低信噪比环境下基于深度网络的方向到达估计算法

DoA估计

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3714 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 69, 2021

Deep Networks for Direction-of-Arrival

Estimation in Low SNR

Georgios K. Papageorgiou , Member, IEEE, Mathini Sellathurai , Senior Member, IEEE,

and Yonina C. Eldar

, Fellow, IEEE

Abstract—In this work, we consider direction-of-arrival (DoA)

estimation in the presence of extreme noise using Deep Learning

(DL). In particular, we introduce a Convolutional Neural Network

(CNN) that predicts angular directions using the sample covariance

matrix estimate. The network is trained from multi-channel data

of the true array manifold matrix in the low signal-to-noise-ratio

(SNR) regime. By adopting an on-grid approach, we model the

problem as a multi-label classiﬁcation task and train the CNN to

predict DoAs across all SNRs. The proposed architecture demon-

strates enhanced robustness in the presence of noise, and resilience

to a relatively small number of snapshots. Moreover, it is able

to resolve angles within the grid resolution. Experimental results

demonstrate signiﬁcant performance gains in the low-SNR regime

compared to state-of-the-art methods and without the requirement

of any parameter tuning in both cases of correlated and uncorre-

lated sources. Finally, we relax the assumption that the number

of sources is known aprioriand present a training method, where

the CNN learns to infer their number and predict the DoAs with

high conﬁdence. The increased robustness of the proposed solution

is highly desirable in challenging scenarios that arise in several

ﬁelds, ranging from wireless array sensors to acoustic microphones

or sonars.

Index Terms—Direction-of-arrival (DoA) estimation,

convolution neural network CNN, deep learning DL, multilabel

classiﬁcation, array signal processing.

I. INTRODUCTION

IRECTION-OF-ARRIVAL (DoA) estimation has been at

the forefront of research activity for many decades, due

to the plethora of applications ranging from radar and wireless

communications to sonar and acoustics [1], with localization

being one of the most signiﬁcant ones. Estimation of the angular

directions is possible with the use of multiple sensors in a

speciﬁed geometric conﬁguration, e.g., linear, rectangular and

circular. Efﬁcient use of the observations from multiple sensors

enables the DoA estimation of several sources, depending on

Manuscript received November 16, 2020; revised April 26, 2021; accepted

June 1, 2021. Date of publication June 16, 2021; date of current version July

13, 2021. The associate editor coordinating the review of this manuscript and

approving it for publication was Dr. Florian Roemer. This work was supported

in part by UK’s EPSRC Grant EP/P009670/1. (Corresponding author: Georgios

Papageorgiou.)

Georgios K. Papageorgiou and Mathini Sellathurai are with the School of

Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, U.K.

(e-mail: g.papageorgiou@hw.ac.uk; m.sellathurai@hw.ac.uk).

Yonina C. Eldar is with the Electrical Engineering, Technion-Israel Institute

of Technology, Haifa 32000, Israel (e-mail: [email protected]).

This article has supplementary downloadable material available at https://doi.

org/10.1109/TSP.2021.3089927, provided by the authors.

Digital Object Identiﬁer 10.1109/TSP.2021.3089927

the number of array sensors. There are two major categories in

DoA estimation: the overdetermined case, where the number

of sources is less than the number of array sensors, and the

underdetermined case, where the number of sources is equal

or greater than the number of sensors [2], [3]. In this work,

we investigate the ﬁrst category. Moreover, we focus on DoA

estimation in low SNR, which poses several challenges and is

critically important in many real-world situations.

One of the ﬁrst methods introduced for DoA estimation is

MUltiple SIgnal Classiﬁcation (MUSIC) [4], with several other

variants following soon after. The MUSIC estimator belongs to

the class of subspace-based techniques, which attempts to sep-

arate the signal and noise sub-spaces; angle estimation follows

from the so-called MUSIC pseudo-spectra over a speciﬁed grid,

where the corresponding peaks of the pseudo-spectra are se-

lected. Estimation of signal parameters via rotational invariance

techniques (ESPRIT) [5] and its variants is another success-

ful example [6]. Unitary ESPRIT [6], [7], which incorporates

forward-backward averaging, is a notable variant that leads

to improved performance compared to ESPRIT, especially in

cases of correlated source signals. A signiﬁcant step towards

improvement in DoA estimation was with the development of

Root-MUltiple SIgnal Classiﬁcation (R-MUSIC), which esti-

mates the angular directions from the solutions of higher-order

polynomials [8]. The aforementioned methods, are covariance-

based techniques that require a sufﬁcient number of data snap-

shots to accurately estimate the DoAs, particularly in low SNRs.

