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Design of sparse arrays via deep learning for enhanced DOA estimation
EURASIP Journal on Advances in Signal Processing volume 2021, Article number: 17 (2021)
Abstract
This paper introduces an enhanced deep learningbased (DL) antenna selection approach for optimum sparse linear array selection for directionofarrival (DOA) estimation applications. Generally, the antenna selection problem yields a combination of subarrays as a solution. Previous DLbased methods designated these subarrays as classes to fit the problem into a classification problem to which a convolutional neural network (CNN) is employed to solve it. However, these methods sample the combination set randomly to reduce computational cost related to the generation of training data, and it often leads to suboptimal solutions due to illsampling issues. Hence, in this paper, we propose an improved DLbased method by constraining the combination set to retain the holefree subarrays to enhance the method’s performance and sparse subarrays rendered. Numerical examples show that the proposed method yields sparser subarrays with better beampattern properties and improved DOA estimation performance than conventional DL techniques.
Introduction
The design of sparse or nonuniform arrays for DOA estimationbased applications has received tremendous attention due to their ability to resolve \(\mathcal {O}(M^{2})\) sources given M sensors only. One of the wellknown techniques for realizing sparse arrays for DOA estimation applications is array thinning or antenna selection [1–3]. Conventionally, the antenna selection problem is cast as either optimum placement of a given number of antenna elements or selecting an optimum subset of antennas or subarray [4]. Regardless of the formulation, it has been reported in [4–6] that an optimum subarray can preserve a large physical aperture and enables high angular resolution of target localization. Moreover, it reduces the energy and computation costs exerted otherwise on the radiofrequency (RF) system frontend [7–9]. Despite the merits, the art of selecting an optimum subarray from a uniform array is not a trivial matter, and it requires careful consideration of the selection technique depending on the choice of the selection criteria [4–11].
The traditional formulation of antenna selection involves either combinatorial or convex optimization problem formulations. As such, convex relaxation and combinatorial optimization algorithms are used to obtain optimum subarray while minimizing or maximizing a specific objective function [4–8]. However, such methods involve searchbased, and greedybased algorithms, which are computationally expensive [7]. Learningbased optimization techniques have recently been adopted to solve optimization problems such as antenna selection due to less computationally complex solutions than their conventional counterparts [9–11]. For instance, a support vector machine (SVM) together with an artificial neural network (ANN) was introduced in [9] to realize sparse array configurations that maximize signaltointerferenceplusnoise ratio (SINR) using sensing environmental features extracted from caponbeamformer.
Moreover, [10] proposed a deep learningbased (DL) sparse array selection technique to realize sparse arrays given sample covariance matrix as an input. The method in [10] utilizes a convolutional neural network (CNN) model to predict the best subarray sensor indices, which minimizes the Cram\(\acute {\mathrm {e}}\)rRao bound (CRB) of DOA estimation. In particular, [10] focused on a single source DOA estimation problem on both 1dimensional (1D) and 2dimensional (2D) arrays. The approach was extended to multiple sources problems in [11]. Moreover, [11] presents further results on the DOA estimation performance of the rendered sparse arrays using two distinctive DOA estimation methods: the fast iterative softthresholding algorithm (FISTA) [12] and multiple signal classification algorithm (MUSIC) [13].
