Virtual array element beamforming algorithm based on LSTM neural network

Virtual array element beamforming technology can improve the angular resolution of conventional beamforming algorithm. However, when the number of real array elements is limited, the beam pattern of the virtual array element beamforming algorithm based on linear prediction(the VAEBF-LP) is slightly distorted, the root mean square error(RMSE) of Direction of Arrival(DOA) is larger and so is the main lobe width. In response to this problem, A virtual array element beamforming algorithm based on the LSTM neural network(the VAEBF-LSTM) is proposed. This algorithm uses the LSTM neural network to learn the temporal characteristics between the received data of real array elements and the virtual array elements. Obtaining the received data of virtual array elements, this algorithm enables the beam pattern to be free of distortion and achieve higher angular resolution detection of targets with a relatively low RMSE of DOA in this context. The simulation results show that the VAEBF-LSTM not only makes the beam pattern distortion disappear but also makes the main lobe smaller in width than that of the conventional beamforming algorithm with the guarantee that the RMSE of DOA remains considerably low when the number of real array elements is limited.


Introduction
Restricted by the Rayleigh criterion, the main lobe of the beam pattern obtained by the conventional beamforming algorithm is wide [1], which affects the target detection accuracy when the array aperture is limited [2].To solve this problem, the method of merely increasing the aperture of the array is often adopted.However, due to the requirements of the application scene and the cost, this method has great limitations in engineering applications.In recent years, virtual array element technology has been proposed to improve the angular resolution of conventional beamforming algorithm.The algorithms for constructing virtual array elements mainly include high-order cumulants [3,4], interpolation transformation [5,6], linear prediction [7], and least square method [8].
However, the beam pattern obtained by the VAEBF-LP is slightly distorted, the RMSE of DOA is larger and so is the main lobe width when the number of real array elements is limited.Through further research, it is found that using the linear prediction algorithm to realize virtual array element technology is equivalent to using the AR model to realize time series prediction and the AR model is an early basic model of time series prediction problem [9].
Considering that deep learning has good nonlinear fitting ability and predictive ability when dealing with nonlinear problems, the complex relationship in sequence data can be discovered by a deep learning model.This method helps extract the features contained in a large amount of data and learn the optimal feature representation.In recent years, time-series prediction methods based on deep learning have developed rapidly and LSTM networks have been successfully applied to time-series data prediction tasks such as water quality prediction [10] and stock index return rate prediction [11].In the field of underwater acoustics, the received data of an array element within a certain period can be regarded as a time series.
Inspired by this, the LSTM network was introduced.The received data of real array elements is used as input and the received data of virtual array elements is used as output and both of them will be processed by conventional beamforming algorithm.The more information conventional beamforming algorithm processes, the more possibility of getting better beamforming effects.
The trained LSTM can provide more precise information on the sound field that can be used by the conventional beamforming algorithm.Compared with the VAEBF-LP, the VAEBF-LSTM not only eliminates the beam pattern distortion but also makes the width of the main lobe smaller than that of the conventional beamforming algorithm under the premise of ensuring a relatively low RMSE of DOA when the number of actual array elements is limited.

Virtual array element beamforming model
Suppose a plane wave is incident at an angle  on a uniform array with the number of array elements N , as shown in figure1.In this figure, solid dots represent real array elements, numbered 1, 2,..., N , and hollow dots represent virtual array elements, numbered 1, 2,...
. The virtual array element technology is based on the received data of real array elements and expands N array elements outward to obtain the received data of the virtual array elements.

Long short-term memory neural network(LSTM)
Hochreiter and Schmidhuber proposed a long short-term memory neural network (LSTM) [12].LSTM proposes the concept of "memory cells" by referring to human memory patterns and decision-making methods for past events and introduces three gating structures to control memory cells.This structure controls information flow and realizes selective updates of past and current information.Therefore, LSTM can better explore long-term dependencies and carry out the processing of sequence data with long-term dependencies.The LSTM network introduces three gating structures: input gate, forget gate, and output gate.The schematic diagram of the hidden layer unit of the LSTM network is shown in figure 2.
Among them, t f is the output of the forget gate.The function of the forget gate is to realize the selective retention of information in the memory cells 1 t c  and to eliminate non-important information.The formula is as follows:  is a candidate memory cell generated according to the activation value 1 t a  and current input t x and as a supplement to the attribute information of the memory cell.The formula is as follows: t i is the output of the input gate.The input gate gives the memory cell the right to maintain the candidate value to a certain extent and determine the proportion of memory cells that need to be updated.The formula is as follows: As for the activation value t a , it is not only determined by the activation value of the previous moment 1 t a  but the current input t x .The formula is as follows: * tanh( )

