# Neural Network-Based Multi-Target Detection within Correlated Heavy-Tailed
Clutter

S. Feintuch Stefan Feintuch, Haim H. Permuter, Igal Bilik, and Joseph
Tabrikian 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>joseph@bgu.ac.il).
This work was partially supported by the Israel Science Foundation under
Grants 2666/19 and 1895/21.  H. Permuter  Senior Member, IEEE I. Bilik  Senior
Member, IEEE and J. Tabrikian  Fellow, IEEE

###### Abstract

This work addresses the problem of range-Doppler multiple target detection in
a radar system in the presence of slow-time correlated and heavy-tailed
distributed clutter. Conventional target detection algorithms assume Gaussian-
distributed clutter, but their performance is significantly degraded in the
presence of correlated heavy-tailed distributed clutter. Derivation of optimal
detection algorithms with heavy-tailed distributed clutter is analytically
intractable. Furthermore, the clutter distribution is frequently unknown. This
work proposes a deep learning-based approach for multiple target detection in
the range-Doppler domain. The proposed approach is based on a unified NN model
to process the time-domain radar signal for a variety of signal-to-clutter-
plus-noise ratios (SCNRs) and clutter distributions, simplifying the detector
architecture and the neural network training procedure. The performance of the
proposed approach is evaluated in various experiments using recorded radar
echoes, and via simulations, it is shown that the proposed method outperforms
the conventional cell-averaging constant false-alarm rate (CA-CFAR), the
ordered-statistic CFAR (OS-CFAR), and the adaptive normalized matched-filter
(ANMF) detectors in terms of probability of detection in the majority of
tested SCNRs and clutter scenarios.

###### Index Terms:

Radar Target Detection, Correlated Heavy-Tailed Clutter, Neural Networks, Deep
Learning, CA-CFAR, OS-CFAR, ANMF, Range-Doppler, LFM, Multiple Target
Detection, Machine Learning.

##  I INTRODUCTION

Target detection in range-Doppler map is one of the major radar tasks [1, 2].
Conventionally, the decision on target presence is made by comparing the
energy within the cell-under-test with a threshold, which is calculated
according to the energy at neighboring cells [3]. The presence of spiky
clutter in the cells used for the detection threshold calculation increases
the threshold level, and thus, compromises the target detection performance
[3].

Considering the detector input as a one-dimensional complex signal that
contains slow-time samples of received radar echoes in each range bin, the
task of radar target detection within correlated heavy-tailed clutter is
conventionally formulated as a binary hypotheses decision task. Under this
formulation, the hypotheses $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ represent
cases where there is no target and the target is present within the cell-
under-test (CUT), respectively [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22]. In [4, 5, 6, 7, 8, 9, 10] the problem of radar
target detection was formulated as a binary hypothesis testing, where the
optimum detectors were derived under certain conditions. The design for a
regularized covariance matrix estimation in the adaptive normalized matched-
filter (ANMF) was introduced in [11] to maximize the asymptotic probability of
detection, while retaining a constant false-alarm rate (CFAR). The properties
of CFAR detectors in the presence of correlated heavy-tailed clutter were
studied in [18]. The problem of range-migrating target detection within heavy-
tailed clutter was addressed in [16], in which a fast-converging amplitude
estimation algorithm for target detection was proposed. An orthogonal-
projection-based approach to suppress the sea clutter at each range cell in
combination with cell-averaging CFAR (CA-CFAR), was proposed in [22]. The
authors in [21] addressed the target detection within heavy-tailed clutter
using massive multiple-input multiple-output (mMIMO) radar. Therein, a
detector was proposed for the asymptotic regime with increasing number of
antennas, and its robustness to the unknown clutter distribution was
demonstrated.

However, these model-based approaches were designed considering a specific
measurement model, and their performance may degrade in the case of model
mismatch. Alternatively, data-driven machine learning (ML) approaches have
been proposed in [12, 13, 14, 15, 19, 23, 20]. In these approaches, target
detection is performed using features extracted from the data. Thus, they
enable detectors’ robustness to environmental and clutter statstics’
variations. K-nearest neighbors (KNN) based approaches using various feature
space transforms of the raw one-dimensional complex signal were proposed in
[12, 13, 14, 15, 19] to address the binary hypothesis decision task. In
particular, the authors in [12, 14] proposed to obtain a KNN-based decision
rule from simulated data, and evaluated the proposed methods using the IPIX
database [24] of recorded radar echoes that contain correlated heavy-tailed
sea clutter. Authors in [23] used support vector machine to switch between
conventional CFAR methods and perform target detection in an environment
containing clutter edges and/or multiple interfering targets under white
Gaussian noise. The work in [20] extended the work in [21] to angle dimension
and proposed a reinforcement learning (RL) based approach to design the
beamforming matrix in a cognitive radar (CR) setup.

The binary hypothesis-based approaches in [4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 20, 22, 19, 21] assume under the $\mathcal{H}_{1}$
hypothesis a) the presence of a single-target within each CUT and b) the
availability of target-free secondary data, which is used for clutter
covariance matrix estimation. However, practical scenarios may include
multiple targets with similar azimuth, range, and Doppler. Therefore, the
performances of these methods degrade in such scenarios. In addition, the
methods in [4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23] use
the data after range matched-filter processing, which linearly projects each
fast-time received pulse to range bins [3]. This linear transformation fails
to suppress the clutter echo signals, since these are not orthogonal to the
projection signals that correspond to each range bin.

Recently, deep neural networks (DNNs) with various network architectures have
been introduced for radar target detection, where the network input consists
of the samples of the received radar echo [17, 25, 26, 27]. Considering a one-
dimensional problem with a-priori known signal, a multi-layer perceptron (MLP)
based detector for binary hypothesis detection within non-Gaussian noise was
proposed in [17]. A fully-connected architecture for multiple target detection
in the presence of homogeneous Rayleigh-distributed clutter was utilized in
[25]. A single-target detection within additive white Gaussian noise (AWGN)
using convolutional neural network (CNN) based architecture for range-Doppler
target detection and azimuth-elevation estimation was proposed in [26].
However, the works in [25, 26, 27] assume white Gaussian-distributed clutter,
whereas a more realistic clutter model would be correlated and non-Gaussian.
Although the work in [17] addresses the non-Gaussian clutter, it also assumes
the binary hypothesis decision task, which has limitations as previously
mentioned.

Neural network (NN) based processing using range-Doppler map input was also
studied in the literature, the majority of these works invoke computer vision
methods for radar target detection within AWGN [28, 29, 30, 31]. A fully
connected NN architecture for multiple target detection within heavy-tailed
clutter was proposed in [32]. A residual block [33] was proposed in [29] for
background noise estimation in the range-Doppler map for the conventional CFAR
detector. In [30] a model-based data augmentation technique was proposed for
linear frequency modulated (LFM) radar detector in the 3D range-Doppler-angle
domain. The proposed technique was used to generate a synthetic dataset for
U-net [34] training, considering a single target in the azimuth-elevation
domain at each range-Doppler region-of-interest (ROI). The work in [31]
extended [30], by utilizing the absolute value of the range-Doppler map for
additional data augmentation.

Contrary to previous works described above, which address the radar target
detection within heavy-tailed clutter as a one-dimensional binary hypothesis
decision task for each range bin, this work addresses the problem of radar
target detection within heavy-tailed clutter as a two-dimensional (range-
Doppler) detection problem with multiple targets in unknown ranges and radial
velocities (Doppler). Furthermore, in practical radar scenarios characterized
by correlated clutter, the conventional range-Doppler transform designed for
AWGN model fails to suppress the clutter, since the clutter signal is
correlated in slow-time and can be spread over multiple range bins. Therefore,
the range-Doppler map-based DNN approaches mentioned above, do not fully
exploit the power of DNNs to learn highly abstract nonlinear transformations
for suppressing the clutter. To that aim, this work proposes to leverage DNN’s
ability to learn highly complex nonlinear functions in order to transform the
complex time-domain radar echo samples into the range-Doppler domain while
suppressing the correlated clutter.

The contributions of this work are:

  1. 1.

A novel neural processing block named dimensional-alternating fully connected
(DAFC) block, is proposed to process raw time-domain radar echoes for the task
of multiple target detection. A DNN architecture that utilizes this block is
proposed to map radar signals to either range or Doppler domains while
suppressing correlated heavy-tailed clutter.

  2. 2.

The proposed DNN architecture is utilized as part of a novel range-Doppler
multiple target detector, that is evaluated in the presence of correlated
heavy-tailed clutter.

  3. 3.

The proposed method significantly outperforms conventional methods and proves
to be more robust in various aspects: multiple targets within AWGN and
correlated heavy-tailed clutter, varying clutter conditions/“spikiness”
measure, and detection threshold sensitivity to clutter “spikiness”.

  4. 4.

The proposed method proves to generalize well to unseen data, based on
experiments involving recorded real data.