The simulation results in [10] show that the proposed DLbased selector yields suboptimal 1D sparse subarray compared to 2D random and circular subarrays. The realized 1D sparse subarray has sensors located at the end of each side of the uniform linear array (ULA) grid. It was observed in [9] that there is a tradeoff between the distribution of sensors in a sparse array and the size of the peak sidelobe levels (PSL), which dictates the variance of DOA estimation. Thus, the linear sparse subarray realized in [10] exhibits high PSLs, hence the poor DOA estimation performance. In this work, we aim to take advantage of the less computation complexity of DLbased algorithms and consider properties of nonuniform sparse linear arrays that control sparse arrays’ performance to improve the DLbased selector and corresponding sparse linear arrays.
In this paper, we introduce an enhanced DLbased antenna selection approach. The proposed approach takes advantage of beampattern properties of sparse arrays with a near holefree or holefree difference coarray and its impact on peak sidelobe levels and variance DOA estimates. Since picking M antennas out of Nelement ULA yields _{N}C_{M} possible subarrays, which are considered classes and form the training samples for DLbased selector, we constrain such subarrays to retain a holefree difference coarray, and the resulting subarrays are used to generate the training dataset. A basic CNN model is employed to classify sparse arrays, which minimize CRB of DOA estimation using the realized dataset. Numerical examples show that the proposed method yields sparser arrays with improved beampattern properties and DOA estimation performance compared to the original large ULA and other wellknown sparse arrays.
Notations Throughout the paper, we use lowercase and uppercase bold characters to denote vectors and matrices, respectively, i.e., I_{K} represents the K×K identity matrix. Operators (·)^{T} and (·)^{H} stand for transpose and the conjugate transpose of a vector or matrix in that order. And vec(·) denotes vectorization operator, and diag(·) represents a diagonal matrix. Moreover, ⊙ and E[·] denote the KhatriRao product and statistical expectation operator respectively. The calligraphic letter \(\mathcal {Z}_{i}\) denotes a position set of ith array whereas \(\boldsymbol {\mathcal {G}}\) and \(\boldsymbol {\mathcal {L}}\) consist of all subarrays and optimized subarrays in that order. Moreover, the cardinalities of set \(\boldsymbol {\mathcal {G}}\) and \(\boldsymbol {\mathcal {L}}\) are given by nonbold capital letters G and L for \(\boldsymbol {\mathcal {G}}\) and \(\boldsymbol {\mathcal {L}}\), respectively. The operator _{(·)}C_{(·)} represents combination whereas functions \(\angle \{ \cdot \}, \mathbb {R}\mathrm {e}\{ \cdot \}\) and \(\mathbb {I}\mathrm {m}\{ \cdot \}\) indicate the phase, real and imaginary parts of a complex argument in that order.
Preliminaries
Consider a ULA with N sensors with interelement spacing of \(z_{n}a_{o}, z_{n} \in \mathcal {Z}\) for n=1,2,…,N where \(\mathcal {Z}\) denotes the set of sensor positions of the physical array. Assuming that D uncorrelated narrowband sources are impinging on the array from far field directions θ_{1},θ_{2},...,θ_{d}, for d=1,2,...,D. Then, the dth source steering vector can be expressed as
where κ=2π/λ,a_{o}=1/λ and λ is the carrier’s frequency wavelength. As such, the received signal vector can be expressed as
where s(t) and n(t) are the source signal vector and the noise vector at tth snapshot. respectively. And, A=[a(θ_{1}),a(θ_{2}),…,a(θ_{D})] is the array steering matrix. Moreover, the noise is assumed to be additive white Gaussian noise (AWGN) and that the source signals and the noise are uncorrelated [14–16]. Thus, the corresponding covariance matrix of x(t) can be expressed as
where R_{s}=diag(ρ_{1},ρ_{2},…,ρ_{D}) is the signal covariance matrix such that ρ_{i} for i=1,2,…,D denote the signal power powers, and \(\sigma _{n}^{2}\) denotes additive noise power. Following [17–21], vectorizing (3) yields
where B_{c}=(A^{∗}⊙A),p is the source signal and y becomes the new received signal vector based on coarray model [17]. Note that B_{c} denotes the new steering matrix of the coarray model whose sensor locations are defined as a difference between the sensor positions of \(\mathcal {Z}\) i.e., \(\mathcal {Q}_{u}=\{n_{1}n_{2} n_{1},n_{2} \in \mathcal {Z} \}\) [18]. Cleaningoff duplicate components of B_{c} yields an extended steering matrix denoting a virtual ULA of size \(\mathcal {O}(N^{2})\) given N sensors. As a result, applying a coarray based DOA estimator, i.e., coarray MUSIC, on (4), it is possible to estimate more sources than number of sensors [17, 19].