The principle of virtual array element beamforming algorithm based on LSTM neural network
For a uniform linear array, the received data of each array element over a period has continuity from the time dimension.At the same time, the received data of adjacent array elements over a period are correlated in the time dimension due to the array setup.the VAEBF-LSTM utilizes the trend of the received data of real array elements to make a nonlinear prediction of the received data of virtual array elements.Specifically, the relationship between the received data of real array elements and the virtual array elements can be mined and more detailed information about the sound field can be obtained by building an LSTM network.
Suppose there is a sound source emitting a continuous acoustic signal in the far field and the sound source is located at some known angles.To reduce the number of array elements and realize the higher angular resolution detection of targets with a relatively low RMSE of DOA, a uniform line array is first arranged to receive the signal in the sound field and then some array elements are removed from one side of the array.
In the early period, the received data of each array element is recorded long enough to ensure that the acoustic signals from all possible angles are recorded.After the LSTM network is trained with the received data in the early period, it will learn the time domain characteristics of the received data between the two parts of array elements.In the later period, only the acoustic signal from a certain angle is recorded.By inputting the received data of the real array element in the early period into the LSTM model after training, the received data of the virtual array elements in the later period is predicted, and the sound field information at the position of the virtual array elements in the later period is obtained.By this means, there is more precise information that can be used by the conventional beamforming algorithm, the distortion removal and the higher angular resolution detection of targets can be achieved with the guarantee that the RMSE of DOA remains considerably low when the number of real array elements is limited.The flowchart of the proposed algorithm is shown in figure 3.

Modeling of LSTM neural network
A. Design of network structure LSTM network has a good effect on time series data prediction.Therefore, this paper uses the LSTM network to realize the prediction of the received data of the virtual array elements.The network structure is shown in figure 4   In this task, the introduced LSTM network consists of three main parts: the input layer, hidden layer, and output layer.The input layer receives the received data from real array elements and transmits the data to the hidden layer behind for nonlinear transformation.The output layer is a fully connected layer with the number of neurons equal to the number of virtual array elements.It processes the output of the nonlinear transformation layer and constructs the predicted received data of virtual array elements.Moreover, To prevent overfitting of the LSTM network, a dropout layer is added to the network.Some neurons in the neural network are temporarily removed to improve the generalization ability of the model.

B. Data set Generation
Given that the received data from real array elements is available for both early and later periods, it serves as input for the network.On the contrary, the data received by virtual array elements, which is intended for prediction, will constitute the network's output.
C. Training process According to a certain ratio, the data set is divided into a training set and a validation set.The structure of the LSTM network is shown in Figure 2. In deep learning, regression problems usually use mean square error as the loss function to update the weights and biases of the model [13].The LSTM network uses the backpropagation algorithm to update and optimize the parameters that need to be trained in the LSTM network layer by layer by reducing the loss function.In the process of model training, the hyperparameters of the network model can be adjusted by observing the error curve and accuracy curve of the training set and validation set which can help find the best parameter configuration of the model and enhance the generalization ability of the model [14].

