WU Shuang ,YUAN Ye ,ZHANG Weike ,and YUAN Naichang

1.Facility Design and Instrumentation Institute,China Aerodynamics Research and Development Center,Mianyang 621000,China;2.State Key Laboratory of Complex Electromagnetic Environment Effects on Electronics and Information System,National University of Defense Technology,Changsha 410073,China

Abstract:This paper develops a deep estimator framework of deep convolution networks (DCNs) for super-resolution direction of arrival (DOA) estimation.In addition to the scenario of correlated signals,the quantization errors of the DCN are the major challenge.In our deep estimator framework,one DCN is used for spectrum estimation with quantization errors,and the remaining two DCNs are used to estimate quantization errors.We propose training our estimator using the spatial sampled covariance matrix directly as our deep estimator’s input without any feature extraction operation.Then,we reconstruct the original spatial spectrum from the spectrum estimate and quantization errors estimate.Also,the feasibility of the proposed deep estimator is analyzed in detail in this paper.Once the deep estimator is appropriately trained,it can recover the correlated signals’ spatial spectrum fast and accurately.Simulation results show that our estimator performs well in both resolution and estimation error compared with the state-of-the-art algorithms.

Keywords:off-grid direction of arrival (DOA) estimation,deep convolution network (DCN),correlated signal,quantization error,super-resolution.

1.Introduction

Direction of arrival (DOA) estimation is an essential branch of array signal processing and has a wide application in radar,communication,and sonar [1-4].Over the past few decades,many algorithms have been developed to address this problem.The subspace-based multiple signal classification (MUSIC) [5] and estimate signal parameters via rotational invariance techniques (ESPRIT) [6]algorithms are the most outstanding,which have achieved a leap towards super-resolution technology [5-8].However,many applications involve DOA estimation of correlated signals in multipath environments [9].The spatial smoothing [10,11] processing for the spatial covariance matrix (SCM) is required,thus reducing the array’s effective aperture and reducing resolution.

Given this,fitting algorithms are developed.Sparsityinducing algorithms [12-14] reconstruct the raw array output via overcomplete dictionaries and the signal sparsity prior,while maximum likelihood estimation algorithms [15-19] directly fit the raw signal subspace or the raw array output from the array output.Although the sparsity-inducing algorithms can directly estimate the correlated signals’ DOA with a small number of snapshots or even a single snapshot,they are unstable under a low signal to noise ratio (SNR).Besides,these algorithms need to discretize the spatial domain into the grid in advance.However,quantization errors will be introduced when the signals are not located on the grid (offgrid signals).Yang et al.[14] proposed a sparse Bayesian learning (SBL) algorithm that estimates DOA and quantization errors simultaneously.However,due to the need to manually adjust the hyper-parameters,the algorithm’s estimation deviation is significant.Although the statistical performance of the maximum likelihood estimation algorithm is closer to the Cramer-Rao lower bound(CRLB) than that of the MUSIC algorithm,their solving process involves a multi-dimensional nonlinear optimization problem with a considerable amount of computation.Then,Choi [16] and Viberg et al.[17] respectively proposed the alternating projection (AP) algorithm and the modified variable projection (MVP) algorithm to reduce the computational complexity.However,the convergence rate of these algorithms depends on the selection of initial values.The iterative quadratic maximum likelihood (IQML) [18] algorithm and the method of direction estimation (MODE) [19] were proposed to calculate the initial value from SCM,but these algorithms are also of computational complexity.

In the past few years,data-driven algorithms have been developed by utilizing deep learning technology.References [20-24] show that the DOA estimation framework based on deep neural networks (DNNs) consumes less execution time and has higher estimation accuracy.However,these algorithms are for a single sound signal,so they may not be suitable for radar,communication base stations,and other environments for multiple signals.In[25],an auto-encoder network and multiple small parallel DNNs were cascaded to achieve DOA estimation.This hierarchical framework effectively reduces the complexity of DNNs with the auto-encoder network,which plays the role of spatial filter banks but cannot superresolve signals due to the non-ideal gain response of the spatial filter banks.Similarly,Elbir [26] proposed a multitarget DOA estimation algorithm based on the assumption that only one signal exists in each spatial subregion.Both [27] and [28] use the prior that signals are sparse in the spatial domain to train the convolutional neural network (CNN).However,the congenital disability of these algorithms is that quantization errors are not taken into account.They also need to perform feature extraction on array output or SCM in advance.Therefore,the advantages of automatic feature extraction of deep learning are not fully utilized.