Note that the unique element in set \(\mathcal {Q}_{u}\) is defined as difference coarray \(\mathcal {Q}\). Thus, the number that element or lag \(q (q \in \mathcal {Q}_{u})\) occurs in set \(\mathcal {Q}\) is known as weight function w(q). If w(q_{1})>1 then q_{1} is a redundant lag (R). Else, if w(q_{1})=0, then the coarray has a hole (H) or missing sensor at lag q_{1}. Figure 1 illustrates the coarray properties of a typical sparse array \(\mathcal {Z}=[0,1,2,6]\). From Fig. 1a, the central contiguous segment of sensors is termed the central ULA segment (\(\mathbb {U}\)). Furthermore, if the holes does not exist in the coarray, then the coarray retains a large \(\mathbb {U}\). Otherwise, the coarray size shrinks [17–20].
Methods
This section introduces the methods used in work. Antenna selection problem formulation is introduced first, followed by a review of the relationship between coarray, sparse arrays, and minimum sidelobe level. This is followed by the proposed approach, corresponding training data generation procedures, and the CNN structure.
Antenna selection problem formulation
Given a ULA with N elements, the number of possible combinations M elements can be picked from the ULA is defined as [7]
As in [10], all possible subarrays in (5) are considered as classes. Assuming that set \(\boldsymbol {\mathcal {G}}\) contains all classes in (5), and that each \(\boldsymbol {g} \in \boldsymbol {\mathcal {G}}\) is associated with x_{m} and y_{m}, for m=1,…,M in xyplane, then the gth class consisting of all antenna elements in gth subarray can be denoted as \(\mathcal {Z}_{g}=\left \{z_{1}^{g}, z_{2}^{g}, \ldots, z_{M}^{g}\right \}\). As a result, \(\boldsymbol {\mathcal {G}}\) can be redefined as \(\boldsymbol {\mathcal {G}}=\left \{\mathcal {Z}_{1}, \mathcal {Z}_{2}, \ldots, \mathcal {Z}_{{G}}\right \}\) [10]. The assumptions above transform the antenna selection problem from a combinatorial optimization framework to a ML classification framework. Any ML or DLbased classification algorithm can be employed to classify the desired class (subarray) using appropriate metrics that characterize the best class (sparse subarray configuration in our case). However, this is possible if labels for the classes are known [11].
Coarray, sparse arrays, and minimum sidelobe level
In sparse array processing, it is wellknown that for efficient spatial sampling purposes, an array whose coarray has no redundancy or holes is considered as a perfect array [7, 19]. Assuming no holes exist (H=0), a perfect array aperture can be defined as
where \(\mathcal {Z}_{a}\) is the array aperture and N is the number of antennas. Unfortunately, such an array does not exist for N>4 [20]. Alternatively, one can construct a sparse array with no holes but retains the largest possible aperture, i.e., \(\mathcal {Z}_{a}\) to approximate the perfect array. Consequently, several methods have been proposed in [7, 19, 20] for designing of minimum redundancy array as well as minimum hole arrays. Here, the former is the array that minimizes (RH=0,N=constant) for a given N elements, whereas the latter minimizes holes in the coarray.
These arrays are attractive due to their good beampattern properties—PSLs and narrow main lobe [7]. For instance, Fig. 2 compares the beampattern responses of ULA, a conventional DL proposed in [10], PSLconstrained array proposed in [8], and sparse array with a holefree difference coarray. It can be observed that the beampattern response of conventional DL shows high PSLs as compared to that of PSLconstrained and sparse array with a holefree difference coarray which shows well suppressed PSLs. Although the relationship between the two concepts is not directly proved here, the connection was thoroughly investigated in [7, 19–21], and sparse array with holefree coarray or with minimum redundancy and minimum holes were recommended as the best solution for array thinning because of their narrow main lobe and minimum PSLs. Hence, inspired by beampattern properties of MRA and MHA as well as the work in [8], we aim to impose a holefree constraining term on the solution set \(\boldsymbol {\mathcal {G}}\) which is used to create the training dataset in a bid to improve the performance DLbased antenna selection technique.
Proposed DLbased antenna selection approach
