# Neural Network-Based DOA Estimation in the Presence of Non-Gaussian
Interference

S. Feintuch Stefan Feintuch, Joseph Tabrikian, Igal Bilik, and Haim H.
Permuter are with the School of Electrical and Computer Engineering, Ben
Gurion University of the Negev, Beer Sheva, Israel. (e-mails:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>haimp@bgu.ac.il).
This work was partially supported by the Israel Science Foundation under
Grants 2666/19 and 1895/21.  J. Tabrikian  Fellow, IEEE I. Bilik  Senior
Member, IEEE and H. Permuter  Senior Member, IEEE

###### Abstract

This work addresses the problem of direction- of-arrival (DOA) estimation in
the presence of non-Gaussian, heavy-tailed, and spatially-colored
interference. Conventionally, the interference is considered to be Gaussian-
distributed and spatially white. However, in practice, this assumption is not
guaranteed, which results in degraded DOA estimation performance. Maximum
likelihood DOA estimation in the presence of non-Gaussian and spatially
colored interference is computationally complex and not practical. Therefore,
this work proposes a neural network (NN) based DOA estimation approach for
spatial spectrum estimation in multi-source scenarios with a-priori unknown
number of sources in the presence of non-Gaussian spatially-colored
interference. The proposed approach utilizes a single NN instance for
simultaneous source enumeration and DOA estimation. It is shown via
simulations that the proposed approach significantly outperforms conventional
and NN-based approaches in terms of probability of resolution, estimation
accuracy, and source enumeration accuracy in conditions of low SIR, small
sample support, and when the angular separation between the source DOAs and
the spatially-colored interference is small.

###### Index Terms:

Array Processing, DOA Estimation, Source Enumeration, Spatially-Colored
Interference, Non-Gaussian Interference, Neural Networks, Deep Learning,
Machine Learning, MVDR, MDL, AIC, Radar.

##  I Introduction

Direction-of-arrival (DOA) estimation using a sensor array is required in
multiple applications, such as radar, sonar, ultrasonic, wireless
communications, and medical imaging [1]. In real-world applications, the
signal received at the sensor array is a superposition of signals from the
sources of interest, interference, and receiver thermal noise. In radars, the
received signal consists of a target echo, clutter, and thermal noise. In
multiple scenarios, the radar clutter has a spatially-colored, heavy-tailed
non-Gaussian distribution [2], which can significantly degrade the performance
of conventional estimators.

Minimum-variance-distortionless-response (MVDR) [3], is a conventional
adaptive beamforming approach for DOA estimation. MVDR estimates the spatial
spectrum and obtains the source DOAs via a one-dimensional peak search on a
predefined grid. The estimation of signal parameters using rotational
invariance techniques (ESPIRIT) [4], multiple signal classification (MUSIC)
[5], and root-MUSIC (R-MUSIC) [6] are additional widely used DOA estimation
approaches. These approaches involve received signal autocorrelation matrix
processing, which conventionally is performed via the sample autocorrelation
matrix estimation [3, 5, 4, 6]. However, the performance of the sample
autocorrelation matrix estimator degrades in small sample support or non-
Gaussian scenarios. Furthermore, these methods use second-order statistics
only and omit the higher-order statistics on non-Gaussian-distributed
interference. In addition, ESPRIT, MUSIC, and R-MUSIC approaches require
a-priori knowledge of the number of sources (or targets), which limits their
practical use.

The problem of DOA estimation in the presence of non-Gaussian interference is
of great practical interest. The maximum likelihood estimator (MLE) for DOA
estimation in the presence of non-Gaussian interference does not have a
closed-form analytical solution [7, 8]. Multiple model-based DOA estimation
approaches have been intensively studied in the literature [7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18].

Robust covariance matrix-based DOA estimation and source enumeration methods
have been studied in the literature. For complex elliptically symmetric (CES)
distributed data, the authors in [9] showed that a scatter matrix-based
beamformer is consistent, and the semiparametric lower bound, and Slepian-
Bangs formula for DOA estimation were derived in [10]. In [11], a generalized
covariance-based (GC) approach for the covariance matrix estimation in
scenarios with impulsive alpha-stable noise was proposed for MUSIC DOA
estimation. However, these methods consider a specific family of
distributions, such as the CES or alpha-stable, and are, therefore, limited in
the case of model mismatch. In [12], a probability measure transform (MT)
based covariance matrix estimator was proposed for MUSIC-based DOA estimation
and minimum descriptive length (MDL) based source enumeration. The MT-based
covariance estimator was also adopted for robust MVDR beamformer [13]. These
methods are usually based on setting a parameter that determines the tradeoff
between the level of robustness and performance.

The problem of DOA estimation in the presence of a mixture of spatially-white
K-distributed and Gaussian-distributed noise under a deterministic and unknown
(conditional) source model was studied in [7]. An iterative MLE-based approach
for the conditional and joint likelihood of interference distribution’s
parameters was derived in [14, 15]. This approach was further extended in [16]
to the marginal likelihood function. However, this approach is computationally
complex due to numerical integral evaluation that involves a $2M$ dimensional
grid search for $M$ targets [8]. Therefore, [8] proposed a kernel minimum
error entropy-based adaptive estimator and a novel criterion to reduce the
estimator’s computational complexity. The expectation-maximization (EM) with a
partial relaxation-based DOA estimation algorithm under the conditional model
assumption was proposed in [17]. In [18], a sparse Bayesian learning (SBL)
approach for outlier rejection of impulsive and spatially-white interference
was proposed. This EM-based approach did not require a-priori knowledge of the
number of sources and was shown to resolve highly-correlated and coherent
sources. However, none of these model-based DOA estimation approaches
considered an a-priori an unknown number of sources and spatially-colored
interference and therefore are limited for real-world applications. Although
source enumeration methods, such as MDL and Akaike information criterion (AIC)
[19] can be used, they assume signal Gaussianity and can therefore be
inaccurate in non-Gaussian scenarios.

Deep learning and machine learning approaches were recently adopted for radar
signal processing. Three types of NN-based DOA estimation approaches have been
introduced in literature [20]. The first approach assumes a-priori known
number of sources and uses a NN, which is optimized to output a vector of the
estimated DOAs [21, 22, 23, 24, 25, 26, 27]. The second approach does not
assume a-priori known number of sources and uses a NN for source enumeration
[25, 26, 27, 28, 29, 30, 31]. The third approach uses a NN to estimate source
presence probability at each DOA on a predefined angular grid and obtains the
source DOAs via a peak search [32, 33, 34, 35, 36, 37, 38, 39, 40, 41].
However, all these approaches have not addressed non-Gaussian and spatially-
colored interference [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41].

The cases of non-Gaussian and/or spatially-colored interference have been
addressed using machine learning-based approaches. For massive MIMO cognitive
radar, a reinforcement learning-based approach for multi-target detection
under heavy-tailed spatially-colored interference was proposed in [42]. In
[43], authors addressed the MIMO radar target detection under non-Gaussian
spatially-colored interference by using a CNN architecture that is optimized
according to a novel loss. A radial-basis-function (RBF) NN [44] and a
convolutional neural network (CNN) [45] architectures were proposed for DOA
estimation in the presence of non-Gaussian impulsive noise. In [46], a CNN-
based architecture that includes denoising NN, source enumeration NN, and DOA
estimation sub-NNs, was introduced. However, [44, 45, 46] consider spatially-
white noise and are suboptimal for scenarios with spatially-colored
interference.

In [47], a novel NN-based approach is introduced for radar target detection
under slow-time correlated and heavy-tailed distributed clutter in the range-
Doppler domain without any spatial processing. This work extends [47] to the
field of array processing, by employing a similar NN-based approach under
performance criteria for source DOA estimation and enumeration. More
specifically, we address the array processing problem with a-priori unknown
number of sources in the presence of non-Gaussian, heavy-tailed, spatially-
colored interference at a low signal-to-interference ratio (SIR) and small
sample size. The contribution of this work includes:

  1. 1.

A novel NN-based processing mechanism is used for array processing within non-
Gaussian spatially-colored interference. The proposed NN architecture utilizes
the structure of information within the set of received complex snapshots.

  2. 2.

The proposed NN is optimized to output an interference-mitigated spatial
spectrum and is used for simultaneous source enumeration and DOA estimation of
sources within non-Gaussian spatially-colored interference.

The proposed approach outperforms conventional adaptive beamforming and
competes for straightforward NN-based methods in terms of probability of
resolution and estimation accuracy in scenarios with non-Gaussian spatially-
colored interference. In addition, the proposed approach outperforms
conventional source enumeration techniques in scenarios characterized by non-
Gaussian spatially-colored interference.