The following notations will be used throughout the paper. Roman boldface
lower-case and upper-case letters represent vectors and matrices,
respectively. Non-bold italic letters stands for scalars. $\mathbf{I}_{N}$ and
$\mathbf{0}_{N}$ are the identity matrix and zero matrix of size $N\times N$,
respectively. $\mathbb{E}$, superscript $T$, and superscript $H$ are the
expectation, transpose, and Hermitian transpose operators, respectively. Vec,
$|\cdot|$, and $\mathbb{I}$ stand for the vectorization, set size, and
indicator operators, respectively. $\left(\cdot\right)$ and
$\left(\cdot,\cdot\right)$ denote single argument and double arguments
functions. $[\mathbf{a}]_{n}$ 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. $[\mathbf{A}]_{[\cdot,:]}$ and $[\mathbf{A}]_{[:,\cdot]}$
represent an arbitrary row and column in the matrix $\mathbf{A}$,
respectively.

The remainder of this paper is organized as follows: The addressed problem is
stated in Section II. Section III presents the proposed DAFC-based radar
target detection approach. The performance of the proposed approach is
evaluated via simulated data and recorded real data in Section IV, and our
conclusions are summarized in Section V.

##  II PROBLEM STATEMENT

The measurement model is described in Subsection II-A, and the multiple target
detection problem in the range-Doppler domain is formulated in Subsection
II-B.

Figure 1: Example of range-Doppler map containing simulated targets, clutter,
and noise. The targets are circled in red, with the marked SCNRs. The non-
homogeneous clutter is present at the vicinity of the $4m/s$ Doppler velocity
in the majority of the range bins. Figure 2: Dataset generation scheme.

###  II-A Measurement Model

Consider the baseband fast-time $\times$ slow-time model of a single received
radar echo:

$\displaystyle\mathbf{X}=\mathbf{S}\left(\mathcal{T}\right)+\mathbf{C}+\mathbf{W}\;$
(1)

where
$\mathbf{X},\mathbf{S}\left(\mathcal{T}\right),\mathbf{C},\mathbf{W}\in\mathbb{C}^{N\times
K}$,
$\mathcal{T}=\left\\{(r_{j},v_{j}):(r_{j},v_{j})\in\left[r_{min},r_{max}\right]\times\left[v_{min},v_{max}\right]\right\\}$
denotes the set of targets present in the frame, and
$\left[r_{min},r_{max}\right]$ and $\left[v_{min},v_{max}\right]$ are
intervals of targets’ ranges and radial velocities, respectively. The matrices
$\mathbf{S}\left(\cdot\right)$, $\mathbf{C}$, and $\mathbf{W}$ represent the
target echo signal, the clutter, and the additive noise. The targets’ matrix
$\mathbf{S}\left(\cdot\right)$ is defined as:

$\displaystyle\mathbf{S}\left(\mathcal{T}\right)=\begin{cases}\sum_{(r,v)\in\mathcal{T}}{\widetilde{\mathbf{S}}(r,v)}\;&,\mathcal{T}\neq\emptyset\\\
\mathbf{0}_{N\times K}&,\mathcal{T}=\emptyset\end{cases}$ (2)

where $\widetilde{\mathbf{S}}(r,v)$ is the radar echo matrix received from a
single target at range $r$ and radial velocity $v$, and is defined as [3]:

$\displaystyle\widetilde{\mathbf{S}}(r,v)=$ $\displaystyle
A_{rv}e^{j\phi_{rv}}\mathbf{r}(r)\mathbf{v}^{T}(v)$ (3)

where $\mathbf{0}_{N\times K}$ dentoes the $N\times K$ zero matrix,
$\phi_{rv}\sim\text{U}\left([0,2\pi]\right)$ is unknown phase,
$A_{rv}\in\mathbb{R}^{+}$ represents the received signal amplitude and depends
on the target radar cross section (RCS) and the propagation path loss.

Notice that the model in (1) represents the radar echo of both the pulse-
Doppler and the LFM-CW radars with the appropriate range and radial velocity
steering vectors, $\mathbf{r}(\cdot)$ and $\mathbf{v}(\cdot)$. Thus, for LFM-
CW radar:

$\displaystyle\mathbf{r}(r)=\left[1\quad
e^{-j2\pi\frac{2Br}{cN}}\quad\dots\quad
e^{-j2\pi\frac{2Br}{cN}(N-1)}\right]^{T}\;,$ (4)
$\displaystyle\mathbf{v}(v)=\left[1\quad
e^{-j2\pi\frac{2f_{c}v}{c}T_{0}}\quad\dots\quad
e^{-j2\pi\frac{2f_{c}v}{c}T_{0}(K-1)}\right]^{T}\;,$

where $N$ is the number of samples per LFM chirp, $K$ is the number of chirps
per dwell, $B$ is the transmit signal bandwidth, $f_{c}$ is the carrier
frequency, and $T_{0}$ stands for the pulse repetition interval (PRI).

Conventionally, slow-time radar clutter is statistically modeled as a random
vector at each range bin [12, 35, 24]. Let
$\\{\mathbf{c}_{r}\in\mathbb{C}^{K}\\}_{r\in\mathcal{R}}$ denote the group of
one-dimensional slow-time clutter vectors. Then, the clutter matrix
$\mathbf{C}$ in (1) can be obtained by converting $\\{\mathbf{c}_{r}\\}_{r}$
to the fast-time $\times$ slow-time representation by:

$\displaystyle\mathbf{C}=\sum_{r\in\mathcal{R}}{\mathbf{r}\left(r\right)\mathbf{c}_{r}^{T}}\;,$
(5)

where $\mathcal{R}$ is the set of range bins, that partition the continuous
range space to grid points spaced by the range resolution $\Delta r=c/(2B)$.
The clutter signal matrix $\mathbf{C}$ in (1) is a sum of $|\mathcal{R}|$
“clutter echoes”, one per range bin. Note that by projecting the fast-time
vectors (i.e. columns) in (5) to the range steering vectors representing the
range bins in $\mathcal{R}$, we will obtain the set of original clutter
vectors $\\{\mathbf{c}_{r}\\}_{r}$, one per range-bin.

The fast-time$\times$slow-time noise matrix $\mathbf{W}$ in (1) is defined by
$\widetilde{\mathbf{w}}=\text{Vec}\left(\mathbf{W}\right)$, where
$\widetilde{\mathbf{w}}$ is modeled as an AWGN vector:

$\displaystyle\widetilde{\mathbf{w}}\sim\mathcal{CN}\left(\mathbf{0}_{NK},\sigma^{2}\mathbf{I}_{NK}\right)\;.$
(6)

Let
$\tilde{\mathbf{s}}\left(r,v\right)\triangleq\text{Vec}(\widetilde{\mathbf{S}}\left(r,v\right))$
and $\tilde{\mathbf{c}}\triangleq\text{Vec}(\mathbf{C})$ be the vectorizations
of a target and clutter matrices in (3) and (5), respectively. The clutter-to-
noise ratio (CNR) for a given frame and signal-to-clutter-plus-noise ratio
(SCNR) for a given target within the frame are defined as:

CNR
$\displaystyle=\frac{\mathbb{E}\left[\|\tilde{\mathbf{c}}\|^{2}\right]}{\mathbb{E}\left[\|\tilde{\mathbf{w}}\|^{2}\right]}\;,$
(7) SCNR
$\displaystyle=\frac{\mathbb{E}\left[\|\tilde{\mathbf{s}}\left(r,v\right)\|^{2}\right]}{\mathbb{E}\left[\|\tilde{\mathbf{c}}+\tilde{\mathbf{w}}\|^{2}\right]}\;.$

###  II-B Range-Doppler Detection Formulation

The sets of range and Doppler bins are denoted by $\mathcal{R}$ and
$\mathcal{V}$, respectively. The range bins defined earlier and the Doppler
bins $\mathcal{V}$ partition the continuous Doppler space to grid points
spaced by the Doppler resolution $\Delta v=c/(2f_{c}KT_{0})$. The set of
range-Doppler bins is obtained by the Cartesian product
$\mathcal{R}\times\mathcal{V}$. A range-Doppler detector can be formulated as
a mapping between the received signal in (1) to a per-bin decision in the
range-Doppler domain:

$\displaystyle\hat{\mathbf{Y}}=\mathbf{H}\left(\mathbf{X}\right)\in\left\\{0,1\right\\}^{d_{R}\times
d_{V}},$ (8)

where $d_{R}=|\mathcal{R}|$, $d_{V}=|\mathcal{V}|$ and $\mathbf{H}(\cdot)$ is
a mapping from a fast-time $\times$ slow-time input frame $\mathbf{X}$ to a
range-Doppler decision matrix $\hat{\mathbf{Y}}$.

Let $[m,l]$ denote a coordinate in the discrete range-Doppler space
$\mathcal{R}\times\mathcal{V}$. The decision on target presence in the range-
Doppler bin corresponding to the coordinate $[m,l]$ is defined using entries
in the range-Doppler decision matrix $\hat{\mathbf{Y}}$:

$\displaystyle\begin{cases}\text{Target},&[\hat{\mathbf{Y}}]_{m,l}=1\\\
\text{No target},&[\hat{\mathbf{Y}}]_{m,l}=0\end{cases}\;.$

An optimum detector, maximizes the probability of detection $P_{D}$ for a
fixed probability of false-alarm $P_{FA}$.

