# BRSR-OpGAN: Blind Radar Signal Restoration using Operational Generative
Adversarial Network

Muhammad Uzair Zahid, Serkan Kiranyaz, , Alper Yildirim, ,
and Moncef Gabbouj Muhammad Uzair Zahid is with the Department of Computing
Sciences, Tampere University, 33100 Tampere, Finland (e-mail:
muhammaduzair.zahid@tuni.fi).Serkan Kiranyaz is with the Department of
Electrical Engineering, College of Engineering, Qatar University, Qatar.Alper
Yildirim is with the Department of Electrical and Electronics Engineering,
School Of Engineering and Natural Sciences, Ankara Medipol University,
Turkey.Moncef Gabbouj is with the Department of Computing Sciences, Tampere
University, Finland.

###### Abstract

Objective: Many studies on radar signal restoration in the literature focus on
isolated restoration problems, such as denoising over a certain type of noise,
while ignoring other types of artifacts. Additionally, these approaches
usually assume a noisy environment with a limited set of fixed signal-to-noise
ratio (SNR) levels. However, real-world radar signals are often corrupted by a
blend of artifacts, including but not limited to unwanted echo, sensor noise,
intentional jamming, and interference, each of which can vary in type,
severity, and duration. This study introduces Blind Radar Signal Restoration
using an Operational Generative Adversarial Network (BRSR-OpGAN), which uses a
dual domain loss in the temporal and spectral domains. This approach is
designed to improve the quality of radar signals, regardless of the diversity
and intensity of the corruption. Methods: The BRSR-OpGAN utilizes 1D
Operational GANs, which use a generative neuron model specifically optimized
for blind restoration of corrupted radar signals. This approach leverages
GANs’ flexibility to adapt dynamically to a wide range of artifact
characteristics. Results: The proposed approach has been extensively evaluated
using a well-established baseline and a newly curated extended dataset called
the Blind Radar Signal Restoration (BRSR) dataset. This dataset was designed
to simulate real-world conditions and includes a variety of artifacts, each
varying in severity. The evaluation shows an average SNR improvement over 15.1
dB and 14.3 dB for the baseline and BRSR datasets, respectively. Finally, even
on resource-constrained platforms, the proposed approach can be applied in
real-time. Significance: This is a pilot study in blind radar restoration with
the capability of time-domain restoration of real-world radar signals at an
exceptional level of performance across a wide range of SNR values and
artifact types. The BRSR-OpGAN method exhibits robust and computationally
efficient restoration of real-world radar signals, significantly outperforming
existing methods. We have made the source code and the real-world radar
dataset publicly available in a GitHub repository111The implementation of
BRSR-OpGAN and BRSR data generation is publicly shared at
https://github.com/MUzairZahid/BRSR-OpGAN. to facilitate further research.

###### Index Terms:

Radar signal processing, restoration, denoising, Operational Generative
Adversarial Networks (OpGANs)

††publicationid: pubid: 0000–0000/00$00.00 © 2021 IEEE

##  I Introduction

Radar signal restoration is an integral process that enhances the detection
and classification of radar signals. This process involves removing noise and
interference from the received signal, which can be caused by various sources,
such as atmospheric conditions, system imperfections, and external sources.
The target signal can be detected and identified more accurately by
effectively restoring the received signal, allowing for more precise
measurements and classifications. This is particularly important in
applications such as weather forecasting, electronic warfare (EW), and vehicle
safety, where the reliability and accuracy of various radar systems are
critical. In EW, the detection and classification of threat radar signals is
of vital importance [1]. EW receivers aim to detect radars from the longest
possible distance. Low probability of intercept (LPI) radars are challenging
to detect by EW receivers due to their low output power and ability to spread
their energy across the frequency spectrum [2]. The SNR improvement, i.e., the
signal processing gain, is vital to facilitate the detection of LPI radars
from longer distances. Therefore, radar signal restoration is an integral part
of radar signal processing and is an active area of research in remote sensing
and radar signal processing.

However, as the number of radar radiation sources increases, the
electromagnetic environment becomes increasingly complex. This complexity
often results in disturbances in the radar signals captured by the receiver,
leading to a relatively low SNR that presents significant challenges for
recognizing radar signals. Existing methods in radar signal restoration often
target specific distortion types with fixed SNR levels, focusing on isolated
tasks such as denoising or echo reduction. Despite their effectiveness in
controlled environments, these traditional techniques are insufficient for
dealing with the complexity of real-world situations.

Real-world scenarios pose a confluence of signal corruption challenges,
including unwanted echo, interference, and sensor noise such as Additive White
Gaussian Noise (AWGN), varying in type, severity, and duration as illustrated
in Figure 1. Given the complex and variable nature of these distortions,
restoring radar signals requires a comprehensive and blind approach capable of
addressing the broad spectrum of signal corruption scenarios typically
encountered in real-world radar environments. Essentially, such an approach
must go beyond the limitations of conventional methods and offer robust
solutions that can adapt to the diverse and dynamic nature of practical radar
signals.

Figure 1: Sample signals from the extended dataset demonstrating the impact of
various artifacts and the effectiveness of the proposed restoration method. In
each example, the clean signal (blue) and the corrupted signal (red) are
overlaid in the top plot (a), while the restored signal (green) is shown in
the bottom plot (b). Common artifacts include Additive White Gaussian Noise
(AWGN), Echo, Interference, and a blend of all with randomized weights.

Time-frequency analysis is a crucial technique for extracting intra-pulse
features of radar emitters, transforming signals into time-frequency images
(TFIs). Notable methods in time-frequency analysis encompass the short-time
Fourier transform (STFT)[3], Wigner-Ville distribution (WVD) [4], and Choi-
Williams distribution (CWD) [5], among others. These methods have gained
popularity for their ability to provide a detailed representation of signals
across both time and frequency domains. Upon generating TFIs, recent
advancements in radar waveform recognition have increasingly incorporated deep
learning (DL) to analyze these images, extracting features and identifying
waveforms with enhanced precision [6, 7, 8]. Multilayer denoising is proposed
in [9] using a sequence of variational mode decomposition (VMD), local mean
decomposition (LMD), and wavelet thresholding (WT) methods for the three-stage
decomposition of the noisy radar signal. The processed signal was then
transformed into a time-frequency image using the Choi-Williams Distribution
(CWD). Subsequently, a neural network with dilated convolution was trained on
time-frequency images, achieving a recognition rate of 75.3% under low SNR
conditions. Similarly, [10] presented a wavelet denoising approach for
suppressing mutual interference in automotive FMCW radar systems,
demonstrating significant enhancements in target detection accuracy.

Recent advancements in deep learning techniques have paved the way for
innovative solutions to radar signal denoising. Fuchs et al. [11] explored
automotive radar interference mitigation using a convolutional autoencoder,
highlighting the effectiveness of deep learning models in denoising and
improving signal-to-noise-plus-interference ratios. Some previous works, such
as [12, 13, 14], first transform raw time-series radar signals to 2D images
using time-frequency analysis methods and then utilize 2D deep networks, both
of which significantly increase the computational complexity. DNCNet was
introduced in [15], a deep neural network designed for efficient denoising and
classification of radar signals. They utilized two subnetworks: a denoising
subnetwork using a U-Net architecture with a noise level estimation module and
a classification subnetwork. The authors proposed a two-phase training
procedure where the denoising subnetwork is initially trained, followed by a
phase to strengthen the mapping between denoising results and perceptual
representation. Although direct evaluations of the restored signals were not
provided, the study focused on the classification performance of the signals
across different SNR levels, demonstrating significant improvements in
classification performance over the restored signals.