In this paper,the proposed estimator framework utilizes the SCM as input without requiring any manual preprocessing.We demonstrate that the proposed estimator framework can learn decorrelation automatically.The proposed framework is trained with uncorrelated and coherent signals but can be applied to the correlated signals scenarios.Since the spatial domain is continuous,quantization errors will be introduced after discretization using the grid.If we use a dense grid to quantize the spatial domain,the number of output units in the last layer of the network will increase dramatically so that the complexity of the network will increase superlinearly with the increase of output dimensions.However,in our framework,we use multiple deep convolution networks(DCNs) to learn the inaccurate spatial spectrum and quantization errors,respectively,without increasing the complexity of the network.Finally,the original spatial spectrum is reconstructed according to the inaccurate spatial spectrum and quantization errors.

The significant contributions of this paper are threefold.

(i) We propose useing the proposed estimator to learn the spatial spectrum directly from SCM.Compared with the data-driven algorithms in [25] and [27],our algorithm does not require SCM preprocessing.Also,compared with physical-driven algorithms [5-6,14,19],eigendecomposition and subspace estimation (signal number estimation) are avoided.

(ii) In [25] and [26],to reduce the complexity of the network,the spatial domain is divided into multiple subregions.These algorithms have limitations when multiple signals are not in the preset spatial domain.However,the proposed algorithm is based on the assumption that the entire spatial domain is continuous.Besides,compared with the scenarios where [28] can only be used for coherent signals,our algorithm can be used for unrelated,correlated,and coherent scenarios.

(iii) To the best of our knowledge,this paper is the first to propose solving the quantization errors of the off-grid signals with DCNs.The quantization errors are not considered in the relevant data-driven algorithms [25-28].

Moreover,compared with the state-of-the-art algorithms,simulations also show that the proposed estimator performs better in estimation accuracy and resolution probability.

2.Signal model

In this paper,anM-element uniform linear array (ULA)with half-wavelength internal antenna spacingdis considered.Assuming that theKfar-filed narrowband signalss(t)=[s1(t),s2(t),···,sK(t)]Timpinge on this ULA from locations of θ=[θ1,θ2,···,θK]T,the array output can be represented as

whereA=[a(θ1),a(θ2),···,a(θK)] andRs=E[s(t)sH(t)]denotes the array manifold matrix and the signal autocorrelation matrix,respectively,IMrepresents theM×Midentity matrix.In practical applications,the spatial sampled covariance matrix (SSCM) is usually used because the available snapshots of array output are finite in length.It is assumed thatLsnapshots are collected,the SSCM can be written as

3.Proposed deep estimator

This section proposes an efficient DOA deep estimator framework for correlated signals using SSCM directly.After off-line training,the proposed deep estimator can output DOA estimates online at a fast speed.Both the subspace estimation and signal number estimation are not required.Then,considering the off-grid signals,three independent parallel DCNs are installed in the deep estimator framework to reduce the quantization errors.By setting different output layers and training labels for each DCN,three spatial spectrums with different category numbers (grid density) can be obtained.Finally,we propose reconstructing the original spatial spectrum using the three spatial spectra with different grid densities.The detailed descriptions of our deep estimator are as follows.

3.1 Structure of DCNs

In this paper,we consider the problem of DOA estimation as a direction classification problem.As shown in Fig.1,our deep estimator contains three DCNs.They have the same structure,while the last layer (output layer)of each has a different dimension.The DCNs have eight layers,with the first four layers being convolution layers and the last four layers being dense (full connection) layers.The convolution kernel size of the convolution layers is (3,3),the number of convolution kernel is 32,32,64,64,in turn,and the dense layers (except for the last layer) have dimensions of 512.Before the output of each layer is nonlinearly processed by the activation functionf(·):=ReLU(·)[29],we add the batch normalization(BN) [30] operation in each layer to accelerate the convergence rate of the network .We also add the Dropou t[31] operation in the dense layers (except for the last layer) to prevent the overfitting in the training process and set the Dropout rate to be 0.5.Similar to image preprocessing,we normalize the SSCM and set the two input channels of the DCNs to be the real part,,and the imaginary part,For the output layer,its dimension is naturally the number of category.By choosing [-θ0,θ0] as the spatial scope of interest and sampling it uniformly at an angle spacing Δθ,the spatial grid can be denoted asΘ(n)=-θ0+(n-1)Δθ(n=1,2,···,N) and can be viewed asNcategories.For the off-grid signals with the direction of θk(k=1,2,···,K),we define its direction as thenk=roundcategory,where round(·) denotes rounding to the nearest integer.Further,we define the normalized spatial spectrum(NSS) of lengthNas