The conventional range-Doppler transform, which maps the received signal in
(1) to the range-Doppler domain, can be obtained by taking the absolute
squared value of the 2D-FFT of $\mathbf{X}$. Fig. 1 shows an example of the
conventional range-Doppler transform of simulated radar signal consisting of
multiple targets, correlated heavy-tailed clutter, and AWGN. Note that there
is a non-homogeneous clutter that is observed around the Doppler velocity of
$4$ $m/s$, and is present in the majority of the range bins. This example
visually exemplifies the limitations of conventional range-Doppler processing
in suppression of correlated clutter. Therefore, suppression of correlated
heavy-tailed clutter involves nonlinear range-Doppler transforms, as proposed
in this work.

Conventional range-Doppler detectors, such as the CA-CFAR and OS-CFAR, operate
on the output of the conventional range-Doppler transform (absolute square of
2D-FFT). The decision for target presence in each range-Doppler bin is based
on calculating an adaptive threshold by utilizing the energy information in
the surrounding bins [3]. Correlated heavy-tailed clutter can induce spikes
and/or high levels in these surrounding cells, and thus compromise the range-
Doppler detector’s performance.

##  III PROPOSED APPROACH

The proposed approach for target detection within heavy-tailed clutter is
detailed in this section. This paper proposes a data-driven approach, and for
this purpose the dataset generation method is described in Subsection III-A.
Next, the pre-processing and innovative DAFC block are introduced in
subsections III-B and III-C. Finally, the NN architecture and the proposed
range-Doppler detector are detailed in subsections III-D and III-E.

###  III-A Dataset Generation

The dataset generation is schematically shown in Fig. 2. First, the simulated
radar echoes of clutter and noise are generated using the clutter signals,
$\\{\mathbf{c}_{r}\\}_{r\in\mathcal{R}}$, to build the matrix $\mathbf{C}$ in
(5) along with AWGN, $\mathbf{W}$. Next, the set of targets, $\mathcal{T}$ is
used to build the signal matrix, $\mathbf{S}\left(\mathcal{T}\right)$ in (2).
The received radar input frame, $\mathbf{X}$, is generated according to (1).
The range-Doppler label matrix $\mathbf{Y}_{true}\in\\{0,1\\}^{d_{R}\times
d_{V}}$ is a binary matrix with the following entries:

$\displaystyle[\mathbf{Y}_{true}]_{m,l}=\begin{cases}1,&[m,l]\in\mathcal{A}\left(\mathcal{T}\right)\\\
0,&\text{else}\end{cases},$ (9)

where $\mathcal{A}\left(\mathcal{T}\right)$ is the set of matrix indices that
represent the closest range-Doppler bins to the true targets in $\mathcal{T}$.

This work proposes to transform an input fast-time$\times$slow-time frame to
range or Doppler domains and detect targets using a NN-based approach. It is
proposed to use two separate and identical NN models for range and Doppler
domains. These models are trained according to the supervised learning
framework. The range label used to train the range model (i.e. range NN) is
$\mathbf{y}_{r,true}=\mathbb{I}\left(\sum_{l}{[\mathbf{Y}_{true}]_{[:,l]}}\right)$
and the Doppler label used to train the Doppler model (i.e. Doppler NN) is
$\mathbf{y}_{v,true}=\mathbb{I}\left(\sum_{m}{[\mathbf{Y}_{true}]_{[m,:]}}\right)$.

The clutter signal matrix $\mathbf{C}$ in (1) consists of $|\mathcal{R}|$
clutter slow-time vectors $\\{\mathbf{c}_{r}\\}_{r\in\mathcal{R}}$. Both
simulated and recorded radar echoes are used for these clutter vectors.

1) Simulated clutter

This work adopts the commonly used K-distributed spherically-invariant random
vector (SIRV) model for correlated heavy-tailed clutter slow-time vector at
each range bin [12, 35]:

$\displaystyle\mathbf{c}\sim\mathcal{K}\left(\nu,\mathbf{M}\right),\quad\mathbf{c}=\sqrt{s}\mathbf{z}\in\mathbb{C}^{K}$
(10) $\displaystyle
s\sim\Gamma(\nu,\nu),\;\mathbf{z}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{M}\right)$
$\displaystyle[\mathbf{M}]_{[p,q]}=\exp{-2\pi^{2}\sigma_{f}^{2}(p-q)^{2}-j(p-q)f_{d}T_{0}}\;,$

where $\Gamma\left(\nu,\nu\right)$ denotes the Gamma distribution with shape
$\nu$ and rate $\nu$ [12, 35], and $f_{d}$ is the clutter’s Doppler frequency
shift [12]. The shape parameter, $\nu$, controls the “spikiness” measure of
the clutter amplitude distribution [12, 35]. A set of independent,
identically-distributed (i.i.d.) simulated clutter vectors for each range bin
in $\mathcal{R}$ is generated as:
$\\{\mathbf{c}_{r}\\}_{r\in\mathcal{R}}\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\mathcal{K}\left(\nu,\mathbf{M}\right)$.

2) Recorded real data clutter:

We use the Grimsby IPIX database of recorded echoes from radar clutter [24].
This database contains high-resolution radar echoes collected using the
McMaster IPIX radar in Grimsby on the shore of Lake Ontario. Each file in the
database contains 60,000 pulses of radar echoes, recorded during 60 seconds.
Each file contains radar echoes from Lake Ontario collected at various dates,
hours, azimuths, and range sections. A set
$\\{\mathbf{c}_{r}\\}_{r\in\mathcal{R}}$ of recorded clutter slow-time vectors
for each range bin in $\mathcal{R}$ is created by cropping $K$ consecutive
pulses from all range bins, at a random offset within a file.

###  III-B Pre-Processing

In this work, the real-valued NNs were considered. The pre-processing is used
to convert the complex-valued input signal $\mathbf{X}$ from (1) to a real-
valued representation that is appropriate for a NN processing and target
detection in range or Doppler and is composed of 3 steps:

Pre-Processing flow

1:• Input: $\mathbf{X}\in\mathbb{C}^{N\times K}$, detection parameter
$p\in\\{\text{``range''},\text{``Doppler''}\\}$

  1. 1.

Reshape input:

$\displaystyle\mathbf{X}_{0}=\begin{cases}\mathbf{X}^{T},&\;p=\text{``range''}\\\
\mathbf{X},&\;p=\text{``Doppler''}\end{cases}$

  2. 2.

Center features over rows:

$\displaystyle[\mathbf{X}_{1}]_{[\cdot,:]}$
$\displaystyle=[\mathbf{X}_{0}]_{[\cdot,:]}-\overline{\mathbf{x}}^{T}.$

  3. 3.

Concatenate real and imaginary terms:

$\displaystyle\mathbf{Z}_{0}=\left[\Re{\mathbf{X}_{1}},\;\Im{\mathbf{X}_{1}}\right]\;.$

2:• Output:
$\mathbf{Z}_{0}=\mathcal{P}\left(\mathbf{X}\right)\in\\{\mathbb{R}^{N\times
2K},\mathbb{R}^{K\times 2N}\\}$

The row vector, $[\mathbf{X}]_{[\cdot,:]}$, represents a row in $\mathbf{X}$
and
$\quad\overline{\mathbf{x}}^{T}=\frac{1}{M_{0}}\sum_{m=1}^{M_{0}}{[\mathbf{X}_{0}]_{[m,:]}}$,
where $M_{0}$ is the number of rows in $\mathbf{X}_{0}$. Note that the target
echo signal in (3) is an outer product of two complex sinusoidal vectors, such
that the range and Doppler information is encoded in the frequency content
along the corresponding axis. Step 1 transposes the input frame $\mathbf{X}$
such that the column axis represents the parameter of interest (i.e. the
“feature axis”) and the row axis contains different realizations of that
feature. Step 2 centers each feature and Step 3 concatenates the real and
imaginary parts of the data  to real-valued representation, $\mathbf{Z}_{0}$,
of dimensions $N\times 2K$, for $p=\text{``Doppler''}$, and $K\times 2N$ for
$p=\text{``range''}$.

Figure 3: Neural network architecture described in Subsection III-D.
$\mathbf{X}$ is the complex-valued fast-time $\times$ slow-time input frame,
$\mathcal{P}\left(\mathbf{X}\right)$ is the pre-processing flow described in
Subsection III-B and
$\mathbf{Z}_{0}\triangleq\mathcal{P}\left(\mathbf{X}\right)$ is the real-
valued matrix output of the pre-processing flow. The row mapping (orange) and
column mapping (magenta) represent steps 1 and 2 from the DAFC in Subsection
III-C with tanh activation function. The dashed lines represent the mapping
operation via a FC transform. The final layer outputs the vector $\mathbf{y}$
which consists of sigmoid activation at each output neuron.

###  III-C Dimensional-Alternating Fully Connected

This Subsection presents a novel neural processing block for radar target
detection in range or Doppler domains.

Dimensional-Alternating Fully Connected

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

  1. 1.

FC transform of each row in $\mathbf{Z}_{in}$:

$\displaystyle\mathbf{Z}_{r}=\mathcal{F}_{\mathbf{h}_{r},\mathbf{W}_{r},\mathbf{b}_{r}}\left(\mathbf{Z}_{in}\right)\triangleq
h_{r}\left(\mathbf{Z}_{in}\mathbf{W}_{r}+\mathbf{1}_{H}\mathbf{b}_{r}^{T}\right)$

  2. 2.