The advent of Generative Adversarial Networks (GANs) [16] has opened new
avenues in the field of signal processing, offering innovative ways to tackle
complex restoration tasks. GANs, through their adversarial training mechanism,
have shown remarkable success in generating high-quality outputs in various
domains, including signal, image, and speech processing. This success prompts
the exploration of their applicability in the domain of radar signal
restoration, particularly in a one-dimensional (1D) context, which is
inherently suited for processing spatio-temporal signals such as radar
signals. Operational GANs (OpGANs) represent a recent evolution of the GAN
framework that enables the model to learn more robust and accurate mappings
from degraded to clean signals by incorporating 1D Self-ONN [17, 18] layers
into the GAN architecture, allowing for handling a more comprehensive set of
signal degradations, addressing real-world artifacts’ diversity and
unpredictability.

Building on these insights, in this study, we aim to tackle this problem by
using a blind restoration approach without making any prior assumptions about
the type and severity of each artifact corrupting the signal. We propose a
novel BRSR-OpGAN model where both generator and discriminator are compact
Self-Organized Operational Neural Networks (Self-ONNs) [19]. In many studies,
Self-ONNs have been shown to outperform CNNs in various regression and
classification tasks [20, 21, 22], thanks to their efficient computational
processing and generative neuron model. With just 11 layers in the generator
network and 7 layers in the discriminator network, this study demonstrates
that relatively shallow OpGAN networks can achieve an unprecedented blind
restoration performance over real-world radar signals. We also introduce a
novel loss function for the generator, incorporating spectrogram-based loss
terms along with the time domain loss, enhancing the generator’s output
waveform in both time and frequency domains. Additionally, we use 1D-SelfONN
layer-based residual blocks to further enhance the restoration performance.

The lack of benchmark datasets and rigorous evaluation protocols for radar
signal restoration remains a significant challenge in the field. Datasets
currently used in the literature are accompanied by disclaimers on their
respective websites, indicating that they may not be suitable for
comprehensive evaluation [23]. Moreover, existing studies predominantly rely
on classification performance metrics to assess the effectiveness of signal
restoration techniques. This focus on classification accuracy, both pre- and
post-restoration, complicates direct comparisons of signal quality improvement
across different approaches. Consequently, there is an urgent need to develop
standardized datasets and evaluation frameworks prioritizing signal quality
metrics to enable more meaningful and consistent benchmarking of radar signal
restoration methods. To generate a large-scale real-world radar signal
dataset, the BRSR benchmark dataset with a random blend of artifacts is
compiled with 85,800 radar modulation signals from the 12-class dataset
referenced in [9]. These signals were then corrupted by randomly selected
artifacts each with randomly varying weights. The applied radar artifacts
include AWGN noise, a set of 100 signals for interference, and an unwanted
echo model with randomized attenuation and delay. The random combination and
artifact severity resulted in radar corruption at different SNR levels,
ranging from $-14$ to $10$ dB.

This study addresses a critical gap in radar signal restoration datasets and
presents the true potential of 1D OpGANs for blind signal restoration. Through
rigorous testing and validation, our research underscores the importance of
innovative approaches in overcoming the challenges of signal degradation,
marking a pioneering advancement in the field of radar signal processing. We
can summarize the novel and significant contributions of this study as
follows:

  1. 1.

This pilot study approaches the radar signal restoration blindly, i.e., no
assumptions were made regarding the types and severities of artifacts on the
received radar signal.

  2. 2.

In this study, we present the first application of 1D OpGANs to the
restoration of a radar signal in real-time by employing the raw signals
directly.

  3. 3.

A novel loss function has been developed for the generator to improve
restoration performance by utilizing information from both time and frequency
domains.

  4. 4.

Unlike previous studies, this research focuses on restoring various types of
distortions, particularly those that are severely corrupted, while the SNR
randomly varies between -14 dB and 10 dB.

  5. 5.

To simulate real-world radar signals with any combination of artifacts and
severities, an extended BRSR dataset has been curated with paired clean and
corrupted signals and is publicly shared along with the source codes.

  6. 6.

The proposed approach has been evaluated on benchmark datasets, achieving an
unprecedented level of average SNR restoration performance ($>15$ dB for the
baseline and $>14$ dB for the BRSR dataset, respectively). Furthermore, the
proposed BRSR-OpGAN encapsulates a compact generator model with an elegant
computational complexity, allowing real-time processing on a single CPU
computer.

The rest of the paper is organized as follows: Section II briefly outlines 1D
Self-ONNs, the proposed approach with BRSR-OpGAN model architecture, and the
experimental setup. Section III presents the baseline data set used and the
creation of a benchmark BRSR dataset. Section IV presents the results over a
wide range of restoration experiments and evaluates the performances achieved.
Finally, Section V concludes the paper and suggests avenues for future
research.

##  II Methodology

In this section, we begin by summarizing the fundamentals of Self-Organized
Operational Neural Networks (Self-ONNs) and their primary attributes.
Subsequently, we outline our novel method for utilizing 1D OpGANs for radar
signal restoration.

###  II-A 1D Self-Organized Operational Neural Networks

In this section, we briefly describe the main network characteristics of 1D
Self-ONNs.222The optimized PyTorch implementation of 1D Self-ONNs is publicly
shared at https://github.com/junaidmalik09/fastonn. While conventional CNNs
have linear convolution operation and ONNs have fixed (static) nodal operators
[24, 25, 26, 27], Self-ONNs with generative neurons can have any arbitrary
nodal function, $\psi$, (including possibly standard types such as linear and
harmonic functions) optimized for each kernel element and for each connection.
Clearly, Self-ONNs offer greater operational flexibility and diversity, as
they can form optimized nodal operator functions during training to maximize
the learning performance.

Each kernel element of the Self-ONN performs a nonlinear transformation,
$\psi$, whose function can be expressed using the Taylor series near the
origin (a=0),

$\psi(x)=\sum_{n=0}^{\infty}\frac{\psi^{(n)}(0)}{n!}x^{n}$ (1)

The $Q^{th}$ order truncated approximation, formally known as the Taylor
polynomial, takes the form of the following finite summation:

$\psi(x)^{(Q)}=\sum_{n=0}^{Q}\frac{\psi^{(n)}(0)}{n!}x^{n}$ (2)

Any nonlinear function $\psi(x)$ near zero can be approximated by the above
formulation. When the activation function bounds the neuron’s input feature
maps in the vicinity of 0 (e.g., tanh), the formulation in Eq (2) can be
exploited to form a composite nodal operator where the power coefficients,
$\frac{\psi^{(n)}(0)}{n!}$, can be the parameters of the network learned
during training.

It was shown in [21] that the nodal operator of the kth generative neuron in
the $l^{th}$ layer can take the following general form:

$\widetilde{\psi_{k}^{l}}\left(w_{ik}^{l(Q)}(r),y_{i}^{l-1}(m+r)\right)\\\
=\sum_{q=1}^{Q}w_{ik}^{l(Q)}(r,q)\left(y_{i}^{l-1}(m+r)\right)^{q}$ (3)

Let $x_{ik}^{l}\in\mathbb{R}^{M}$ denote the contribution of the $i^{th}$
neuron at the $(l-1)^{th}$ layer to the input map of the $l^{th}$ layer, which
can be articulated as:

$\widetilde{x_{lk}^{l}}(m)=\sum_{r=0}^{K-1}\sum_{q=1}^{Q}w_{ik}^{l(Q)}(r,q)\left(y_{i}^{l-1}(m+r)\right)^{q}$
(4)

In this formulation, $y_{i}^{l-1}\in\mathbb{R}^{M}$ is the output map of the
$i^{th}$ neuron at the $(l-1)^{th}$ layer, and $w_{ik}^{l(Q)}$ represents a
learnable kernel matrix of dimensions $K\times Q$, i.e.,
$w_{ik}^{l(Q)}\in\mathbb{R}^{K\times Q}$, formed as,