Furthermore, they often assume that the number of sources is

known, which is not the case in many practical applications.

During the past decade, Compressed Sensing (CS) method-

ologies have also been used to address DoA estimation [9],

[10]. These methods exploit the sparse characteristic of the

signal sources in the spatial domain (angles). CS techniques

are generally separated into three main categories: a) on-grid,

b) off-grid and c) grid-less methods [11], [12]. Grid-less ap-

proaches achieve better performance at the expense of very high

computational complexity [13]. Off-grid and on-grid methods

offer a more balanced solution with less computational demands

at the expense of negligible loss in performance due to the grid

mismatch problem [14]. DoA estimates are obtained after the

solution of sparse minimization tasks, for which two major

approaches are identiﬁed: i) greedy methods based on the 

pseudo-norm and ii) convex relaxations based on the 

-norm.

Notable is the method known as 

2,1

-SVD, which ﬁrst performs

a dimensionality reduction technique to the received signal data

This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/

PAPAGEORGIOU et al.: DEEP NETWORKS FOR DIRECTION-OF-ARRIVAL ESTIMATION IN LOW SNR 3715

and then solves the 

2,1

-norm minimization task in the reduced

dimension with signiﬁcantly less computational burden. It was

introduced in [15] and was also employed later in [16]. One of

the major disadvantages that all these methods have in common

is that tuning of one or more parameters (which depends on the

number of snapshots, the SNR or both) is required to guaran-

tee good performance; moreover, the DoA estimates are often

extremely sensitive to the tuning of these parameters.

A very recent approach to DoA estimation is via the use of

Deep Learning (DL) [17], [18]. DL-based methods enjoy several

advantages over optimization-based ones: a) after training the

network no optimization is required and the solution is the result

of simple operations (multiplications and additions); b) they

do not require any speciﬁc tuning of parameters, in contrast

to optimization-based techniques, whose solution strongly de-

pends on the tuning of those parameters, and c) they demonstrate

resilience to data imperfections, e.g., using fewer snapshots,

performing well in low SNR. A deep neural network (DNN)

with fully connected (FC) layers was employed in [19] for DoA

classiﬁcation of two targets using the signal covariance matrix.

However, the reported results indicate poor DoA estimation

results in high SNR. The authors in [20] proposed a DNN

for channel estimation in massive MIMO systems; however,

they focus on the high SNR regime and on channel estimation

performance. A multilayer autoencoder with a series of parallel

multilayer classiﬁers, i.e., a multi-layer perceptron (MLP), was

employed in [21] with focus on the robustness to array imper-

fections. The MLP architecture addresses DoA estimation of

only two sources via the use of a multitask autoencoder that

acts as a group of spatial ﬁlters followed by a series of paralllel

multi-layer DNNs for spatial spectrum estimation. The network

is trained at each individual SNR.

A deep Convolutional Neural

Network (CNN) that was also trained in low SNRs was proposed

in [22]. However, the method did not demonstrate signiﬁcant

performance gains in terms of DoA estimation, due to the adop-

tion of 1-dimensional (1D) ﬁlters (convolutions). A CNN for

broadband DoA estimation in the context of speech processing

was proposed in [23], [24]. In contrast to these works, we study

the case of narrow-band DoA estimation. The authors in [25],

[26] employed DNNs for the range-based localization of ships.

Their approach is based on distance estimation, whereas we

focus on the estimation of the signal’s directions. A DL-based

method for pseudo-spectra estimation was published in [27],

where the authors also proposed an extension to estimate the

angles. However, the demonstrated results were in high SNR and

the number of sources was assumed to be known. In the context

of acoustics, the authors in [28] proposed a DNN for beamform-

ing with a single-snapshot sample covariance matrix (SCM),

which was later extended in [29], [30] to include slightly more

snapshots and sources. Such an approach cannot be adopted in

low SNR scenarios, where t he number of snapshots and sensors

needs to be considerably higher.