FC transform of each column in $\mathbf{Z}_{r}$:

$\displaystyle\mathbf{Z}_{out}=\mathcal{G}_{\mathbf{h}_{c},\mathbf{W}_{c},\mathbf{b}_{c}}\left(\mathbf{Z}_{r}\right)\triangleq
h_{c}\left(\mathbf{W}_{c}^{T}\mathbf{Z}_{r}+\mathbf{b}_{c}\mathbf{1}_{W^{\prime}}^{T}\right)$

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

Let $\mathcal{F}_{\mathbf{h}_{r},\mathbf{W}_{r},\mathbf{b}_{r}}$ denote a
fully connected (FC) transform applied to each row in $\mathbf{Z}_{in}$ and
$\mathcal{G}_{\mathbf{h}_{c},\mathbf{W}_{c},\mathbf{b}_{c}}$ denote a FC
transform applied to each column in $\mathbf{Z}_{r}$.
$\mathbf{W}_{r}\in\mathbb{R}^{W\times
W^{\prime}},\mathbf{b}_{r}\in\mathbb{R}^{W^{\prime}},\mathbf{W}_{c}\in\mathbb{R}^{H\times
H^{\prime}},\mathbf{b}_{c}\in\mathbb{R}^{H^{\prime}}$ represent the DAFC
block’s learnable parameters and
$h_{r}\left(\cdot\right),h_{c}\left(\cdot\right)$ are nonlinear activation
functions, which are applied element-wise. This block is repeatedly used in a
pipeline structure, such that the input to the first block is the output of
the pre-processing flow $\mathbf{Z}_{0}=\mathcal{P}\left(\mathbf{X}\right)$.

The DAFC block is designed according to the three following ideas: 1)
structured transformation, 2) sparsity, and 3) non-linearity.

1) Structured transformation

The proposed radar signal processing approach is specifically tailored to the
conventional radar with the fast-time $\times$ slow-time data structure. From
(1), (3), and the pre-processing procedure described in Subsection III-B, each
row in $\mathbf{Z}_{0}$ encodes the information about the parameter of
interest (i.e. range or Doppler) in its frequency content. The variation
between the rows in $\mathbf{Z}_{0}$ is a result of the interaction between
multiple target echo signals and the statistical variation of the clutter and
noise, and the random phase for each single target echo signal.

With that view of information structure in mind,
$\mathcal{F}_{\mathbf{h}_{r},\mathbf{W}_{r},\mathbf{b}_{r}}$ is a “feature-
extracting” transform that aims to extract the parameter of interest related
features from each row, that will contribute to the detection capability of
the NN. $\mathcal{G}_{\mathbf{h}_{c},\mathbf{W}_{c},\mathbf{b}_{c}}$, is a
“re-calibration” transform that inserts nonlinear interaction between
transformed rows, in analogy to the excitation stage in [36]. This interaction
provides an additional degree of nonlinear processing that can learn to
mitigate clutter-induced interference and enhance the contribution of the
transformed rows. The DAFC operation also resembles the two-parts separable
convolution [37] where an image’s channel dimension and spatial dimension are
processed consecutively.

2) Sparsity

The data processed by the proposed approach is matrix-shaped. The straight-
forward approach to process this type of data is to vectorize each matrix and
invoke a vanilla FC layer. In contrast, the proposed approach performs a
matrix-to-matrix transform by consecutively processing the data along the two
axes (rows and columns) while using a sparse set of parameters. For example,
consider a transform from $\mathbf{Z}_{in}\in\mathbb{R}^{H_{1}\times W_{1}}$
into $\mathbf{Z}_{out}\in\mathbb{R}^{H_{2}\times W_{2}}$ and assume for
simplicity that $H_{1}=W_{1}=M_{1}$ and $H_{2}=W_{2}=M_{2}$. The number of
parameters in such a transform is:

$\displaystyle\text{FC: }H_{2}W_{2}(H_{1}W_{1}+1)=O(M_{1}^{2}M_{2}^{2})\;,$
(11) $\displaystyle\text{DAFC:
}W_{2}(W_{1}+1)+H_{2}(H_{1}+1)=O(M_{1}M_{2})\;.$

Notice that the DAFC’s parameter complexity grows linearly with the data
dimensions, compared with the quadratic parameter complexity growth of the FC
layer. Since NNs are typically optimized using gradient-based learning [38],
this lower parameter complexity significantly reduces the dimensionality of
the NN’s learnable parameter space and therefore can aid in the convergence of
the gradient-based learning procedure.

In addition, the parameter complexity coincides with the computational
complexity. Therefore, the design of the DAFC block contributes to the
applicability of the NN. The lower complexity also enables higher dimensional
representations, which can contribute to the capacity [38] of the NN, hence
enabling the NN to represent more complex and abstract representations that
will enhance the detection capabilities.

3) Non-linearity

Another interesting attribute of the DAFC block, is the additional degree of
non-linearity. The straight-forward FC-based approach will process the input
matrix via an affine transformation that is followed by a nonlinear activation
function. In contrast, the DAFC inserts an additional nonlinear activation
after applying an affine map on each row. Although the affine maps in the DAFC
block are of lower dimension, the additional degree of non-linearity can
contribute to the capacity [38] of the NN, thus enabling more abstract and
complex representations to be learned.

Figure 4: Range-Doppler detection scheme. $\mathbf{X}$ is the complex input
signal. The NN-based processing (light purple) contains separate instances of
NN architectures described at Subsection III-D for range and Doppler. The
model-based processing projects the inputs signal onto the steering matrices
as described in Subsection III-E. The range-Doppler detection (light gray)
contains the integration scheme between the NN and model-based outputs as
detailed in Subsection III-E and provides the range-Doppler decision matrix
$\hat{\mathbf{Y}}$.

###  III-D The NN Architecture and Training

The proposed NN architecture is selected to provide a mapping from the complex
fast-time $\times$ slow-time input to a probability mass function (pmf) of a
target presence in each range or Doppler bin. To that aim, the pre-processing
flow described in Subsection III-B and the DAFC neural processing described in
Subsection III-C are utilized to compose a NN architecture that delivers the
desired mapping. The selected NN architecture is formulated as:

$\displaystyle\mathbf{y}=\mathbf{h}\left(\mathbf{X}\right)\triangleq\mathcal{F}\left(\text{Vec}\left(\mathcal{S}_{3}\left(\mathcal{S}_{2}\left(\mathcal{S}_{1}\left(\mathcal{P}\left(\mathbf{X}\right)\right)\right)\right)\right)\right)\;,$
(12)

where $\mathcal{P}$ is the pre-processing procedure described in Subsection
III-B, $\mathcal{S}_{b}$ denotes the enumerated $b$-th DAFC block and
$\mathcal{F}$ is a FC transform (i.e. layer) with sigmoid activation at each
output neuron. Fig. 3 shows the proposed NN architecture with parameters
specified in Table I.

Block |  Output Dimension |  Activation
---|---|---
$\mathbf{Z}_{0}=\mathcal{P}\left(\mathbf{X}\right)$ |  $K\times 2N$ or $N\times 2K$ |  -
$\mathbf{Z}_{1}=\mathcal{S}_{1}\left(\mathbf{Z}_{0}\right)$ |  $128\times 1024$ |  tanh
$\mathbf{Z}_{2}=\mathcal{S}_{2}\left(\mathbf{Z}_{1}\right)$ |  $16\times 256$ |  tanh
$\mathbf{Z}_{3}=\mathcal{S}_{3}\left(\mathbf{Z}_{1}\right)$ |  $4\times 128$ |  tanh
Vec$\left(\mathbf{Z}_{3}\right)$ |  512 |  -
$\mathbf{y}=\mathcal{F}\left(\text{Vec}\left(\mathbf{Z}_{3}\right)\right)$ |  $d_{R}$ or $d_{V}$ |  sigmoid
Table I:  NN architecture. Each row contains the specifications of each block
in the NN. The number of parameters in the NN for $N=K=64$ and
$d=d_{R}=d_{V}=64$ is 470,676.

The motivation for the dimensionality expansion in the early stages is similar
to the high-dimensional feature space rationale used in kernel-SVM methods
[39]. Namely, the NN learns a high-dimensional mapping that transforms the
input data to a high-dimensional space, in which the expressive
characteristics are enhanced and the detection ability is increased. The
dimensionality reduction in later stages is aimed to reduce the dimension of
the latent representation to a dimension that is closer to the label space
dimension. This trend of dimensionality expansion by early layers and
reduction in later layers is also presented in [40], where the authors show
that the intrinsic dimensionality of CNNs’ layers follows the same trend.