$w_{ik}^{l(Q)}(r)=\left[w_{ik}^{l(Q)}(r,1),w_{ik}^{l(Q)}(r,2),\ldots,w_{ik}^{l(Q)}(Q)\right]$
(5)

The commutativity of summation operations in Eq. (4) allows for an alternative
representation:

$\widetilde{x_{lk}^{l}}(m)=\sum_{q=1}^{Q}\sum_{r=0}^{K-1}w_{ik}^{l(Q)}(r,q-1)\left(y_{i}^{l-1}(m+r)\right)^{q}$
(6)

which can be simplified as:

$\widetilde{x_{lk}^{l}}=\sum_{q=1}^{Q}\text{Conv1D}\left(w_{ik}^{l(Q)},\left(y_{i}^{l-1}\right)^{q}\right)$
(7)

Hence, the formulation can be accomplished by applying $Q$ 1D convolution
operations. Finally, the output of this neuron can be formulated as follows:

$x_{k}^{l}=b_{k}^{l}+\sum_{i=0}^{N_{(l-1)}}x_{ik}^{l}$ (8)

Figure 2: The proposed blind radar restoration approach using BRSR-OpGAN.
After training, the encoder and decoder networks are employed together as a
composite filter $G=f_{D}\cdot f_{E}$ for denoising unseen radar signals.

where $b_{k}^{l}$ is the bias associated with the neuron. The $0^{th}$ order
term (q=0), representing the DC bias, is omitted as the neuron’s learnable
bias parameter can compensate for each individual bias term. With a setting of
$Q=1$, a generative neuron simplifies to a convolutional neuron.

The efficient formulations of the forward propagation and Backpropagation (BP)
training in a raw-vectorized form can be found in[28].

###  II-B Proposed Approach

The proposed approach for blind radar signal restoration using the BRSR-OpGAN
model leverages a generative model architecture using paired corrupted
(received) and clean (transmitted) radar signals. The primary objective of
this study is to develop a supervised neural network adept at removing any
blend of artifacts from the received radar signals. Utilizing a paired signal
training dataset $\\{(\hat{x},x)_{i}\\}_{i=1}^{N}$, where $\hat{x}$ denotes
the distorted version and $x$ the clean radar signal for each frame, we aim to
fine-tune the network’s parameters to effectively map distorted inputs
$\hat{x}$ to their clean counterparts $x$. Once these parameters are
optimized, the generator network of the BRSR-OpGAN is capable of restoring the
radar signals. As detailed earlier, our approach integrates 1D Self-ONN layers
into the GAN’s generator and discriminator, specifically designed to process
complex valued radar signal waveforms. The overall framework of our proposed
radar signal restoration method is depicted in Figure 2.

The proposed BRSR-OpGAN model executes time-domain, segment-based restoration
on complex-valued radar signals, each with a length of $1024$ samples and $2$
channels where in-phase and quadrature (I/Q) samples are placed separately in
each channel. To ensure unbiased training across different artifact types and
severities, each segment is deliberately corrupted with a random blend of
artifacts, including AWGN, echo, and interference. Each artifact is randomized
at varying severity levels, controlled by randomly assigned weights as
detailed in Section III. The primary goal is for the trained model to
consistently transform a corrupted segment ($\hat{x}$) into a clean segment
($x$), independent of its content, type, and severity of each artifact present
as well as the modulation type.

As illustrated in Figure 2, a raw radar signal segment from each batch is
randomly selected as the input pair for the BRSR-OpGan. The normalization
process applied to each segment is volume-independent, ensuring the model’s
performance is not biased by signal amplitude variations. The normalization is
performed independently for each channel (I/Q) as follows:

$X_{N}^{s}(i)=\frac{X^{s}(i)}{X_{\text{max}}^{s}}$

where $X^{s}(i)$ represents the original signal value at the $i$-th point in
segment $s$, $X_{\text{max}}^{s}$ is the absolute maximum value in segment
$s$, and $X_{N}^{s}(i)$ is the normalized signal value. This formula ensures
that each segment is scaled relative to its absolute maximum value, thus
standardizing the input for more effective processing by the BRSR-OpGAN.

####  II-B1 Model Architecture

The proposed approach consists of two main networks. The first two sub-
networks, the encoder $f_{E}$ and the decoder $f_{D}$, together constitute the
generator network $G$. These are highlighted in grey and brown, respectively,
in Figure 2 and are integral components of the final network, which is used to
restore the signals at inference time. The last network, highlighted in
orange, is the discriminator network $D$.

Both generator and discriminator networks are 1-D Self-ONNs. Furthermore, the
encoder network is specifically designed to handle a 2-channel Radar signal,
accommodating the real and imaginary parts of complex-valued radar signals.
This design allows us to take advantage of the spatial information from each
channel, leading to better noise reduction and signal clarity while preserving
the radar components’ morphology.

The encoder network $f_{E}$ comprises five Residual downsampling blocks
(ResDownBlock) blocks, each executing a sequence of four operations. These
operations include a residual block utilizing 1-D Self-ONNs, instance
normalization, downsampling and a hyperbolic tangent (tanh) activation
function. The encoder processes the corrupted radar signal $\hat{x}$,
compressing it into a lower-dimensional vector $\hat{z}$, which captures the
essential features of $x$ and is thus referred to as the latent vector of $x$.
This compression indicates that $f_{E}$ retains only the useful features of
the signal, effectively filtering out noise.

Conversely, the decoder network $f_{D}$ upscales the latent representation
$\hat{z}$ back to a signal $\tilde{x}$ that matches the dimensions of $x$,
facilitating a direct comparison. It consists of six blocks, where the first
five are residual downsampling blocks (ResDownBlock) that replicate the
sequence of operations found in the encoder blocks, except that the
downsampling is substituted with an upsampling operation. The final block of
the decoder is a layer that performs a convolution operation without any
activation function or normalization operation. The generated signal
$\tilde{x}$, along with the clean signal $x$, is fed into the discriminator,
which assists the decoder in enhancing the signal quality by discriminating
between the generated and clean signals, ultimately aiming to fake the
discriminator.

The discriminator network $D$ consists of six ResDownBlock, each primarily
performing a downsampling operation using residual blocks similar to those in
the decoder’s downsampling blocks, except for the last one. The final block is
the classification block, as it classifies the signal $\tilde{x}$ generated by
the decoder against the clean signal $x$. This classification block comprises
an adaptive average pooling layer followed by a dense layer, which categorizes
the output as either real or fake. This setup is essential for the adversarial
learning component of the proposed approach, where $D$ actively challenges $G$
to produce increasingly realistic outputs.

Figure 3 depicts the detailed architectures of the encoder, decoder, and
discriminator networks. This illustration highlights that each ResDownBlock
block in the encoder and its corresponding ResUpBlock block in the decoder are
interconnected through dashed arrows. These dashed arrows represent skip
connections, which are crucial for mitigating the problem of vanishing
gradients often encountered in GAN architectures such as the one proposed. The
figure also displays the input dimensionality of features at each layer, as
well as the number of filters/neurons for each layer, denoted by $N$.
Additionally, some specific layers incorporate dropout (with a rate of 0.25),
indicated as $d=True$. The kernel sizes in the 1D Self-ONN layers are
uniformly set to $1\times 3$.

Figure 3: The generator and discriminator architectures of the BRSR-OpGAN. The
ResDownBlock, ResUpBlock, and ClassificationBlock components are also
detailed.

####  II-B2 Objective Function

Our approach utilizes an objective function similar to conditional Generative
Adversarial Networks (cGANs) for image-to-image translation [29], tailored
explicitly for training with paired noisy and clean radar signals. This
structure facilitates a cGAN setup where the generator (denoiser) is trained
to generate outputs that depend on the noisy inputs, intending to match the
clean reference signals. This conditioning ensures that the generated outputs
are statistically plausible and specific reconstructions of the input noise
profiles.

