#  Hybrid-Layers Neural Network Architectures for Modeling the Self-
Interference in Full-Duplex Systems

Mohamed Elsayed, Student Member, IEEE, Ahmad A. Aziz El-Banna, Member, IEEE,
Octavia A. Dobre, Fellow, IEEE, Wanyi Shiu, and Peiwei Wang M. Elsayed, A. A.
A. El-Banna, and O. A. Dobre are with the Faculty of Engineering and Applied
Science, Memorial University, St. John’s, NL A1B 3X5, Canada (e-mail:
{memselim, aaelbanna, odobre}@mun.ca).A. A. A. El-Banna is also with the
Faculty of Engineering at Shoubra, Benha University, Banha, Egypt.W. Shiu and
P. Wang are with Huawei Canada Research Centre, Huawei Technologies Canada
Co., Ltd., Ottawa, ON K2K 3J1, Canada (e-mail: {wanyi.shiu,
peiwei.wang}@huawei.com).

###### Abstract

Full-duplex (FD) systems have been introduced to provide high data rates for
beyond fifth-generation wireless networks through simultaneous transmission of
information over the same frequency resources. However, the operation of FD
systems is practically limited by the self-interference (SI), and efficient SI
cancelers are sought to make the FD systems realizable. Typically, polynomial-
based cancelers are employed to mitigate the SI; nevertheless, they suffer
from high complexity. This article proposes two novel hybrid-layers neural
network (NN) architectures to cancel the SI with low complexity. The first
architecture is referred to as hybrid-convolutional recurrent NN (HCRNN),
whereas the second is termed as hybrid-convolutional recurrent dense NN
(HCRDNN). In contrast to the state-of-the-art NNs that employ dense or
recurrent layers for SI modeling, the proposed NNs exploit, in a novel manner,
a combination of different hidden layers (e.g., convolutional, recurrent,
and/or dense) in order to model the SI with lower computational complexity
than the polynomial and the state-of-the-art NN-based cancelers. The key idea
behind using hybrid layers is to build an NN model, which makes use of the
characteristics of the different layers employed in its architecture. More
specifically, in the HCRNN, a convolutional layer is employed to extract the
input data features using a reduced network scale. Moreover, a recurrent layer
is then applied to assist in learning the temporal behavior of the input
signal from the localized feature map of the convolutional layer. In the
HCRDNN, an additional dense layer is exploited to add another degree of
freedom for adapting the NN settings in order to achieve the best compromise
between the cancellation performance and computational complexity. The
complexity analysis of the proposed NN architectures is provided, and the
optimum settings for their training are selected. The simulation results
demonstrate that the proposed HCRNN and HCRDNN-based cancelers attain the same
cancellation of the polynomial and the state-of-the-art NN-based cancelers
with an astounding computational complexity reduction. Furthermore, the
proposed cancelers show high design flexibility for hardware implementation,
depending on the system demands.

###### Index Terms:

Full-duplex (FD) technology, self-interference (SI) modeling, convolutional
layer, recurrent layer, polynomial-based cancelers, NN-based cancelers,
complexity analysis.

##  I Introduction

Recently, the evolution of Internet-of-Everything, supporting massive
connectivity among billions of users and billions of devices, has imposed a
radical shift towards the next generation of wireless networks, such as beyond
the fifth-generation (B5G) [References]. These modern networks aim to provide
high reliability, low latency, and high data rates, in the order of tens
Gbits/s, to enable extended reality applications, live multimedia streaming,
and autonomous systems in smart cities and factories, such as drone swarms,
cars, and robotics [References], [References]. As such, to cater to this novel
breed of applications and support such a plethora of services, the next
generation of wireless systems should be inherently tailored to simultaneously
deliver higher data rates with lower communication delays for both uplink and
downlink.

In this regard, full-duplex (FD) has emerged as one of the key enabling
technologies for B5G wireless networks by providing high data rates through
simultaneous transmission of information over the same frequency resources
[References]-[References]. Efficient exploitation of the resources enables the
FD systems to meet the high quality-of-service requirements in terms of
spectral efficiency, which represents a major factor in designing B5G wireless
networks. On the other hand, the main challenge in implementing the FD systems
is the self-interference (SI), which comes out from the transmitter of the
same device on its own receiver [References]. This undesirable interference
significantly hinders the proliferation of FD systems in the next generation
of wireless networks [References]-[References].

Over the past decade, a flurry of research interest has been directed for
canceling the interference in FD systems to make them realizable [References],
[References]. Typically, canceling the SI can be implemented in analog radio
frequency (RF) and/or digital domains to bring the SI signal’s power down to
the receiver’s noise level. The analog RF suppression is implemented at the
very first stage of the receiver chain to refrain the SI signal from
saturating the analog components of the receiver, such as the low-noise
amplifier (LNA), variable-gain amplifier (VGA), and analog-to-digital
converter (ADC) [References]. In particular, the analog RF cancellation can be
classified into passive and active cancellations [References], [References].
Passive cancellation is implemented using techniques such as antenna
separation [References], circulators [References], polarized antennas
[References], and balanced hybrid-junction networks [References]. On the other
side, the active suppression is performed using analog circuits, which
generate a copy of the SI signal in order to be subtracted from the original
SI signal at the receiver chain [References]. In general, the analog
suppression techniques are insufficient to entirely remove the SI at the
receiver side, and a non-negligible residual SI still exists after the analog
cancellation process. Hence, digital cancellation approaches are utilized in
order to mitigate the residual interference [References].

Digital domain cancellation uses the same notion of active suppression where a
processed copy of the baseband transmitted signal is subtracted from the
residual SI signal, but in the digital domain [References]. In principle, the
digital cancelers could effectively eliminate this SI signal since it stems
from a transmit signal that is obviously known to the receiver. However, this
is not the case in practice, as the SI signal is significantly distorted by
the SI coupling channel and the impairments of the transceiver components,
such as non-linear distortion of the power amplifier (PA), in-phase and
quadrature-phase (IQ) imbalance of the mixer, phase noise of imperfect
transceiver’s oscillators, and digital-to-analog converter (DAC) and ADC’s
quantization noise [References].

In order to efficiently cancel the SI in the digital domain, the digital
cancelers should properly model the distortion incorporated into the input
signal due to the imperfection of the hardware and the SI channel. Generally,
modeling the transceiver’s impairments is based on the polynomial
approximation of the SI signal at the receiver side [References]. The
polynomial-based models have excellent modeling capabilities to mimic the SI
signal; however, they suffer from high complexity [References]. Accordingly,
low-complexity modeling approaches are sought for approximating the SI signal
in FD systems.

Applying neural networks (NNs) and deep learning has gained significant
momentum in the field of signal processing and wireless communications in the
last few years [References]-[References]. NNs have been recently employed to
replace the model-based approaches in numerous communication areas in order to
approximate the non-linearities with good performance and low implementation
complexity. For instance, NNs have brought breakthroughs in signal detection
[References], signal classification [References], channel estimation
[References], [References], channel equalization [References], channel coding
[References], PA modeling [References]-[References], digital pre-distortion
[References]-[References], and non-linearity compensation in optical fiber
systems [References].

In addition, there has been a surge of interest in applying NNs for SI
cancellation in FD systems [References]-[References]. More specifically, the
first attempt of using NNs for canceling the SI has been reported in
[References], where a real-valued time delay NN (RV-TDNN)111We note that the
RV feed-forward NN (RV-FFNN) in [References] has a similar structure to the
RV-TDNN in [References] since both employ the input signal’s buffered samples
at the input layer. Henceforth, we will use RV-TDNN instead of RV-FFNN for
accurate referring. is introduced to model the SI signal with computational
complexity lower than the polynomial-based canceler. In [References], a
recurrent NN (RNN) and a complex-valued TDNN (CV-TDNN) have been investigated
for SI mitigation; it is shown that the CV-TDNN has excellent modeling
capabilities to approximate the SI with lower computational complexity than
the polynomial and RNN-based cancelers. In [References], [References], the
hardware design of the polynomial and NN-based cancelers introduced in
[References] have been provided. Furthermore, in [References], the ladder-wise
grid structure (LWGS) and moving-window grid structures (MWGS), two low-
complexity NN models, have been introduced for SI cancellation. It is
demonstrated that the LWGS and MWGS attain a similar cancellation performance
to the polynomial and CV-TDNN-based cancelers with a significant complexity
reduction. The previous works shed light on the few attempts that target
applying low-complexity NN models for SI cancellation in FD systems. However,
further enhancements in the complexity are required to build energy-efficient
NN-based cancelers, which can be suitable for hardware implementation in
mobile communication platforms. As such, this study fills in this gap by
providing efficient NN-based SI cancelers, which achieve a similar
cancellation performance to that of the polynomial and the state-of-the-art
NN-based cancelers while attaining a remarkable complexity reduction.