The last activation is chosen to be sigmoid since we choose the NN to produce
a pmf of target presence in each range or Doppler bin. Thus, the multi-
dimensional pmf, $\mathbf{y}\in[0,1]^{d}$, where $d\in\\{d_{R},d_{V}\\}$,
encodes the target presence probability in each bin. The proposed NN
architecture enables detection of multiple targets at a single inference
cycle, without the need for ROI pooling of range or Doppler subspaces,
similarly to the object detection approach in [41]. Similarly to [30], this
architecture considers a sparse label. Therefore, the class-balanced cross-
entropy [42] is used as the following loss function:

$\displaystyle\mathcal{L}\left(\mathbf{y},\mathbf{y}_{true}\right)=-\frac{1}{d}\sum_{j=0}^{d-1}$
$\displaystyle\frac{1-\beta}{1-\beta^{n_{0}}}\left(1-[\mathbf{y}_{true}]_{j}\right)\log{\left(1-[\mathbf{y}]_{j}\right)}$
(13)
$\displaystyle+\frac{1-\beta}{1-\beta^{n_{1}}}\left([\mathbf{y}_{true}]_{j}\right)\log{\left([\mathbf{y}]_{j}\right)}\;,$

where $\beta\rightarrow 1$, $\mathbf{y}\in[0,1]^{d}$ and
$\mathbf{y}_{true}\in\\{0,1\\}^{d}$ are the NN output and true label vector,
respectively, $d$ is the dimension of label vector ($d_{R}$ or $d_{V}$). Terms
$n_{1}$, $n_{0}$ are proportional to the number of the target bins and the
target-free bins, respectively, and represent density of the target labels
within the training data set.

###  III-E NN-Based Range-Doppler Detector

Let the range and Doppler steering matrices be:
$\mathbf{R}=\begin{bmatrix}\mathbf{r}(0)&\mathbf{r}(\Delta
r)&\dots&\mathbf{r}(\Delta r(d_{R}-1))\end{bmatrix}$ and
$\mathbf{V}=\begin{bmatrix}\mathbf{v}(0)&\mathbf{v}(\Delta
v)&\dots&\mathbf{v}(\Delta v(d_{V}-1))\end{bmatrix}$, where $\mathbf{r}(r)$
and $\mathbf{v}(v)$ are defined in (4). Let $\mathbf{h}_{r}(\cdot)$ and
$\mathbf{h}_{v}(\cdot)$ denote the range and Doppler NN models, which are in
fact separate instances of the NN described in Subsection III-D. Fig. 4 shows
the range-Doppler detection scheme detailed in Algorithm 1.

Algorithm 1: Range-Doppler Detection

1:• Input: $\mathbf{X}\in\mathbb{R}^{N\times K}$, detection threshold $t$

  1. 1.

NN feed-forward, range-Doppler projection

$\displaystyle\mathbf{y}_{r}=\mathbf{h}_{r}\left(\mathcal{\mathbf{X}}\right),\;\mathbf{y}_{v}=\mathbf{h}_{v}\left(\mathcal{\mathbf{X}}\right),\;\mathbf{Z}=\text{Abs}\left(\mathbf{R}^{H}\mathbf{X}\mathbf{V}^{*}\right)$

  2. 2.

Apply threshold to NN-yielded pmf’s and use them to filter the projected
matrix:

$\displaystyle\hat{\mathbf{y}}_{r}$
$\displaystyle=\mathbb{I}\left(\mathbf{y}_{r}>t\right),\;\hat{\mathbf{y}}_{v}=\mathbb{I}\left(\mathbf{y}_{v}>t\right)\;,$
$\displaystyle\mathbf{U}$
$\displaystyle=\mathbf{Z}\odot\mathbf{y}_{r}\mathbf{y}_{v}^{T}\odot\hat{\mathbf{y}}_{r}\hat{\mathbf{y}}_{v}^{T}$

  3. 3.

Apply threshold to obtain range-Doppler detection matrix:

$\displaystyle\hat{\mathbf{Y}}=\mathbb{I}\left(\frac{1}{\text{max}\mathbf{U}}\mathbf{U}>t\right)$

  * •

Output: $\hat{\mathbf{Y}}\in\mathbb{R}^{d_{R}\times d_{V}}$

In Step 1, $\mathbf{y}_{r}$ and $\mathbf{y}_{v}$ are NN-yielded multi-
dimensional pmf’s of target presence in each range and Doppler bin, and
$\mathbf{Z}$ is the projection of the input frame on the steering vectors
representing the range and Doppler bins. Step 2 uses $\mathbf{y}_{r}$ and
$\mathbf{y}_{v}$ to detect targets, mitigate clutter and factorize
$\mathbf{Z}$ at each range-Doppler bin to obtain $\mathbf{U}$. In Step 3,
$\mathbf{U}$ is normalized to values in the interval $[0,1]$, such that the
detection threshold $t$ can be used. Note that $\mathbf{U}$ is the combination
of the model-based transform (i.e. projection to range-Doppler steering
matrices, which is equivalent to conventional range-Doppler transform), NN-
based transform (to obtain a $[0,1]$ score for each range-Doppler bin in
$d_{R}\times d_{V}$), and range-Doppler bins, which exceed the threshold, $t$,
at each NN output (i.e. targets detected by the NNs).

The detection threshold $t$ determines the trade-off between the probability
of detection, $P_{D}$, and the probability of false-alarm $P_{FA}$. In this
work, the threshold is set empirically for the desired $P_{FA}$. The
experiment in Subsection IV-C4 shows that the CFAR property is satisfied under
clutter statistics variations.

The range-Doppler detector in Algorithm 1 contains a combination of NN-based
processing and model-based processing. The goal of the NNs is to output a pmf
for the presence of target in each range or Doppler bin ($\mathbf{y}_{r}$ and
$\mathbf{y}_{v}$), that attains low probability for clutter-containing and/or
target-free bins, whereas high probability for target-containing bins.
Therefore, the proposed NN has the ability to mitigate the clutter in the
detector output, since low-probability bins are filtered by the detection
threshold $t$, whereas the model-based projection is not able to suppress the
correlated heavy-tailed clutter’s energy, as explained in Subsection II-B. On
the other hand, solely relying on the NNs as range-Doppler detectors suffers
from an inherent ambiguity in the combined range-Doppler space, since the
operation $\mathbf{y}_{r}\mathbf{y}^{T}_{v}$ results in a Cartesian product of
all possible range-Doppler combinations. To that aim, the model-based
projection delivers a signal that contains information regarding energy
presence in each range-Doppler bin, and thus compensating for the NN-based
ambiguity. The experiment described in Subsection IV-E, shows the significant
gain obtained by this combination.

##  IV PERFORMANCE EVALUATION

For all experiments, we have used synthetic simulated targets according to the
target signal model described in Subsection II-A. A training dataset of
$10,000$ frames containing targets ($\mathcal{T}\neq\emptyset$) and $10,000$
frames without targets ($\mathcal{T}=\emptyset$), was used in each experiment
to keep a balanced training dataset. For each target-containing frame (i.e.
$\mathcal{T}\neq\emptyset$) the number of targets is
$|\mathcal{T}|\sim\text{Unif}\left(\\{1,\dots,8\\}\right)$ with targets’
parameters (range and Doppler) sampled uniformly over the continuous range-
Doppler space:

$\displaystyle
r\sim\text{Unif}\left([r_{min},r_{max}]\right),\;v\sim\text{Unif}\left([v_{min},v_{max}]\right)\;.$
(14)

For each target in $\mathcal{T}$, the SCNR is sampled from
$\text{SCNR}_{[dB]}\sim\text{Unif}\left(-5,10\right)$. In order to enrich the
training dataset, each batch is simulated independently to increase the number
of frames for each NN training. Adam optimizer [43] with learning rate
$10^{-3}$ and $\beta=0.99$ parameter were used to train the network, together
with $L_{2}$ regularization factor of $5\cdot 10^{-4}$. Batch size of $256$,
$300$ epochs, and a plateau learning rate scheduler with $0.905$ factor were
selected.

The test dataset in each evaluation scenario contains $10,000$ fast-
time$\times$slow-time frames with $\mathcal{T}\neq\emptyset$ and $5,000$
frames with $\mathcal{T}=\emptyset$. For each frame with
$\mathcal{T}\neq\emptyset$, the number of targets is, unless stated else, a
random number in $\\{1,\dots,8\\}$ with random $\text{SCNR}_{[dB]}$ for each
target, sampled from
$\text{SCNR}_{[dB]}\in\\{-10,-7.5,\dots,0,\dots,7.5,10\\}$. Range and Doppler
values of targets are sampled in the same manner as the training dataset.

The rest of the datasets’ parameters are detailed in Table II.

Notation |  Description |  Value
---|---|---
$B$ |  Chirp bandwidth |  $50\;[MHz]$
$T_{0}$ |  PRI |  $1\;[msec]$
$N$ |  Samples per pulse (fast-time) |  $64$
$K$ |  Pulses per frame (slow-time) |  $64$
$f_{c}$ |  Carrier frequency |  $9.39\;[GHz]$
$\sigma_{f}$ |  K-distribution correlation coefficient |  $0.05$
$[r_{min},r_{max}]$ |  Range interval |  $[0,93]\;[m]$
$[v_{min},v_{max}]$ |  Doppler interval |  $[-7.5,7.5]\;[m/sec]$
$\mathcal{R}$ |  Range bins |  $\\{0,3,\dots,93\\}[m]$
$\Delta r$ |  Range resolution |  $3\;[m]$
$\mathcal{V}$ |  Doppler bins |  $\\{-7.73,-7.48,\dots,7.73\\}$ $[m/sec]$
$\Delta v$ |  Doppler resolution |  $0.249\;[m/sec]$
$d_{R}=|\mathcal{R}|$ |  Number of range bins |  $32$
$d_{V}=|\mathcal{V}|$ |  Number of Doppler bins |  $63$
CNR |  Clutter to Noise ratio |  $15$ [dB]
Table II:  Parameters of generated datasets.