However,for off-grid signals,the normalized spatial spectrum ρNis on-grid with quantization errors.Obviously,the quantization errors depend on the sampling interval Δθ,and ideally,there are no quantization errors when Δθ is an infinitesimal increment.In most practical applications,the small sampling spacing Δθ′=αΔθ is enough to satisfy the estimation accuracy requirements,where α ∈(0,1] denotes the scaling factor.Therefore,the number of category becomesNα=(N-1)+1 under Δ θ′.For the category number increasing fromNtoNα,the complexity of the DCN will increase dramatically,and the output layer’s dimensionNαis also much larger than the input layer’s dimensionM,which requires enormous resources to store the training labels.Given these problems,we propose using two DCNs to learn the grid mismatch information for off-grid signals without increasing the label dimension.ForN,we choose two integers,PandQ,wherePis the largest prime less thanNandQis the smallest prime greater thanN.Then,we define the two folded normalized spatial spectrums (FNSS) as

whereDdenotes the size of the training set.Then,we set the activation function of the output layer as the softmax activation functiong(·),which can be expressed as

wherezjdenotes the value of thejt h neuron andNoutdenotes the output layer’s dimension.Suggest that the final output of the DCN is the normalized result of all output neurons.Then,we set the loss function for training to the cross-entropy [32] function of ρ and,which can be depicted as

3.2 Feasibility analysis of our deep estimator

For DOA estimation algorithms (such as MUSIC,SBL,and MODE),estimation accuracy and resolution are the two main performance metrics.Because the proposed deep estimator can estimate both the DOAs and the quantization errors to meet estimation accuracy,we describe the estimator’s resolution in this subsection.

According to the information theory,if we useP-categories classifier to solve anNα-categories classification problem,we need at leastsuch classifiers,where·denotes rounding up to an integer.Consequently,the maximum number of categories of classification problems that can be solved by our deep estimator isP2,i.e.,Nα＜P2.Since bothPandQare prime numbers,we can conclude thatPandQare co-prime.Consider the DOA estimation problem as a classification problem,then super-resolution means that the classifier can screen any two off-grid signals arriving from different directions.With the set of categories being expressed as Ω={1,2,···,Nα} the opposite can be described as

3.3 Spatial spectrum reconstruction

and the final DOA estimates are the categories corresponding to the peaks ofPDEEP.

4.Simulations

This section presents the proposed estimator’s training process and compares its performance with the state-ofart algorithms.We consider an 8-element ULA with halfwavelength internal antenna spacing.We show that the proposed deep estimator yields better resolution and estimation accuracy.

4.1 Training setup

For tr aining,two signals and the interested spatial scope[ -60°,60°] are set to generate the training sets.We consider the two signals’ potential angular separation set{0.2°,0.4°,···,40°}.For each potential angular separation Δ?,the two signals’ location set we generate with a step of 0.2°in the spatial scope is{(-60°,-60°+Δ?),(-59.8°,-59.8°+Δ?),···,(60°-Δ?,60°)}.Then,for each source location scenario,we repeatedly generate ten SSCMs in the case of random noise by setting the SNR of the two signals to be independent random distribution in [0 dB,10 dB].We calculate the SSCM by collecting 256 snapshots.Besides,we also consider two cases where two signals are coherent and uncorrelated.The label length of ρN,ρP,and ρQare respectivelyN=121,P=119,andQ=127 with choosing Δθ=1°and α=0.2.The labels of the three DCNs are generated according to (4)-(6).Before we train the DCNs,we shuffle the training sets and randomly select 80% for training and 20% for validation.We also train the networks with the batch size of 512 and shuffle the order of samples for each training epoch.The adaptive stochastic gradient optimization algorithm (Adam) [33] is adopted for backward gradient propagation,and the learning rate is set at 10-4.All the training processes and experiments are carried out on a PC with Intel Xeon E3-1535M CPU and NVIDIA Quadro M2200 GPU.

In the first experiment,the convergence rate of the proposed deep networks is tested.Fig.2(a) and Fig.2(b)show the training process and the validation process of the DCN1,respectively.To illustrate the superiority of the proposed activation function ReLU,we also show the loss convergence curves when DCN1 is combined with activation functions sigmoid [34],tanh,and LReLU [35].It can be seen that,compared with other activation functions,the proposed ReLU activation function can minimize the loss of training and validation of the network and accelerate network convergence.Then,we display the training and validation process of DCN2 and DCN3 in Fig.3(a) and Fig.3(b),respectively.We can see that the validation process curves level off as the training epoch increases,implying no overfitting in our proposed DCNs.