The following notations are used throughout the paper. Roman boldface lower-
case and upper-case letters represent vectors and matrices, respectively,
while Italic letters stand for scalars. $\mathbf{I}_{N}$ is the identity
matrix of size $N\times N$ and $\mathbf{1}_{N}$ is a column vector of length
$N$ whose entries are equal to one. $\mathbb{E}(\cdot)$, $(\cdot)^{T}$, and
$(\cdot)^{H}$ are the expectation, transpose, and Hermitian transpose
operators, respectively. $\text{Vec}(\cdot)$, $\text{diag}(\cdot)$, and
$|\cdot|$ stand for the vectorization, diagonalization, and absolute value
operators, respectively. $[\mathbf{a}]_{n}$ and $[\mathbf{A}]_{n,m}$ are the
$n$-th and $n,m$-th elements of the vector $\mathbf{a}$ and the matrix
$\mathbf{A}$, respectively.

The remainder of this paper is organized as follows. The addressed problem is
stated in Section II. Section III introduces the proposed NN-based DOA
estimation approach. The proposed approach is evaluated via simulations in
Section IV. Our conclusions are summarized in Section V.

##  II Problem Definition

This work considers the problem of DOA estimation using an array of $L$
receiving elements and $M$ distinct and unknown sources with DOAs,
$\bm{\Theta}=\\{\theta_{1},\dots,\theta_{M}\\}$. The measurements contain $K$
spatial snapshots, $\\{\mathbf{x}_{k}\\}_{k=1}^{K}$:

$\displaystyle\mathbf{x}_{k}$
$\displaystyle=\mathbf{A}\left(\bm{\Theta}\right)\mathbf{s}_{k}+\sigma_{c}\mathbf{c}_{k}+\mathbf{n}_{k}\;,$
(1)
$\displaystyle=\sum_{m=1}^{M}{\mathbf{a}\left(\theta_{m}\right)s_{k,m}}+\sigma_{c}\mathbf{c}_{k}+\mathbf{n}_{k}\;,\;k=1,\dots,K\;,$

where
$\mathbf{A}\left(\bm{\Theta}\right)=\begin{bmatrix}\mathbf{a}\left(\theta_{1}\right)&\cdots&\mathbf{a}\left(\theta_{M}\right)\end{bmatrix}$,
with $\mathbf{a}\left(\theta_{m}\right)\in\mathbb{C}^{L}$ denoting the
steering vector for source at direction $\theta_{m}$, and
$\mathbf{s}_{k}\triangleq\begin{bmatrix}s_{k,1}&\cdots&s_{k,M}\end{bmatrix}^{T}$
is the source signal vector. We assume an unconditional model [48], where
$\\{\mathbf{s}_{k}\\}\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\mathcal{CN}\left(\mathbf{0}_{M},\text{diag}\left(\sigma_{1}^{2},\dots,\sigma_{M}^{2}\right)\right)$,
is temporally uncorrelated between pulses. The targets are assumed to be
spatially distinct. The receiver thermal noise, denoted by $\mathbf{n}_{k}$,
is considered to be complex Gaussian-distributed
$\\{\mathbf{n}_{k}\\}\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\mathcal{CN}\left(\mathbf{0}_{L},\sigma_{n}^{2}\mathbf{I}_{L}\right)$.
The heavy-tailed non-Gaussian and spatially-colored interference is modeled by
the interference amplitude $\sigma_{c}$, and the interference component
$\mathbf{c}_{k}\in\mathbb{C}^{L}$. The considered compound-Gaussian
distributed interference, $\\{\mathbf{c}_{k}\\}\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\mathcal{K}\left(\nu,\theta_{c}\right)$ represents a non-
Gaussian interference with angular spread around an unknown direction
$\theta_{c}$, such that $\mathbf{c}\sim\mathcal{K}\left(\nu,\theta_{c}\right)$
implies

$\displaystyle\mathbf{c}=\sqrt{\tau}\mathbf{z}\;,$ (2)
$\displaystyle\tau\rotatebox[origin={c}]{90.0}{$\models$}\mathbf{z},\;\tau\sim\Gamma\left(\nu,\nu\right),\;\mathbf{z}\sim\mathcal{CN}\left(\mathbf{0}_{L},\mathbf{M}_{\theta_{c}}\right)\;.$

The compound-Gaussian statistical model is conventionally used in the
literature to model heavy-tailed non-Gaussian interference [8, 14, 16, 7, 49,
43]. The texture component, $\tau\in\mathbb{R}_{+}$, determines the heavy-
tailed behavior and is characterized by $\nu$. The speckle component,
$\mathbf{z}\in\mathbb{C}^{L}$, determines the spatial distribution of the
interference and is characterized by the covariance matrix,
$\mathbf{M}_{\theta_{c}}$. The spatial covariance matrix of the interference
upholds:

$\displaystyle\mathbb{E}\left[\sigma_{c}^{2}\mathbf{c}\mathbf{c}^{H}\right]=$
$\displaystyle\sigma_{c}^{2}\mathbb{E}\left[\tau\right]\mathbb{E}\left[\mathbf{z}\mathbf{z}^{H}\right]=\sigma_{c}^{2}\mathbf{M}_{\theta_{c}}\;,$
(3)

where $\mathbf{M}_{\theta_{c}}$ can be modeled as [14, 15, 16, 49, 43]:

$\displaystyle\left[\mathbf{M}_{\theta_{c}}\right]_{m,l}$
$\displaystyle=\rho^{\left|m-l\right|}e^{j(m-l)\pi\sin\theta_{c}}\;.$ (4)

The model in (3) and (4), represents the spatial interference, characterized
by $\rho$, with a spread around the interference DOA, $\theta_{c}$.

##  III The Proposed DAFC-Based Neural Network

In the following, the data pre-processing and the proposed NN-based processing
mechanism are introduced in Subsections III-A and III-B. The proposed NN
architecture and loss function are detailed in Subsections III-C and III-D,
respectively.

The proposed approach is  similar to the NN architecture that was introduced
for linear-frequency-modulated (LFM) radar target detection in the range-
Doppler domain [47]. The proposed architecture in this work utilizes the
dimensional alternating fully-connected (DAFC) mechanism introduced in [47].
However, since this work addresses a different problem, we have empirically
found that the pre-processing, NN architecture, and loss function described in
Subsections III-A, III-C, and III-D, respectively, allow to attain better
results.

###  III-A Pre-Processing

The input matrix, $\mathbf{X}\in\mathbb{C}^{L\times K}$ is constructed from
the set of $K$ snapshots in (1), $\\{\mathbf{x}_{k}\\}$:

$\displaystyle\mathbf{X}=\begin{bmatrix}\mathbf{x}_{1}&\mathbf{x}_{2}&\cdots&\mathbf{x}_{K}\end{bmatrix}\;,$
(5)

where the $k$-th column of $\mathbf{X}$ contains the $k$-th snapshot. The
variation between the columns of $\mathbf{X}$ is induced by the statistical
characteristics of the source signal $\mathbf{s}_{k}$, interference signal
$\mathbf{c}_{k}$, and thermal noise $\mathbf{n}_{k}$. Therefore, each column
in $\mathbf{X}$ can be interpreted as a complex “feature” vector containing
essential information for DOA estimation. The set of columns in $\mathbf{X}$
can be interpreted as “realizations” of that feature.

The complex-valued matrix, $\mathbf{X}$, is converted into a real-valued
representation needed for the NN-based processing. To keep consistency with
[47], we apply a transpose operator to the input matrix, such that the
snapshots are stacked in rows. The output of the pre-processing denoted by
$\mathbf{Z}_{0}\in\mathbb{C}^{K\times 2L}$, is:

$\displaystyle\mathbf{Z}_{0}=\left[\Re{\mathbf{X}^{T}},\;\Im{\mathbf{X}^{T}}\right]\;.$
(6)

###  III-B Dimensional Alternating Fully-Connected

The DAFC block was introduced to process measurements in a form similar to the
model in Section II [47]. Fig. 1 schematically shows the DAFC mechanism.

For arbitrary dimensions $D_{1},D_{2},D_{3}$, the formulation of a general
fully-connected (FC) layer applied to each row in a given matrix
$\mathbf{Z}\in\mathbb{R}^{D_{1}\times D_{2}}$ can be represented by the
transform $\mathcal{F}\left(\cdot\right)$:

$\displaystyle\mathcal{F}:\mathbb{R}^{D_{1}\times
D_{2}}\rightarrow\mathbb{R}^{D_{1}\times D_{3}}\;,$ (7)
$\displaystyle\mathcal{F}\left(\mathbf{Z}\right)\triangleq
h\left(\mathbf{ZW}+\mathbf{1}_{D_{1}}\mathbf{b}^{T}\right)\;.$

This matrix-to-matrix transformation is characterized by the “learnable”
weight matrix, $\mathbf{W}\in\mathbb{R}^{D_{2}\times D_{3}}$, the bias vector,
$\mathbf{b}\in\mathbb{R}^{D_{3}}$, and a scalar element-wise activation
function, $h(\cdot)$.