In the following subsections, the performance of the proposed approach is
evaluated in a clutter-free scenario and in various scenarios with both
simulated and recorded radar clutter. The evaluation is via the probability of
detection ($P_{D}$) for a fixed probability of false-alarm ($P_{FA}$). $30$
independent test datasets are generated in each experiment in order to
evaluate the $P_{D}$. For each generated dataset, the detection threshold is
set to determine a predefined $P_{FA}$. Note that in each experiment, the same
range NN instance and the same Doppler NN instance are used for various SCNRs
and various clutter conditions, as further detailed in the experiments below.

In the following experiments, the performance of the proposed approach was
evaluated and compared to a) the ANMF, on slow-time signals at each range bin
and b) the CA-CFAR and OS-CFAR, on the 2D FFT-based range-Doppler energy map.
The NMF [4] is the GLRT approximation for scenarios with a single target
within a CUT. However, NMF requires the true clutter covariance matrix and
therefore, is impractical. The ANMF, is the adaptive NMF that utilizes the
estimated clutter covariance matrix [12]. The $\Sigma$-ANMF [8, 12] estimates
the clutter covariance matrix using the $\Sigma$ estimator and therefore, is
suited for scenarios with non-Gaussian clutter [8].

The CA-CFAR is an optimal detector for targets within a homogeneous
environment in the range-Doppler domain, and the OS-CFAR is a robust CFAR
method that is designed to operate in a heterogeneous environment in the
range-Doppler domain [3]. The selected window size is $9\times 15$ with
$3\times 3$ guard cells for the CA-CFAR, and $17\times 33$ with $3\times 5$
guard cells for the OS-CFAR. The ordered statistic rank used for the OS-CFAR
is the window median. Similarly to the proposed method, the detection
threshold for the $\Sigma$-ANMF, CA-CFAR, and OS-CFAR is determined
empirically according to the desired $P_{FA}$.

###  IV-A Performance Evaluation Metrics

The metrics for the following performance evaluation, $P_{D}$ and $P_{FA}$,
are defined in this Subsection. Consider a dataset generated according to the
guidelines in Subsection III-A, and let $(i,j)$ denote the index of the $j$-th
target in the $i$-th example in the dataset. For each target
$\left(r_{i,j},v_{i,j}\right)$ denote the indices of the “closest” range and
Doppler bins by $[m_{i,j},l_{i,j}]$. Define the “neighboring” box
$B\left(r_{i,j},v_{i,j}\right)$ as:

$\displaystyle
B\left(r_{i,j},v_{i,j}\right)=[m_{i,j},l_{i,j}]\bigcup\left\\{\left([m_{i,j}\pm
1,l_{i,j}\pm 1]\right)\right\\}\;.$ (15)

The event of successfully detecting the target $\left(r_{i,j},v_{i,j}\right)$
is defined as:

$\displaystyle D\left(r_{i,j},v_{i,j}\right)=\mathbb{I}\left(\exists[m,l]\in
B\left(r_{i,j},v_{i,j}\right):[\hat{\mathbf{Y}}]_{m,l}=1\right)\;.$ (16)

In words, we define successful target detection if the true target is located
at most one bin away from a predicted target position. The probability of
detection, $P_{D}$, is evaluated by the ratio between the number of
successfully detected targets, according to (16), to the total number of
targets in the dataset:

$\displaystyle
P_{D}=\frac{1}{\left|\bigcup_{i,j}{\left(r_{i,j},v_{i,j}\right)}\right|}\sum_{i,j}{D\left(r_{i,j},v_{i,j}\right)}\;.$
(17)

The $P_{FA}$ is evaluated as the number of range-Doppler bins without true
targets where the entry of the predicted targets matrix $\hat{\mathbf{Y}}$ is
$1$:

$\displaystyle P_{FA}$
$\displaystyle=\frac{1}{\left|B_{0}\right|}\sum_{[m,l]\in
B_{0}}{[\hat{\mathbf{Y}}]_{m,l}}\;,$ (18) $\displaystyle B_{0}$
$\displaystyle=\bigcup_{i}{\left(\\{1,\dots,d_{R}\\}\times\\{1,\dots,d_{V}\\}\setminus\bigcup_{j}{B\left(r_{i,j},v_{i,j}\right)}\right)}\;,$

where $B_{0}$ is the “no-target” space, defined as the set of indices of all
range-Doppler bins in the dataset, excluding the true targets and their
corresponding “neighboring” boxes.

###  IV-B Clutter-Free Scenario

In this scenario, the performance of the proposed approach is evaluated in a
clutter-free environment. In the clutter-free, single-target scenario the CA-
CFAR is an approximation of the GLRT [3]. The OS-CFAR is designed for multi-
target scenario and heterogeneous environments [3]. The NN models in the
proposed method are trained using data with half of the training frames
containing AWGN ($\mathbf{C}=\mathbf{0}$) and half containing clutter,
simulated with shape parameter $\nu$, sampled from
$\nu\sim\text{Unif}\left(0.1,1.5\right)$.

Fig. 5 compares the receiver operating characteristics (ROC) of the proposed
approach to the conventional CA-CFAR and OS-CFAR detectors in scenarios with
$\\{1,2,4,8\\}$ targets. Notice that in this scenario, the optimal single-
target CA-CFAR detector attains $P_{D}=1$ for all $P_{FA}$ values. However,
with increasing the number of targets, its $P_{D}$ is degraded as a result of
mutual target masking [3], which is more prominent at lower $P_{FA}$. At the
low $P_{FA}$ values, the OS-CFAR performance is degraded due to the strong
target proximity to the weak target [44]. The proposed approach is robust to
variation in the number of targets compared to the CA-CFAR and OS-CFAR
detectors. Furthermore, the NN models used in this experiment were trained
using a dataset that contains also correlated heavy-tailed clutter. This
evidence demonstrates the generalization capability of the proposed NN-based
detection approach to variation in the interference behavior.

Figure 5: ROC of evaluated detectors in clutter-free scenarios with
$\\{1,2,4,8\\}$ targets and $SCNR=0\;dB$. Figure 6: ROC of multiple targets
with simulated clutter, $\text{SCNR}=0\;dB$.

(a) Clutter with targets, $P_{FA}=5\cdot 10^{-5}$

(b) Clutter with targets, $\text{SCNR}=0\;dB$

Figure 7: Detection results with simulated clutter with targets for various
clutter “spikiness” $\nu\in\\{0.2,0.5,1.0\\}$. Subplot (a) considers constant
$P_{FA}=5\cdot 10^{-5}$ and subplot (b) considers SCNR $=0\;dB$.

(a) Clutter with embedded targets, $P_{FA}=5\cdot 10^{-5}$

(b) Clutter with embedded targets, $\text{SCNR}=0\;dB$

Figure 8: Detection results with simulated clutter with embedded targets for
various clutter “spikiness” $\nu\in\\{0.2,0.5,1.0\\}$. Subplot (a) considers
constant $P_{FA}=5\cdot 10^{-5}$ and subplot (b) considers SCNR $=0\;dB$.

###  IV-C Simulated Clutter

In this experiment, the performance of the proposed approach is evaluated in
scenarios with simulated correlated heavy-tailed clutter, as described in
Subsection III-A. For the proposed approach, the NN models were trained using
training sets with shape parameter $\nu$ sampled from
$\nu\sim\text{Unif}\left(0.1,1.5\right)$. The evaluation is performed using
three different test sets, generated with shape parameters,
$\nu\in\\{0.2,\;0.5,\;1.0\\}$.

####  IV-C1 Comparison to ANMF

The ANMF detector is the GLRT approximation for a single target within SIRV
clutter. For $\mathcal{H}_{1}$ hypothesis, it considers a single target within
the range CUT and the availability of target-free secondary data. Therefore,
the ANMF is implemented via a sequence of binary hypothesis tests per range
bin by scanning the Doppler values, $\mathcal{V}$. The range-Doppler bins with
energy exceeding the detection threshold are declared as containing detected
targets. In this work, we consider a modification of the $\Sigma$-ANMF
detector, denoted as $\Sigma W$-ANMF, which aims to suppress target-presence
in neighbouring range cells. The $\Sigma W$-ANMF involves Hanning window
applied to each fast-time signal, and taking $4$ guard cells around the range
CUT, prior to the range-transform and $\Sigma$-ANMF. Targets’ Doppler values
are drawn from a continuous interval, and thus, the binary hypothesis test
formulation is mismatched in the desired slow-time signal model.