A key innovation in our model is the inclusion of both time and frequency
domain losses in the reconstruction loss component, which provides a more
holistic fidelity assessment than standard approaches that only focus on the
time domain. This dual-domain strategy allows the model to correct
discrepancies effectively across both temporal and spectral characteristics,
which is crucial for maintaining the integrity of radar signals.

This formulation ensures that the generator is trained to produce outputs that
are both realistic (as judged by the discriminator) and accurate (as measured
by the reconstruction error). The objective function for training our model is
formulated as follows:

$L_{tot}(G,D,\hat{x},x)=\alpha L_{adv}(G,D,\hat{x},x)\\\ +\beta\cdot
L_{time}(G,D,\hat{x},x)+\gamma\cdot L_{freq}(G,D,\hat{x},x)$ (9)

where $L_{tot}$ is the total loss for the generator, $L_{adv}$ represents the
adversarial loss, which measures how indistinguishable the denoised outputs
are from real, clean signals in terms of adversarial criteria, $L_{time}$
represents the time-domain reconstruction loss, specifically an L1 loss that
measures the mean absolute difference between the clean and the restored
signals, $L_{freq}$ is the frequency-domain reconstruction loss, involving the
L1 loss between the normalized spectrograms of the clean and restored signals
obtained via Short-Time Fourier Transform (STFT).

The coefficients $\alpha$, $\beta$, and $\gamma$ are tuning parameters that
balance the relative importance of each loss component within the overall
objective function. These parameters are adjustable and should be optimized
based on empirical performance during validation phases to ensure the best
fidelity and perceptual quality combination in the restored radar signals.

In GAN training, the generator (G) and discriminator (D) networks are trained
simultaneously in an adversarial manner. Specifically, the weights are updated
based on competing objectives for the generator and discriminator: the
generator seeks to minimize the loss function $L_{A}(G,D,\hat{x},x)$, while
the discriminator seeks to maximize the corresponding loss function
$L_{A}(G,D,\hat{x},x)$. In this setup, the generator is tasked with
transforming the corrupted (received) signal $\hat{x}$ into the target
(transmitted) version $\tilde{x}$. $x$. The adversarial training ensures that
the generator produces highly accurate and indistinguishable signal from the
target signal over time.

####  II-B3 Training Procedure of the Generator Networks

In the BRSR-OpGAN model, we focus on a structured training approach that
leverages adversarial training alongside domain-specific reconstruction
metrics to optimize performance. Here, we define the adversarial loss and
dual-domain reconstruction losses utilized in our model.

Adversarial Loss: This component encourages the generator to produce outputs
that the discriminator $D$ will classify as real (close to clean signals).
This loss is designed to decrease the discriminator’s ability to distinguish
generated signals from real signals:

$L^{\star}_{adv}(D,\tilde{x})=(D(\tilde{x})-1)^{2}$ (10)

Here, the generator $G$ tries to minimize this loss by generating signals
$(\tilde{x})$ that are classified by $D$ as real, thus aiming for
$D(G(\hat{x}))$ to be close to 1.

Reconstruction Loss: To ensure that the generated signals can fake the
discriminator and accurate reconstructions of the target signals can be
accomplished, we integrate both time-domain and frequency-domain metrics based
on the L1 norm. This dual-domain approach ensures content accuracy across
different representations of the signal.

Time Domain Reconstruction Loss: This loss measures the accuracy of the
generated signal $\tilde{x}$ in replicating the real, clean signal $x$ in the
time domain:

$L^{\star}_{\text{time}}(\tilde{x},x)=\|\tilde{x}-x\|_{1}$ (11)

where $\|\cdot\|_{1}$ denotes the L1 norm, which is the sum of the absolute
differences between the generated signal and the true clean signal.

Frequency Domain Performance Measurement: In addition to the time domain loss,
this component evaluates the accuracy of the generated signal in the frequency
domain:

$L^{\star}_{\text{freq}}(\tilde{x},x)=\left\|S(\tilde{x})-S(x)\right\|_{1}$
(12)

Where $S(\cdot)$ represents the spectrogram obtained by the squared magnitude
of STFT, which converts the time domain signals into their frequency
components. The L1 norm is used to measure the absolute differences between
the spectrograms of the generated and the real (target) signals.

The N-point discrete Short-Time Fourier Transform (STFT) is computed as
follows:

$\text{STFT}[X,w,n]=X(n,w)=\sum_{m}X[m]W[n-m]e^{-jwm}$ (13)

where the N-point discrete STFT, which is complex-valued, is obtained by
sampling at each discrete radial frequency, $w=\frac{2\pi k}{N}$.

The spectrogram $S(X)$ is then calculated as:

$S(X)=\left|\text{STFT}(X)\right|^{2}$ (14)

Algorithm 1 Training BRSR-OpGAN Model

1:Input: Training samples $\\{\hat{x}_{i},x_{i}\\}_{i=1}^{N}$, balance
coefficients $\alpha$, $\beta$, maximum iterations maxIter, learning rates
$\eta_{G}$, $\eta_{D}$.

2:Output: Trained generator and discriminator weights:
$\Theta_{G},\Theta_{D}$.

3:Initialize trainable parameters, $\Theta_{G},\Theta_{D}$;

4:Normalize the training samples $\hat{x}_{i}$ and $x_{i}$;

5:while $t<\text{maxIter}$ do

6: Sample a batch $(\hat{x},x)$ from the training dataset; $\triangleright$
Generator Training

7: Generate denoised output $\tilde{x}=G(\hat{x})$;

8: Compute: $L_{1}=L^{\star}_{adv}(D,\tilde{x})$ (10);

9: Compute: $L_{2}=L^{\star}_{\text{time}}(\tilde{x},x)$ (11);

10: Compute: $L_{3}=L^{\star}_{\text{freq}}(\tilde{x},x)$ (12)

11: Compute total generator loss:  $L_{G}=L_{1}+\alpha L_{2}+\beta L_{3}$

12: Update generator weights $\Theta_{G}$ using Adam optimizer to minimize
$L_{G}$. $\triangleright$ Discriminator Training

13: Recompute $\tilde{x}=G(\hat{x})$ using updated $\Theta_{G}$.

14: Compute: $L_{D}=L_{adv}(D,\tilde{x},x)$ (16)

15: Update discriminator weights $\Theta_{D}$ using Adam optimizer to minimize
$L_{D}$.

16: $t\leftarrow t+1$

17:end while

18:return $\Theta_{G},\Theta_{D}$

In the generator training, the minimized total loss $L_{G}$ is the summation
of the losses computed in (10), (11), (12), expressed as:

$L_{G}=(D(\tilde{x})-1)^{2}+\lambda_{\text{time}}\cdot\|\tilde{x}-x\|_{1}\\\
+\lambda_{\text{freq}}\cdot\left\|S(\tilde{x})-S(x)\right\|_{1}$ (15)

####  II-B4 Training the Discriminator Networks

The training of the discriminator networks in the BRSR-OpGAN model adheres to
the standard protocol used in generative adversarial networks (GANs). Unlike
traditional GANs, which employ cross-entropy, we utilize Mean Squared Error
(MSE) to calculate the adversarial loss, effectively addressing the vanishing
gradient problem often encountered during the GAN training. For each pair of
signals, consisting of a clean signal $x$ and its corresponding noisy version
$\hat{x}$, the adversarial losses are computed as follows:

$L_{adv}(D,\hat{x},x)=(D(x)-1)^{2}+(D(\tilde{x}))^{2}$ (16)

In the following step, the discriminator weights are calculated by minimizing
$L_{D}=L_{adv}(D,\hat{x},x)$. In this formulation, $D(x)$ is the
discriminator’s assessment of the authenticity of the clean signal $x$, which
should ideally approach a value of 1, indicating real. Conversely,
$D(\tilde{x})$ represents the discriminator’s evaluation of the denoised
signal produced by the generator $G$, with the aim of this output approaching
0, indicating fake. This setup ensures that the discriminator is tuned to
distinguish effectively between the clean and restored signals.