Let $\mathcal{F}_{r}\left(\cdot\right)$ and
$\mathcal{F}_{c}\left(\cdot\right)$ be two separate, and not necessarily
identical instances of $\mathcal{F}\left(\cdot\right)$ from (7), and
$\mathbf{Z}_{in}$ be an arbitrary input matrix. The DAFC mechanism is
formulated by the following operations:

Dimensional Alternating Fully Connected

• Input: $\mathbf{Z}_{in}\in\mathbb{R}^{H\times W}$

$\mathcal{F}_{r}:\mathbb{R}^{H\times W}\rightarrow\mathbb{R}^{H\times
W^{\prime}}$

$\mathcal{F}_{c}:\mathbb{R}^{W^{\prime}\times
H}\rightarrow\mathbb{R}^{W^{\prime}\times H^{\prime}}$

  1. 1.

Apply a single FC layer to each row in $\mathbf{Z}_{in}$:

$\displaystyle\mathbf{Z}_{r}=\mathcal{F}_{r}\left(\mathbf{Z}_{in}\right)$

  2. 2.

Apply a single FC layer to each column in $\mathbf{Z}_{r}$:

$\displaystyle\mathbf{Z}_{c}=\mathcal{F}_{c}\left(\mathbf{Z}_{r}^{T}\right)$

  3. 3.

Transpose to keep orientation:

$\displaystyle\mathbf{Z}_{out}=\mathbf{Z}_{c}^{T}$

• Output:
$\mathbf{Z}_{out}\triangleq\mathcal{S}\left(\mathbf{Z}\right)\in\mathbb{R}^{H^{\prime}\times
W^{\prime}}$

In the following, three DAFC design principles are detailed.

1) Structured transformation

The input to the first DAFC block is the pre-processed, $\mathbf{Z}_{0}$,
given in (6). Therefore, the first FC layer, $\mathcal{F}_{r}$, of the first
DAFC block is capable of extracting spatial-related features from each row in
$\mathbf{Z}_{0}$. The second FC layer, $\mathcal{F}_{c}$, of the first DAFC
block, introduces an interaction between transformed rows. This implies that
a) $\mathcal{F}_{r}$ can perform “spatial-feature” extraction by transforming
each pre-processed snapshot (each row of $\mathbf{Z}_{0}$) to a high-
dimensional feature space, and b) $\mathcal{F}_{c}$ can perform a nonlinear
transformation of the extracted features (the columns of
$\mathcal{F}_{r}\left(\mathbf{Z}_{0}\right)$) from each snapshot. In this way,
the DAFC utilizes both spatial and statistical information. Thus, the DAFC
mechanism can contribute to estimating the source DOAs and mitigating the
interference when incorporated into a NN.

2) Dimension reduction of learnable parameters

Conventional DOA estimation considers the input data as the collection of
measurement vectors (the snapshots $\\{\mathbf{x}_{k}\\}$) in a matrix form.
One straightforward approach to processing the input data using a NN is to
reshape it and process it via an FC-based architecture. In this way, each
neuron in the layer’s output interacts with every neuron in the input. On the
other hand, the DAFC block transforms the data using a structured
transformation, which significantly reduces the number of learnable parameters
compared to the straightforward FC-based approach.

This parameter reduction can be observed in the following typical case.
Consider an input matrix $\mathbf{Z}_{1}\in\mathbb{R}^{D_{1}\times D_{1}}$,
which is transformed to an output matrix
$\mathbf{Z}_{2}\in\mathbb{R}^{D_{2}\times D_{2}}$. The number of learnable
parameters in the FC- and the proposed DAFC-based approaches is of the order
of $\mathcal{O}\left(D_{1}^{2}D_{2}^{2}\right)$, and
$\mathcal{O}\left(D_{1}D_{2}\right)$, respectively. Notice that the DAFC-based
transformation complexity grows linearly with the number of learnable
parameters compared to the quadratic complexity growth of the straightforward,
FC-based approach.

The contribution of learnable parameters dimension reduction is twofold.
First, the conventional NN optimization is gradient-based [50]. Therefore, a
significant reduction in the learnable parameter dimension reduces the degrees
of freedom in the optimizable parameter space and improves the gradient-based
learning algorithm convergence rate. Second, reduction in the learnable
parameter dimension can be interpreted as increasing the “inductive bias” of
the NN model [51], which conventionally contributes to the NN statistical
efficiency and generalization ability, thus, reducing the NNs tendency to
overfit the training data.

3) Nonlinearity

The proposed DAFC considers an additional degree of nonlinearity compared to
the straightforward FC-based approach. A straightforward matrix-to-matrix
approach includes an interaction of every neuron in the output matrix with
every neuron in the input matrix, followed by an element-wise nonlinear
activation function. On the other hand, the proposed DAFC consists of two
degrees of nonlinearity in $\mathcal{F}_{r}$ and $\mathcal{F}_{c}$. Although
the weight matrices applied as part of $\mathcal{F}_{r}$ and $\mathcal{F}_{c}$
are of lower dimension than the weight matrix used in the straightforward
approach, the extra degree of nonlinearity can increase the NN’s capacity
[50]. Therefore, a NN architecture with the proposed DAFC is capable of
learning a more abstract and rich transformation of the input data.

Figure 1:  The DAFC mechanism concept. Each row of dimension $W$ in
$\mathbf{Z}_{in}$, represented by the red color, is transformed by
$\mathcal{F}_{r}$ to a row of dimension $W^{\prime}$ in the middle matrix,
represented by the transparent red color. Next, each column of dimension $H$
in the middle matrix, represented by the blue color, is transformed by
$\mathcal{F}_{c}$ to a column of dimension $H^{\prime}$ in $\mathbf{Z}_{out}$,
represented by the transparent blue color. Figure 2: Proposed NN architecture.
The pre-processing $\mathcal{P}$ is described in Section III-A and appears in
yellow. The purple matrices denote the concatenation of DAFC blocks, which is
detailed in Section III-B. The blue vector represents a vectorization of the
last DAFC output, and the orange vector stands for FC layers with tanh
activation function. The last green vector is the output of the last FC layer,
which consists of the sigmoid activation function and yields the estimated
spatial spectrum $\hat{\mathbf{y}}$.

###  III-C NN Architecture

The continuous DOA space is discretized into a $d$-dimensional grid:
$\bm{\phi}=\begin{bmatrix}\phi_{1}&\phi_{2}&\cdots&\phi_{d}\end{bmatrix}^{T}$.
This implies that the entire field-of-view (FOV) is partitioned into $d$ DOAs,
$\\{\phi_{i}\\}_{i=1}^{d}$, determined by the selected grid resolution,
$\Delta\phi\triangleq\phi_{i+1}-\phi_{i}$. The proposed NN is designed to
represent a mapping from the input set of snapshots, $\\{\mathbf{x}_{k}\\}$
given in (1), into the probability of source present in the DOAs
$\\{\phi_{i}\\}_{i=1}^{d}$. The proposed NN architecture is formulated as
follows:

$\displaystyle\mathbf{Z}_{0}$
$\displaystyle=\mathcal{P}\left(\mathbf{X}\right)\;,$ (8)
$\displaystyle\mathbf{z}_{\text{vec}}$
$\displaystyle=\text{Vec}\left(\mathcal{S}_{6}\left(\dots\mathcal{S}_{1}\left(\mathbf{Z}_{0}\right)\right)\right)\;,$
$\displaystyle\hat{\mathbf{y}}$
$\displaystyle=\mathcal{G}_{3}\left(\mathcal{G}_{2}\left(\mathcal{G}_{1}\left(\mathbf{z}_{\text{vec}}\right)\right)\right)\;,$

where $\mathbf{Z}_{0}$ is the output of the pre-processing procedure, denoted
as $\mathcal{P}\left(\cdot\right)$ and detailed in Section III-A, and
$\mathbf{X}$ is the input matrix in (5).

In the next stage, six DAFC instances, represented by
$\mathcal{S}_{1}\left(\cdot\right),\dots,\mathcal{S}_{6}\left(\cdot\right)$,
of different dimensions with tanh activation for the row transform
($\mathcal{F}_{r}$ in Section III-B) and ReLu activation for the column
transform ($\mathcal{F}_{c}$ in Section III-B), are used to generate the
vectorized signal $\mathbf{z}_{\text{vec}}$. Our experiments showed that this
configuration of row and column activation functions provides the best
performance. At the last stage, the signal, $\mathbf{z}_{\text{vec}}$, is
processed by three FC layers, where the first two use tanh activation, and the
final (output) layer of equal size to the DOA grid dimension, $d$, use sigmoid
activation function to output $\hat{\mathbf{y}}\in\left[0,1\right]^{d}$. Thus,
$\\{\left[\hat{\mathbf{y}}\right]_{i}\\}_{i=1}^{d}$ represent the estimated
probabilities of a source presence at $\\{\phi_{i}\\}_{i=1}^{d}$. Table I and
Fig. 2 summarize the parameters and architecture of the proposed NN-based
approach.