Based on the aforementioned, in this article, two novel low-complexity NN
architectures referred to as the hybrid-convolutional recurrent NN (HCRNN) and
hybrid-convolutional recurrent dense NN (HCRDNN) are proposed. The proposed
NNs exploit hybrid hidden layers (e.g., convolutional, recurrent, and/or
dense) to efficiently model the memory effect and non-linearity incorporated
into the SI signal, with low complexity. The key idea behind using hybrid
layers is to build an NN model, which makes use of the characteristics of the
different layers employed in its architecture. In particular, the proposed NNs
exploit, in a novel manner, the feature extraction characteristics of the
convolutional layer along with the sequence modeling capabilities of the
recurrent layer and/or the learning abilities of the dense layer in order to
model the SI with lower computational complexity than the polynomial and the
state-of-the-art NN-based cancelers. To the best of the authors’ knowledge,
applying hybrid-layers NN architectures for SI cancellation has not been
previously reported in the literature, and it is introduced for the first time
in this paper. More specifically, in the proposed HCRNN, the input data
containing the I/Q components of the input samples is formulated into a two-
dimensional (2D) graph for the sake of suitable processing by the
convolutional layer. The convolutional layer is then applied to the 2D graph
to extract the input features (e.g., memory effect and non-linearity) at a
reduced network scale. Moreover, a recurrent layer is then utilized to help in
learning the temporal behavior of the input signal from the output feature map
of the convolutional layer. In the proposed HCRDNN, a dense layer is added
after the convolutional and recurrent layers to build a deeper NN model with
low computational complexity. Working with hybrid-layers NN architectures
enables adjusting the hidden layers’ settings to achieve a certain
cancellation performance with a considerable computational complexity
reduction.

The contributions of this article are summarized as follows:

  * •

Two novel hybrid-layers NN architectures, termed as the HCRNN and HCRDNN, are
proposed for the first time to model the SI in FD systems with low
computational complexity. In contrast to the state-of-the-art NNs that
directly apply the traditional dense or recurrent layers for SI modeling, the
proposed NNs exploit, in a novel manner, a combination of hidden layers (e.g.,
convolutional, recurrent, and/or dense) in order to achieve high learning
capability while maintaining low computational complexity.

  * •

The computational complexity and memory requirements of the proposed HCRNN and
HCRDNN-based cancelers are derived in terms of the number of floating-point
operations (FLOPs) and network parameters, respectively, and analyzed compared
to those of the polynomial and the state-of-the-art NN-based cancelers.

  * •

The optimum settings for training the proposed HCRNN and HCRDNN architectures
(e.g., number of convolutional filters, filter size, number of neurons in
recurrent and dense layers, activation functions, learning rate, batch size,
and optimizer) are selected to achieve an acceptable cancellation performance
with a considerable computational complexity reduction.

  * •

Performance analysis of the two proposed NNs is provided in terms of their
prediction capabilities, mean square error (MSE), achieved SI cancellation,
computational complexity, and memory requirements. Both NNs demonstrate
excellent prediction capabilities in modeling the interference in FD systems
with reduced complexity.

The rest of this article is organized as follows. Section II presents the FD
transceiver system model. Section III introduces the proposed HCRNN and
HCRDNN-based cancelers. In Section IV, the complexity of the proposed NN
architectures is analyzed, whereas in Section V, the optimum settings for
their training are selected. Finally, simulation results, future research
directions, and conclusions are presented in Sections VI, VII, and VIII,
respectively.

##  II  System Model

(a) Detailed system model. (b) Digital canceler.

Fig. 1: Full-duplex transceiver system model.

An FD transceiver consisting of a local transmitter, local receiver, and two
SI cancellation techniques is illustrated in Figs. 1(a) and (b). Specifically,
the FD system’s design, shown in Fig. 1(a), employs an analog RF cancellation
and a training-based digital cancellation in order to suppress the SI signal
to the receiver noise level. The RF cancellation is applied at the first stage
of the receiver chain to prevent the SI signal from saturating the receiver’s
analog components (e.g., LNA, VGA, and ADC). However, the digital cancellation
is employed after the ADC to remove the residual SI signal.

Let us denote the digital transmitted samples before the DAC by $x(n)$, with
$n$ representing each sample index. The transmitted samples are converted to
analog, filtered, and up-converted to the carrier frequency using the DAC, low
pass filter (LPF), and IQ mixer, respectively. The IQ mixer, undesirably,
distorts the transmitted signal due to the amplitude and gain imbalances
between the I/Q components (i.e., IQ imbalance). Subsequently, the digital
equivalent of the mixer’s output signal can be expressed as [References],
[References]

$x_{IQ}(n)=\frac{1}{2}(1+\psi e^{j\theta})\,x(n)+\frac{1}{2}(1-\psi
e^{j\theta})\,x^{*}(n),$ (1)

where $\psi$ and $\theta$ represent the gain and phase imbalance coefficients
of the transmitter, respectively. The mixer’s output signal is then amplified
by the PA, which further distorts the transmitted signal due to its non-
idealities. The PA’s output signal can be expressed using the conventional
parallel-Hammerstein (PH) model, described by (48) in the Appendix, as
[References], [References], [References]

$x_{PA}(n)=\sum_{\underset{p\,odd}{p=1,}}^{P}\sum_{m=0}^{M_{PA}}h{}_{m,p}\thinspace
x_{IQ}(n-m)^{\frac{p+1}{2}}x_{IQ}^{*}(n-m)^{\frac{p-1}{2}},$ (2)

where $h_{m,p}$ indicates the PA’s impulse response. In addition, $P$ and
$M_{PA}$ represent the non-linearity order and the PA’s memory depth,
respectively.

The PA’s output signal leaks to the receiver through the SI channel, forming
the SI signal. Accordingly, at the receiver side of the FD node, there are
three signals: an SI signal, a noise signal, and a far-end desired signal from
another FD node. In this work, we assume, for simplicity of analysis, that
there is no thermal noise, and there are no far-end desired signals from any
other FD nodes [References], [References]. As such, the residual SI signal
after the RF cancellation process is filtered, amplified, down-converted, and
digitized using the band-pass (BPF), LNA, IQ mixer, and ADC, respectively, and
can be expressed as [References]

$y_{{}_{SI}}(n)=\sum_{\underset{p\,odd}{p=1,}}^{P}\sum_{q=0}^{p}\sum_{m=0}^{M-1}h{}_{m,q,p}\,x(n-m)^{q}x^{*}(n-m)^{p-q},$
(3)

where $h_{m,q,p}$ represents the impulse response of a channel, including the
composite effect of the PA, IQ mixer, and SI channel, whereas $M$ indicates
the memory effect incorporated into the input signal by the PA and SI coupling
channel.

The estimation of the aforementioned channel is performed using the least-
squares (LS) approach as follows. Firstly, we formulate the matrix
$\boldsymbol{X}$ with dimensions $N\times M$ from the input data $x$, where
$N$ is the length of the training data. Then, the output signal
$\boldsymbol{y}$ with dimension $N\times 1$ can be obtained as
$\boldsymbol{y=Xh}$, where $\boldsymbol{h}$ is a vector of size $M\times 1$.
In the LS method, the aim is to minimize the cost function
$J(\boldsymbol{\boldsymbol{\hat{h}}})$, which can be expressed as

$J(\boldsymbol{\boldsymbol{\hat{h}}})=\left\|\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\hat{h}}\right\|^{2}=(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\hat{h}})^{H}(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\hat{h}})$
$\hskip
29.02179pt=\boldsymbol{y^{H}\boldsymbol{y}}-\boldsymbol{y^{H}\boldsymbol{X}\boldsymbol{\hat{h}}}-\boldsymbol{\hat{h}}^{H}\boldsymbol{\boldsymbol{X}^{H}}\boldsymbol{\boldsymbol{y}}+\boldsymbol{\hat{h}}^{H}\boldsymbol{\boldsymbol{X}^{H}}\boldsymbol{\boldsymbol{X}}\boldsymbol{\boldsymbol{\hat{h}}},$
(4)

where $\left(.\right)^{H}$denotes the Hermitian transpose operator. By setting
the derivative of this cost function with respect to $\boldsymbol{\hat{h}}$ to
zero, the LS solution can be given as

$\boldsymbol{\boldsymbol{\hat{h}}}=\left(\boldsymbol{X^{H}}\boldsymbol{X}\right)^{-1}\boldsymbol{X^{H}}\boldsymbol{y}.$
(5)