It is important to note that the generator and discriminator are trained
adversarially, as described in Algorithm1. One training iteration updates the
generator model to fake the discriminator by forcing it to output 1 when fed
with the restored signals. This behavior results from the minimization of Eq.
(10). Using the updated generator model, the restored signals are then
computed again. The discriminators are then updated to produce the output of 1
for the clean signals, i.e., minimizing Eq. (16).

###  II-C Experimental Setup

Throughout our experiments, we adopt a training scheme with a maximum of 1000
back-propagation (BP) epochs with a batch size of 64. The Adam optimizer is
employed, with the generator and discriminator’s initial learning rates set at
$5\cdot 10^{-4}$. Furthermore, we set the loss weights $\lambda_{\text{time}}$
and $\lambda_{\text{freq}}$ in Eq. (15) to 100 and 200, respectively. We
maintained uniformity in network architecture and training parameter settings
across all radar restoration experiments on both datasets. The order of the
Taylor polynomials in all Self-ONN layers, Q, is set as 3 for all BRSR-OpGAN
configurations.

To implement the proposed BRSR-OpGAN architecture, we utilized the FastONN
library [28], built upon Python and PyTorch. This setup ensures consistency
and reliability across our experiments, facilitating robust evaluation and
comparison of results.

Figure 4: The illustration of extended real-world BRSR dataset generation with
a random choice of artifacts and their severities.

##  III Signal Modeling and Dataset Generation

As discussed earlier, real-world radar signals are corrupted by various
artifacts and noise when received, severely impacting their quality and
interpretability. Understanding and modeling these distortions accurately is
essential for developing and testing signal restoration algorithms under real-
world conditions. This section presents the baseline dataset and the
methodology for creating an extended radar dataset that mimics real-world
situations by incorporating various artifacts with randomized parameters.

###  III-A Major Artifact Types of the Received Radar Signals

The received radar signals are susceptible to various artifacts that can
significantly degrade their quality and interpretability. This section
outlines the major artifacts encountered in radar signal processing:

####  III-A1 Additive White Gaussian Noise (AWGN)

Additive White Gaussian Noise (AWGN) is characterized by a constant power
spectral density (PSD) across all frequencies. This type of noise can be
mathematically modeled as:

$\hat{x}(t)=x(t)+n(t)$ (17)

where $\hat{x}(t)$ represents the received signal, $x(t)$ is the original
radar signal, and $n(t)$ is the AWGN. The noise $n(t)$ follows a Gaussian
distribution with zero mean and variance $\sigma^{2}$:

####  III-A2 Echo Signal Distortion

Echo signal distortion occurs when a signal overlaps with its delayed replica:

$\hat{x}(t)=x(t)+\alpha x(t-\tau)$ (18)

where $\alpha$ is the attenuation factor, and $\tau$ is the delay of the echo.

####  III-A3 Interference

Interference arises when another signal interferes or disrupts the original
radar signal:

$\hat{x}(t)=x(t)+i(t)$ (19)

where $i(t)$ is the interfering signal.

####  III-A4 Composite Signal Distortion

This complex model combines all major artifacts with randomly assigned weights
based on the desired SNR value as follows:

$\hat{x}(t)=x(t)+w_{1}\cdot n(t)+w_{2}\cdot x(t-\tau)+w_{3}\cdot i(t)$ (20)

###  III-B Baseline Dataset

As part of our study, we utilize the 12-class Signal Dataset proposed by the
authors in [9] as the baseline dataset. This dataset has a range of SNR from
-14 dB to 10 dB, with steps of 2 dB. Each signal is corrupted by adding AWGN
as the only artifact, resulting in the generation of 400 samples at each SNR
level with a total number of 62,400 samples. The dataset is divided into an
8:2 ratio, with 80% of the samples allocated for training and 20% for
validation. To ensure the reliability of the training results, 150 test
samples are generated for each signal at each SNR, resulting in a total of
23,400 signals for testing.

###  III-C BRSR Benchmark Dataset Generation

In this study, we aim to generate radar modulation samples that simulate the
complex and varied conditions encountered in real-world radar environments.
Figure 4 illustrates the methodology used to create the BRSR dataset that can
accurately mimic actual received radar signals corrupted by any blend of
common artifacts. To accomplish this, we corrupt clean (transmitted) radar
signals with three major artifact types: Additive White Gaussian Noise (AWGN),
Unwanted Echo, and Interference.

Each artifact type (AWGN, Echo, Interference) is additive, allowing the use of
random weights to control its impact. This enables the creation of diverse
scenarios by varying the severity of each artifact. Additionally, scenarios
with either single or dual artifacts active are included to represent
conditions where one or more types of distortions are absent so that a wide
range of real-world signal corruptions can be simulated.

We utilize 85,800 clean data samples from the baseline dataset to generate the
BRSR dataset, which is also split into training, validation, and test sets,
maintaining the same proportions as the baseline dataset. Each sample consists
of a 2x1024 segment, with two channels representing the real and imaginary
parts of the complex-valued radar signals. All signals are processed through
the real-world corrupted radar generation setup (illustrated in Figure 4). The
final dataset includes 85,800 samples with a mix of all artifact types,
referred to as the BRSR dataset throughout this paper. Henceforth, this
benchmark dataset can serve as a robust platform for developing and evaluating
radar signal restoration techniques.

Algorithm 2 BRSR Dataset Generation with Randomized Artifacts

1:Input: Set of clean radar signals $\\{x(t)\\}$, desired SNR range $[-14,10]$
dB, echo delay range $[\tau_{\min},\tau_{\max}]$, set of interference signals
$\\{i_{k}(t)\\}$

2:Output: Pairs of clean and distorted signals $\\{(x(t),\hat{x}(t))\\}$

3:for each clean radar signal $x(t)$ do

4: Randomly select a desired SNR value $\text{SNR}_{\text{rand}}\in[-14,10]$
dB;

5: Compute desired noise power $P_{n}$ using (21);

6: Randomly choose a combination of artifacts
$A\subset\\{\text{AWGN},\text{Echo},\text{Interference}\\}$;

7: for each artifact $a_{i}\in A$ do

8: Generate artifact using (17), (18), (19);

9: end for

10: Assign random pseudo weights using (22);

11: Calculate actual weights using (23);

12: Combine artifacts to form the distorted signal using (24):
$\hat{x}(t)=x(t)+\sum_{i=1}^{n}w_{i}\cdot a_{i}(t)$

13: Save the pair $(x(t),\hat{x}(t))$

14:end for

###  III-D Mathematical Modeling

The generation of the BRSR dataset that simulates real-world radar signal
conditions involves introducing the described artifacts into clean radar
signals with randomized parameters. The algorithm for dataset creation is
presented in Algorithm 2. Given a clean radar signal $x(t)$ from the baseline
dataset, the process of the signal corruption with a blend of artifacts is
modeled as follows:

  1. 1.

Random SNR Assignment: For each selected signal $x(t)$, the desired SNR level
is randomly chosen within the range of $-14$ dB to $10$ dB, represented as
$\text{SNR}_{\text{rand}}$.

  2. 2.

Calculation of Desired Noise Power: Desired noise power is calculated as

$P_{\text{rand}}=\frac{P_{s}}{10^{\text{SNR}_{\text{rand}}/10}}$ (21)

where $P_{s}$ is the power of the clean signal $x(t)$.

  3. 3.

Selection of Artifacts: The dataset generation algorithm randomly selects a
combination of artifacts from the set {AWGN, Echo, Intereference}, resulting
in seven possible scenarios, including individual artifacts, pairs, and the
trio.