Operator |  Output Dimension |  Activation |  # Parameters
---|---|---|---
$\mathcal{P}$ |  $K\times 2L$ |  - |  -
$\mathcal{S}_{1}$ |  $64\times 256$ |  tanh-ReLu |  9,536
$\mathcal{S}_{2}$ |  $128\times 512$ |  tanh-ReLu |  139,904
$\mathcal{S}_{3}$ |  $256\times 1024$ |  tanh-ReLu |  558,336
$\mathcal{S}_{4}$ |  $64\times 512$ |  tanh-ReLu |  541,248
$\mathcal{S}_{5}$ |  $16\times 256$ |  tanh-ReLu |  132,368
$\mathcal{S}_{6}$ |  $4\times 128$ |  tanh-ReLu |  32,964
vec |  512 |  - |  -
$\mathcal{G}_{1}$ |  1024 |  tanh |  525,312
$\mathcal{G}_{2}$ |  256 |  tanh |  262,400
$\mathcal{G}_{3}$ |  $d$ |  sigmoid |  31,097
Table I:  Specification of the proposed NN architecture for
$K=16,\;L=16,\;d=121$. “tanh-ReLu” activation stands for tanh in
$\mathcal{F}_{r}$ and ReLU in $\mathcal{F}_{c}$ of each DAFC block. The number
of total learnable parameters is $2,233,165$.

The estimated source DOAs are extracted from the spatial spectrum via
peak_search and applying $0.5$ threshold:

$\displaystyle\\{i_{1},\dots,i_{\hat{N}}\\}$
$\displaystyle=\textit{peak\\_search}\left(\\{\left[\hat{\mathbf{y}}\right]_{i}\\}_{i=1}^{d}\right)$
(9) $\displaystyle\hat{\mathbf{\Theta}}$
$\displaystyle=\left\\{\phi_{i_{n}}:\left[\hat{\mathbf{y}}\right]_{i_{n}}>0.5\right\\}_{n=1}^{\hat{N}}\;.$

Namely, the set of estimated DOAs, $\hat{\mathbf{\Theta}}$, consists of the
grid points corresponding to the peaks of $\hat{\mathbf{y}}$ that exceed the
$0.5$ threshold. The number of peaks that exceed this threshold is used for
source enumeration, and therefore the proposed NN can be utilized as a source
enumeration method as well.

The dimensionality of the hidden layers in the proposed NN architecture
expands in the first layers and then reduces. This trend resembles the NN
architecture presented in [47] and characterizes both the DAFC-based and FC-
based processing stages. This expansion-reduction structure can be explained
by a) the early NN stages need to learn an expressive and meaningful
transformation of the input data by mapping it to a higher dimensional
representation and b) the late stages need to extract significant features
from the early mappings and are therefore limited in dimensionality. In
addition, the late stages are adjacent to the output vector and therefore need
to be of similar dimensions.

###  III-D Loss Function

The label used for the supervised learning process,
$\mathbf{y}\in\\{0,1\\}^{d}$, is defined as a sparse binary vector with the
value $1$, at the grid points that correspond to the source DOAs, and $0$,
otherwise. In practice, the DOAs in $\bm{\Theta}$ do not precisely correspond
to the grid points. Therefore, for each DOA in $\bm{\Theta}$, the nearest grid
point in $\\{\phi_{i}\\}_{i=1}^{d}$ is selected as the representative grid
point in the label. Each training example is determined by the input-label
pair, $\left(\mathbf{X},\mathbf{y}\right)$. Using the NN feed-forward in (8),
$\mathbf{X}$ is used to generate the output spatial spectrum,
$\hat{\mathbf{y}}$, which is considered as the estimated label.

The loss function, $\mathcal{L}$, is a weighted mean of the binary cross
entropy (BCE) loss computed at each grid point:

$\displaystyle\mathcal{L}\left(\mathbf{y},\hat{\mathbf{y}},t\right)$
$\displaystyle=\frac{1}{d}\sum_{i=1}^{d}{w_{i}^{(t)}\text{BCE}\left(\left[\mathbf{y}\right]_{i},\left[\hat{\mathbf{y}}\right]_{i}\right)}\;,$
(10) $\displaystyle\text{BCE}\left(y,\hat{y}\right)$
$\displaystyle=-y\log\left(\hat{y}\right)-\left(1-y\right)\log\left(1-\hat{y}\right)\;,$

where $w_{i}^{(t)}$ represents the loss weight of the $i$-th grid point at the
$t$-th epoch. The loss value for equally-weighted BCEs evaluated per grid
point ($w_{i}^{(t)}=1$ in (10)) does not significantly increase in the case of
a large error in source/interference estimated probability due to the sparsity
of the label $\mathbf{y}$. This forces the NN convergence into a sub-optimal
solution that is prone to “miss” the sources. Therefore, the loss weights,
$\\{w_{i}^{(t)}\\}_{i=1}^{d}$, are introduced to “focus” the penalty on
source/interference grid points.

The loss weight of the $i$-th grid point, $w_{i}^{(t)}$, is determined by the
presence of source or interference in the corresponding label entry
$\left[\mathbf{y}\right]_{i}$. This relation is defined using the epoch and
label dependent factors $e_{0}^{(t)},e_{1}^{(t)}$, according to:

$\displaystyle w_{i}^{(t)}$
$\displaystyle=\begin{cases}1/e_{1}^{(t)},&\text{if $\phi_{i}$ contains source
or interference}\\\ 1/e_{0}^{(t)},&\text{else}\end{cases}\;.$ (11)

For $t=0$, the factor $e_{1}^{(0)}$ is determined by the fraction of label
grid points that contain source or interference out of the total label grid
points in the training set, and $e_{0}^{(0)}$ is the corresponding complement.
For subsequent epochs, the factors are updated according to a predefined
schedule, similar to a predefined learning rate schedule. The loss weights are
updated $N_{w}$ times with spacing of $\Delta t$ epochs during training. The
update values are determined by updating $e_{0}^{(t)},e_{1}^{(t)}$, according
to the following decaying rule:

$\displaystyle e_{q}^{(t)}$ $\displaystyle=(1-\beta^{(l)})e_{q}^{(l\Delta
t)}+\beta^{(l)},\;l\Delta t\leq t<(l+1)\Delta t$ (12) $\displaystyle q$
$\displaystyle=0,1,\;l=1,\dots,N_{w},$

where $l$ is the loss weight update iteration, and
$\\{\beta^{(l)}\\}_{l=1}^{N_{w}}$ represent the loss weight update factors
which uphold, $0\leq\beta^{(l)}\leq 1$. Note that for $N_{w}\Delta t\leq t$,
the weight factor remains $e_{i}^{(N_{w}\Delta t)}$ during the rest of the
training stage. Notice that as $\beta^{(l)}\rightarrow 1$, the corresponding
loss weights will tend to be equally distributed across the grid points, i.e.,
$e_{1}^{(t)}\approx e_{0}^{(t)}$. In this case, an erroneously estimated
probability for source/interference containing grid point is equally weighted
to a neither-containing grid point. On the other hand, as
$\beta^{(l)}\rightarrow 0$, the corresponding factors will uphold
$e_{1}^{(t)}\ll e_{0}^{(t)}$, yielding a significantly larger contribution of
source/interference containing grid points to the loss value. The rule in (12)
enables a “transition of focus” throughout the training. That is, during the
early epochs $\beta^{(l)}\rightarrow 0$, which contributes more weight to the
source/interference containing areas in the estimated label $\hat{\mathbf{y}}$
(i.e., the estimated spatial spectrum) to focus the NN to being correct for
source/interference. During the later epochs, $\beta^{(l)}$ is incrementally
increased, which relaxes the focus on source/interference from early epochs.
Thus, reducing erroneously estimated sources in areas that do not contain
source/interference (i.e., “false-alarms”).

##  IV Performance Evaluation

This section evaluates the performance of the proposed DAFC-based NN approach
and compares it to the conventional approaches, summarized in Subsection
IV-A1. The data for all considered scenarios are simulated using the
measurement model from Section II.

