapt to the scenarios of array imperfections. However, in order to satisfy the additive property when multiple sources are distributed in different subregions, this algorithm replaces the activation function with unit operation, which makes it impossible to approximate the nonlinear model perfectly. Furthermore, the deep convolution network (DCN) [9] is utilized to learn the features of the spectrum proxy and reconstruct the spatial spectrum, which is more accurate and efficient for AOA estimation.
Recent researches have focused on the near-field source localization [10][11][12], and the regression approach of the CNN structure [13] is proposed to estimate the AOA and range parameters of near-field sources, which has better performance in the case of low signal-to-noise ratios (SNRs) and small number of snapshots. However, this method regards the elements of the covariance matrix as the input of the network, which is only suitable for the scenarios with the fixed number of near-field sources. In contrast, the incoherent sources can be separated in the frequency spectrum, employing the phase matrix as the input of the network can be suitable for the different number of sources.
In the case of the same number of sensors, the nested array [14][15][16] has a larger aperture than the uniform array, which can improve the accuracy of parameter estimation. In [17], the sensor's location of the symmetric doublenested array (SDNA) is given in a closed form. The vectorization of multiple fourth-order cumulant matrices in [18] is applied to achieve the localization of underdetermined near-field sources via a nested array, which can enhance degrees-of-freedom. However, the process of the fourthorder cumulant matrices requires high computational cost.
Accordingly, the motivation of the work is to employ the CNN to achieve near-field source localization and improve the accuracy of parameter estimation. We first calculate the phase difference matrix via the symmetric double-nested array (SDNA). Then, the autoencoders are used to divide the AOA subregions. Next, the corresponding classification CNNs of subregions are constructed to obtain the AOAs of near-field sources. Finally, we employ the regression CNN to determine the range of near-field sources. Simulation results illustrate that the proposed method is robust to the off-grid parameters and suitable for the scenarios with the different number of sources. Moreover, the proposed method outperforms the existing method for near-field source localization.

Signal Model
As shown in Figure 1, the geometry of SDNA is composed of two subarrays, where the solid circles denote the first subarray with spacing d and locate in M is even, and the hollow circles denote the second subarray with spacing ðM/2 + 1Þd and locate in Assuming that the center of the array is regarded as the reference point and the SDNA is impinged by K incoherent narrowband near-field sources, the location of the kth nearfield source can be described as ðθ k , r k Þ, where θ k denotes the AOA and r k denotes the range parameter. Note that the near-field region of the array is between 0:62ðD 3 /λÞ 1/2 and 2D 2 /λ, where D = MðM/2 + 1Þd denotes the aperture of the array and λ denotes the wavelength of the near-field sources. Hence, the received data of the mth sensor at the nth snapshot can be modeled as for m = −M, ⋯, −1, 0, 1, ⋯, M, n = 1, 2, ⋯, N, where N denotes the number of snapshots, s k ðnÞ denotes the kth transmitted near-field source at the nth snapshot, w m ðnÞ stands for the additive white complex Gaussian noise, and φ k,m stands for the phase which is related to the time delay between the kth near-field source at the mth sensor and that at the center of SDNA. In addition, φ k,m can be expressed as for k = 1, ⋯, K, where 2π/λ denotes the wavenumber, r k,m denotes the range measured from the kth near-field source to the mth sensor. Applying the second-order Taylor series expansion [17,18] of r k,m at the point d/r k , the signal model can be written as where η k = −2π sin θ k /λ with ϕ k = π cos 2 θ k /λr k .

Proposed Algorithm
In this section, we consider the phase difference as the input of the networks, utilize the autoencoder to divide the AOA subregions, and exploit the CNN to achieve the localization of the near-field sources.

Preprocessing of Phase Difference.
It is generally known that the number of incoherent sources can be estimated by the number of peaks in the frequency spectrum. Assuming that the gth training data contains a single near-field source, the phase difference between the m 1 th sensor and m 2 th sensor can be calculated as where G denotes the number of the training dataset, and ϑ m 1 ,g and ϑ m 2 ,g , respectively, denote the phase of the peak in frequency spectrum at the m 1 th sensor and the m 2 th sensor. Accordingly, the phase difference is converted between -2π and 2π, which can be expressed as Therefore, the ð2M + 1Þ × ð2M + 1Þ dimensional phase 2 Wireless Communications and Mobile Computing difference matrix of the gth training data can be written as where the element u m 1 ,m 2 ,g is calculated by (6). Note that the localization parameters can be determined from the corresponding phase difference matrix.