Fig.2 Training and validation process for DCN1 with different activation functions

Fig.3 Training and validation process with ReLu activation function

4.2 Spatial spectrum and DOA estimates

To implement the decorrelation via the proposed network,we consider using a training set containing uncorrelated signals ( ξ=0) and coherent signals ( ξ=1) to train the network.ξ denotes the correlation coefficient between two signals.After being trained,there is no overfitting in our networks.Therefore,we can test the network’s decorrelation ability in the case of correlated signals ( 0 ＜ξ ＜1).In the first experiment,we generate testing samples with correlation coefficients of ξ=0.3,0.6,and 0.9,respectively.The source locations of the two signals are set to -2.67°and 3.33°,respectively,which are also not included in the training set as well.The SNR and snapshots numbers are set to 10 dB and 256,respectively.The spatial spectrum of correlated signals of our network is shown in Fig.4,where the red lines denote the actual signal locations.As shown in Fig.4,although the proposed deep estimator is not trained under the samples of correlated signals,it can output a sharp spatial spectrum,and the spectral peaks are well-matched with the actual source locations,which indicates that our network has the satisfactory performance of decorrelation.

Fig.4 Testing spatial spectrum of correlated signals

We then change the correlated coefficients of two signals to 0.75 and compare the spatial spectrum with SBL and the forward and backward spatial smoothing MUSIC(FBSS-MUSIC) algorithm [10].In the FBSS operation,the number of subarray elements is set asM-K+1.The experimental results are shown in Fig.5.Although the three approaches can super resolve two signals under the above conditions,our deep estimator’s spatial spectrum is sharper than that of MUSIC,and the estimation deviation is smaller than that of SBL.

Fig.5 Test spatial spectrum of correlated signals ( ξ=0.75)

Finally,we compare the DOA estimates and corresponding estimation errors of deep estimator with that of the MOD E algorithm.Consider four angular separations 5°,10°,15°,and 20°.For each angular separation Δ?,the two signals’ location pairs increase from(-59.33°,59.33°+Δ? ) to ( 59.33°-Δ?,59.33°) in the spatial scope [ -59.33°,59.33°] with a step of 1°.In this experiment,the correlation coefficient between the two signals is set to 0.9.The SNR and snapshots numbers are set to 10 dB and 256,respectively.The MODE can be applied to correlated signals directly,but Fig.6 indicates that the resolution of MODE degrades obviously with correlated signals.Besides,the MODE estimation errors increase as the distance of signal locations from the array’s normal increases.However,our deep estimator’s DOA estimates are well-matched with the truth,and the estimation errors have no apparent relationship with the signal location.

Fig.6 DOA estimates and the corresponding errors of two correlated signals ( ξ=0.9)

4.3 Generalization performance

In this subsection,we test the generalization ability of our deep estimator.First,we use the deep estimator trained with two signals to estimate three and four signals’ spatial spectrum.For three signals,the locations of signal1,signal2 and signal3 are -20.33°,0.33°,and 20.33°,respectively.For four signals,the locations of signal1,signal2,signal3 and signal4 are -30.33°,-10.33°,10.33°,and 30.33°,respectively.We can conclude from Fig.7 and Fig.8 that for three or four signals (including correlated and uncorrelated signals),the estimator has generalization capability and can output a satisfactory spatial spectrum.

Fig.7 Spatial spectrum for three signals

Fig.8 Spatial spectrum for four signals

Then,we test the generalization ability of our deep estimator for lower and higher SNR.We train the deep estimator with SNR selected randomly in [0 dB,10 dB]but test it with the SNR of two signals increasing from-3 dB to 15 dB in Fig.9(a).We train the deep estimator with 256 snapshots but test it with the snapshots increasing logarithmically from 100 to 100 000 in Fig.9(b).We set the locations of two correlated ( ξ=0.9) signals to-10.23°and 10.57°,respectively.In Fig.9(a) and Fig.9(b),we set the number of the snapshot to 256 and carry out 10 000 trials for each SNR setting.In Fig.9(c) and Fig.9(d),we set the SNR of two signals to 0 dB.