###  IV-A Setup & Training

This work considers a uniform linear array (ULA) with half-wavelength-spaced
$L$ elements. Each simulated example consists of the input-label pair,
$\left(\mathbf{X},\mathbf{y}\right)$, where the input $\mathbf{X}$ is defined
in (5), and the label $\mathbf{y}$ is defined in Section III-D. The simulation
configurations are detailed in Table II. The performance of the proposed
approach is evaluated using a single NN instance. Therefore, a single NN model
is used for various signal-to-interference ratios (SIRs), signal-to-noise
ratios (SNRs), interference-to-noise ratios (INRs), DOAs, interference
distribution, and the number of sources for joint DOA estimation and source
enumeration. The following definitions for the $m$-th source are used in all
experiments:

INR
$\displaystyle=\frac{\mathbb{E}\left[\left\|\mathbf{c}\right\|^{2}\right]}{\mathbb{E}\left[\left\|\mathbf{n}\right\|^{2}\right]}=\sigma_{c}^{2}/\sigma_{n}^{2}\;,$
(13) $\displaystyle\text{SNR}_{m}$
$\displaystyle=\frac{\mathbb{E}\left[\left\|\mathbf{a}\left(\theta_{m}\right)s_{m}\right\|^{2}\right]}{\mathbb{E}\left[\left\|\mathbf{n}\right\|^{2}\right]}=\sigma_{m}^{2}/\sigma_{n}^{2}\;,$
(14) $\displaystyle\text{SIR}_{m}$
$\displaystyle=\frac{\mathbb{E}\left[\left\|\mathbf{a}\left(\theta_{m}\right)s_{m}\right\|^{2}\right]}{\mathbb{E}\left[\left\|\mathbf{c}\right\|^{2}\right]}=\sigma_{m}^{2}/\sigma_{c}^{2}\;.$
(15)

The NN optimization for all evaluated architectures is performed using the
loss function in (10) and Adam optimizer [52] with a learning rate of
$10^{-3}$, and a plateau learning rate scheduler with a decay of $0.905$. The
set of loss weight update factors, $\\{\beta^{(l)}\\}_{l=1}^{N_{w}}$, in (12)
is chosen as the evenly-spaced logarithmic scale between $10^{-5}$ and
$10^{-2}$ with $N_{w}=6$, that is $\\{10^{-5},7.25\cdot 10^{-5},5.25\cdot
10^{-4},3.8\cdot 10^{-3},2.78\cdot 10^{-2},0.2\\}$. The chosen batch size is
$512$, the number of epochs is $500$, and early stopping is applied according
to the last $200$ epochs.

Notation |  Description |  Value
---|---|---
$M_{max}$ |  Maximal number of sources |  $4$
$L$ |  Number of sensors |  $16$
$K$ |  Number of snapshots |  $16$
$d$ |  Angular grid dimension |  $121$
$\Delta\phi$ |  Angular grid resolution |  $1\degree$
FOV |  Field of view |  $\left[-60\degree,60\degree\right]$
$\sigma_{n}^{2}$ |  Thermal noise power |  $1$
Table II:  Simulation Configurations.

####  IV-A1 DOA Estimation Approaches

This subsection briefly summarizes the conventional DOA estimation approaches.
The performance of the proposed approach is compared to the conventional MVDR,
CNN, and FC-based NN. All the NN-based approaches were implemented using a
similar number of layers and learnable parameters. In addition, the FC-based
NN and CNN were optimized using the same learning algorithm and
configurations.

(a) Conventional Adaptive Beamforming
The MVDR [3] estimator is based on adaptive beamforming, and it is the maximum
likelihood estimator in the presence of unknown Gaussian interference [53].
The MVDR estimates DOAs by a peak search on the MVDR spectrum:

$\displaystyle
P_{MVDR}\left(\phi\right)=\frac{1}{\mathbf{a}^{H}\left(\phi\right)\hat{\mathbf{R}}_{x}^{-1}\mathbf{a}\left(\phi\right)}\;,$
(16)

where
$\hat{\mathbf{R}}_{x}=\frac{1}{K}\sum_{k=1}^{K}{\mathbf{x}_{k}\mathbf{x}^{H}_{k}}$
is the sample covariance matrix estimator. Notice that the MVDR spectrum
utilizes only second-order statistics of the received signal $\mathbf{x}_{k}$.
For Gaussian-only interference (i.e., $\mathbf{c}_{k}=0$ in (1)), the second-
order statistics contain the entire statistical information. However, for non-
Gaussian interference, information from higher-order statistics is needed.

(b) CNN Architecture
We consider a CNN-based DOA estimation approach using a CNN architecture that
is similar to the architecture provided in [38]. The input to the CNN of
dimension $L\times L\times 3$ consists of the real, imaginary, and angle parts
of $\hat{\mathbf{R}}_{x}$. The CNN architecture consists of $4$ consecutive
CNN blocks, such that each block contains a convolutional layer, a batch
normalization layer, and a ReLu activation. The convolutional layers consist
of $\left[128,256,256,128\right]$ filters. Kernel sizes of $3\times 3$ for the
first block and $2\times 2$ for the following three blocks are used. Similarly
to [38], $2\times 2$ strides are used for the first block and $1\times 1$ for
the following three blocks. Next, a flatten layer is used to vectorize the
hidden tensor, and $3$ FC layers of dimensions $1024,\;512,\;256$ are used
with a ReLu activation and Dropout of $30\%$. Finally, the output layer is
identical to the proposed DAFC-based NN as detailed in Subsection III-C. The
considered loss function is identical to the proposed DAFC-based approach in
(10). The number of trainable parameters in the considered CNN architecture
accounts for $3,315,449$. Notice that the CNN-based architecture utilizes the
information within the sample covariance matrix and, therefore, is limited to
second-order statistics only.

(c) FC Architecture
A straightforward implementation of an FC-based architecture, as mentioned in
Subsection III-B, was implemented. The data matrix, $\mathbf{X}$, is
vectorized, and the real and imaginary parts of the values were concatenated
to obtain a $2KL$-dimension input vector. The selected hidden layers are of
sizes: $\left[512,512,1024,1024,512,256\right]$ where each hidden layer is
followed by a tanh activation function. The output layer is identical to the
proposed DAFC-based NN approach as detailed in Subsection III-C. The
considered loss function is (10), and the number of trainable parameters in
the FC-based NN accounts for $2,787,449$. Notice that the FC-based NN
architecture utilizes all the measurements by interacting with all samples in
the input data. However, this processing is not specifically tailored to the
structure of information within the measurements. On the other hand, the
proposed DAFC-based NN utilizes the information structure to process the input
data. Therefore, for the considered DOA estimation problem, the “inductive
bias” [51] for this approach is improper and can result in under-fitted NN
architecture.

####  IV-A2 Performance Evaluation Metrics

This subsection discusses the criteria for the performance evaluation of the
proposed DOA estimation approach. In this work, similarly to [38], the DOA
estimation accuracy of a set of sources is evaluated by the Hausdorff distance
between sets. The Hausdorff distance, $d_{H}$ between the sets, $\mathcal{A}$,
and $\mathcal{B}$, is defined as:

$\displaystyle d_{H}\left(\mathcal{A},\mathcal{B}\right)$
$\displaystyle=\max\left\\{d\left(\mathcal{A},\mathcal{B}\right),d\left(\mathcal{B},\mathcal{A}\right)\right\\}\;,$
(17) $\displaystyle d\left(\mathcal{A},\mathcal{B}\right)$
$\displaystyle=\sup\left\\{\inf\left\\{\left|\alpha-\beta\right|:\beta\in\mathcal{B}\right\\}:\alpha\in\mathcal{A}\right\\}\;.$

Notice that $d\left(\mathcal{A},\mathcal{B}\right)\neq
d\left(\mathcal{B},\mathcal{A}\right)$. Let
$\mathbf{\Theta}=\\{\theta_{m}\\}_{m=1}^{M}$ and
$\hat{\mathbf{\Theta}}=\\{\hat{\theta}_{m}\\}_{m=1}^{\hat{M}}$ be the sets of
true and estimated DOAs, respectively. The estimation error is obtained by
evaluating the Hausdorff distance,
$d_{H}(\mathbf{\Theta},\hat{\mathbf{\Theta}})$. We define the root mean
squared distance (RMSD) for an arbitrary set of $N$ examples (e.g., test set),
$\left\\{\mathbf{X}^{\left(n\right)},\mathbf{y}^{\left(n\right)}\right\\}_{n=1}^{N}$,
with the corresponding true and estimated DOAs,
$\left\\{\mathbf{\Theta}^{\left(n\right)},\hat{\mathbf{\Theta}}^{\left(n\right)}\right\\}_{n=1}^{N}$
as:

RMSD
$\displaystyle\triangleq\sqrt{\frac{1}{N}\sum_{n=1}^{N}{d_{H}^{2}\left(\mathbf{\Theta}^{\left(n\right)},\hat{\mathbf{\Theta}}^{\left(n\right)}\right)}}\;.$
(18)

Angular resolution is one of the key criteria for DOA estimation performance.
The probability of resolution is commonly used as a performance evaluation
metric for angular resolution. In the considered problem, resolution between
two sources and between source and interference are used for performance
evaluation. For an arbitrary example with $M$ sources, the resolution event
$A_{res}$ is defined as:

$\displaystyle A_{res}\left(\mathbf{\Theta},\hat{\mathbf{\Theta}}\right)$
$\displaystyle\triangleq\begin{cases}1,&\bigcap_{m=1}^{M}{\xi_{m}\leq
2\degree}\text{ and }|\hat{\mathbf{\Theta}}|\geq M\\\
0,&\text{else}\end{cases}\;,$ (19) $\displaystyle\xi_{m}$
$\displaystyle\triangleq\min_{\hat{\theta}\in\hat{\mathbf{\Theta}}}|\theta_{m}-\hat{\theta}|,\;m=1,\dots,M\;.$

For example, a scene with $M$ sources is considered successfully resolved if
for each true DOA a) there exists a close-enough estimated DOA,
$\hat{\theta}\in\hat{\mathbf{\Theta}}$, that is at most $2\degree$ apart, and
b) there exists at least $M$ DOA estimations. According to (18), the
probability of resolution can be defined as:

$\displaystyle P_{res}$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}{A_{res}\left(\mathbf{\Theta}^{\left(n\right)},\hat{\mathbf{\Theta}}^{\left(n\right)}\right)}\;.$
(20)

####  IV-A3 Data Sets

This subsection describes the structure and formation of Training & Test sets.