To the best of our knowledge no speciﬁc DL-based technique

was developed for robust DoA estimation in the low SNR regime.

Another disadvantage of methods such as [21], [22] is that they

https://github.com/LiuzmNUDT/DNN-DOA

are trained for a speciﬁc number of snapshots, which leads

to signiﬁcant deviations for different amount of snapshots and

varying SNR.

The scope of this work is to ﬁll in the gap in t he literature

of DoA estimation in the low SNR with the use of DL. Due to

large deviations of the SCM estimates from the true manifold

matrix in low SNR, DoA estimation becomes really challenging

and the majority of the methods fail to demonstrate the desired

robustness. In this work, we contribute towards this direction by:

a) exploiting a deep network with 2D convolutional layers, which

are well known for their excellent feature extraction properties;

b) using multi-channel data, i.e., the real part, imaginary part, and

phase of the complex valued covariance matrix entries (similar

to how they are used in image processing); and c) employing

dropout layers as a means of regularization, thus, improving the

generalization of the estimator and avoiding over-ﬁtting. Our

contributions are summarized as follows:

We introduce a deep CNN trained on multi-channel data,

which are explicitly formed from the complex-valued data

of the true covariance matrix. The ﬁrst and second channels

are formed by taking the real and imaginary parts, respec-

tively, from the complex-valued entries of the covariance

matrix. The third channel includes the phase from the com-

plex valued entries of the covariance matrix (in [−π, π]).

The proposed CNN employs 2D convolutional layers and is

trained to directly predict the angular directions of multiple

sources. Testing is performed using the SCM estimate for

any amount of snapshots. The use of multi-channel data

along with the adoption of 2D convolutional layers enables

the extraction of features from the input data leading to

more robust DoA estimation in low SNR. A discretization

approach (on-grid) is adopted for the desirable angular

(spatial) region and the direction estimation task is modeled

as a multi-label classiﬁcation one.

Efﬁcient methods for training the proposed network are

presented. In particular, we train the CNN across a range of

low SNRs and demonstrate that it can successfully predict

DoAs in high SNR as well.

We introduce a training method for a varying number

of sources. Subsequently, the proposed CNN infers the

number of sources from the received data, while predicting

the DoAs.

The performance of the proposed solution is evaluated

over an extensive set of simulated experiments, where it

is compared against state-of-the-art methods in various

experimental set-ups with off-grid angles. Additionally,

comparison to the Cramér-Rao lower bound (CRLB) is

provided as benchmark.

The results indicate that the proposed CNN: a) outperforms

its competitors in DoA estimation in the low SNR regime; b) is

resilient in estimation even for a small collection of snapshots

regardless of the angular separation of the sources; c) demon-

strates enhanced robustness in case of SNR mismatches, and d)

is able to infer both the number of sources and DoAs with very

small errors and high conﬁdence level.

The rest of the paper is organized as follows: in Section II,

we present the signal model. In Section III, we introduce the

3716 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 69, 2021

proposed CNN for DoA prediction and in Section IV we discuss

the adopted training approach. Section V presents s imulation

results and in Section VI, we summarize and highlight our

conclusions.

Notation: Throughout the paper the following notation is

adopted: X denotes a set and |X | its cardinality; X is a matrix,

x is a vector and x is a scalar. The i, j-th element of a matrix X

is denoted (X)

i,j

and the i-th entry of a vector x is x(i). The

i-th example of the vector x is denoted as x(i). The imaginary

unit is j (so that j

= −1). The conjugate transpose of a matrix

is (·)

; its conjugate is (·)

∗

and its transpose is (·)

. The

N × N identity matrix is I

. The white circularly-symmetric

Gaussian distribution with mean m and covariance C is denoted

by CN(m, C). The convolution operator is denoted as ∗.The

ﬂoor of a number α is written as α. Functions are denoted by

lower case italics, e.g., f(·). The symbol E[·] is the expectation

operator; Re{·}, Im{·} denote the real and the imaginary parts

of a complex scalar / vector / matrix, respectively. Finally, the

phase of the complex-valued variable α is denoted by ∠{α}.