Fig. 6 shows the ROC of the proposed approach compared to the $\Sigma$-ANMF
[8] in several scenarios. It shows that in on-gird scenarios, where the target
range and Doppler values are drawn from $\mathcal{R}\times\mathcal{V}$, the
$\Sigma$-ANMF performance is significantly higher comparing to the scenarios
where the target range and Doppler values do not exactly match the range-
Doppler gird. In these scenarios, the high range side-lobes may compromise the
target-free secondary data assumption. Fig. 6 shows that the proposed approach
significantly outperforms the $\Sigma$-ANMF and $\Sigma W$-ANMF detectors.

####  IV-C2 Detection Performance

Using the simulated correlated heavy-tailed clutter, the performance of the
proposed detection approach is evaluated in a) clutter with targets and b)
clutter with embedded targets scenarios. In the first scenario, the Doppler of
targets and clutter is independent, whereas in the second scenario the
targets’ Doppler are randomly set to be at most $\pm$ 1 $m/sec$ offset from
the clutter’s Doppler, represented by $f_{d}$ in (10). Figs. 7 and 8 show the
$P_{D}$ of the proposed approach compared to the CA-CFAR and OS-CFAR detectors
in scenarios with simulated heavy-tailed clutter with
$\nu\in\\{0.2,\;0.5,\;1\\}$ as a function of SCNR and $P_{FA}$ for the
embedded and not embedded scenarios.

The parameter $\nu$ introduced in the model in Subsection III-A, controls the
“spikiness” of the simulated clutter amplitude. Figs. 7 and 8 show the
robustness of the proposed approach to the “spikiness” of the simulated
clutter. Notice the difference between the conventional CA-CFAR and OS-CFAR
detectors whose performance significantly degrades in the presence of spiky
clutter (decreasing $\nu$). The scenario in Fig. 7 contains both homogeneous
and inhomogeneous cases and therefore shows that the CA-CFAR outperforms the
OS-CFAR in most cases. In Fig. 8, the targets’ environment is inhomogeneous
(due to clutter-induced spikes and clutter edges), and therefore the OS-CFAR
outperforms the CA-CFAR. In both scenarios, the proposed approach outperforms
the CA-CFAR and OS-CFAR detectors and shows a minor degradation between the
homogeneous and inhomogeneous scenarios. These observations demonstrate the
ability of the proposed approach to generalize to various clutter distribution
types (various $\nu$ values), and the ability of the proposed approach to
suppress correlated clutter, since it succeeds in “extracting” embedded
targets from the clutter.

####  IV-C3 Number of Targets

The sensitivity of the proposed approach to the number of targets
$\left|\mathcal{T}\right|$ is evaluated in this experiment using clutter with
embedded targets as defined in IV-C2. Fig. 9 shows that the CA-CFAR and OS-
CFAR performances degrade with decreasing $\nu$ and increasing the number of
targets (due to the mutual target masking [3] and strong interfering targets
[44]). On the other hand, The proposed approach shows robustness to clutter
“spikiness” variation and demonstrates generalization capability to multiple
targets.

Figure 9: Detection performance with simulated clutter and embedded targets as
a function of the number of targets with $\text{SCNR}=0dB$ and $P_{FA}=5\cdot
10^{-5}$ for various clutter “spikeness” $\nu\in\\{0.2,\;0.5,\;1\\}$. Figure
10: $P_{FA}$ verification for simulated clutter in various clutter “spikeness”
$\nu\in\\{0.2,\;0.5,\;1\\}$. $\hat{P}_{FA}$ denotes the false-alarm
probability obtained using the validation dataset, $P_{FA}$ denotes the false-
alarm probability obtained using the test set when the threshold was set
according to the validation dataset.

(a) Clutter with targets, $P_{FA}=5\cdot 10^{-5}$

(b) Clutter with targets, $\text{SCNR}=0\;dB$

Figure 11: Detection results in scenarios with targets and recorded radar
clutter for $3$m and $15$m range resolutions, and VV and HH polarizations.
Subplot (a) considers constant $P_{FA}=5\cdot 10^{-5}$ and subplot (b)
considers SCNR $=0\;dB$.

(a) Clutter with embedded targets, $P_{FA}=5\cdot 10^{-5}$

(b) Clutter with embedded targets, $\text{SCNR}=0\;dB$

Figure 12: Detection results in scenarios with embedded targets and recorded
radar clutter for $3$m and $15$m range resolutions, and VV and HH
polarizations. Subplot (a) considers constant $P_{FA}=5\cdot 10^{-5}$ and
subplot (b) considers SCNR $=0\;dB$.

####  IV-C4 $P_{FA}$ Verification

The CFAR property of the proposed approach is examined in this subsection via
simulations of various types of clutter statistics determined by the
parameter, $\nu$. In addition, the sensitivity of the proposed approach to
mismatched detection threshold is investigated.

For each $P_{FA}$, the detection threshold for the proposed, the CA-CFAR  and
OS-CFAR approaches, is selected using a validation dataset of $5,000$ frames
with $\mathcal{T}\neq\emptyset$, and $2,500$ frames with
$\mathcal{T}=\emptyset$. For frames with $\mathcal{T}\neq\emptyset$, the
following parameters were selected:
$\text{SCNR}_{[dB]}\sim\text{Unif}\left(-10,10\right)$,
$|\mathcal{T}|\sim\text{Unif}\left(1,\dots,8\right)$, and
$\nu\sim\text{Unif}\left(0.1,1.5\right)$. The thresholds obtained according to
this validation dataset are evaluated on test datasets with various clutter
“spikeness” parameters, $\nu\in\\{0.2,0.5,1\\}$.

Fig. 10 shows that the proposed approach is more robust than the conventional
CA-CFAR and OS-CFAR detectors, in terms of variation in actual $P_{FA}$. This
can be explained by the fact that the clutter “spikeness” affects the adaptive
detection threshold level in the CA-CFAR and OS-CFAR [3], and thus, their
performances degrade in the $P_{FA}$ mismatch. In contrast, the NN models in
the proposed approach learn a mapping (specifically a pmf) which is well
generalized to variations in clutter statistics and therefore do not vary
substantially when the clutter “spikeness” measure is changed.

###  IV-D Recorded Radar Clutter

The performance of the proposed approach is evaluated in this Subsection using
real recorded radar measurements. The radar clutter signals in each frame
$\\{\mathbf{c}_{r}\\}_{r}$ are generated using recorded radar clutter from the
McMaster IPIX database [24], as detailed in Subsection III-A. Records that
definitely do not contain any targets and contain only clutter-plus-noise
echoes were selected. The following data files with HH and VV polarizations
have been chosen for the $3m$ and $15m$ range resolution, respectively:
$34,36,49,52,57,86,87,88,90,98,102,103,104,105,106$, $156,165,166$ and
$35,48,55,154$. The training and evaluation process was performed in a
$k-fold$ cross-validation manner. The files with $3m$ range resolution were
split into $6$ groups of $3$, and files with $15m$ range resolution were split
into $4$ groups of $1$, where at each cross-validation iteration, a different
group was left-out from training and used for test.

We have found it beneficial to add a standardization step in the pre-
processing flow described in Subsection III-B, which divides each element in
$\mathbf{X}_{1}$ by the sample standard deviation computed over the element’s
column. For the $15m$ range resolution files, the targets’ ranges are sampled
from the corresponding ranges and the chirp bandwidth $B$ is changed
accordingly. A different pair of NN models for each range resolution and
polarization were trained and evaluated on all considered SCNRs. In addition,
the clutter-plus-noise statistics is unknown, since this is real recorded
radar data, hence the SCNR and CNR definitions in (7) are unavailable.
Therefore, we resort to empirical methods to approximate the clutter-plus-
noise energy within each sampled frame, by using the norm of the sampled real
recorded data.

Figs. 11 and 12 show the detection performance of the proposed approach and
the CA-CFAR and OS-CFAR, in a) clutter with targets and b) clutter with
embedded targets, similarly to the experiment detailed in Subsection IV-C2.
Note that all tested approaches achieve better performance for the VV
polarization. This observation can be explained by the analysis of the
polarization effects in the IPIX database [24] in [45], where it was shown
that the HH amplitude is spikier than the VV amplitude [18]. Both Figs. 11 and
12 show that the proposed approach outperforms the CA-CFAR and OS-CFAR for the
majority of tested SCNR values. Comparing Figs. 11 and 12 to Figs. 7 and 8, it
can be observed that the OS-CFAR outperforms the CA-CFAR in both scenarios.
This performance difference can be explained by the difference between the
energy pattern in the range-Doppler map of the real-world scenarios from the
simulation model. Since the clutter data is obtained from various dates,
hours, azimuths, and range sections, it is reasonable that the targets’
environment is heterogeneous, which leads to the superior performance of the
OS-CFAR over the CA-CFAR.

Furthermore, since these results are obtained from a cross-validation
experiment on real data, it is a strong evidence for the generalization
capability of the proposed approach to unseen data. Each file in the database
contains clutter recordings from various dates, hours, azimuths, and range
sections, hence the interference statistics contained in each file are
different. Therefore, these results show that the proposed approach has an
ability of generalization to unseen data and clutter statistics.