(a) Training Set
The considered training set contains $N_{train}=10,000$ examples re-generated
at each epoch. For each example, i.e. an input-label pair
$\left(\mathbf{X},\mathbf{y}\right)$, the number of DOA sources, $M$, is
generated from uniform and $i.i.d.$ distribution, $\\{1,\dots,M_{max}\\}$. The
training set contains $10\%$ of interference-free examples and $90\%$ of
interference-containing. Out of the interference-containing examples, $90\%$
generated such that the source DOAs, $\\{\theta_{m}\\}_{m=1}^{M}$, and the
interference’s DOA, $\theta_{c}$, are distributed uniformly over the simulated
FOV. The remaining $10\%$ are generated such that $\theta_{c}$ is distributed
uniformly over the FOV, and the source DOAs, $\\{\theta_{m}\\}_{m=1}^{M}$, are
distributed uniformly over the interval
$\left[\theta_{c}-8\degree,\theta_{c}+8\degree\right]$. This data set
formation enables to “focus” the NN training on the challenging scenarios
where the source and interference DOAs are closely spaced. The generalization
capabilities of the proposed NN to variations in interference statistics are
achieved via the interference angular spread parameter, $\rho$, from the
uniform distribution, $\text{U}\left(\left[0.7,0.95\right]\right)$, and the
interference spikiness parameter, $\nu$, from the uniform distribution,
$\text{U}\left(\left[0.1,1.5\right]\right)$. The INR for each interference-
containing example and $\\{\text{SIR}_{m}\\}_{m=1}^{M}$ or
$\\{\text{SNR}_{m}\\}_{m=1}^{M}$ are drawn independently according to Table
III.

(b) Test Set
The test set consists of $N_{test}=20,000$ examples. The results are obtained
by averaging the evaluated performance over $50$ independent test set
realizations. Considering the low-snapshot support regime, the number of
snapshots is set to $K=16$, except for experiment (c) in IV-B2. Considering
heavy-tailed interference, the spikiness parameter is set to $\nu=0.2$. The
INR is set to $\text{INR}=5\;dB$, and the interference angular spread
parameter is set to $\rho=0.9$. The signal amplitude was set to be identical
for all sources, $\sigma_{1}=\dots=\sigma_{m}$, except for experiment (b) in
IV-B2.

Notation |  Description |  Value
---|---|---
$\rho$ |  Interference angular spread parameter |  $\sim\text{U}\left(\left[0.7,0.95\right]\right)$
$\nu$ |  Interference spikiness parameter |  $\sim\text{U}\left(\left[0.1,1.5\right]\right)$
INR |  INR |  $\sim\text{U}\left([0,10]\right)\;[dB]$
$\text{SIR}_{m}$ |  SIR of $m$-th source |  $\sim\text{U}\left([-10,10]\right)\;[dB]$
$\text{SNR}_{m}$ |  SNR of $m$-th source |  $\sim\text{U}\left([-10,10]\right)\;[dB]$
Table III:  Training set parameters. $\text{SNR}_{m}$ distribution applies to
interference-free examples.

(a)

(b)

Figure 3: Scenario with a single source at $\theta_{1}=0.55\degree$ and
interference located at $\theta_{c}=\theta_{1}+\Delta\theta_{c}$. (a)
probability of resolution and (b) RMSD.

###  IV-B Experiments

####  IV-B1 Single Source Within Interference

In this scenario, the ability to resolve a single source from interference is
evaluated. Let $M=1$ with $\theta_{1}=0.55\degree$, and
$\theta_{c}=\theta_{1}+\Delta\theta_{c}$ such that $\Delta\theta_{c}$ is the
angular separation between the single source and interference. The
$0.55\degree$ offset is considered to impose a realistic off-grid condition.
Fig. 3 shows the RMSD and probability of resolution for all evaluated
approaches.

Fig. 3(a) shows that the FC-based NN approach does not manage to resolve the
single source from the interference for all evaluated angular separations.
This result supports the under-fitting limitation of the FC-based NN approach
for the DOA estimation, which can be explained by the architecture that
processes the input data as-is, without any structured transformation or
model-based pre-processing. The MVDR and CNN performance in terms of
resolution are similar since both of them rely only on second-order
statistics, which is sufficient in scenarios with widely separated sources and
interference. It can be seen in Fig. 3(a) that for
$2\degree<\Delta\theta_{c}<5\degree$ the MVDR shows a drop, which is a result
of the source’s proximity to the interference. When the angular distance
between the interference and the source reduces below $5^{\circ}$, the source
and interference peaks coincide, and thus the probability of resolution is
reduced. When the angular distance further decreases, the probability that the
united peak falls within the resolution window increases, resulting in higher
$P_{res}$. Fig. 3(a) shows that the proposed DAFC-based NN approach
outperforms all other considered approaches in low angular separation
scenarios. This observation serves as an evidence of the capability of the
DAFC to utilize the information in the input data that is needed for
resolution between closely spaced sources and interference.

Fig. 3(b) shows the RMSD of all considered DOA estimation approaches. The
proposed DAFC-based NN approach outperforms the other tested approaches in low
SIR. At high SIR and small angular separation, $\Delta\theta_{c}=5\degree$,
the interference is negligible with respect to the strong source signal, and
therefore, the DAFC-based, CNN, and MVDR approaches obtain similar
performance. For large angular separation, $\Delta\theta_{c}=30\degree$, the
source and the interference are sufficiently separated, and therefore, DOA
estimation errors are mainly induced by the interference DOA, $\theta_{c}$.
The MVDR spectrum contains a peak at $\theta_{c}=30.55\degree$, and therefore,
MVDR’s $RMSD=30\degree$ is approximately constant. The NNs are trained to
output a $0$-probability for the interference. Therefore, the NN-based
approaches: FC, CNN, and DAFC achieve a smaller DOA estimation error. The
DAFC-based NN and CNN utilize structured transformations, which better fit the
input data, and therefore, they outperform the FC-based NN approach in terms
of RMSD.

(a)

(b)

Figure 4:  Probability of resolution for two sources located at
$\theta_{1,2}=\theta_{c}\pm\Delta\theta/2$, and interference located at
$\theta_{c}=0.55\degree$. (a) AWGN-only scenario and (b) interference-
containing scenario.

(a) FC

(b) MVDR

(c) CNN

(d) DAFC

Figure 5:  Spatial spectrum, two sources with $\text{SIR}=-5\;dB$ located at
$\theta_{1,2}=\theta_{c}\pm\Delta\theta/2$ with $\Delta\theta=12\degree$ and
$\theta_{c}=0.55\degree$. The dashed blue lines represent the mean spatial
spectrum, and the color fill represents the standard deviation around the mean
obtained from $2,000$ i.i.d. examples. The solid vertical orange lines
represent the true source DOAs, and the dashed vertical green line represents
the interference DOA.

####  IV-B2 Resolving Two Sources from Interference

This subsection evaluates the performance of the tested DOA estimation
approaches in scenarios with two sources within AWGN and interference.

(a) Resolution of Equal-Strength Sources
In the following experiment, the resolution between two equal-power sources,
$M=2$, with $\theta_{1}=-\frac{\Delta\theta}{2}+0.55\degree$, and
$\theta_{2}=\frac{\Delta\theta}{2}+0.55\degree$, is evaluated. The off-grid
additional $0.55\degree$ offset to the $\Delta\theta$ angular separation
between the sources represents the practical scenario. The interference at
$\theta_{c}=0.55\degree$ influences the two sources similarly. Fig. 4 shows
the probability of resolution of the tested approaches in scenarios with (a)
the AWGN only and (b) spatially-colored interference.

The FC-based NN approach does not resolve the two targets in both evaluated
scenarios. Subplot (a) in Fig. 4 shows that the proposed DAFC-based NN
approach outperforms the MVDR and the CNN at low-SNR and small angular
separation scenarios due to its generalization ability to spatially-white
interference. Subplot (b) in Fig. 4 shows that at low SIR of
$\text{SIR}=-5\;dB$, the performances of MVDR and CNN significantly degrade
compared to the proposed DAFC-based NN approach. Comparing subplots in Fig. 4,
notice that at $\text{SIR}=-5\;dB$, the MVDR fails to resolve the sources with
angular separation $\Delta\theta<20\degree$ due to the presence of the heavy-
tailed spatially-colored interference in the proximity of the sources.
However, the proposed DAFC-based NN approach mitigates this interference and
resolves the sources, and hence, outperforms other tested approaches at both
$\text{SIR}=0\;dB$ and $\text{SIR}=-5\;dB$.