Motivated by beampattern properties of MHAs and MRAs [7], we extend and enhance the DLbased antenna selection technique proposed in [10]. By taking advantage of essential sensor properties of the array as introduced in [19, 20], we constrain the subarrays in the solution set (5) to retain a holefree difference coarray as a means of enforcing sensor distribution within subarrays that form the feature space.
The idea is given a solution set \(\boldsymbol {\mathcal {G}}\) which consists of subarrays as possible solutions to a _{N}C_{M} antenna selection problem. We intend to use all \(\boldsymbol {g} \in \boldsymbol {\mathcal {G}}\) which retains a holefree difference coarray only to generate the training dataset and discard the rest. As a result, we implement a basic search algorithm to search through \(\boldsymbol {\mathcal {G}}\) and reserve all \(\boldsymbol {g} \in \boldsymbol {\mathcal {G}}\) with a holefree difference array, i.e., \({\mathcal {Q}_{g}} = \mathcal {Q}_{ULA}\) and omit those without, i.e., \({\mathcal {Q}_{g}}\ne \mathcal {Q}_{ULA}\) where \({\mathcal Q}_{g} \) and \({\mathcal {Q}_{ULA}} \) are difference coarrays of a M sensor subarray \(\boldsymbol {g} \in \boldsymbol {\mathcal {G}}\) and N sensor ULA respectively. The steps above are summarized in Algorithm 1.
In other words, the difference coarray as a constraint on every \(\boldsymbol {g} \in \boldsymbol {\mathcal {G}}\) can be expressed in terms of essential property ofsensors of an array. For multiple sensors failure or omission, the essential property states that
Property 1
(kessential property [19]) \(\mathcal {S} \subset \mathcal {Z}\) is said to be kessential when (1) \(\mathcal {S}=k\), and (2) the difference coarray changes when \(\mathcal {S}\) is removed from \(\mathcal {Z}\) i.e. \(\hat {\mathcal {Q}} \ne \mathcal {Q}\) where \(\hat {\mathcal {Q}} \) and \({\mathcal {Q}} \) are difference coarrays of \( \mathcal {Z}\backslash \mathcal {S}\) and \( {\mathcal {Z}} \) respectively.
This entails that N−M sensors, which are not essential for the preservation of the Nsensor array’s difference coarray, can be discarded out Nsensor array without changing the array aperture and difference coarray. Therefore, we can reformulate the property 1 and define it with respect to the antenna selection problem as follows
Property 2
Let \(\mathcal {Q}\) be the difference coarray of a physical subarray \(\mathcal {Z}_{g}\) such that \(\mathcal {Z}_{g} \triangleq \boldsymbol {g} \in \boldsymbol {\mathcal {G}}\). If \(\boldsymbol {\mathcal {G}}\) consists of all possible subarrays as solutions to an (N,M) antenna selection problem, then for all \(\boldsymbol {g} \in \boldsymbol {\mathcal {G}}, \boldsymbol {g}\) is essential with respect to \(\mathcal {Z}_{ULA}\) if the difference coarray of the large array \(\mathcal {Z}_{N}\) changes when g is removed, that is, if \({\mathcal {Z}_{g}}=\mathcal {Z}_{ULA} \backslash \boldsymbol {g}\), then \(\grave {\mathcal {Q}} \ne \mathcal {Q}\) where \(\grave {\mathcal Q} \) and \({\mathcal {Q}} \) are difference coarrays of \(\mathcal {Z}_{g}\) and \( {\mathcal {Z}}_{ULA} \) respectively.
Note that the use of holefree subarrays will not only assist in the realization of sparser subarrays with the welldistributed sensors but also sparse arrays with improved beampattern characteristics [7]. As a result, instead of using (5) as in [10] when preparing the training dataset, we resolve to use \(\boldsymbol {\mathcal {L}}\), output from Algorithm 1. Henceforth, for clarity sake, we refer to the implementation using \(\boldsymbol {\mathcal {L}}\) as the proposed method and the one using \(\boldsymbol {\mathcal {G}}\) or a portion of (5) as conventional method [10].
Training dataset generation for antenna selection problem
In this section, we consider training dataset generation–input data samples and corresponding labels or ground truths. Basically, the feature space is comprised of angle, real and imaginary components of a sample covariance matrix \(\boldsymbol {\hat {R}}\). Thus, the input data is N×N×3 realvalued matrices \(\{ \boldsymbol {H}\}_{i=1}^{3}\) whose (i,j)−th entry consists of \([\boldsymbol {H}_{1}]_{i,j}=\angle [\boldsymbol {\hat {R}}]_{i,j}, [\boldsymbol {H}_{2}]_{i,j}=\mathbb {R}\mathrm {e} [\boldsymbol {\hat {R}}]_{i,j}\) and \([\boldsymbol {H}_{3}]_{i,j}=\mathbb {I}\mathrm {m} [\boldsymbol {\hat {R}}]_{i,j}\) denoting the phase, real and imaginary components of sample covariance matrix \(\boldsymbol {\hat {R}}\) [10].