###  IV-E Combination of NN-Based and Model-Based

The proposed approach combines NNs with model-based processing (projection to
steering vector matrices). This Subsection demonstrates the significant
performance gain provided by this combination. Fig. 13 shows the ROC of the
proposed approach using simulated radar clutter with $\nu=0.5$. Plots in Fig.
13 show contributions of various components of the proposed approach. The blue
line represents Algorithm 1, the orange line represents using only the NN-
based performance in Algorithm 1, i.e.
$\mathbf{Z}=\mathbf{1}_{d_{R}}\mathbf{1}^{T}_{d_{V}}$ (where $\mathbf{1}_{d}$
is a column vector of size $d$ whose entries are equal to one) without
normalizing by max in Step 3 and the green line represents detection using
only projection-based signals in Algorithm 1:
$\mathbf{y}_{r}=\mathbf{1}_{d_{R}},\mathbf{y}_{v}=\mathbf{1}_{d_{R}}$ which
results in $\mathbf{U}=\mathbf{Z}$.

Notice that the NN-only-based detection suffers from inherent ambiguity since
the operation $\mathbf{y}_{r}\mathbf{y}_{v}^{T}$ results in a Cartesian
product of all possible range-Doppler combinations. On the other hand,
projection-based processing (transform) does not mitigate the correlated
clutter. These results show the importance of combining the two approaches
(NN- and model-based), since each of them alone performs poorly, whereas the
combination delivers a substantial gain in the detection performance.

Figure 13: ROC of various portions of the proposed algorithm with
$\text{SCNR}=0\;dB$, and simulated clutter with $\nu=0.5$.

###  IV-F Label Sparsity

Evaluating the binary cross-entropy (BCE) criterion on target/non-target bins
and loss function in (13), over the training and validation sets at each epoch
can be used to evaluate the capabilities of the proposed approach. Fig. 14
shows the average BCE and loss functions in range and Doppler domains, trained
on the IPIX data and using the last $5\%$ of each file ($3000$ pulses) as
validation data. The BCE for target-free bins converges, with lower values for
training data. The BCE for target bins shows similar performance for
validation and training sets. Fig. 14 clearly shows that the proposed NN
architecture in combination with the class-balanced cross-entropy [42]
generalizes to unseen data and overcomes the true labels sparsity challenge.

Figure 14: Label sparsity effect investigation. Average learning curves of NNs
training using recorded radar clutter echoes.

###  IV-G Computational Complexity

The computational complexity of the proposed approach is evaluated in this
subsection considering $n=N=K=64$ and the range-Doppler map of size $N\times
K$. The computational complexity of the $N\times K$-sized 2D-FFT is the
complexity of applying 1D-FFT per row followed by 1D-FFT per column, which is
$\mathcal{O}\left(2n\cdot n\log n\right)$. Considering $n=64$, the 2D-FFT
complexity is $\mathcal{O}\left(768n\right)$. The absolute squared value
operator applied per range-Doppler bin, with a complexity of
$\mathcal{O}\left(n^{2}\right)$ adds additional complexity of
$\mathcal{O}\left(64n\right)$. The total complexity of the conventional range-
Doppler transform is $\mathcal{O}\left(832n\right)$ for $n=64$. For the CA-
CFAR, the additional complexity is associated with averaging the cells in the
window, which results in the computational complexity of
$\mathcal{O}\left(2n\right)$. For the OS-CFAR, the additional computation
complexity is associated with sorting cells in the window. The OS-CFAR window
contains $m=546$ cells, and the sorting operation takes
$\mathcal{O}\left(m\log m\right)$ complexity, which is equivalent to
$\mathcal{O}\left(78n\right)$ for $n=64$. Assuming parallel execution of the
CA-CFAR and OS-CFAR per range-Doppler bin, their complexity is
$\mathcal{O}\left(832n+2n\right)=\mathcal{O}\left(834n\right)$ and
$\mathcal{O}\left(832n+78n\right)=\mathcal{O}\left(910n\right)$, respectively.

The proposed approach in Algorithm 1 consists of $3$ steps. Step 1 contains NN
feed-forward, and model-based projection, executed in parallel. The DAFC
operation consists of two consecutive matrix multiplications and applying
activation functions. Assuming parallel execution of matrix-vector
multiplication, the complexity of $\mathbf{Z}\mathbf{w}$, where
$\mathbf{Z}\in\mathbb{R}^{d_{1}\times d_{2}}$ and
$\mathbf{w}\in\mathbb{R}^{d_{2}}$ is $\mathcal{O}\left(d_{2}\right)$. Namely,
we assume that the dot product between the rows of $\mathbf{Z}$ and
$\mathbf{w}$ is executed in parallel. As detailed in Subsection III-C, the
DAFC operation consists of two consecutive matrix multiplications. Consider
$\mathbf{Z}\in\mathbb{R}^{d_{1}\times d_{2}}$ and
$\mathbf{W}=\begin{bmatrix}\mathbf{w}_{d_{1}}&\dots&\mathbf{w}_{d_{3}}\end{bmatrix}\in\mathbb{R}^{d_{1}\times
d_{3}}$. Then
$\mathbf{ZW}=\begin{bmatrix}\mathbf{Zw}_{d_{1}}&\dots&\mathbf{Zw}_{d_{3}}\end{bmatrix}$
consists of $d_{3}$ matrix-vector multiplications of
$\mathcal{O}\left(d_{2}\right)$ complexity, which is
$\mathcal{O}\left(d_{2}d_{3}\right)$. Therefore, the complexity of
transforming an $H\times W$ input to an $H^{\prime}\times W^{\prime}$ output
is $\mathcal{O}\left(WW^{\prime}+HH^{\prime}\right)$. According to Table I,
The complexity of the DAFC blocks is $\mathcal{O}\left(1024\cdot
2n+128n\right)$ for the first block, $\mathcal{O}\left(256\cdot 16n+16\cdot
2n\right)$ for the second block, and $\mathcal{O}\left(128\cdot 4n+4\cdot
0.25n\right)$ for the third block. The total complexity of the DAFC operations
sums up to $\mathcal{O}\left(6817n\right)$. The final FC layer’s complexity is
$\mathcal{O}\left(8n\right)$. The complexity of bias addition activation
functions was neglected in this analysis since these can also be executed in
parallel. Therefore, the computational complexity of a single NN instance
consists of $\mathcal{O}\left(6825n\right)$.

Two NN instances can be executed in parallel, and the 2D-FFT plus absolute
operation exceeds a single NN feed-forward computational complexity.
Therefore, the computational complexity of Step 1 in Algorithm 1 is
$\mathcal{O}\left(6825n\right)$. The computational complexity of Step 2 and
Step 3 is negligible, since it contains per range-Doppler bin operations,
which can be executed in parallel. In addition, the pre-processing complexity,
$\mathcal{O}\left(n\right)$, is relatively low and can be executed during the
following frame. Thus, the total computational complexity of the proposed
approach is $\mathcal{O}\left(6825n\right)$.

The computational complexities of the proposed approach, the CA-CFAR, and the
OS-CFAR are summarized in Table III.

CA-CFAR |  OS-CFAR |  DAFC
---|---|---
$\mathcal{O}\left(834n\right)$ |  $\mathcal{O}\left(910n\right)$ |  $\mathcal{O}\left(6825n\right)$
Table III:  Computational complexity in terms of $n=N=K=64$.

Although the computational complexity of the proposed approach is $8$ times
higher than the complexity of the CA-CFAR and the OS-CFAR, the proposed NN
architecture consists of consecutively performed matrix multiplications and
element-wise activation functions. Therefore, the proposed method’s
computational complexity can be significantly reduced using an appropriate
implementation of basic linear algebra operations, implemented using hardware
accelerators such as graphics processing unit (GPU) or digital signal
processor (DSP).

##  V CONCLUSION

This work addressed the problem of multiple target detection in the range-
Doppler domain in the presence of correlated heavy-tailed clutter. A NN-based
approach was proposed to learn a complex nonlinear transform for clutter
mitigation in LFM radar, and its superiority over the conventional CA-CFAR,
OS-CFAR, and ANMF detectors was demonstrated. A novel DAFC processing block
was introduced to utilize the structure of information encoded in each complex
fast-time $\times$ slow-time radar echo and to transform it using a sparse set
of parameters. A unified NN-based architecture incorporating the DAFC block
was proposed for the problem of multiple target detection in range or Doppler
domains, separately. The generalization capability of the unified DAFC-based
NNs to various SCNRs and clutter conditions was demonstrated via multiple
tests. This generalization capability simplifying the NN training and reduces
implementation complexity.

The proposed architecture was used to design a range-Doppler detector that
uses the NNs outputs as a pmf to re-weight a conventional projection into
steering vectors. The performance of the proposed approach was evaluated using
simulated correlated heavy-tailed clutter and a database of recorded heavy-
tailed radar clutter echoes. The superiority of the proposed approach over the
ANMF, the conventional CA-CFAR and the robust OS-CFAR detectors was
demonstrated in multiple tested scenarios. Robustness to increasing the number
of targets was observed for the AWGN and correlated heavy-tailed clutter
cases. A significant performance advantage was demonstrated for various
clutter “spikiness” conditions in terms of probability of detection and
detection threshold sensitivity. Real data-based experiments demonstrated the
strong generalization capabilities of the proposed approach to unseen data,
containing various clutter statistics.

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