  4. 4.

Echo Delay: For echo artifacts, a delay parameter $\tau$ is randomly set
within the predefined range, [128, 512].

  5. 5.

Interference Signal Selection: A signal from a predefined set of 100 signals
is randomly chosen to simulate interference.

  6. 6.

Artifact Weighting: When multiple artifacts are selected, random pseudo
weights $\tilde{w}_{i}$ are assigned to each artifact such that

$\sum_{i=1}^{n}\tilde{w}_{i}=1$ (22)

where $\tilde{w}_{i}$ is set to zero for the artifacts not present in the
selected set.

  7. 7.

Calculation of Actual Weights: Actual weights are calculated using

$w_{i}=\tilde{w}_{i}\times\sqrt{\frac{P_{\text{rand}}}{P_{i}}}$ (23)

where $P_{i}$ is the actual noise power of the corresponding artifact. This
equation ensures a proportional blend of artifacts based on the overall
$\text{SNR}_{\text{rand}}$.

  8. 8.

Distorted Signal: The final degraded signal $\hat{x}(t)$ is obtained by
applying the selected artifacts to the clean signal $x(t)$ according to their
respective weights and parameters. This can be mathematically expressed as:

$\hat{x}(t)=x(t)+\sum_{i=1}^{n}w_{i}\cdot a_{i}(t)$ (24)

where $a_{i}(t)$ represents the individual artifacts.

This process ensures the generation of a comprehensive dataset that accurately
represents the variety and unpredictability of real-world radar signal
conditions. As a result, each signal in this dataset is subjected to a unique
combination of artifacts.

##  IV Experimental Results

In this section, we first present the performance evaluation metrics. Then, we
evaluate the proposed BRSR-OpGAN model’s performance over the baseline and
extended BRSR datasets. Subsequently, a comparative analysis is presented,
followed by a thorough discussion of the results obtained. We conclude this
section with both quantitative and qualitative assessments and a detailed
examination of the computational efficiency of the proposed approach.

###  IV-A Evaluation Metrics

The performance of signal restoration can be evaluated using several metrics,
including SNR, Peak Signal-to-Noise Ratio (PSNR), and Mean Squared Error
(MSE). These metrics provide a quantitative assessment of the restoration
quality. The definitions and mathematical formulations of these metrics are as
follows:

####  IV-A1 SNR

The SNR is used to measure the performance of signal restoration. It is
defined as:

$\text{SNR}_{\text{out}}=10\log_{10}\left(\frac{\sum_{i=1}^{N}x(t_{i})^{2}}{\sum_{i=1}^{N}[x(t_{i})-\tilde{x}(t_{i})]^{2}}\right)$
(25)

where $x(t)$ and $\tilde{x}(t)$ represent the original clean signal and the
restored signal, respectively, and $N$ is the number of sampling points.

####  IV-A2 PSNR

The PSNR is another metric for assessing signal restoration quality,
particularly in peak error. It is defined as:

$\text{PSNR}_{\text{out}}=10\log_{10}\left(\frac{P_{\text{max}}^{2}}{\frac{1}{N}\sum_{i=1}^{N}[x(t_{i})-\tilde{x}(t_{i})]^{2}}\right)$
(26)

where $P_{\text{max}}$ is the maximum possible signal value in the original
signal $x(t)$, and $\frac{1}{N}\sum_{i=1}^{N}[x(t_{i})-\tilde{x}(t_{i})]^{2}$
represents the mean squared error (MSE) between the clean and the restored
signals.

####  IV-A3 MSE

The MSE measures the average squared difference between the original
(transmitted) and restored signals. It is expressed by:

$\text{MSE}=\frac{1}{N}\sum_{i=1}^{N}[x(t_{i})-\tilde{x}(t_{i})]^{2}$ (27)

The standard metrics, SNR, PSNR, and MSE, collectively provide a comprehensive
evaluation of the signal restoration performance, allowing us to quantify the
performance of the restoration algorithms under various conditions.

TABLE I: SNR values by different restoration methods over the baseline
dataset.

Restoration Algorithms | Average SNR (dB)
---|---
Corrupted Signal (Reference) | -14 | -8 | -2 | 4 | 10
EEMD | -11.57 | -5.53 | 0.21 | 4.23 | 10.79
VMD-WT | -2.42 | -0.24 | 3.75 | 8.29 | 13.03
VMD-LMD-WT | -2.24 | 0.48 | 4.77 | 9.51 | 14.31
CGAN | 1.29 | 6.00 | 11.40 | 15.71 | 18.64
BRSR-OpGAN | 1.31 | 6.72 | 12.11 | 16.53 | 19.49

###  IV-B Quantitative Evaluations

The quantitative evaluations aim to assess the effectiveness of the BRSR-OpGAN
model in enhancing radar signal quality. We first utilize a baseline dataset
involving radar signals corrupted only with AWGN to compare our model with the
recent denoising methods. This setup allows us to benchmark our framework
against traditional methods using published datasets under controlled
conditions, highlighting the potential improvements offered by our approach.
As the main objective of this study, we then introduce results over the
extended BRSR dataset comprising more challenging real-world radar data with
various artifacts to test the true performance of the proposed blind
restoration approach.

For comparative evaluations and benchmarking, over the baseline dataset we
used three recent denoising algorithms: ensemble empirical mode decomposition
(EEMD) [30], the combination of variational mode decomposition and wavelet
denoising algorithm (VMD-WT) [31], and a multilayer denoising method termed as
VMD-LMD-WT [9]. Despite the existence of numerous GAN-based methods for radar
signal enhancement, most focus primarily on denoising and often assume a
limited number of degradation sources with predefined SNR levels. A
significant issue in the current literature is the lack of publicly available
datasets, as most existing approaches do not share their data. Furthermore,
these studies use a single noise type such as AWGN and evaluate their results
based on the classification accuracy before and after restoration rather than
the restoration performance. Consequently, it was challenging to utilize pre-
trained weights, particularly for the BRSR dataset with an unknown blend of
several artifacts. To address this limitation and facilitate comparative
evaluations, we also conducted experiments using the conventional (CNN-based)
GAN (CNN-GAN) with the same architecture as BRSR-OpGAN. This allows us to
incorporate deep learning-based methods into our results, comprehensively
evaluating radar signal restoration techniques.

In TABLE I, we present the performance of the proposed BRSR-OpGAN against the
three recent denoising methods on the baseline dataset. Average SNR values are
computed over all independent test samples at 5 different SNR levels. As the
input SNR increases, the improvement in SNR generally decreases. This is
typical in denoising applications because higher initial SNR values leave less
room for enhancement. Notably, at an SNR of -14 dB, the SNR improvement peaks.
At this level, the proposed BRSR-OpGAN method enhances the SNR by 15.31 dB,
significantly surpassing the performance of the other recent denoising
methods. It is interesting that, three competing methods show limited
performance improvements despite the fact that AWGN noise is the only artifact
corrupting the radar signals at predefined SNR levels, which aligns with our
earlier discussion. This performance confirms our approach’s efficacy in
managing a comparatively simpler noise scenario and sets a solid baseline for
further evaluations.

TABLE II: Performance comparison of different BRSR-OpGAN configurations

Restoration Algorithms | Average SNR (dB) | Average PSNR (dB) | Average MSE
---|---|---|---
Corrupted Signal (Reference) | -1.94 | -1.27 | 9.65
VMD-LMD-WT [9] | 3.36 | 4.04 | 3.25
CNN-GAN (Time Domain) | 8.86 | 9.54 | 0.39
CNN-GAN (Dual Domain) | 9.04 | 9.71 | 0.42
BRSR-OpGAN (Q=3, Time Domain) | 9.53 | 10.20 | 0.34
BRSR-OpGAN (Q=3, Dual Domain) | 10.30 | 10.98 | 0.30
BRSR-OpGAN (Q=3, Dual Domain, 2nd Pass) | 12.36 | 13.03 | 0.22

Over the BRSR dataset, Table II summarizes the average SNR, PSNR, and MSE
values for both corrupted and restored signals under different restoration
algorithms. As discussed earlier, these are realistic restoration results
since the restoration methods are blindly applied to the radar signals
corrupted by unknown artifacts with randomized types and severities.