To generate inputout training dataset pairs, we need to determine subarrays with the best performance within the solution set \(\boldsymbol {\mathcal {L}}\) to act as ground truths or labels. For simplicity, like [10], we assume the CRB as a benchmark of determining the best array configurations. Therefore, we assume that the received signal vector at lth subarray with M elements is defined as
where A_{l} is the subarray steering matrix, s_{l}(t) denotes the signal vector and n_{l}(t) is the noise vector corresponding to the lth subarray position set \(\mathcal {Z}_{l}\) at the tth snapshot. Like (2), we assume that s_{l}(t) and n_{l}(t) are spatially and temporarily uncorrelated [14, 16]. Furthermore, we assume constant signal variance \(\sigma _{s}^{2}\) and noise variance \(\sigma _{n}^{2}\). Hence, the signaltonoise ratio (SNR) in dB is expressed as \(\text {SNR}=10~\text {log}_{10} \left ({\sigma _{s}^{2}}/{\sigma _{n}^{2}}\right)\).
Therefore, following assumptions in [15], the CRB_{θ} for every \(\boldsymbol {l} \in \boldsymbol {\mathcal {L}}\) can be expressed as
where \(\boldsymbol {P}^{\bot }_{A} = \boldsymbol {I}\boldsymbol {A}_{l}\left (\boldsymbol {A}_{l}^{H}\boldsymbol {A}_{l}\right)^{1}\boldsymbol {A}^{H}_{l}\) is the orthogonal projection onto the null space of \(\boldsymbol {A}_{l}^{H}, \boldsymbol {B}=\left [ \boldsymbol {b}(\theta _{1}), \boldsymbol {b}(\theta _{2}), \ldots, \boldsymbol {b}(\theta _{D})\right ] \) such that \( \boldsymbol {b}(\theta _{i}) =\frac {\partial }{\partial \theta _{i}} \boldsymbol {A}_{l}(\theta _{i})~~~\text {for}~ i=1,2, \ldots,D\) and
Next, for various DOAs, we construct sample covariance matrices R_{l} for l=1,2,…,L and compute CRBs for all \(\boldsymbol {l} \in \boldsymbol {\mathcal {L}}\). Then, subarrays with lowest CRBs for various DOAs selected and save into \(\boldsymbol {\mathcal {W}}\). Here, \(\boldsymbol {w}_{i} \in \boldsymbol {\mathcal {W}}\) for i=1,2,…,W represents class labels such that w is defined as
Following (10) and realization of \(\boldsymbol {\mathcal {W}}\), we construct inputoutput data pairs as (H,w) where H is the realvalue input data obtained from the covariance matrix and \(\boldsymbol {w} \in \boldsymbol {\mathcal {W}}\) is the label representing the best subarrays sensor positions for the sample covariance matrix \(\boldsymbol {\hat {R}}\) [10]. The above training dataset generation procedures are summarized in Algorithm 2.
Convolutional neural network architecture
In this work, we adopt a general CNN structure consisting of 9 sections as in [10]. In general terms, the first layer (1st layer) accepts the 2D input and the last output layer (9th layer) is a classification layer with l units where a softmax function is used to obtain the probability distribution of the classes [22]. The second (2nd layer) and the fourth (4th layer) layers are maxpooling layers with 2×2 kernel to reduce the dimension whereas the third (3rd layer) and the fifth (5th layer) layers are convolutional layers with 64 filters of size 3×3.
Finally, the seventh (7th layer) and the eighth (8th layer) layers are fully connected layers with 1024 units. Note that the rectified linear units (ReLU) are used after each convolutional and fully connected layers such that ReLU(x)=max(x,0) [11]. Furthermore, during the training phase, 90% and 10% of the data are allocated for training and validation purposes, respectively. The stochastic gradient descent with momentum (SGD) is used with a learning rate of 0.03 and a minibatch of 500 for 50 epochs [10].
Results and discussion
In this section, we perform a series of numerical simulations to evaluate the performance of the proposed antenna selection approach as well as the performance of the realized DLbased sparse linear arrays. First, we train our CNN model and predict sparse arrays. This is followed by sparse array performance evaluations in terms of array configuration, beampattern characteristics, and DOA estimation performance.
We measure the DOA estimation performance of the arrays using rootmeansquareerror (RMSE), which can be expressed as
where, \(\theta _{d}^{(c)}\) is the dth DOA in the cth simulation trial, and \(\hat {\theta }_{d}^{(c)} \) is the corresponding angle estimate. Moreover, for performance comparison purposes, we consider the following conventional sparse linear arrays with sensor positions defined as