In the digital canceler, the goal is to estimate the distortion caused by the
imperfection of the transceiver hardware components and SI channel to generate
an accurate replica of the SI signal $\tilde{y}_{{}_{SI}}(n)$ at the receiver.
This is attained by feeding the baseband transmitted samples before the
digital-to-analog conversion to a trainable-based digital canceler in order to
produce such a replica. This replica is then subtracted from the SI signal
after the ADC to remove the interference, and the residual SI after the
digital cancellation is given by $y(n)=y_{{}_{SI}}(n)-\tilde{y}_{{}_{SI}}(n)$.
The achieved SI cancellation in the digital domain can be quantified in dB as

$\mathbb{\mathcal{C}}_{dB}=10\log_{10}\left(\frac{\sum_{n}\left|y_{{}_{SI}}(n)\right|^{2}}{\sum_{n}\left|y(n)\right|^{2}}\right)\negmedspace.$
(6)

In this work, the digital canceler is formed by linear and non-linear
trainable-based cancelers, as depicted in Fig. 1(b). The former is utilized to
estimate the linear part of the SI signal based on the conventional LS channel
estimation [References], whereas the latter is used to mimic the non-linear
part of the SI signal using an NN model. The SI signal is then reconstructed
by combining the linear and non-linear components as follows:

$\tilde{y}_{{}_{SI}}(n)=\tilde{y}_{{}_{SI,lin}}(n)+\tilde{y}_{{}_{SI,nl}}(n),$
(7)

where $\tilde{y}_{{}_{SI,lin}}(n)$ is the linear part of the SI signal, which
can be obtained by substituting $p=1$ and $q=1$ in (3) as

$\tilde{y}_{{}_{SI,lin}}(n)=\sum_{m=0}^{M-1}h{}_{m,1,1}\,x(n-m),$ (8)

while $\tilde{y}_{{}_{SI,nl}}(n)$ is the non-linear part, which can be given
as

$\tilde{y}_{{}_{SI,nl}}(n)=g\left\\{x\left(n\right),x\left(n-1\right),...,x\left(n-M+1\right)\right\\},$
(9)

where $g\left\\{.\right\\}$ represents the NN mapping function.

##  III Proposed Hybrid-Layers NN Architectures

Fig. 2: Proposed HCRNN architecture.

The main function of the NN-based canceler is to provide an accurate behavior
model to mimic the non-linearities and memory effect attributed to the input
signal. As such, to account for such effects, TDNN or RNN-based models were
employed in the literature. While the aforementioned models attain good
modeling capabilities for approximating the SI in FD systems, their complexity
is a considerable issue that should be addressed. A good choice for designing
a low-complexity NN model is to exploit the high feature extraction along with
the parameter reduction capabilities of the convolutional layer. Further,
adding a recurrent layer with a sufficient number of neurons after the
convolutional layer can enhance the modeling capabilities with no much
increase in the computational complexity. Lastly, exploiting an additional
dense layer after the recurrent layer can help to add another degree of
freedom for adapting the NN settings in order to achieve the best compromise
between the model performance and the computational complexity. In the next
subsections, we will show how the proposed NN architectures exploit the
aforementioned layers in order to build low-complexity NN-based cancelers.

###  III-A HCRNN Architecture

Fig. 3: Example of the basic operation of the proposed HCRNN.

The proposed HCRNN architecture is shown in Fig. 2. In the HCRNN, hybrid
hidden layers are employed to detect the memory effect and non-linearity
incorporated into the input signal due to the impairments of the FD
transceiver and SI channel. The HCRNN is arranged in five layers: the input
layer, convolutional layer, reshape layer, recurrent layer, and output layer.
At the input layer, the input data is formulated into a 2D graph consisting of
the I/Q components of the instantaneous and delayed versions of the input
samples. Arranging the data into a 2D graph allows the input samples to be in
an appropriate form that can be efficiently processed by the next
convolutional layer. The convolutional layer, the first hidden layer of the
HCRNN, is applied to the 2D grid in order to detect the input signal’s
features with a reduced network scale. Applying the convolutional layer to the
input graph comes with a considerable reduction in the computational
complexity due to the weight-sharing characteristics and dimensionality
reduction capabilities of the convolutional filters [References]. The
convolutional layer’s output is then reordered using a reshape layer, to be
processed by the next recurrent layer, which constitutes the second hidden
layer of the HCRNN. The aim of the recurrent layer is to assist in learning
the temporal behavior of the input signal from the localized feature map of
the convolutional layer. Finally, at the output layer of the HCRNN, the I/Q
components of the SI signal are estimated.

An example showing the basic operation of the proposed HCRNN, using $M=13$,
three convolutional filters, and $8\times 1\times 1$ filter size, is
illustrated in Fig. 3. At the convolutional layer, the input data is convolved
with the filters, acting as optimizable-feature extractors, in order to detect
the patterns of the input attributes. The output feature map of the $l^{th}$
filter after applying the convolution operation can be expressed as

$\boldsymbol{F}_{c}^{l}=\begin{bmatrix}I_{in}(n)&Q_{in}(n)\\\
I_{in}(n-1)&Q_{in}(n-1)\\\ I_{in}(n-2)&Q_{in}(n-2)\\\ .&.\\\ .&.\\\
I_{in}(n-M+1)&Q_{in}(n-M+1)\end{bmatrix}\otimes\begin{bmatrix}k_{1,1}^{l}&k_{1,S}^{l}\\\
k_{2,1}^{l}&k_{2,S}^{l}\\\ k_{3,1}^{l}&k_{3,S}^{l}\\\ .&.\\\ .&.\\\
k_{R,1}^{l}&k_{R,S}^{l}\end{bmatrix},$ (10)

where $l=1,2,...,L$, with $L$ denoting the number of convolutional filters.
$I_{in}(n)$ and $Q_{in}(n)$ represent the I/Q components of the current sample
$x(n)$, respectively, while $[I_{in}(n-1),...,I_{in}(n-M+1)]$ and
$[Q_{in}(n-1),...,Q_{in}(n-M+1)]$ indicate the I/Q components of the delayed
samples, respectively. In addition, $R\times S\times Z$ and $k_{i,j}^{l}$
represent the dimensions and $(i^{th},j^{th})$ entry of the $l^{th}$
convolutional filter, respectively. Finally, $\otimes$ denotes the convolution
operation. It is worth mentioning that for all convolutional filters,
$R\in\left\\{1,2,...,M\right\\}$, $S\in\left\\{1,2\right\\}$, and $Z=1$ since
the input data is arranged into a 2D graph consisting of two columns to
represent the I/Q components of the instantaneous and delayed versions of the
input samples.

After performing the convolution, a bias term is added, and a non-linear
activation function is applied to each element of $\boldsymbol{F}_{c}^{l}$.
Accordingly, the output feature map of the $l^{th}$ convolutional filter can
be expressed as

$\boldsymbol{F}^{l}=\begin{bmatrix}f^{c}(a_{1,1}^{l}+b_{l})&f^{c}(a_{1,C}^{l}+b_{l})\\\
f^{c}(a_{2,1}^{l}+b_{l})&f^{c}(a_{2,C}^{l}+b_{l})\\\
f^{c}(a_{3,1}^{l}+b_{l})&f^{c}(a_{3,C}^{l}+b_{l})\\\ .&.\\\ .&.\\\
f^{c}(a_{B,1}^{l}+b_{l})&f^{c}(a_{B,C}^{l}+b_{l})\end{bmatrix},$ (11)

where $f^{c}(.)$ denotes the convolutional layer’s activation function, while
$b_{l}$ represents the bias term associated with the $l^{th}$ convolutional
filter. $B\times C$ indicates the output feature map’s dimensions, whereas
$a_{i,j}^{l}$ denotes the $(i^{th},j^{th})$ entry of the feature map just
after applying the convolution. With unity-stride and without zero-padding,
$a_{i,j}^{l}$ can be given as

$a_{i,j}^{l}=\sum_{r=0}^{R-1}\sum_{s=0}^{S-1}\boldsymbol{X}_{r,s+j}\boldsymbol{K}_{r,s}^{l},$
(12)

where $\boldsymbol{X}$ and $\boldsymbol{K}^{l}$ represent the input and the
$l^{th}$ convolutional filter matrices with entries described in (10),
respectively.