Autoencoder of AOA Subregions.
Since the counterdiagonal elements of the phase difference matrix in (8) are centrosymmetric and only contain the AOAs but not the range parameters of near-field sources, we first employ the upper right counter-diagonal elements to estimate the AOA of each near-field source.
In order to guarantee the precision of the AOA estimation, we divide the AOA scope into P subregions by utilizing the structure of the autoencoder. The autoencoder of the pth subregion for the gth training data is shown in Figure 2, where the input of the encoder u ð1,0Þ g is an M-dimensional vector, the output of the encoder u ð1,p,1Þ g is an M/2-dimensional vector, and the output of the decoder u ð1,p,2Þ g is an M -dimensional vector.
Based on the phase difference matrix in (8), the input of the autoencoder is given by for m 3 = M, M − 1, ⋯, 1, ð·Þ T denotes the transpose operation. To be specific, the output of the encoder is and the output of the decoder is where ReLU(·) denotes the activation function of a rectified linear unit, Q ð1,p,1Þ g and Q ð1,p,2Þ g , respectively, denote the feedforward weight matrix of the encoder and decoder, and b ð1,p,1Þ g and b ð1,p,2Þ g , respectively, denote the bias vector of the encoder and decoder. Noticeably, the DNN algorithm in [8] replaces the activation function with the unit operation to meet the additive property when multiple sources are distributed in different subregions. In contrast, the proposed method employs the phase difference as the input of the autoencoder and applies the ReLU function, which can approximate the nonlinear model and is suitable for the scenarios of multiple sources.
The dataset Ψ    = arg min 3.3. Classification CNN for AOA Estimation. After the auto-encoder is trained, the prediction for the gth training data of the pth autoencoder iŝ Based on the prediction of the autoencoder, we employ the classification CNN to obtain the spatial spectrum and estimate the AOAs of the near-field sources. Herein, the CNN of the pth subregion for the gth data is depicted in Figure 4: Regression CNN for the range estimation of the gth data. Algorithm for near-field source localization 1.
Construct the phase difference matrix of each near-field source in (8).

2.
Extract the upper right counter-diagonal elements in (8) and perform the trained pth autoencoder to obtain the output of the kth subregionû 1,p,2 ð Þ k .

3.
Perform the trained classification CNN to obtain the corresponding output of the kth spatial spectrumû 2,p,5 ð Þ k .

4.
Splice the output of the P AOA spatial spectrum and search the spatial peaks to determine the AOAs of the near-field sources.

5.
Construct a particular range vector in (17) to eliminate the estimated AOA of the kth near-field source.

6.
Perform the trained regression CNN with u 3,0 ð Þ k to determine the corresponding range of the kth near-field source. Clearly, the output of the lth CL is where u ð2,p,0Þ g =û ð1,p,2Þ g , * denotes the convolution operation, Ωð·Þ denotes the zero-padding operation to make the output size equal to the input size, and Q ð2,p,lÞ g and b ð2,p,lÞ g , respectively, denote the convolution kernel and bias. Moreover, the ζ p -dimensional vector u ð2,p,5Þ g denotes the output of the FCL and the pth AOA subregion, which is given by where Tanh(·) denotes the activation function of tangent hyperbolic and Q ð2,p,5Þ g and b ð2,p,5Þ g , respectively, denote the feed-forward weight matrix and bias vector of FCL. Herein, MSE is also used as the loss function and the dataset Ψ denotes the label (i.e., the corresponding spatial spectrum) of the pth AOA subregion for the gth training data.
Therefore, the AOA spatial spectrum of the near-field sources can be obtained by splicing the output of the P AOA subregions, and the AOA of the near-field sources can be determined from the position of the spatial peaks.