II. S

IGNAL MODEL

The standard model for a N-element sensor array in the

narrow-band mode, with K far-ﬁeld sources present, is:

y(t)=



k=1

a(θ

(t)+e(t)=

= A(θ)s(t)+e(t),t=1,...,T. (1)

Here A(θ)=[a(θ

), a(θ

),...,a(θ

)] is the N × K array

manifold matrix, θ =[θ

,...,θ

]

is the vector of (unknown)

source directions and T is the total number of collected snap-

shots, s(t)=[s

(t),...,s

(t)]

and e(t) denote the trans-

mitted signal and additive noise vectors at sample index t,

respectively. The model (1) is generic in the sense that it does

not depend on the array geometry; however, in this work we

will consider a uniform linear array (ULA) conﬁguration for

simplicity.

Thus, the columns of the array manifold matrix are

expressed as

a(θ

)=[1,e

2πd

sin(θ

)

,...,e

2πd

sin(θ

)(N−1)

]

, (2)

where d is the array interelement distance and λ = c/f is the

wavelength at carrier frequency f with c the speed of light/sound.

In this case, A(θ) becomes a Vandermonde matrix. The follow-

ing assumptions are typical in the literature of narrow-band DoA

estimation:

A1) The source DoAs are distinct.

A2) Each source signal follows the unconditional-model

assumption (UMA) in [31], which assumes that the

transmitted signal is randomly generated (Gaussian sig-

naling). Moreover, the sources are uncorrelated, lead-

ing to a diagonal source covariance matrix: R

E[s(t)s

(t)] = diag(σ

,...,σ

The analysis and methodology also holds for any other array conﬁguration,

e.g., non-uniform linear or rectangular.

A3) The additive noise values are independent and iden-

tically distributed (i.i.d.) zero-mean white circularly-

symmetric Gaussian, i.e., e(t) ∼CN(0,σ

) and un-

correlated from the sources.

A4) There is no temporal correlation between each snapshot.

We are interested in the estimation of the unknown DoAs θ

from measurements y(1),...,y(T ). Considering A1–A4, the

received signal’s covariance matrix is given by:

= E [y(t)y

(t)] = A(θ)R

(θ)+σ

. (3)

The statistical richness of R

in (3) allows for the estimation

of up to K ≤ N − 1 distinct DoAs. However, in practice, the

matrix in (3) is unknown and is replaced by its sample estimate





t=1

y(t)y

(t), (4)

which is an unbiased estimator of R

We note that assumptions A2–A4 are only used for generating

the data in the training procedure of the proposed CNN. The

network can make predictions even if these assumptions are vi-

olated. However, as the deviation of the test from the training data

becomes higher, estimation becomes less accurate. Violation of

these assumptions has different impact on classical estimators

in the ﬁeld of DoA estimation as well.

III. A D

EEP CONVOLUTIONAL NEURAL NETWORK

FOR

DOAESTIMATION

In this section, we formulate DoA estimation as a multilabel

classiﬁcation task. In Section III-A, we present the data manage-

ment and labeling approach, whereas Section III-B is devoted to

the description of the CNN’s architecture that learns to predict

the DoAs. The convolution layers perform the feature extraction

from the multi-channel input data, and, subsequently, the FC

layers use the output of the convolution layers to infer the DoA

estimates using a pre-selected grid.

A. Data Management and Labeling

DoA prediction is modeled as a multilabel classiﬁcation

task.Forφ

max

∈{1

◦

,...,90

◦

} we consider 2G +1 discrete

points of resolution ρ (in degrees), which deﬁne a grid

G = {−Gρ,...,−ρ, 0

◦

,ρ,...,Gρ}⊂[−90

◦

, 90

◦

], such that

max

= Gρ. At each SNR level, K angles are selected from

the set G and the respective covariance matrix is calculated

according to (3). The input data X of the proposed CNN is

a real-valued N × N × 3 matrix, whose third dimension rep-

resents different “channels.” In particular, the ﬁrst and second

channels are the real and imaginary parts of R

, i.e., X

:,:,1

Re{R

} and X

:,:,2

=Im{R

}, whereas the third channel cor-

responds to phase entries, i.e., X

:,:,3

= ∠{R

}. Thus, the in-

put to the CNN is a collection of D data points deﬁned as

X = {X

(1)