where DA is the conventional DLbased array proposed in [10] for a (20,10) antenna selection problem, PSA is the PSLconstrained array proposed in [8], M_{1}=3 and M_{2}=7 nested array [18] and 10−element MRA [21] in that order. Note that all conventional sparse arrays share the same number of sensors M=10 but differs in aperture sizes.
Sparse array selection using DLbased method
In this example, we aim to select a 10−element (M=10) sparse array from a 20−element ULA (N=20) using the proposed DLbased technique. The problem yields _{20}C_{10}=184756 subarrays,i.e, \(\boldsymbol {\mathcal {G}}\). However, after applying Algorithm 1, the instances dropped to 14791 subarrays,i.e., \(\boldsymbol {\mathcal {L}}\). This is followed by selection of the best subarray to calculate \(\boldsymbol {\mathcal {W}}\), which was found to consists of 67 subarrays for the proposed method and 71 for the conventional method. As pointed out in [9–11], the size of \(\boldsymbol {\mathcal {W}}\) is much less as compared to that of \(\boldsymbol {\mathcal {L}}\).
Table 1 briefly compares the solution sets against the number of labels realized between the conventional and the proposed methods. A closer look shows that the number of labels generated in each case is comparably the same. This indicates that the proposed method does not only helps to enhance the DLbased antenna selection approach proposed in [10] but also reduces computational load associated with training dataset annotation.
The CNN model as defined in Section (3.5) was trained for M=10,N=20. The training dataset was generated using SNR_{TRAIN}=10 dB,T_{TRAIN}=100 snapshots, Q_{TRAIN}=120 signal, and noise realisation and (D_{θ})_{TRAIN}=120 DOAs spaced uniformly within θ∈(−90^{∘},90^{∘}). During testing, M, N, and T were kept constant whereas SNR_{TEST}=0 dB,Q=1, and D_{θ})_{TEST}=2 DOAs picked randomly in within the same range of (−90^{∘},90^{∘}).
Figure 3 shows the array configurations of the predicted sparse arrays. We observed that the array configurations of the proposed method are as sparse as most conventional sparse arrays. Moreover, the predicted sparse arrays exhibit holefree difference coarrays identical to the size of the difference coarray of the original 20element ULA, i.e., \(\mathcal {O}(N)\). This large difference coarray \(\mathcal {Q}\) enables the predicted sparse arrays to estimate more uncorrelated sources than the number of sensors.
Beampattern response of the proposed sparse arrays
In this section, we evaluate the beampattern responses of the proposed DLbased sparse arrays in comparison to the responses of conventional sparse arrays. In the example, the look angle is assumed to be located at θ=0^{∘}. Figure 4 shows the computed beampattern responses.
It can be observed in Fig. 4 that the beampattern response of the conventional DL array shows high PSLs as compared to the PSLconstrained array, which shows well suppressed PSLs. Moreover, the proposed arrays with holefree difference coarray yield beampatterns with well suppressed PSLs such that the PSLs are closer to that of the PSLconstrained array. This indicates the proposed method’s effectiveness in yielding sparse arrays with enhanced beampattern properties [7].
DOA estimation performance of the proposed sparse arrays
In this section, we examine the DOA estimation performance of the predicted sparse arrays in comparison to conventional DLbased arrays, original 20−element ULA, 10−element ULA, and 10−element PSLconstrained array. We consider D=3 sources case where the sources are uniformly distributed between [−60^{∘} to 60^{∘}], and rootMUSIC estimator is utilized to calculate RMSE as a function of SNR and number of snapshots [12]. In the first scenario, we compute RMSE versus SNR with D=3, 500 snapshots over 1000 trials while varying the SNR from −20 dB to 10 dB. In the second scenario, we compute RMSE versus the number of snapshots with D=3, SNR =0 dB over 1000 trials while varying the number of snapshots from 50 to 600. Figure 5 shows the plot of RMSE versus SNR (left) and RMSE versus number of snapshots (right).