Simulation experiment and result analysis
In this section, the performance of the VAEBF-LSTM proposed in this paper and the VAEBF-LP are compared under the following experimental conditions.Firstly, the distortions of the beam patterns of the VAEBF-LP and the VAEBF-LSTM, when the number of real array elements is limited, are verified, and then the variation of the RMSE of DOA and the width of the main lobe of the two algorithms under different numbers of virtual array elements are discussed, and finally these two algorithms are evaluated.
In the following experiment, a uniform 4-element linear array is selected.The received data consists of a 1000Hz single-frequency signal combined with a Gaussian signal, and the signal-to-noise ratio stands at 9dB.Additionally, the range of virtual array elements varies from 1 to 4 incrementing by 1 step each time and 100 Monte Carlo experiments are performed respectively.
In the training model stage, simulate the received data of real array elements and virtual array elements within the period 1 T with azimuths of [ 45 , 30 , 10 , 0 ,10 , 30 , 45 ]             respectively in a far field.Among them, 80% of the samples are divided into the training set to train the LSTM network, and the remaining 20% of the samples are used as the verification set to evaluate the effect of the model constructed by different hyperparameters.When it comes to the prediction in the using model stage, take a period called 2 T after period 1 T and simulate the received data of real array elements within the period 2 T with azimuths of 30 in the same field as a testing set.Once this set is input into the trained LSTM network, it generates predictions for the received data of virtual array elements within the period 2 T .Figure 5 and figure 6 show the comparison of the beam patterns of the VAEBF-LP and the VAEBF-LSTM when the number of virtual array elements is 1 to 4, respectively.In these two figures, the solid, dash-dotted, dashed, and dotted lines indicate the beam patterns when the number of virtual array elements is 1, 2, 3, and 4, respectively.It can be seen that the beam pattern is slightly distorted when the number of virtual array elements is limited, especially when the number of virtual array elements increases to 4. On the contrary, as the number of virtual array elements increases, the beam patterns for the VAEBF-LSTM were never distorted.Therefore, the VAEBF-LSTM can eliminate the distortion in beam patterns, which the VAEBF-LP can not do.
Figure 7 shows the comparison of the RMSE of DOA of the three algorithms: conventional beamforming, the VAEBF-LP, and the VAEBF-LSTM.In this figure, the dotted line with data points shaped as pentagrams, the dash-dotted line with data points shaped as circles, and the dashed line with data points shaped as rectangles indicate the variation of the RMSE of DOA of the conventional beamforming algorithm, the VAEBF-LP, the VAEBF-LSTM under different numbers of virtual array elements, respectively.
It can be seen that the RMSE of DOA of the VAEBF-LP is smaller than that of the conventional beamforming algorithm in general.However, the RMSE of DOA of the VAEBF-LSTM is much smaller than that of the VAEBF-LP.By comparing these two figures, it can be inferred that the VAEBF-LSTM has a significantly lower RMSE of DOA, so more accurate DOA estimation results can be acquired.Figure 8 shows the comparison of the main lobe width of the three algorithms: conventional beamforming, the VAEBF-LP, and the VAEBF-LSTM.In this figure, the dotted line with data points shaped as pentagrams, the dash-dotted line with data points shaped as circles, and the dashed line with data points shaped as rectangles indicate the variation of the main lobe width of the conventional beamforming algorithm, the VAEBF-LP, the VAEBF-LSTM under different numbers of virtual array elements, respectively.
It can be seen that the main lobe width of the VAEBF-LP is smaller than that of the conventional beamforming algorithm.However, the VAEBF-LSTM always has the smaller main lobe width than the VAEBF-LP.And there is a continuous decrease in the main lobe width with the increase of the number of virtual array elements.Furthermore, this is consistent with the relationship between the main lobe width and the number of array elements in beamforming.
Therefore, the conclusion can be drawn based on the above experiments.Derived from figure 5 to figure 8, it can be concluded that when the number of real array elements is limited, the VAEBF-LSTM does not suffer from beam pattern distortion, and is capable of reducing the main lobe width based on conventional beamforming with a relatively low RMSE of DOA compared with the VAEBF-LP.

Conclusion
In this paper, the VAEBF-LSTM is proposed.It uses the LSTM neural network to learn the temporal characteristics between the received data of real array elements and virtual array elements and obtains the received data of the virtual array elements.Utilizing simulation experiments and analysis, the algorithm proposed in this paper not only guarantees undistorted beam patterns but also excels in narrowing the width of the main lobe based on conventional beamforming while maintaining a relatively low RMSE of DOA, which distinguishes the VAEBF-LP.
In the current simulation experiment, both the receiving array and the far-field sound source remain stationary.In future endeavors, we aim to make the algorithm applicable to scenarios where both the receiving array and the far-field sound source are in motion.This will broaden the scope of potential application scenarios.

Figure 2 .
Figure 2. Schematic diagram of the hidden layer unit of the LSTM network. 1

to
is the output of the output gate.It controls the proportion of the memory cell value flowing into the current activation value.The formula is as follows: Journal of Physics: Conference Series 2718 (2024) 012093 IOP Publishing doi:10.1088/1742-6596/2718/1/0120934

Figure 3 .
Figure 3. Flowchart of the proposed algorithm. below.

Figure 5 .
Figure 5. CBF Spectrum of The Virtual Array Element Beamforming Algorithm Based On Linear Prediction.

Figure 6 .
Figure 6.CBF Spectrum of The Virtual Array Element Beamforming Algorithm Based On LSTM Neural Network.

Figure 7 .
Figure 7. Variation of RMSE of DOA Under Different Numbers of Virtual Array Elements.

Figure 8 .
Figure 8. Variation of Main Lobe Width Under Different Numbers of Virtual Array Elements.