,...,X

(D)

Next, for each example X

(i)

,theK training angles

in G are transformed into a binary vector with K ones

(the rest are zeros). For example, if φ

max

= 60

◦

and

the desired resolution is ρ = 1

◦

, the grid becomes G =

PAPAGEORGIOU et al.: DEEP NETWORKS FOR DIRECTION-OF-ARRIVAL ESTIMATION IN LOW SNR 3717

{−60

◦

,...,−1

◦

, 0

◦

, 1

◦

,...,60

◦

} with |G| = 121 grid points;

moreover, the angle pair {−60

◦

, −59

◦

} corresponds to the

121 × 1 binary vector z =[1, 1, 0,...,0]

, which serves as the

corresponding label/output of the proposed CNN. Thus, the i-th

label z

(i)

belongs to the set Z = {0, 1}

2G+1

according to the

described process. Hence, the i-th training example consists of

pairs in the form (X

(i)

, z

(i)

) leading to the training data set

D = {(X

(1)

, z

(1)

), (X

(2)

, z

(2)

),...,(X

(D)

, z

(D)

)} of size D.

According to the well-known universal approximation theo-

rem [32] a feed-forward network with a single hidden layer pro-

cessed by a multilayer perceptron can approximate continuous

functions on a compact subset of R

. The goal of this multilabel

classiﬁcation task is to induce a ML hypothesis deﬁned as

a function f from the input space to the output space, i.e.,

f : R

N×N×3

→Z. Although the true covariance matrix is used

for training the network, for its testing and evaluation the sample

covariance in (4) is used, since the former is unknown. To this

end, during the testing phase of the CNN all input examples can

be considered as “unseen data” to the training.

B. The Proposed CNN’s Architecture

The nonlinear function f is parametrized by a CNN of 8

layers, i.e.,

f(X)=f

(...f

(X))) = z. (5)

The architecture of the proposed CNN is based on the stan-

dard convolutional structures used in the literature of image

processing [33], [34] with some modiﬁcations that are required,

due to the nature of our problem. Each function {f

(·)}

i=1,...,4

represents a series of convolutional layers: a 2D convolutional

layer of n

= 256 ﬁlters, followed by a batch normalization

layer [35] and a rectiﬁed linear unit ReLU layer, i.e., the nonlin-

ear activation function ReLU(x)=max(0,x) applied element

wise to the variables of the previous layer. Additionally, after

the ReLU layer of f

(·) a ﬂatten layer is used, which shapes the

tensor-valued output of the ﬁnal convolutional layer to a vector.

For the kernel of size κ × κ,weusedκ =3for f

(·) and κ =2

for the rest of the convolutional layers. The stride δ that we used

is δ =2for f

(·) and δ =1for the rest of the convolutional

layers (no padding). Hence, for each one of the n

ﬁlters the

mathematical expression of the convolution operation at the ﬁrst

layer with input data X ∈ R

N×N×3

and kernel K ∈ R

κ×κ×3

a 2D matrix of dimension M × M (output of the layer per ﬁlter)

given by:

(X ∗ K)

m,n



i=1



j=1



k=1

i,j,k

δ(m−1)+i,δ(n−1)+j,k

, (6)

m, n =1,...,M, where M = (N − κ)/δ +1. Thus, the

convolution operation of the q-th ﬁlter at the -th convolutional

layer has the following parameters:

Input: X

[−1]

of size M

[−1]

× M

[−1]

× n

[−1]

with

[0]

= X and M

[0]

= N , n

[0]

=3;

Filter: kernel K

[]

of dimension κ

[]

× κ

[]

× n

[−1]

;

Stride: δ

[]

;

Bias: b

[]

;

Output: X

[]

= X

[−1]

∗ K

[]

of size M

[]

× M

[]

and is the M

[]

× M

[]

matrix given by:

[−1]

∗ K

[]

)

m,n

[]



i=1

[]



j=1

[−1]



k=1

q,[]

i,j,k

[−1]

δ(m−1)+i,δ(n−1)+j,k

+ b

[]

, (7)

for m, n =1,...,M and q =1, 2,...,n

[]

. The collection of

the outputs in (7) for all ﬁlters q =1, 2,...,n

[]

leads to the

tensor X

[]

of dimension M

[]

× M

[]

× n

[]

. Thus, the total

number of learned parameters at the -th layer are (κ

[]

× κ

[]

[−1]

) × n

[]

for the ﬁlters plus n

[]

for the biases. Pooling layers

were not used (although tested), since the loss of information

resulted in poor performance.

Thereafter, the FC layers follow. Each function {f

(·)}

i=5,6,7

is a dense layer with 4096, 2048 and 1024 neurons, respec-

tively, followed by a ReLU layer and a Dropout layer. The

latter randomly sets weights to zero with probability 30% (non

trainable parameters), so the network is forced to learn instead

of memorizing the data. The Dropout layers also act as regular-

ization t o the learning process. The -th FC layer maps its input

[−1]

∈ R

[−1]

to the output c

[]

∈ R

[]

via a set of weights

[]

∈ R

[]

×V

[−1]

and biases b

[]

∈ R

[]

. Thus, the output

of the -th FC layer (before nonlinear activation and dropout) is

given by

[]

= W

[]

[−1]

+ b

[]

, (8)

where the set of parameters ϑ

[]

= {W

[]

, b

[]

} with V

[−1]

[]

+ V

[]

entries (in total) is optimized during the training of

the neural network. The ﬁnal (output) l ayer, f

(·), consists of a

dense layer with 2G +1neurons followed by a Sigmoid layer,

which applies the function s(x)=e

/(e

+1)element-wise to

the values of the previous layer and returns values in [0, 1]. The

selection of the sigmoid function over the softmax is due to the

presence of K labels, which could independently receive a value

equal (during the training) or close (during the inference) to 1.

Thus, the output of the CNN is a probability at each entry of

the predicted label, which for the input data X

(i)

is expressed as

(i)

= f (X

(i)

⎛

⎜

⎝

ˆp

2G+1

⎞

⎟

⎠

. (9)

The layout of the proposed CNN is depicted in Fig. 1.

The training of the CNN is performed ofﬂine in a supervised

manner over the training data set D. In particular, since the

adopted approach is a multilabel classiﬁcation task, we attempt

to optimize the set of all trainable parameters ϑ whose updates

are carried out via back-propagation by minimizing the recon-

struction error, i.e.:

∗

= arg min



i=1



(i)

; z

(i)



, (10)

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基于互素数组的时空空间平滑MUSIC算法进行DOA估计

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语音信号处理论文及教材

基于USB2.0的MEMS数字麦克风阵列采集系统设计.pdf

UWB channel model.zip

分布式MIMO雷达单脉冲测角.pdf

yasuo.zip_文章/文档_matlab_

DeepSeek从入门到精通-清华大学-202502.pdf

YOLOv8-deepsort 实现智能车辆目标检测+车辆跟踪+车辆计数

pdf转为扫描件，免费

YOLOv8网络结构图，自制visio文件，yolov8.vsds，需要的自取，在原有的基础上直接改就行了

yolov8(2023年8月版本),已经下好yolov8s.pt和yolov8n.pt

DEEP SEEK 本地部署（Ollama + ChatBox）+ 私有知识库（cherry studio）教程

Transformer模型实现长期预测并可视化结果（附代码+数据集+原理介绍）

社交平台上经济类话题的文章热度信息，数据是真实的，但不是真实日期

行人跌倒数据集（VOC格式）

CIFAR10数据集免费下载

DeepSeek从入门到精通-清华大学

大作业05-YOLOV5口罩检测数据集+代码+模型 2000张标注好的数据+教学视频.zip

Deep Learning Tuning Playbook（中译版）

LabVIEW AI Vision(LabVIEW AI视觉工具包)

zotero翻译插件.xpi

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etgate：以太坊-Tendermint令牌发送网关

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