For efficient processing of the forwarded data through the network, the output
feature maps of all filters are reshaped before they are passed to the
recurrent layer. More specifically, in the output feature map of each filter,
there are dependencies between each column’s elements due to the temporal
behavior existing in the input data. Therefore, in the proposed HCRNN, the
output feature maps of all filters are reformulated using the reshape layer in
order to take these dependencies into account and pass them in a sequence to
the recurrent layer. Reshaping the feature maps with this mechanism enables
the recurrent layer to detect the aforementioned sequence for proper modeling
of the system’s temporal behavior. Based on this, the resultant feature map
after the reshaping process can be expressed as

$\boldsymbol{F}_{out}=\begin{bmatrix}\boldsymbol{F}^{1}&\boldsymbol{F}^{2}&...&\boldsymbol{F}^{L}\end{bmatrix},$
(13)

where $\boldsymbol{F}^{1}$, $\boldsymbol{F}^{2}$, and $\boldsymbol{F}^{L}$
represent the output feature maps of the $1^{st}$, $2^{nd}$, and $L^{th}$
convolutional filters, respectively. The reshaped feature map is then passed
to the recurrent layer, and the output at any time step $t$ can be expressed
as

$\boldsymbol{y}_{r}(t)=f^{r}\left(\boldsymbol{f}_{out}(t)\boldsymbol{W}_{x}+\boldsymbol{y}_{r}(t-1)\boldsymbol{W}_{y}+\boldsymbol{b}_{r}\right),$
(14)

where $\boldsymbol{y}_{r}(t)\in\mathbb{R}^{1\times n_{hr}}$ represents the
recurrent layer output at any time step $t$, with $n_{hr}$ denoting the number
of recurrent layer’s neurons. Similarly,
$\boldsymbol{y}_{r}(t-1)\in\mathbb{R}^{1\times n_{hr}}$ indicates the
recurrent layer’s output at the previous time step $t-1$.
$\boldsymbol{f}_{out}(t)\in\mathbb{R}^{1\times n_{i}}$ represents a row vector
of $\boldsymbol{F}_{out}$, which is passed to the recurrent layer at time step
$t$, with $n_{i}$ as the number of input features.
$\boldsymbol{W}_{x}\in\mathbb{R}^{n_{i}\times n_{hr}}$ denotes the weight
matrix for the connections between the input and hidden units at the current
time step. Further, $\boldsymbol{W}_{y}\in\mathbb{R}^{n_{hr}\times n_{hr}}$
indicates the weight matrix for the feedback connections from the hidden units
at the previous time step. Finally, $f^{r}(.)$ represents the activation
function operation of the recurrent layer, whereas
$\boldsymbol{b}_{r}\in\mathbb{R}^{1\times n_{hr}}$ indicates a row vector
containing the bias terms of the recurrent layer neurons.

The recurrent layer’s output is then passed to the output layer, which is
formed by a fully-connected (dense) layer containing two neurons. The output
layer neurons are utilized to map the extracted features by the recurrent
layer to the final output (i.e., estimated I/Q components of the SI signal) as
follows:

$I_{out}^{{}^{\prime}}(n)=f^{o}\left(\sum_{i=1}^{n_{hr}}\boldsymbol{W}_{o}^{1,i}\mathbf{\boldsymbol{y}}_{r}^{i}+\boldsymbol{b}_{o}^{1}\right),$
(15)

$Q_{out}^{{}^{\prime}}(n)=f^{o}\left(\sum_{i=1}^{n_{hr}}\boldsymbol{W}_{o}^{2,i}\boldsymbol{y}_{r}^{i}+\boldsymbol{b}_{o}^{2}\right),$
(16)

where $f^{o}(.)$ represents the output layer’s activation function, while
$\\{\boldsymbol{W}_{o}^{1,i},\boldsymbol{b}_{o,}^{1}\\}$ and
$\\{\boldsymbol{W}_{o}^{2,i},\boldsymbol{b}_{o}^{2}\\}$ indicate the weight
and bias terms associated with the first and second neurons of the output
layer, respectively.

###  III-B HCRDNN Architecture

Fig. 4: Proposed HCRDNN architecture.

The network architecture of the proposed HCRDNN is depicted in Fig. 4. The key
difference between the HCRDNN and HCRNN architectures is that an additional
dense layer is employed after the convolutional and recurrent layers as
increasing the number of hidden layers can enable building more complex NN
models with reduced complexity [References]. Additionally, adding a dense
layer to the HCRDNN architecture provides another degree of freedom to adapt
the NN settings in order to achieve the optimum cancellation-complexity trade-
off. In particular, in the HCRDNN, the convolutional layer’s hyper-parameters
(e.g., number of filters, filter size) and the number of neurons in the
recurrent and dense layers are jointly adjusted to achieve a considerable
cancellation performance with a significant computational complexity
reduction.

The output of the $u^{th}$ neuron at the dense layer of the HCRDNN can be
expressed as

$y_{d}^{u}=f^{d}\left(\sum_{i=1}^{n_{hr}}\boldsymbol{W}_{d}^{u,i}\boldsymbol{y}_{r}^{i}+\boldsymbol{b}_{d}^{u}\right),$
(17)

where $f^{d}(.)$ indicates the activation function of the dense layer, while
$\\{\boldsymbol{W}_{d}^{u,i},\boldsymbol{b}_{d,}^{u}\\}$ represent the weight
and bias terms associated with the $u^{th}$ neuron in the dense layer,
respectively. The dense layer’s output is then passed to the output layer to
estimate the I/Q components of the SI signal as (15) and (16).

After predicting $I_{out}^{{}^{\prime}}(n)$ and $Q_{out}^{{}^{\prime}}(n)$,
the MSE is calculated as a cost function to measure how well the proposed NN
models predict the actual outputs as follows:

$E=\frac{1}{2N}\sum_{n=1}^{N}\left(\left[I_{out}(n)-I_{out}^{{}^{\prime}}(n)\right]^{2}+\left[Q_{out}(n)-Q_{out}^{{}^{\prime}}(n)\right]^{2}\right),$
(18)

where $\\{I_{out}(n),$ $I_{out}(n)\\}$ and $\\{I_{out}^{{}^{\prime}}(n),$
$Q_{out}^{{}^{\prime}}(n)\\}$ represent the actual and predicted values of the
I/Q components of the SI signal, receptively, whereas $N$ denotes the number
of training observations, as stated before in Section II. During the HCRNN and
HCRDNN models’ training, the convolutional, recurrent, and dense layers’
weights and biases are set to minimize this cost function.

After the training process, the proposed NNs are employed to provide non-
linear SI cancellation as part of the digital cancellation in the FD
transceiver, as illustrated in Fig. 1(b).

##  IV  Complexity Analysis

In this section, the computational complexity and memory requirements of the
polynomial and proposed NN-based cancelers are analyzed in terms of the number
of computations and memory usage of the linear and non-linear cancelers
required to provide the total SI cancellation (i.e., the summation of linear
and non-linear cancellations). In particular, in this work, the computational
complexity is assessed in terms of the number of FLOPs used to perform the
linear and non-linear cancellation processes. Moreover, the memory usage
requirements are assessed in terms of the number of stored parameters utilized
to achieve the total cancellation. In this analysis, we focus on evaluating
the complexity of the polynomial and proposed NN-based cancelers in the real-
time inference stage since powerful processing units can be employed to
perform the training process offline.

###  IV-A Linear Canceler Complexity

In this subsection, we calculate the number of FLOPs required to implement the
linear canceler in terms of the number of RV multiplications and additions.
From (8), the linear canceler requires $M$ complex multiplications and $M-1$
complex additions to perform the linear cancellation. Using the assumption
that each complex addition is implemented by two real additions, and each
complex multiplication is executed using three real multiplications and five
real additions [References], the number FLOPs of the linear canceler can be
expressed as

$\mathscr{\mathcal{F}_{\mathit{lin}}}=\underset{\chi_{mul,lin}^{\Re}}{\underbrace{3M}}+\underset{\text{$\chi_{add,lin}^{\mathfrak{\mathbb{\Re}}}$}}{\underbrace{7M-2}},$
(19)

where $\chi_{mul,lin}^{\Re}$ and $\chi_{add,lin}^{\mathfrak{\mathbb{\Re}}}$
represent the number of real multiplications and additions required for the
linear canceler, respectively, while $M$ denotes the memory effect
incorporated into the input signal by the PA and SI channel, as stated before.
Similarly, from (8), the memory requirements of the linear canceler is
evaluated as a function of the number of stored parameters, which can be
expressed as

$\mathcal{P}_{lin}=2M.$ (20)

###  IV-B Non-Linear Canceler Complexity

In this subsection, the number of FLOPs and parameters required for non-linear
cancellation using polynomial and NN-based cancelers are analyzed.