As shown in Fig. 5, the proposed sparse arrays show better performances comparable to PSLconstrained and closer to the performance of the original 20element ULA throughout the SNR and number of snapshots levels. Moreover, the predicted sparse arrays performed better than 10−element ULA and conventional DL. The results demonstrate that the proposed antenna selection approach can be used to thin or select sparse arrays with few sensors instead of a full array without degrading the estimation accuracy considerably while reducing the computation cost.
DOA estimation performance comparison with conventional sparse arrays
In this example, we consider DOA estimation of more sources than the number of sensors. Specifically, we compare the performance of the predicted sparse arrays with MRA, NA, and the original 20element ULA. To that end, we consider D=11 sources case and the RMSE of DOA estimation as a function of SNR and number of snapshots. Like the previous example, we compute RMSE versus SNR with D=11, 1000 snapshots over 500 trials while varying the SNR from −20 dB to 10 dB. Then, we compute RMSE versus the number of snapshots with D=11, SNR =0 dB over 500 trials while varying the number of snapshots from 50 to 500. Note that the sources were assumed to be distributed uniformly between [− 60^{∘},72^{∘}]. Figure 6 shows the plot of RMSE versus (left) SNR and (right) number of snapshots.
Figure 6 shows that the predicted sparse arrays exhibit better performances slightly higher than the performance of the original 20element ULA, MRA, and NA but better than the performance of PSLconstrained despite sharing the same size of the aperture. However, the same is not the case with MRA and NA because MRA and NA have large array aperture as compared to the predicted sparse arrays. Nonetheless, the results indicate the potential and the effectiveness of the proposed antenna selection technique.
Computation complexity analysis
The time complexity of DL neural network can be approximated using the complexity of the convolutional and fully connected layers as [10, 22]
where for the first term \(\mathcal {D}^{r}, v^{r}, \mathcal {J}^{r1}_{cl}\), and \(\mathcal {J}^{r}_{cl}\) denote the size of output feature map, 2D filter size, number of input, and output features of the rth convolutional layers, respectively. And for the second term, \(\mathcal {D}^{r}\) and \(\mathcal {J}^{r}_{fl}\) represents the size of 2D input and total number of units of rth fully connected layer, respectively. Assuming 2 convolutional layers with 64 feature maps with 3×3 kernel and 2 fully connected layers, operations with respect to the first and second terms of (12) can be approximated as (N^{2}·(2·9·64^{2})) and (2·64^{2}(N^{2}+2)) in that order. Thus, combining the two terms yield (64^{2}(22N^{2}+4)) and the corresponding \(\mathcal {T}_{DL}\) is \(\mathcal {O}\left ({64}^{2}\left (22{N}^{2}+4\right)\right)\) which can be further simplified to \(\mathcal {O}\left (22\cdot {64}^{2}{N}^{2}\right)\) [22].
In comparison, the order of a convex relaxation algorithm (through the difference of two convex sets, which is a polynomialtime algorithm) used to design PSLconstrained array in [8] is almost \(\mathcal {O}\left (N^{3}+N^{2}L\right)\). However, once trained, the DLbased selector requires very few matrix computations to yields the best solution. For instance, running the models in MATLAB using a PC with Intel(R) Core (TM)i5 at 2.60 GHz with 4 GB RAM, the proposed DLbased method required only 0.0270s (prediction phase only) to predict a sparse array, whereas the approach in [8] takes almost 0.157s for N=20 and M=10 case.
Conclusion
This paper presented an enhanced deep learningbased antenna selection approach. The approach employs a convolutional neural network algorithm to select a sparse subarray given a sample covariance matrix as input. Motivated by beampattern characteristics of arrays with holefree or near holefree coarrays, we constrained the subarrays used to generate training target data consisting of holefree difference coarray to achieve sparse arrays with large aperture and welldistributed sensors. It has been demonstrated through numerical examples that the proposed method yields sparser arrays with improved beampattern properties and retain holefree or near holefree coarrays with welldistributed sensors. Moreover, the rendered sparse arrays show enhanced DOA estimation performance comparable to that of the original large array and other wellknown sparse arrays.
Availability of data and materials
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Declarations
Abbreviations
 DOA:

Directionofarrival
 DL:

Deep learning
 CNN:

Convolutional neural network
 ULA:

Uniform linear array
 RF:

Radiofrequency
 SINR:

Signaltointerferenceplusnoise ratio
 MSE:

Meansquarederror
 PSL:

Peak sidelobe levels
 ML:

Machine learning
 ANN:

Artificial neural network
 SVM:

Support vector machine
 1D:

Onedimensional
 2D:

Twodimensional
 CRB:

CramérRao bound
 MUSIC:

Multiple signal classification algorithm
 FISTA:

Fast iterative softthresholding algorithm
 MRA:

Minimum redundancy array
 NA:

Nested array, AWGN: Additive white Gaussian noise
 KR:

KhatriRao
 PSA:

PSL constrained sparse array
 SNR:

Signaltonoise ratio
 MC:

Monte Carlo
 KRMUSIC:

KhatriRao MUSIC
 RMSE:

Rootmeansquareerror
 ReLU:

Rectified linear unit
 SGD:

Stochastic gradient descent
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Funding
This work was supported in part by Japan Society for the Promotion of Science (JSPS) GrantinAid for Scientific Research no. 20K04500. The authors are sincerely grateful for their support.
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Wandale, S., Ichige, K. Design of sparse arrays via deep learning for enhanced DOA estimation. EURASIP J. Adv. Signal Process. 2021, 17 (2021). https://doi.org/10.1186/s13634021007275
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Keywords
 Antenna selection
 Directionofarrival estimation
 Deep learning