Starting with multiple decomposition-based denoising (VMD-LMD-WT), we observe
minimal improvement compared to the noisy reference signal. However, the CNN-
GAN only with the temporal loss function shows significant improvement in the
SNR, PSNR, and MSE metrics. This model serves as a solid baseline, achieving
an average SNR of 8.86 dB and an average PSNR of 9.54 dB.

Further enhancements are established when using the CNN-GAN model with dual
domain loss (both temporal and spectral losses). The average SNR increases to
9.04 dB, and the PSNR reaches 9.71 dB, with a slightly increased MSE of 0.42.
This highlights the advantage of leveraging time and frequency domain
information during training.

The proposed BRSR-OpGAN using only the time domain loss further improves
performance compared to the CNN-GAN. The average SNR and PSNR values increase
to 9.53 dB and 10.20 dB, respectively, and the MSE further decreases to 0.34.
However, when the BRSR-OpGAN is trained with dual domain loss, it achieves
state-of-the-art (SOTA) performance levels, with an average SNR of 10.30 dB,
PSNR of 10.98 dB, and an MSE of 0.30.

The most significant improvements are observed with the BRSR-OpGAN (Q=3, Dual
Domain, 2nd Pass) configuration. By utilizing a second-pass BRSR-OpGAN
training, where the restored signal from the first pass is used as the input,
the second restoration pass by the generator of the second BRSR-OpGAN achieves
the highest average SNR of 12.36 dB and PSNR of 13.03 dB, with the lowest MSE
of 0.22. Such a two-pass restoration approach significantly boosts the model’s
performance and achieves the highest restoration performance of above 14 dB
average SNR improvement.

Figure 5 illustrates the SNR of all test data signals before and after
restoration using the BRSR-OpGAN model. The joint distribution highlights the
relationship between corrupted SNR values (x-axis) and restored SNR values
(y-axis). Notably, even for the corrupted signals with the lowest SNR values,
down to -14 dB with a randomized blend of artifacts, the model shows a
significant improvement in all SNR levels. Additionally, the marginal
histograms on the top and right sides of the plot display the distribution of
corrupted and restored SNR values, respectively. The restored SNR values are
skewed towards higher ranges, confirming the model’s robustness in enhancing
signal quality under challenging conditions.

Figure 5: Joint distribution of corrupted vs restored SNR. Figure 6: Histogram
of SNR improvements.

Figure 6 presents the histogram of the SNR improvement achieved by the BRSR-
OpGAN model across the test dataset. The SNR improvement values follow a
roughly normal distribution centered around a mean improvement of 14.30 dB, as
indicated by the red dashed line, with the median slightly lower at 13.66 dB,
marked by the green dashed line. This close alignment suggests a symmetrical
distribution of SNR improvement values. The peak of the histogram occurs
around the mean, showing that most test signals experienced an SNR improvement
of approximately 14 dB. The spread of the data, with improvements ranging from
slightly negative values to over 30 dB, demonstrates the model’s robustness in
handling a wide variety of corrupted signals. While a few instances show
negative SNR improvement, indicating some severe artifacts corrupting the
signal beyond any restoration; however, their occurrence is minimal. The
substantial mean and median SNR improvements highlight the model’s
effectiveness in enhancing signal quality, confirming its potential for
reliable blind radar signal restoration in practical applications. Overall,
the significant SNR enhancements validate the BRSR-OpGAN model’s capability to
restore radar signals effectively in real-world scenarios.

These findings underscore the importance of dual-domain loss function during
training and iterative enhancement techniques in improving radar signal
restoration. The 2-pass restoration approach, particularly, shows substantial
promise in achieving superior restoration quality, setting a new benchmark for
future research in this domain. The BRSR-OpGAN (Q=3, Dual Domain, 2nd Pass)
model is the most effective configuration, providing a robust high-fidelity
radar signal restoration framework.

###  IV-C Qualitative Evaluation

Figures 7 to 10 illustrate examples of radar signal restoration by BRSR-OpGAN.
The top plot displays the original (transmitted) radar signal $x(t)$ followed
by a sequence of three plots, each exhibiting individual components of the
major artifacts, AWGN, Echo, and Interference. The fifth plot shows the
overall distorted signal $\tilde{x}(t)$, which combines the artifacts, and the
bottom plot demonstrates the restored signal $\hat{x}(t)$ produced by the
BRSR-OpGAN model.

###  IV-D Computational Complexity

To conduct computational complexity analysis, we calculate the total number of
parameters (PARs) and the inference time required to restore a radar signal
(2x1024 samples) in a batch size of 64 for each network configuration. The
detailed formulations for PARs calculations in Self-Organizing Neural Networks
(Self-ONNs) are provided in [28]. All experiments were executed on a system
equipped with a 2.4 GHz Intel Core i7 processor, 32 GB of RAM, and NVIDIA
GeForce RTX A1000 6 GB graphics card. The BRSR-OpGAN was implemented using
Python with the PyTorch library, and the classifier’s training and testing
phases were performed on GPU cores. Notably, the inference time was also
measured on a single CPU. As shown in TABLE 4, the BRSR-OpGAN network requires
more multiply-accumulate operations. However, a higher percentage of these
operations are independent and, thus, parallelizable. Therefore, an efficient
implementation using the vectorized formulation [21] results in an actual
running time that is only marginally higher than the equivalent CNN-based GAN,
which is negligible considering the crucial gain in the restoration
performance.

TABLE III: Models and Their Computational Complexities

Model | Parameters (PARs) (K) | Inference Time (ms)
---|---|---
| G | D | Total | CPU | GPU
BRSR-OpGAN | 275.52 | 82.61 | 358.13 | 0.710 | 0.039
CNN-GAN | 103.30 | 30.19 | 133.49 | 0.440 | 0.036

Figure 7: Sample radar signals corrupted with random blend artifacts and the
corresponding BRSR-OpGAN restored signal. Figure 8: Sample radar signals
corrupted with random blend artifacts and the corresponding BRSR-OpGAN
restored signal. Figure 9: Sample radar signals corrupted with random blend
artifacts and the corresponding BRSR-OpGAN restored signal. Figure 10: Sample
radar signals corrupted with random blend artifacts and the corresponding
BRSR-OpGAN restored signal.

##  V Conclusion

The main challenge in real-world radar signal restoration is the degradation
of acquired radar signals by various artifacts such as echo, sensor noise,
signal jamming, and atmospheric disturbances. These artifacts can vary greatly
in type, intensity, and duration, leading to dynamically and non-uniformly
corrupted signals that are difficult to clean effectively. Despite extensive
research in radar signal denoising and interference mitigation, there has not
been a comprehensive solution targeting these multifaceted challenges in blind
radar signal restoration.

In this context, this study introduces a pioneering approach that employs 1D
Operational GANs iteratively for the blind restoration of real-world radar
signals. The proposed model was evaluated using a benchmark radar signal
dataset and an extended BRSR dataset by incorporating all major artifact types
in a randomized manner, thus mimicking real-world scenarios. Our approach
diverges from traditional methods by not making any prior assumptions about
the type or severity of the artifacts corrupting the signal.

We developed the BRSR-OpGAN model as a compact, efficient end-to-end system
for radar signal restoration, introducing a unique loss function for the
generator that leverages both temporal and spectral characteristics of radar
signals. This enables our model to adeptly suppress various artifacts while
maintaining the integrity and characteristics of the original signal. Our
model demonstrates unprecedented restoration performance over the most recent
methods, indicating its potential for enhancing the clarity and usability of
severely corrupted radar signals across various applications.