####  IV-B1 Polynomial-based Canceler Complexity

Using the previous assumptions, it can be deduced from (3) that the number of
RV multiplications and additions required for the non-linear polynomial-based
canceler can be expressed as [References], [References]

$\chi_{mul,poly}^{\mathfrak{\mathbb{\Re}}}=3M\left\\{\left(\frac{P+1}{2}\right)\left(\frac{P+1}{2}+1\right)-1\right\\},$
(21)

$\chi_{add,poly}^{\mathfrak{\mathbb{\Re}}}=7M\left\\{\left(\frac{P+1}{2}\right)\left(\frac{P+1}{2}+1\right)-1\right\\}.$
(22)

Similarly, the number of parameters of the non-linear polynomial-based
canceler can be expressed as [References], [References]

$\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\mathcal{P}_{poly}=2M\left\\{\left(\frac{P+1}{2}\right)\left(\frac{P+1}{2}+1\right)-1\right\\}.$
(23)

####  IV-B2 NN-based Canceler Complexity

For the proposed non-linear HCRNN-based canceler, the number of FLOPs can be
expressed as

$\mathscr{\mathcal{F}_{\mathit{HCRNN}}}=\mathcal{F^{\mathit{c}}}+\mathcal{F^{\mathit{r}}}+\mathcal{F^{\mathit{o}}},$
(24)

where $\mathcal{F}^{\mathit{c}}$, $\mathcal{F}^{\mathit{r}}$, and
$\mathcal{F}^{\mathit{o}}$ represent the number of FLOPs required for the
convolutional, recurrent, and output layers of the HCRNN, respectively.
Firstly, for the convolutional layer, $\mathcal{F}^{\mathit{c}}$ can be
expressed as

$\underset{\text{
\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace}\textrm{Ter$\textrm{m}_{1}^{c}$}}{\mathcal{F^{\mathit{c}}}=\underbrace{\left(2\left\\{R\times
S\times Z\right\\}-1\right)\left(B\times C\times L\right)}}$

$\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\underset{\text{\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace}\textrm{Ter$\textrm{m}_{2}^{c}$}}{+\underbrace{\mathcal{F^{\mathit{act}}}\times\left(B\times
C\times
L\right)}}+\underset{\text{\thinspace\thinspace}\textrm{Ter$\textrm{m}_{3}^{c}$}}{\underbrace{\left(B\times
C\times L\right)}},$ (25)

where $\textrm{Ter}\textrm{m}_{1}^{c}$ represents the number of RV
multiplications and additions required to convolve the 2D graph of the input
data with $L$ convolutional filters. $\textrm{Ter}\textrm{m}_{2}^{c}$ denotes
the number of RV operations (multiplications + additions) required for
applying the activation functions at each element of the output feature map
after the convolution process. $\textrm{Ter}\textrm{m}_{3}^{c}$ indicates the
number of real additions required for adding the bias values. Finally,
$\mathcal{F^{\mathit{act}}}$ represents the number of real operations needed
to evaluate each activation function, which mainly depends on the activation
function type.

In this work, we consider the rectified linear unit ($ReLU$), $Sigmoid$, and
hyperbolic tangent ($tanh$) activation functions, which are defined as
[References]

$\displaystyle ReLU(z)$ $\displaystyle=\max(0,z),$ (26a) $\displaystyle
Sigmoid(z)$ $\displaystyle=\frac{1}{e^{-z}+1},$ (26b) $\displaystyle tanh(z)$
$\displaystyle=\frac{e^{z}-e^{-z}}{e^{z}+e^{-z}}.$ (26c)

By assuming that each real operation (e.g., multiplication, division,
addition, subtraction, and exponentiation) costs one FLOP [References],
[References], the number of FLOPs required to implement each of the
aforementioned activation functions can be expressed as

$\mathcal{F^{\mathit{act}}}=\begin{cases}1,&\textrm{
\thinspace\thinspace\thinspace\thinspace if }ReLU\\\ 4,&\textrm{
\thinspace\thinspace\thinspace\thinspace if }Sigmoid\\\ 6,&\textrm{
\thinspace\thinspace\thinspace\thinspace if }tanh\end{cases}.$ (27)

Secondly, the number of FLOPs required for the recurrent layer can be
expressed as

$\underset{\text{ }}{\underset{\text{
\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace{Ter$\textrm{m}_{1}^{r}$}}}{\mathcal{F^{\mathit{r}}}=\underbrace{2n_{hr}\left(n_{i}+n_{hr}-\frac{1}{2}\right)}}}\underset{\text{\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace{Ter$\textrm{m}_{2}^{r}$}}}{+\underbrace{\mathcal{F^{\mathit{act}}}\times
n_{hr}}}+\underset{\text{\thinspace\thinspace{Ter$\textrm{m}_{3}^{r}$}}}{\underbrace{n_{hr}}},$
(28)

where $\textrm{Ter}\textrm{m}_{1}^{r}$ represents the number of RV operations
associated with multiplying the weight matrices $\boldsymbol{W}_{x}$ and
$\boldsymbol{W}_{y}$ in (14) with their corresponding inputs, while
$\textrm{Ter}\textrm{m}_{2}^{r}$ and $\textrm{Ter}\textrm{m}_{3}^{r}$ indicate
the number of real operations required for applying the activation functions
and adding the biases at each neuron of the recurrent layer, respectively.

Finally, the output layer’s FLOPs can be expressed as

$\underset{\text{\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace{Ter$\textrm{m}_{1}^{o}$}}}{\mathcal{F^{\mathit{o}}}=\underbrace{n_{ho}\left(2n_{hr}-1\right)}}\underset{\text{\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace{Ter$\textrm{m}_{2}^{o}$}}}{+\underbrace{\mathcal{F^{\mathit{act}}}\times
n_{ho}}}+\underset{\text{\thinspace\thinspace{Ter$\textrm{m}_{3}^{o}$}}}{\underbrace{n_{ho}}},$
(29)

where $n_{ho}$ represents the number of output layer’s neurons.
$\textrm{Ter}\textrm{m}_{1}^{o}$ denotes the number of RV operations required
to calculate the weighted sum of inputs at the output layer neurons.
$\textrm{Ter}\textrm{m}_{2}^{o}$ and $\textrm{Ter}\textrm{m}_{3}^{o}$ indicate
the number of real operations associated with employing the activation
functions and adding the bias terms at each neuron of the output layer,
respectively.

The number of parameters of the proposed HCRNN architecture can be given as

$\mathcal{P}_{HCRNN}=\mathcal{P^{\mathit{c}}}+\mathcal{P^{\mathit{r}}+\mathcal{P^{\mathit{o}}}},$
(30)

where $\mathcal{P^{\mathit{c}}}$, $\mathcal{P^{\mathit{r}}}$, and
$\mathcal{P^{\mathit{o}}}$ represent the number of parameters of the
convolutional, recurrent, and output layers, respectively, which can be
expressed as

$\displaystyle\mathcal{P^{\mathit{c}}}$ $\displaystyle=L\left(R\times S\times
Z+1\right),$ (31a) $\displaystyle\mathcal{P}^{\mathit{r}}$
$\displaystyle=n_{hr}\left(n_{i}+n_{hr}+1\right),$ (31b)
$\displaystyle\mathcal{P^{\mathit{o}}}$
$\displaystyle=n_{ho}\left(n_{hr}+1\right).$ (31c)

Using the same mathematical formulation, the number of FLOPs of the proposed
HCRDNN can be calculated as

$\mathscr{\mathcal{F}_{\mathit{HCRDNN}}}=\mathcal{F^{\mathit{c}}}+\mathcal{F^{\mathit{r}}}+\mathcal{F^{\mathit{d}}}+\mathcal{F^{\mathit{o}}},$
(32)

where $\mathcal{F}^{\mathit{c}}$ and $\mathcal{F}^{\mathit{r}}$ represent the
convolutional and recurrent layers’ FLOPs, which can be determined using (25)
and (28), respectively. $\mathcal{F}^{\mathit{o}}$ indicates the output
layer’s FLOPs, which can be calculated by replacing $n_{hr}$ by $n_{hd}$ in
(29), whereas $\mathcal{F}^{\mathit{d}}$ denotes the dense layer’s FLOPs,
which can be expressed as

$\underset{\text{\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace{Ter$\textrm{m}_{1}^{d}$}}}{\mathcal{F^{\mathit{d}}}=\underbrace{n_{hd}\left(2n_{hr}-1\right)}}\underset{\text{\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace{Ter$\textrm{m}_{2}^{d}$}}}{+\underbrace{\mathcal{F^{\mathit{act}}}\times
n_{hd}}}+\underset{\text{\thinspace\thinspace{Ter$\textrm{m}_{3}^{d}$}}}{\underbrace{n_{hd}}},$
(33)

where $\textrm{Ter}\textrm{m}_{1}^{d}$, $\textrm{Ter}\textrm{m}_{2}^{d}$, and
$\textrm{Ter}\textrm{m}_{3}^{d}$ represent the number of real operations
required for calculating the weighted sum of inputs, applying the activation
functions, and summing the biases at the dense layer neurons, respectively.

The number of parameters of the HCRDNN can be given as

$\mathcal{P}_{HCRDNN}=\mathcal{P^{\mathit{c}}}+\mathcal{P^{\mathit{r}}+\mathcal{P^{\mathit{d}}}}+\mathcal{P^{\mathit{o}}},$
(34)

where $\mathcal{P}^{\mathit{c}}$ and $\mathcal{P}^{\mathit{r}}$ represent the
convolutional and recurrent layers’ parameters given by (31a) and (31b),
respectively, while $\mathcal{P}^{\mathit{o}}$ indicates the output layer’s
parameters, which can be evaluated by replacing $n_{hr}$ by $n_{hd}$ in (31c).
Lastly, $\mathcal{P}^{\mathit{d}}=n_{hd}\left(n_{hr}+1\right)$ is the number
of dense layer parameters.