Regression CNN for Range Estimation.
Since the second part of the elements in (8) contains the range parameter of the near-field sources, and the first part of the counterdiagonal symmetric elements (e.g., u m 4 ,m 5 ,g and u -m 5 ,-m 4 ,g ) are equal, we eliminate the AOA in (8) and apply the regression CNN to determine the range of the near-field sources. Herein, the CNN for the range estimation is depicted in Figure 4, which contains three CLs, one FL, and three FCLs.
To be specific, the output of the lth CL is where the input of the 1st CL is Similarly, MSE is used as the loss function for each layer, and the dataset Ψ With respect to the computational complexity, we consider the major multiplications and additions in the networks. As for the autoencoder of each subregion, the computational complexity of encoder requires M × ðM/2Þ multiplications plus M/2 additions, and that of decoder requires ðM/2Þ × M multiplications plus M additions. As for the classification CNN for AOA estimation, the computational complexity of multiplications in CLs is h 1,1 × 12M, h 1,2 × 6M and h 1,3 × 3M, that of additions in CLs are 12M, 6M, and 3M, that of multiplication in FCL is ζ p × 3M, and that of addition in FCL is ζ p , where h 1,1 , h 1,2 , and h 1,3 , respectively, denote the filter length of CLs in the classification CNN. As for the regression CNN for range estimation, the computational complexity of multiplications in CLs is h 2,1 × 3MðM + 1Þ, h 2,2 × 1:5MðM + 1Þ, and h 2,3 × 0:5MðM + 1Þ, that of additions in CLs are 3MðM + 1Þ, 1:5MðM + 1Þ, and 0:5MðM + 1Þ, that of multiplications in FCL are ðMðM + 1Þ /3Þ × ðMðM + 1Þ/2Þ, ðMðM + 1Þ/7Þ × ðMðM + 1Þ/3Þ, and 1 × ðMðM + 1Þ/7Þ, and that of addition in FCL is MðM + 1Þ/ 3, MðM + 1Þ/7, 1, where h 2,1 , h 2,2 , and h 2,3 , respectively, denote the filter length of CLs in the regression CNN.
Once the networks have been trained, the main steps of the proposed method to achieve near-field source localization are as shown in Table 1.

Simulation Results
In the simulations, the sensor number of SDNA is 13 (M = 6 ), the AOA scope is from 60°to 60°which is divided into 12 subregions, and the near-field region is from 2.4λ to 11.5λ. The AOA scope and near-field region are, respectively, uniformly sampled with intervals of 1°and 0.1λ to generate 11132 training groups, where the AOAs are set as {-60°, -59°,…-1°, 0°, 1°,…, 60°} and the range parameters are set as {2.4λ, 2.5λ,…, 11.5λ}. Each group contains 10 training dataset with different SNR, where the snapshot number is 1024 and the SNR is randomly picked from 5 dB to 25 dB. The   Figure 6 displays that the estimation error is within 8e -2 λ, which proves that the proposed method is robust to parameter estimation.

Effectiveness for Different Number of Sources.
Compared with the algorithm in [13] that utilizes the covariance matrix to train the networks in the case of the fixed number of sources, the proposed method employs the phase difference matrix of each source to train the networks. When the SNR is set as 10 dB, Table 2 shows the real location and estimated location in the case of 1, 3, and 5 sources. Besides, Figure 7 displays the AOA spatial spectrum, where the diamondshaped points are the real AOAs and correspond to the spectrum peaks. Therefore, it can conclude that the proposed method is suitable for the localization scenarios with different number of sources.

RMSEs of Parameter Estimation.
In order to further demonstrate the performance of near-field source localization, we compare the proposed method via the geometry of SDNA to that via symmetric double uniform array (SDUA) in the case of the same sensor number. Besides, we compare the proposed method to the state of the art fourth-order cumulant (FOC) method in [18] via the geometry of SDNA and the theoretical Cramér-Rao lower Bound (CRLB) via the geometry of SDNA.
The root mean square errors (RMSEs) of AOA and range parameters are given in Figure 8, where the locations of the near-field sources in the simulations are set as (10°,4λ) and (20°,6.1λ), and the RMSE at each SNR is determined by 500 independent Monte Carlo simulations. It can be seen that the RMSE of the proposed method via SDNA is lower than that via SDUA. Moreover, the proposed method outperforms the FOC method and can improve the accuracy of near-field source localization.

Conclusions
This letter exploits the geometry of SDNA to achieve the localization of near-field sources, where the autoencoders and classification CNNs are utilized to jointly determine the AOAs of near-field sources, and the regression CNN is employed to determine the range of near-field sources. Simulation results illustrate that the proposed method is robust to the parameter estimation of off-grid near-field sources. Besides, the proposed method applies the phase difference as the input of networks, which leads to the advantages for the localization scenarios with different number of nearfield sources. In addition, the proposed method via SNDA outperforms the FOC method and has higher accuracy than that via SNUA for near-field source localization.

Data Availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.