Fig.9 Boxplot of DOA estimates

It can be seen from Fig.9(a) and Fig.9(b) that as the SNR increases from low to high,the distribution of DOA estimates approaches the actual signals’ location,which implies that our deep estimator is robust to noise.Besides,when the SNR is -3 dB,although some DOA estimates deviate slightly from the actual signals’ location,most of the estimates are close to the actual signals’location.We can also see from Fig.9(c) and Fig.9(d) that as the number of snapshots increases,the distribution of DOA estimates becomes more and more concentrated in the actual location.When the number of test snapshots is less than the number of training snapshots,a few estimates deviate slightly from the actual location,but most estimates are close to the actual location.Therefore,our estimator has satisfactory generalization performance concerning SNR and snapshots.

4.4 Statistical performance

In this subsection,the root mean square error (RMSE) is adopted to compare the statistical performance of different algorithms.The RMSE is defined as

First,considering two correlated signals (ξ=0.9) incident into the ULA from -10.23°and 10.57°,we uniformly change the SNR of the two signals from -3 dB to 15 dB.The snapshots number is set to 256.In each SNR scenario,500 trials are carried out.It can be seen from Fig.10(a) that the estimation accuracy of the proposed DCN1 is better than that of DNN in [25] and CNN in [27].Although SBL can reduce the quantization errors,but our deep estimator’s estimation accuracy can be significantly improved.Our estimator’s estimation accuracy is superior and comparable to the state-of-the-art algorithms(FBSS-ROOT-MUSIC,FBSS-ESPRIT,and MODE) when the SNR is lower than 10 dB and higher than 10 dB,respectively.

Fig.10 RMSE

Then,we fix the SNR at 0 dB and change the number of snapshots logarithmically from 100 to 100 000.It can be seen from Fig.10(b) that,compared with both physicaldriven and data-driven algorithms,our deep estimator has a better estimation accuracy when the number of snapshots is small.Although the estimation accuracy of FBSSROOT-MUSIC and FBSS-ESPRIT is close to the CRLB with the increase of the number of snapshots,our deep estimator still has satisfactory estimation accuracy.

4.5 Resolution probability

In this subsection,we compare the relationship between the resolution probabilities of different algorithms and angle separation.For two uncorrelated and correlated signals ( ξ=0.9),we uniformly increase the angle separation δ from 0°to 10°and 0°to 25°,respectively.For each δ,the two signals’ locations are θ1=-δ/2 and θ2=δ/2.Further,the conditions for the successful resolution of the two signals are defined as the DOA estimates of the algorithms satisfy θ1∈(-δ,0) and θ2∈(0,δ).For each angle separation,we perform 500 trials.As shown in Fig.11,for both uncorrelated and correlated signals,the resolution probability of our estimator is significantly better than that of the state-of-the-art algorithms,which is due to the enhanced resolution of the estimator by the proposed co-prime structure.Because the FBBS technique leads to the loss of the array aperture,the resolution probabilities of FBSS-ROOT-MUSIC and FBSSESPRIT degenerate in the case of correlated signals.Although the MODE algorithm has a high estimation accuracy under the correlation signals,the resolution probability is also reduced at small angle separation.

Fig.11 Resolution probability vs.angle separation

4.6 Algorithm execution time

Finally,we record the algorithms’ execution time,the average elapsed time of 1 000 trials in Table 1.The first line represents the algorithms’ execution time under two uncorrelated signals,and the second line represents the algorithms’ execution time for two correlated signals (ξ=0.9).It can be seen that our estimator’s execution time is about 16%,25% and 30% of those of the MODE,ROOTMUSIC and ESPRIT algorithms,respectively.Moreover,our deep estimator’s execution time is only 1/5 000 of SBL.Therefore,the proposed estimator has prominent advantages in computational efficiency,especially in realtime DOA estimation applications.

Table 1 Time of different algorithms s

5.Conclusions

In this paper,we propose a DCNs framework to address the DOA estimation problem.We show that the deep estimator delivers high accurate DOA estimates for correlated signals and improves the resolution probability under small angle separation.Moreover,our deep estimator has satisfactory generalization ability in signal number,SNR and snapshots number.Simulation results also show that the proposed algorithm is superior to the traditional super-resolution algorithms in resolution and execution efficiency.

In future work,we plan to conduct in-depth studies on at least two aspects.The first one is about the generalization performance of the deep estimator with array imperfection.Another problem to be solved is 2D DOAs estimation for 2D arrays.We believe that these two problems are related to the deep estimator framework’s design and positively associated with the deep estimator’s learning manner.