For the RV-TDNN [References], [References], the number of FLOPs and parameters
can be expressed as

$\mathscr{\mathcal{F}_{\mathit{RV-
TDNN}}}=\sum_{j=2}^{\mathcal{L}}2n_{j-1}n_{j}+\mathcal{F^{\mathit{act,j}}}n_{j},$
(35)

$\mathcal{P}_{RV-TDNN}=\sum_{j=2}^{\mathcal{L}}n_{j-1}n_{j}+n_{j},$ (36)

where $\mathcal{L}$ indicates the number of layers of the RV-TDNN, including
the input, hidden, and output layers. $n_{j}$ is the number of neurons in the
$j^{th}$ layer, with $n_{1}$ and $n_{\mathcal{L}}$ representing the number of
input and output layers’ neurons, respectively.
$\mathcal{\mathcal{F^{\mathit{act,j}}}}$ denotes the number of operations
required to apply the activation function at each of the $j^{th}$ layer’s
neurons.

Similarly, for the RNN [References], the number of FLOPs and parameters can be
given by

$\mathscr{\mathcal{F}_{\mathit{RNN}}}=\sum_{j=2}^{\mathcal{L}}2n_{j-1}n_{j}+2n_{j}^{2}+\mathcal{F^{\mathit{act,j}}}n_{j}-2n_{\mathcal{L}}^{2},$
(37)

$\mathcal{P}_{RNN}=\sum_{j=2}^{\mathcal{L}}n_{j-1}n_{j}+n_{j}^{2}+n_{j}-n_{\mathcal{L}}^{2},$
(38)

with $n_{1}$ denotes the number of RNN inputs at each time step.

For the CV-TDNN [References], using the previous implementation assumptions
for the CV additions and multiplications, the number of FLOPs and parameters
can be given by

$\mathscr{\mathcal{F}_{\mathit{CV-
TDNN}}}=10\left\\{\sum_{j=2}^{\mathcal{L}}n_{j-1}n_{j}\right\\}+\mathcal{F^{\mathit{act,j}}}n_{j},$
(39)

$\mathcal{P}_{CV-
TDNN}=2\left\\{\sum_{j=2}^{\mathcal{L}}n_{j-1}n_{j}+n_{j}\right\\}.\hskip
17.07182pt$ (40)

For the LWGS [References] employing $n_{i}$ CV inputs and a single hidden
layer with $n_{h}$ neurons, the number of FLOPs and parameters can be
expressed as

$\mathscr{\mathcal{F}_{\mathit{LWGS}}}=10\left\\{\left(\sum_{j=1}^{\mathcal{\mathit{n_{h}}}}j\right)+\mathcal{\mathit{n_{i}}}\right\\}+\mathcal{F^{\mathit{act}}}n_{h},$
(41)

$\mathcal{P}_{LWGS}=2\left\\{\left(\sum_{j=1}^{\mathcal{\mathit{n_{h}}}}j\right)+\mathcal{\mathit{n_{i}}}+\mathcal{\mathit{n_{h}}+}1\right\\}.$
(42)

Likewise, for the MWGS [References] employing $n_{i}$ CV inputs, a
$\mathcal{W}$ window size, and a single hidden layer with $n_{h}$ neurons, the
number of FLOPs and parameters can be obtained as follows:

$\mathscr{\mathcal{F}_{\mathit{MWGS}}}=10\left\\{\mathcal{\mathit{n_{i}}}+\mathcal{W}\left(\mathcal{\mathit{n_{h}}}-1\right)+\mathcal{\mathit{n_{h}}}\right\\}+\mathcal{F^{\mathit{act}}}n_{h},$
(43)

$\mathcal{P}_{MWGS}=2\left\\{\mathcal{\mathit{n_{i}}}+\mathcal{W}\left(\mathcal{\mathit{n_{h}}}-1\right)+2\mathcal{\mathit{n_{h}}}+1\right\\}.$
(44)

Finally, the total number of FLOPs of any of the aforementioned NN-based
cancelers can be expressed as

$\mathscr{\mathcal{F}_{\mathit{NN-\mathcal{C}}}}=\mathscr{\mathcal{F}_{\mathit{lin}}}+\mathscr{\mathcal{F}_{\mathit{NN}}}+2,$
(45)

where $\mathscr{\mathcal{F}_{\mathit{lin}}}$ represents the linear canceler
complexity, which can be calculated by (19), while
$\mathscr{\mathscr{\mathcal{F}_{\mathit{NN}}}}$ indicates the NN complexity
(e.g., $\mathscr{\mathcal{F}_{\mathit{HCRNN}}}$,
$\mathscr{\mathcal{F}_{\mathit{HCRDNN}}}$, $\mathscr{\mathcal{F}_{\mathit{RV-
TDNN}}}$). The two FLOPs added in (45) represent the number of real additions
required to sum the outputs of the linear and non-linear cancelers in (8) and
(9), respectively [References].

##  V Optimum Settings of the Proposed HCRNN and HCRDNN Architectures

In this section, the optimum settings for training the proposed HCRNN and
HCRDNN architectures are obtained. Specifically, the number of convolutional
filters, filter size, number of recurrent and dense layers’ neurons,
activation functions in each layer, learning rate, batch size, and training
optimizer are selected in such a way that results in a proper cancellation
performance while maintaining low computational complexity. In the following,
we firstly describe the dataset employed to train the proposed NN
architectures, then the network settings for the training process are
optimized.

###  V-A Training Dataset

In the following experiments, the public dataset employed in [References] and
[References] is utilized to train and verify the proposed NN architectures.
The dataset is produced using a realistic FD test-bed, which generates a 10
MHz quadrature phase-shift keying modulated-orthogonal frequency division
multiplexing (OFDM) signal with an average transmit power of 10 dBm. The OFDM
signal employs 1024 sub-carriers and is sampled at 20 MHz. The generated
dataset contains 20,480 time-domain baseband samples, which we split into two
distinct parts. The first part is used for training and consists of 90% of
samples, whereas the second is employed for verification and contains the rest
10%. The employed hardware test-bed provides 53 dB passive analog RF
cancellation using physical separation between the transmit and receive
antennas; hence, in this work, no further active analog cancellation
techniques are employed since the use of digital cancellation after passive
analog suppression is adequate to bring the SI signal’s power down to the
receiver’s noise level [References], [References]. A summary of the parameters
employed to generate the aforementioned dataset is presented in Table. I.