Subplot (b) in Fig. 4 shows the non-monotonic trend of CNN and MVDR
performance at $4\degree<\Delta\theta<18\degree$ and $\text{SIR}=-5\;dB$,
which was observed also in Fig. 3(a). For $4\degree<\Delta\theta<8\degree$,
the sources are closer to the peak of the interference’s lobe and are,
therefore, less mitigated by it. As $\Delta\theta$ increases in the range
$8\degree<\Delta\theta<12\degree$, the sources reach DOAs which are in the
proximity of the interference lobe’s “nulls” which explains the reduction in
resolution, and as $\Delta\theta$ further increases, $16\degree<\Delta\theta$,
the sources are sufficiently separated from the interference such that the
resolution increases. As a result, MVDR and CNN-based approaches that use
second-order statistics only can not resolve the sources in the vicinity of a
stronger interference.

Fig. 5 shows the average spatial spectrum of all tested approaches for
$\Delta\theta=12\degree$ and $\text{SIR}=-5\;dB$. The average spatial spectrum
of the FC-based NN approach does not show two prominent peaks, which results
in its poor probability of resolution in Fig. 4. The MVDR “bell-shaped”
spatial spectrum does not contain the two prominent peaks at $\theta_{1,2}$
since the interference “masks” the two sources. The CNN and proposed DAFC-
based NN approaches show two peaks at the average spatial spectrum. The peaks
at the CNN’s average spatial spectrum are lower, resulting in a low-resolution
probability. The average spatial spectrum of the proposed DAFC-based NN
approach contains two high peaks, resulting in a superior probability of
resolution in Fig. 4.

(b) Resolution of Unequal-Power Sources
Fig. 6 shows the probability of resolution in a scenario with two sources,
$M=2$, at $\theta_{1}=-\Delta\theta/2+0.55\degree$, and
$\theta_{2}=+\Delta\theta/2+0.55\degree$ with interference located between the
sources at $\theta_{c}=0.55\degree$. The signal strength of the second source
is set to $\text{SIR}_{1}=\text{SIR}_{2}+10\;dB$. Comparing Fig. 6 to Fig.
4(b), the competing methods show similar trends, except the degradation of the
CNN’s probability of resolution for the $\text{SIR}=0\;dB$ case. On the other
hand, the proposed DAFC-based NN approach outperforms other tested approaches
in terms of the probability of resolution. Therefore, Fig. 6 demonstrates the
generalization ability of the proposed DAFC-based NN approach to a variance
between source strengths. The reason for the drop of $P_{res}$ of CNN and MVDR
in $10\degree<\Delta\theta<20\degree$ is similar to the same phenomenon that
appears in Figs. 3(a) and 4(b), which was explained in Subsection IV-B2(a).

Figure 6: Probability of resolution for two sources located at
$\theta_{1,2}=\theta_{c}\pm\Delta\theta/2$, and interference located at
$\theta_{c}=0.55\degree$. The SIR in the legend represents the SIR of the
first source, $\text{SIR}_{1}$. The SIR of the second source is set to
$\text{SIR}_{2}=\text{SIR}_{1}+10\;dB$.

(c) Effect of the Number of Snapshots on the Resolution
This experiment investigates the influence of the number of snapshots, $K$, on
the ability to resolve two proximate sources from heavy-tailed spatially-
colored interference. The equal-strength resolution scenario is repeated using
$K=4,\;8,\;16,\;32,\;64$ with different instances of NN training for each $K$
value. Fig. 7 shows the probability of resolution for two equal-strength
sources at $\theta_{1,2}=\theta_{c}\pm\Delta\theta/2$ for
$\Delta\theta=12\degree$ and $\theta_{c}=0.55\degree$.

The FC-based NN approach fails to resolve the two sources. For
$\text{SIR}=0\;dB$, the MVDR, CNN, and DAFC-based NN approaches achieve a
monotonic increasing probability of resolution with increasing $K$. The
proposed DAFC-based NN approach slightly outperforms other tested approaches.
At low SIR of $\text{SIR}=-5\;dB$, the proposed DAFC-based NN approach
significantly outperforms the other tested approaches. This can be explained
by the fact that increasing $K$ increases the probability for outliers to be
present in the input data matrix, $\mathbf{X}$. Therefore, the estimated
autocorrelation matrix, $\hat{\mathbf{R}}_{x}$, is more likely to be biased by
the interference-related outliers, which results in interference “masking” the
sources. The proposed DAFC-based NN approach is immune to these outliers and
successfully exploits the information from the additional snapshots to improve
the probability of resolution.

Figure 7:  Probability of resolution for two sources located at
$\theta_{1,2}=\theta_{c}\pm\Delta\theta/2$ with $\Delta\theta=12\degree$, and
interference located at $\theta_{c}=0.55\degree$, as a function of the number
of snapshots, $K$.

Figs. 4, 5, 6, and 7 show the ability of the proposed DAFC-based NN approach
to utilize the information structure of the input data by performing the
domain-fitted transformation in order to provide superior resolution ability
in the case of proximate heavy-tailed spatially-colored interference, low SIR
and small sample size.

####  IV-B3 Multiple Source Localization

The performances of the tested DOA estimation approaches are evaluated and
compared in a multi-source scenario. Four sources, ($M=4$) were simulated with
angular separation, $\Delta\theta$:
$\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\\}=\theta_{c}+\\{-2\Delta\theta,-\Delta\theta,\Delta\theta,2\Delta\theta\\}$,
where $\theta_{c}\sim\text{U}\left(-15^{\circ},15^{\circ}\right)$ represents a
realistic condition for source and interference DOAs. The RMSD of evaluated
methods is depicted in Fig. 8. The proposed DAFC-based NN approach outperforms
the other tested approaches at low SIR ($\text{SIR}<0\;dB$) for large and
small angular separations. For high SIR and low angular separation,
$\Delta\theta=5\degree$, the MVDR achieves the lowest RMSD. The reason is
that, for this case, the interference is negligible with respect to the lobe
of the strong source in the MVDR’s spectrum. However, at high angular
separation, $\Delta\theta=20\degree$, the proposed DAFC-based NN approach
significantly outperforms the other tested approaches. This is explained by
Fig. 9, which shows the spectrum of the tested DOA estimation approaches.
Notice that the proposed DAFC-based NN mitigates interference, while the
spectra of other tested approaches contain high peaks at the interference DOA,
$\theta_{c}$. These peaks increase the Hausdorff distance in (17), increasing
the RMSD of other tested approaches in Fig. 8.

Figure 8: RMSD in scenarios with $M=4$ sources located at
$\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\\}=\theta_{c}+\\{-2\Delta\theta,-\Delta\theta,\Delta\theta,2\Delta\theta\\}$,
where $\theta_{c}\sim\text{U}\left(-15^{\circ},15^{\circ}\right)$.

(a) FC

(b) MVDR

(c) CNN

(d) DAFC

Figure 9:  Spatial spectrum, four sources with $\text{SIR}=0\;dB$ located at
$\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\\}=\theta_{c}+\\{-2\Delta\theta,-\Delta\theta,\Delta\theta,2\Delta\theta\\}$,
where $\theta_{c}=0.51\degree$ and $\Delta\theta=20\degree$. The dashed blue
lines represent the mean spatial spectrum, and the color fill represents the
standard deviation around the mean obtained from $2,000$ i.i.d. examples. The
solid vertical orange lines represent the true source DOAs, and the dashed
vertical green line represents the interference DOA.

(a) MDL

(b) AIC

(c) DAFC

Figure 10: Confusion matrix for source enumeration, $\text{SIR}=0\;dB$,
sources located at
$\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\\}=\theta_{c}+\\{-2\Delta\theta,-\Delta\theta,\Delta\theta,2\Delta\theta\\}$,
with $\theta_{c}\sim\text{U}\left(-15^{\circ},15^{\circ}\right)$, and
$\Delta\theta=15\degree$. (a) MDL, (b) AIC, (c) proposed DAFC-based NN.

####  IV-B4 Multiple Source Enumeration

The source enumeration performance is evaluated in this experiment. The DOAs
of the sources are selected from the set of following values:
$\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\\}=\theta_{c}+\\{\Delta\theta,-\Delta\theta,-2\Delta\theta,2\Delta\theta\\}$
such that for $M$ sources, the DOAs are selected to be the first $M$ DOAs. The
interference is located at
$\theta_{c}\sim\text{U}\left(-15^{\circ},15^{\circ}\right)$. The proposed
DAFC-based NN approach is compared to the MDL and AIC [19]. Fig. 10 shows the
source enumeration confusion matrices for the MDL, AIC, and the proposed DAFC-
based NN with $\text{SIR}=0\;dB$.