Finally, the proposed approach has an elegant computational efficiency which
further underscores its effectiveness, making it suitable for real-time
applications on resource-constrained platforms. Future research will explore
further optimizations and extensions of the proposed Op-GAN model, i) to
incorporate the classification objective into the restoration to restore the
signals to maximize the classification performance, and ii) to handle even
more complex scenarios and improve robustness against a wider variety of
signal distortions.

## Acknowledgments

This work was supported by the Business Finland project AMaLIA and NSF Center
for Big Learning (CBL).

## References

  * [1] A. De Martino, _Introduction to modern EW systems_. Artech house, 2018.
  * [2] P. E. Pace, _Detecting and classifying low probability of intercept radar_. Artech house, 2009.
  * [3] Z. Zhou, G. Huang, and X. Wang, “Ensemble convolutional neural networks for automatic fusion recognition of multi-platform radar emitters,” _ETRI Journal_ , vol. 41, no. 6, pp. 750–759, 2019.
  * [4] T. R. Kishore and K. D. Rao, “Automatic intrapulse modulation classification of advanced lpi radar waveforms,” _IEEE Transactions on Aerospace and Electronic Systems_ , vol. 53, no. 2, pp. 901–914, 2017.
  * [5] Z. Seddighi, M. R. Ahmadzadeh, and M. R. Taban, “Radar signals classification using energy-time-frequency distribution features,” _IET Radar, Sonar & Navigation_, vol. 14, no. 5, pp. 707–715, 2020.
  * [6] M. Jan and D. Pietrow, “Artificial neural networks in the filtration of radiolocation information,” in _2020 IEEE 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET)_. IEEE, 2020, pp. 680–685.
  * [7] Z. Zhou, G. Huang, H. Chen, and J. Gao, “Automatic radar waveform recognition based on deep convolutional denoising auto-encoders,” _Circuits, Systems, and Signal Processing_ , vol. 37, pp. 4034–4048, 2018.
  * [8] M. Zhang, M. Diao, and L. Guo, “Convolutional neural networks for automatic cognitive radio waveform recognition,” _IEEE access_ , vol. 5, pp. 11 074–11 082, 2017.
  * [9] M. Jiang, F. Zhou, L. Shen, X. Wang, D. Quan, and N. Jin, “Multilayer decomposition denoising empowered cnn for radar signal modulation recognition,” _IEEE Access_ , 2024.
  * [10] S. Lee, J.-Y. Lee, and S.-C. Kim, “Mutual interference suppression using wavelet denoising in automotive fmcw radar systems,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 22, no. 2, pp. 887–897, 2019.
  * [11] J. Fuchs, A. Dubey, M. Lübke, R. Weigel, and F. Lurz, “Automotive radar interference mitigation using a convolutional autoencoder,” in _2020 IEEE International Radar Conference (RADAR)_. IEEE, 2020, pp. 315–320.
  * [12] C. Wang, J. Wang, and X. Zhang, “Automatic radar waveform recognition based on time-frequency analysis and convolutional neural network,” in _2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2017, pp. 2437–2441.
  * [13] S.-H. Kong, M. Kim, L. M. Hoang, and E. Kim, “Automatic lpi radar waveform recognition using cnn,” _Ieee Access_ , vol. 6, pp. 4207–4219, 2018.
  * [14] Z. Zhang, C. Wang, C. Gan, S. Sun, and M. Wang, “Automatic modulation classification using convolutional neural network with features fusion of spwvd and bjd,” _IEEE Transactions on Signal and Information Processing over Networks_ , vol. 5, no. 3, pp. 469–478, 2019.
  * [15] M. Du, P. Zhong, X. Cai, and D. Bi, “Dncnet: Deep radar signal denoising and recognition,” _IEEE Transactions on Aerospace and Electronic Systems_ , vol. 58, no. 4, pp. 3549–3562, 2022.
  * [16] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial nets,” _Advances in neural information processing systems_ , vol. 27, 2014.
  * [17] T. Ince, S. Kiranyaz, O. C. Devecioglu, M. S. Khan, M. Chowdhury, and M. Gabbouj, “Blind restoration of real-world audio by 1d operational gans,” _arXiv preprint arXiv:2212.14618_ , 2022.
  * [18] S. Kiranyaz, O. C. Devecioglu, T. Ince, J. Malik, M. Chowdhury, T. Hamid, R. Mazhar, A. Khandakar, A. Tahir, T. Rahman _et al._ , “Blind ecg restoration by operational cycle-gans,” _IEEE Transactions on Biomedical Engineering_ , vol. 69, no. 12, pp. 3572–3581, 2022.
  * [19] S. Kiranyaz, J. Malik, H. B. Abdallah, T. Ince, A. Iosifidis, and M. Gabbouj, “Self-organized operational neural networks with generative neurons,” _Neural Networks_ , vol. 140, pp. 294–308, 2021.
  * [20] M. U. Zahid, S. Kiranyaz, T. Ince, O. C. Devecioglu, M. E. Chowdhury, A. Khandakar, A. Tahir, and M. Gabbouj, “Robust r-peak detection in low-quality holter ecgs using 1d convolutional neural network,” _IEEE Transactions on Biomedical Engineering_ , vol. 69, no. 1, pp. 119–128, 2021.
  * [21] M. Gabbouj, S. Kiranyaz, J. Malik, M. U. Zahid, T. Ince, M. E. Chowdhury, A. Khandakar, and A. Tahir, “Robust peak detection for holter ecgs by self-organized operational neural networks,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 34, no. 11, pp. 9363–9374, 2022.
  * [22] M. U. Zahid, S. Kiranyaz, and M. Gabbouj, “Global ecg classification by self-operational neural networks with feature injection,” _IEEE Transactions on Biomedical Engineering_ , vol. 70, no. 1, pp. 205–215, 2022.
  * [23] “Datasets - deepsig.” [Online]. Available: https://www.deepsig.ai/datasets/
  * [24] S. Kiranyaz, T. Ince, A. Iosifidis, and M. Gabbouj, “Operational neural networks,” _Neural Computing and Applications_ , vol. 32, no. 11, pp. 6645–6668, 2020.
  * [25] S. Kiranyaz, J. Malik, H. B. Abdallah, T. Ince, A. Iosifidis, and M. Gabbouj, “Exploiting heterogeneity in operational neural networks by synaptic plasticity,” _Neural Computing and Applications_ , vol. 33, pp. 7997–8015, 2021.
  * [26] S. Kiranyaz, T. Ince, A. Iosifidis, and M. Gabbouj, “Generalized model of biological neural networks: Progressive operational perceptrons,” in _2017 International Joint Conference on Neural Networks (IJCNN)_. IEEE, 2017, pp. 2477–2485.
  * [27] ——, “Progressive operational perceptrons,” _Neurocomputing_ , vol. 224, pp. 142–154, 2017.
  * [28] J. Malik, S. Kiranyaz, and M. Gabbouj, “Fastonn–python based open-source gpu implementation for operational neural networks,” _arXiv preprint arXiv:2006.02267_ , 2020.
  * [29] P. Isola, J.-Y. Zhu, T. Zhou, and A. A. Efros, “Image-to-image translation with conditional adversarial networks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 1125–1134.
  * [30] Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise-assisted data analysis method,” _Advances in adaptive data analysis_ , vol. 1, no. 01, pp. 1–41, 2009.
  * [31] T. Qi, X. Wei, G. Feng, F. Zhang, D. Zhao, and J. Guo, “A method for reducing transient electromagnetic noise: Combination of variational mode decomposition and wavelet denoising algorithm,” _Measurement_ , vol. 198, p. 111420, 2022.