TABLE I:  Summary of the parameters employed to generate the dataset utilized for training and verifying the proposed NN architectures. Parameter |  Value
---|---
Type of modulation |  QPSK-modulated OFDM
Pass-band bandwidth |  10 MHz
Number of carriers |  1024
Sampling frequency |  20 MHz
Average transmit power |  10 dBm
Passive analog suppression |  53 dB
Active analog suppression |  N/A
Total analog cancellation |  53 dB
Dataset size |  20,480 samples
TABLE II: Selection of the optimum configuration of the proposed HCRNN. Config. # |  # Filters |  Filter size |  # Rec. neurons |  SI Canc. (dB) | Complexity |  Config. # |  # Filters |  Filter size |  # Rec. neurons |  SI Canc. (dB) | Complexity
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  # Par. |  # FLOPs |  |  |  |  |  |  # Par. |  # FLOPs
1 |  2 |  4$\times$1$\times$1 |  6 |  43.01 |  116 |  640 |  17 |  2 |  4$\times$1$\times$1 |  10 |  44.18 |  208 |  820
2 |  2 |  8$\times$1$\times$1 |  6 |  42.60 |  124 |  688 |  18 |  2 |  9$\times$1$\times$1 |  10 |  44.05 |  218 |  840
3 |  2 |  6$\times$1$\times$1 |  7 |  42.92 |  140 |  735 |  19 |  2 |  10$\times$1$\times$1 |  10 |  44.06 |  220 |  796
4 |  2 |  4$\times$1$\times$1 |  8 |  43.75 |  158 |  722 |  20 |  2 |  11$\times$1$\times$1 |  10 |  44.19 |  222 |  736
5 |  2 |  5$\times$1$\times$1 |  8 |  43.81 |  160 |  758 |  21 |  2 |  12$\times$1$\times$1 |  10 |  44.31 |  224 |  660
6 |  2 |  8$\times$1$\times$1 |  8 |  43.89 |  166 |  770 |  22 |  3 |  11$\times$1$\times$1 |  6 |  43.66 |  154 |  718
7 |  2 |  10$\times$1$\times$1 |  8 |  43.89 |  170 |  698 |  23 |  3 |  12$\times$1$\times$1 |  6 |  43.97 |  157 |  604
8 |  2 |  11$\times$1$\times$1 |  8 |  43.68 |  172 |  638 |  24 |  3 |  11$\times$1$\times$1 |  7 |  44.11 |  176 |  761
9 |  2 |  5$\times$1$\times$1 |  9 |  44.03 |  184 |  805 |  25 |  3 |  12$\times$2$\times$1 |  7 |  43.54 |  194 |  599
10 |  2 |  6$\times$1$\times$1 |  9 |  44.03 |  186 |  825 |  26 |  3 |  11$\times$1$\times$1 |  8 |  44.10 |  200 |  808
11 |  2 |  7$\times$1$\times$1 |  9 |  44.02 |  188 |  829 |  27 |  3 |  12$\times$1$\times$1 |  9 |  44.44 |  229 |  745
12 |  2 |  9$\times$1$\times$1 |  9 |  43.93 |  192 |  789 |  28 |  3 |  13$\times$1$\times$1 |  9 |  43.94 |  232 |  607
13 |  2 |  10$\times$1$\times$1 |  9 |  44.08 |  194 |  745 |  29 |  3 |  11$\times$2$\times$1 |  9 |  43.85 |  232 |  796
14 |  2 |  12$\times$1$\times$1 |  9 |  44.23 |  198 |  609 |  30 |  3 |  12$\times$2$\times$1 |  9 |  44.09 |  238 |  685
15 |  2 |  12$\times$2$\times$1 |  9 |  43.61 |  204 |  569 |  31 |  3 |  12$\times$1$\times$1 |  10 |  44.49 |  257 |  800
16 |  2 |  3$\times$1$\times$1 |  10 |  44.06 |  206 |  768 |  32 |  3 |  12$\times$2$\times$1 |  10 |  44.12 |  299 |  734
TABLE III: Selection of the optimum configuration of the proposed HCRDNN. Config. # |  # Filters |  Filter size |  # Rec. neurons |  # Dense neurons |  SI Canc. (dB) | Complexity |  Config. # |  # Filters |  Filter size |  # Rec. neurons |  # Dense neurons |  SI Canc. (dB) | Complexity
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  # Par. |  # FLOPs |  |  |  |  |  |  |  # Par. |  # FLOPs
1 |  2 |  7$\times$1$\times$1 |  5 |  9 |  44.10 |  166 |  789 |  17 |  2 |  12$\times$1$\times$1 |  8 |  10 |  44.42 |  268 |  740
2 |  2 |  9$\times$1$\times$1 |  5 |  10 |  44.10 |  178 |  755 |  18 |  2 |  12$\times$1$\times$1 |  9 |  7 |  44.42 |  264 |  734
3 |  2 |  12$\times$1$\times$1 |  5 |  11 |  44.29 |  192 |  590 |  19 |  2 |  12$\times$1$\times$1 |  9 |  8 |  44.50 |  276 |  757
4 |  2 |  9$\times$1$\times$1 |  5 |  12 |  44.14 |  194 |  785 |  20 |  2 |  12$\times$1$\times$1 |  9 |  9 |  44.42 |  288 |  780
5 |  2 |  10$\times$1$\times$1 |  5 |  12 |  44.18 |  196 |  741 |  21 |  2 |  12$\times$1$\times$1 |  10 |  6 |  44.46 |  282 |  770
6 |  2 |  12$\times$1$\times$1 |  5 |  12 |  44.27 |  200 |  605 |  22 |  3 |  12$\times$1$\times$1 |  5 |  6 |  44.11 |  175 |  635
7 |  2 |  10$\times$1$\times$1 |  6 |  9 |  44.09 |  197 |  745 |  23 |  3 |  12$\times$1$\times$1 |  5 |  7 |  44.10 |  183 |  650
8 |  2 |  12$\times$1$\times$1 |  6 |  9 |  44.22 |  201 |  609 |  24 |  3 |  12$\times$1$\times$1 |  5 |  8 |  44.15 |  191 |  665
9 |  2 |  11$\times$1$\times$1 |  6 |  10 |  44.11 |  208 |  702 |  25 |  3 |  12$\times$1$\times$1 |  5 |  10 |  44.24 |  207 |  695
10 |  2 |  10$\times$1$\times$1 |  6 |  11 |  44.25 |  215 |  779 |  26 |  3 |  12$\times$1$\times$1 |  5 |  11 |  44.17 |  215 |  710
11 |  2 |  11$\times$1$\times$1 |  6 |  11 |  44.23 |  217 |  719 |  27 |  3 |  12$\times$1$\times$1 |  5 |  12 |  44.41 |  223 |  725
12 |  2 |  12$\times$1$\times$1 |  6 |  11 |  44.26 |  219 |  643 |  28 |  3 |  12$\times$1$\times$1 |  6 |  6 |  44.28 |  199 |  682
13 |  2 |  11$\times$1$\times$1 |  7 |  11 |  44.21 |  246 |  776 |  29 |  3 |  12$\times$1$\times$1 |  6 |  8 |  44.22 |  217 |  716
14 |  2 |  12$\times$1$\times$1 |  7 |  11 |  44.44 |  248 |  700 |  30 |  3 |  12$\times$1$\times$1 |  6 |  9 |  44.36 |  226 |  733
15 |  2 |  12$\times$1$\times$1 |  7 |  12 |  44.44 |  258 |  719 |  31 |  3 |  12$\times$1$\times$1 |  6 |  11 |  44.40 |  244 |  767
16 |  2 |  12$\times$1$\times$1 |  8 |  9 |  44.41 |  257 |  719 |  32 |  3 |  12$\times$1$\times$1 |  6 |  12 |  44.39 |  253 |  784

###  V-B Optimum Configuration

The process of selecting the optimum configuration of the proposed HCRNN
(e.g., number of convolutional filters, filter size, number of recurrent
layer’s neurons) is analyzed as given in Table II.222The results in Tables II
and III are obtained using the following default settings: $ReLU$ activation
function in all hidden layers, 5$\times$1$0^{-3}$ learning rate, 158 batch
size, and Adam optimization algorithm over 15 random initializations. The
aforementioned settings are optimized in the next subsections. To that end, we
test two values for the convolutional filters $L\in\\{2,3\\}$ and change the
filter size dimensions such that $R\in\\{2,3,...,13\\}$, $S\in\\{1,2\\}$, and
$Z=1$ for all filters. Moreover, we consider $n_{hr}\in\\{4,5,...,10\\}$ for
the number of neurons in the recurrent layer. Since there is a large number of
combinations between $L$, $R$, $S$, $Z$, and $n_{hr}$, we only show the NN
configurations that achieve the best cancellation-complexity trade-off in
Table II. Specifically, in this work, choosing the optimum configuration of
the HCRNN is based on the criterion of achieving a similar or comparable
cancellation performance to the polynomial canceler with $P=5$ while
maintaining a lower implementation complexity.333We note that at $P=5$, the
polynomial canceler attains 44.45 dB cancellation [References], while from
(19), (20), (21), (22), and (23), it requires 1558 FLOPs and 312 parameters to
achieve this cancellation.  Based on this, it can be observed from Table II
that the HCRNN configuration with three convolutional filters, $12\times
1\times 1$ filter size, and nine neurons in the recurrent layer achieves the
target cancellation of the polynomial canceler. Besides, it attains the best
compromise between the cancellation performance and implementation complexity
compared to its other counterparts. Similarly, the selection of the optimum
configuration of the proposed HCRDNN is explored in Table III.2 In this case,
we test the number of filters $L\in\\{2,3\\}$, filter dimensions
$R\in\\{2,3,...,13\\}$, $S\in\\{1,2\\}$, and $Z=1$, recurrent layer’s neurons
$n_{hr}\in\\{4,5,...,10\\}$, and dense layer’s neurons
$n_{hd}\in\\{4,5,...,12\\}$. As can be seen from Table III, two configurations
of the proposed HCRDNN achieve the best compromise among the cancellation
performance, number of FLOPs, and parameters. The first configuration (i.e.,
HCRDNN 1) employs two convolutional filters, $12\times 1\times 1$ filter size,
seven neurons in the recurrent layer, and eleven neurons in the dense layer.
However, the second (i.e., HCRDNN 2) utilizes three convolutional filters,
$12\times 1\times 1$ filter size, five neurons in the recurrent layer, and
twelve neurons in the dense layer. It is worth mentioning here that from
Tables II and III, the filter size of $12\times 1\times 1$ is shown to be the
best size for the optimizable convolutional filters employed in both the HCRNN
and HCRDNN architectures. Moreover, it is worth noting that using an
additional dense layer in the proposed HCRDNN 1 and HCRDNN 2 reduces the
number of neurons in the recurrent layer from 9 to 7 and 5 neurons,
respectively, compared to the HCRNN architecture; this significantly affects
the computational complexity of the HCRDNN model.