Figs. 10(a) and 10(b) show that in both the MDL and the AIC, the predicted
number of sources has a constant bias for each true $M$ due to the spatially-
colored interference. Fig. 10(c) shows the source enumeration performance of
the proposed DAFC-based NN approach in the presence of spatially colored
interference. The DAFC-based NN identifies the interference and does not count
it as one of the sources by outputting a low probability for angular grid
points near $\theta_{c}$, resulting in a better source enumeration
performance.

####  IV-B5 Loss Weights

This experiment evaluates the effect of the loss weight update factors,
$\\{\beta^{(l)}\\}_{l=1}^{N_{w}}$, introduced in (12), on the confidence level
in the spatial spectrum. Let $\widetilde{\mathbf{B}}$ denote the set of
$\\{\beta^{(l)}\\}_{l=1}^{N_{w}}$ values used in the proposed approach. The
loss weights, $\\{w_{i}^{(t)}\\}_{i=1}^{d}$, are defined by the factors
$e_{0}^{(t)},e_{1}^{(t)}$ according to (11), and are introduced to provide a
trade-off between the penalty obtained on source/interference and the penalty
obtained for the rest of the output spatial spectrum.

For comparison, we set $\mathbf{B}_{0}=\\{10^{-6},3.98\cdot 10^{-6},1.58\cdot
10^{-5},6.31\cdot 10^{-5},2.51\cdot 10^{-4},10^{-3}\\}$, and
$\mathbf{B}_{1}=\\{10^{-3},3.98\cdot 10{-3},0.0158,0.063,0.25,0.1\\}$ as two
sets of loss weight update factors. For $\mathbf{B}_{0}$, the loss weight
update factors are closer to $0$. Hence the loss weights emphasize the
source/interference, since $e_{1}^{(t)}\ll e_{0}^{(t)}$ which, according to to
(11), translates to larger $w_{i}^{(t)}$ for source/interference grid points.
For $\mathbf{B}_{1}$ the values are closer to $1$. Hence the loss weights are
more equally distributed among grid points since $e_{1}^{(t)}\approx
e_{0}^{(t)}$. The experiment in IV-B1 is repeated here for the DAFC-based NN
approach with the two additional $\mathbf{B}_{0},\mathbf{B}_{1}$ values
mentioned above.

Let $\hat{p}_{1}$ represent the probability assigned for the source-containing
grid point in the estimated label $\hat{\mathbf{y}}$. Let $\hat{p}_{0}$
represent the maximum over probabilities assigned for non-source grid points
in $\hat{\mathbf{y}}$, excluding a $5$-grid point guard interval around the
source. Fig. 11 shows $\hat{p}_{1}$ and $\hat{p}_{0}$ for various angular
separations between the source and interference for $\text{SIR}=-5\;dB$. For
$\mathbf{B}_{0}$, the source’s contribution to the loss value is substantially
higher, which results in a higher probability for the source-containing grid
point. However, this results in a higher probability obtained for non-source
grid points since their contribution to the loss value is negligible compared
to the source-containing grid point, increasing “false-alarm” peaks in the
spatial spectrum, subsequently increasing the estimation error.
Correspondingly, for $\mathbf{B}_{1}$ the source’s contribution to the loss
value is less significant, which results in a low probability assigned for the
source-containing grid points, as well as a low probability for non-source
grid points.

Figure 11: Loss weight update factor impact on probability levels obtained in
the DAFC-based NN’s spatial spectrum, single target at
$\theta_{1}=0.55\degree$ with interference at
$\theta_{c}=\theta_{1}+\Delta\theta_{c}$, $\text{SIR}=-5\;dB$. $\hat{p}_{1}$
represents the probability obtained for source-containing grid points.
$\hat{p}_{0}$ represents the probability obtained for non-source grid points.

##  V Conclusion

This work addresses the problem of DOA estimation and source enumeration of an
unknown number of sources within heavy-tailed, non-Gaussian, and spatially
colored interference. A novel DAFC-based NN approach is proposed for this
problem. The DAFC mechanism applies a structured transformation capable of
exploiting the interference non-Gaussianity for its mitigation while retaining
a low complexity of learnable parameters. The proposed DAFC-based NN approach
is optimized to provide an interference-mitigated spatial spectrum using a
loss weight scheduling routine, performing DOA estimation and source
enumeration using a unified NN.

The performance of the proposed approach is compared to MVDR, CNN-based, and
FC-based approaches. Simulations showed the superiority of the proposed DAFC-
based NN approach in terms of probability of resolution and estimation
accuracy, evaluated by RMSD, especially in weak signal power, a small number
of snapshots, and near-interference scenarios. The source enumeration
performance of the proposed DAFC-based NN approach was compared to the MDL and
AIC. It was shown that in the considered scenarios, the proposed approach
outperforms the MDL and the AIC in the source enumeration accuracy.

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|  Stefan Feintuch received B.sc. and M.Sc. in electrical and computer
engineering from the Ben-Gurion University of the Negev, Beer Sheva, Israel,
in 2022 and 2023, respectively. His research interests include machine
learning, deep learning, radars, and signal processing.
---|---
|  Joseph Tabrikian (Fellow, IEEE) received the B.Sc., M.Sc., and Ph.D.
degrees in Electrical Engineering from the Tel-Aviv University, Tel-Aviv,
Israel, in 1986, 1992, and 1997, respectively. During 1996–1998 he was with
the Department of Electrical and Computer Engineering (ECE), Duke University,
Durham, NC as an Assistant Research Professor. In 1998, he joined the
Department of ECE, Ben-Gurion University of the Negev, Beer-Sheva, Israel, and
served as the department head during 2017-2019. In May 2019 he established the
school of ECE and served as its head till August 2021. He served as an
Associate Editor (AE) for the IEEE Transactions on Signal Processing during
2001–2004 and 2011-2015, and is currently a Senior Area Editor (SAE) for these
transactions. He served as AE and SAE of the IEEE Signal Processing Letters
during 2012-2015 and 2015-2018, respectively. He was a member of the IEEE
Sensor Array and Multichannel (SAM) technical committee during 2010-2015 and
was the technical program co-chair of the IEEE SAM 2010 workshop. During
2015-2021 he served as a member of the Signal Processing for Multisensor
Systems (SPMuS) of EURASIP and during 2017-2022 he was a member IEEE SPTM
technical committee. He is co-author of 7 award-winning papers in IEEE
conferences and workshops. His research interests include estimation and
detection theory, learning algorithms, and radar signal processing.
---|---
|  Igal Bilik (S’03-M’06-SM’21) received B.Sc., M.Sc., and Ph.D. degrees in
electrical and computer engineering from the Ben-Gurion University of the
Negev, Beer Sheva, Israel, in 1997, 2003, and 2006, respectively. During
2006–2008, he was a postdoctoral research associate in the Department of
Electrical and Computer Engineering at Duke University, Durham, NC. During
2008-2011, he has been an Assistant Professor in the Department of Electrical
and Computer Engineering at the University of Massachusetts, Dartmouth. During
2011-2019, he was a Staff Researcher at GM Advanced Technical Center, Israel,
leading automotive radar technology development. Between 2019-2020 he was
leading Smart Sensing and Vision Group at GM R&D, where he was responsible on
development state-of-art automotive radar, lidar and computer vision
technologies. Since Oct. 2020, Dr. Bilik is an Assistant Professor in the
School of Electrical and Computer Engineering at the Ben-Gurion University of
the Negev. Since 2020, he is a member of IEEE AESS Radar Systems Panel and a
vice-Chair of Civilian Radar Committee. Dr. Bilik is an Acting Officer of IEEE
Vehicular Technology Chapter, Israel. Dr. Bilik has more than 170 patent
inventions, authored more than 60 peer-reviewed academic publications,
received the Best Student Paper Awards at IEEE RADAR 2005 and IEEE RADAR 2006
Conferences, Student Paper Award in the 2006 IEEE 24th Convention of
Electrical and Electronics Engineers in Israel, and the GM Product Excellence
Recognition in 2017.
---|---
|  Haim H. Permuter (M’08-SM’13) received his B.Sc. (summa cum laude) and
M.Sc.(summa cum laude) degrees in Electrical and Computer Engineering from the
Ben-Gurion University, Israel, in 1997 and 2003, respectively, and the Ph.D.
degree in Electrical Engineering from Stanford University, California in 2008.
Between 1997 and 2004, he was an officer at a research and development unit of
the Israeli Defense Forces. Since 2009 he is with the department of Electrical
and Computer Engineering at Ben-Gurion University where he is currently a
professor, Luck-Hille Chair in Electrical Engineering. Haim also serves as
head of the communication,cyber/ and information track in his department.
Prof. Permuter is a recipient of several awards, among them the Fullbright
Fellowship, the Stanford Graduate Fellowship (SGF), Allon Fellowship, and and
the U.S.-Israel Binational Science Foundation Bergmann Memorial Award. Haim
served on the editorial boards of the IEEE Transactions on Information Theory
in 2013-2016 and has been reappointed again in 2023.
---|---