TABLE IV: Optimum activation functions of the proposed HCRNN. Config. # | Conv. layer | Rec. layer | SI Canc. (dB) | FLOPs
---|---|---|---|---
1 | $tanh$ | $tanh$ | 44.61 | 850
2 | $tanh$ | $ReLU$ | 44.50 | 805
3 | $tanh$ | $Sigm$ | 44.15 | 832
4 | $ReLU$ | $tanh$ | 44.37 | 790
5 | $\boldsymbol{ReLU}$ | $\boldsymbol{ReLU}$ | 44.44 | 745
6 | $ReLU$ | $Sigm$ | 44.41 | 772
7 | $Sigm$ | $tanh$ | 44.27 | 826
8 | $Sigm$ | $ReLU$ | 43.65 | 781
9 | $Sigm$ | $Sigm$ | 43.12 | 808
TABLE V: Optimum activation functions of the proposed HCRDNN. Config. # | Conv. layer | Rec. layer | Dense layer | SI Canc. (dB) | FLOPs
---|---|---|---|---|---
HCRDNN 1 | HCRDNN 2 | HCRDNN 1 | HCRDNN 2
1 | $tanh$ | $tanh$ | $tanh$ | 44.41 | 44.33 | 830 | 870
2 | $tanh$ | $tanh$ | $ReLU$ | 44.49 | 44.57 | 775 | 810
3 | $tanh$ | $tanh$ | $Sigm$ | 44.21 | 43.98 | 808 | 846
4 | $tanh$ | $ReLU$ | $tanh$ | 44.40 | 44.30 | 795 | 845
5 | $tanh$ | $ReLU$ | $ReLU$ | 44.36 | 44.34 | 740 | 785
6 | $tanh$ | $ReLU$ | $Sigm$ | 44.14 | 43.98 | 773 | 821
7 | $tanh$ | $Sigm$ | $tanh$ | 43.98 | 44.05 | 816 | 860
8 | $tanh$ | $Sigm$ | $ReLU$ | 44.13 | 44.34 | 761 | 800
9 | $tanh$ | $Sigm$ | $Sigm$ | 43.96 | 44.09 | 794 | 836
10 | $ReLU$ | $tanh$ | $tanh$ | 44.48 | 44.24 | 790 | 810
11 | $ReLU$ | $tanh$ | $ReLU$ | 44.54 | 44.48 | 735 | 750
12 | $ReLU$ | $tanh$ | $Sigm$ | 44.17 | 43.88 | 768 | 786
13 | $ReLU$ | $ReLU$ | $tanh$ | 44.36 | 44.14 | 755 | 785
14 | $\boldsymbol{ReLU}$ | $\boldsymbol{ReLU}$ | $\boldsymbol{ReLU}$ | 44.44 | 44.41 | 700 | 725
15 | $ReLU$ | $ReLU$ | $Sigm$ | 44.24 | 43.92 | 733 | 761
16 | $ReLU$ | $Sigm$ | $tanh$ | 44.34 | 43.89 | 776 | 800
17 | $ReLU$ | $Sigm$ | $ReLU$ | 44.31 | 44.26 | 721 | 740
18 | $ReLU$ | $Sigm$ | $Sigm$ | 44.11 | 43.95 | 754 | 776
19 | $Sigm$ | $tanh$ | $tanh$ | 44.08 | 43.89 | 814 | 846
20 | $Sigm$ | $tanh$ | $ReLU$ | 44.04 | 44.07 | 759 | 786
21 | $Sigm$ | $tanh$ | $Sigm$ | 43.90 | 43.59 | 792 | 822
22 | $Sigm$ | $ReLU$ | $tanh$ | 43.46 | 42.78 | 779 | 821
23 | $Sigm$ | $ReLU$ | $ReLU$ | 43.95 | 43.63 | 724 | 761
24 | $Sigm$ | $ReLU$ | $Sigm$ | 43.84 | 43.21 | 757 | 797
25 | $Sigm$ | $Sigm$ | $tanh$ | 42.53 | 41.98 | 800 | 836
26 | $Sigm$ | $Sigm$ | $ReLU$ | 43.02 | 43.54 | 745 | 776
27 | $Sigm$ | $Sigm$ | $Sigm$ | 42.97 | 42.94 | 778 | 812

###  V-C Optimum Activation Functions

In this subsection, we test the cancellation performance and computational
complexity of the proposed HCRNN and HCRDNN using different activation
functions in the convolutional, recurrent, and dense (if any) layers and
select the optimum combination of activation functions that results in the
best cancellation-complexity trade-off. Specifically, in Table IV, we evaluate
the performance of the optimum HCRNN configuration, obtained in the previous
subsection, using different activation functions. As seen from Table IV, the
HCRNN with the $ReLU$ activation function in both convolutional and recurrent
layers achieves the target cancellation of the polynomial canceler and
provides the best compromise between the cancellation and complexity
performances. It can also be inferred from Table IV that using the $tanh$
activation function in the hidden layers of HCRNN only enhances the SI
cancellation by 0.17 dB at the cost of augmenting the FLOPs by 14% compared to
the $ReLU$ activation function. Hence, in this work, we employ the $ReLU$
activation function for the HCRNN as it provides the best cancellation-
complexity trade-off.

Likewise, in Table V, we test the optimum configurations of the HCRDNN using
various activation functions. As can be observed, using the $ReLU$ activation
function in the convolutional, recurrent, and dense layers attains the best
cancellation-complexity trade-off for both HCRDNN architectures.

###  V-D Optimum Learning Rate

The effect of varying the learning rate on the cancellation performance of the
proposed HCRNN and HCRDNN-based cancelers is analyzed in Table VI. It can be
inferred that using a learning rate of 5$\times$1$0^{-3}$ achieves the best
cancellation performance for both the proposed HCRNN and HCRDNN architectures
compared to the other learning rates.

###  V-E Optimum Batch Size

Similarly, in this subsection, we test the effect of varying the batch size on
the performance of the proposed HCRNN and HCRDNN-based cancelers. Herein, we
consider many values for the batch size, and we only show the batch sizes that
result in the best cancellation performance in Table VII. As can be observed,
employing batch size values of 62 and 158 provide the best cancellation
performance for the proposed HCRNN and HCRDNN architectures, respectively.

###  V-F Selected Optimizer

The cancellation performance of the proposed HCRNN and HCRDNN-based cancelers
is analyzed using different optimizers, as illustrated in Table VIII. In this
work, we test the stochastic gradient descent (SGD), adaptive moment
estimation (Adam), root mean square propagation (RMSprop), adaptive delta
(Adadelta), and adaptive max-pooling (Adamax) optimizers. As seen from Table
VIII, the Adam optimization algorithm attains the best performance for the
HCRNN and HCRDNN compared to the other optimization techniques.

Based on the aforementioned subsections, the optimum settings for training the
proposed HCRNN and HCRDNN architectures are summarized in Table IX.

TABLE VI:  Optimum learning rate of the proposed HCRNN and HCRDNN. Learning rate | $\mathbf{1\times 10^{-2}}$ | $\mathbf{5\times 10^{-3}}$ | $\mathbf{1\times 10^{-3}}$ | $\mathbf{5\times 10^{-4}}$ | $\mathbf{1\times 10^{-4}}$
---|---|---|---|---|---
Network | SI Cancellation (dB)
HCRNN | 44.36 | 44.44 | 44.12 | 43.64 | 38.90
HCRDNN 1 | 44.28 | 44.44 | 43.65 | 42.92 | 39.11
HCRDNN 2 | 44.22 | 44.41 | 44.01 | 43.26 | 39.22
TABLE VII: Optimum batch size of the proposed HCRNN and HCRDNN. Batch size | 22 | 62 | 158 | 256 | 512
---|---|---|---|---|---
Network | SI Cancellation (dB)
HCRNN | 44.37 | 44.50 | 44.44 | 44.40 | 44.44
HCRDNN 1 | 44.17 | 44.38 | 44.44 | 44.39 | 44.28
HCRDNN 2 | 44.22 | 44.29 | 44.41 | 44.37 | 44.31
TABLE VIII: Selected optimizer of the proposed HCRNN and HCRDNN. Optimizer | SGD | Adam | RMSprop | Adadelta | Adamax
---|---|---|---|---|---
Network | SI Cancellation (dB)
HCRNN | 41.11 | 44.50 | 44.23 | 37.80 | 44.36
HCRDNN 1 | 38.37 | 44.44 | 43.97 | 37.84 | 43.93
HCRDNN 2 | 38.73 | 44.41 | 43.89 | 37.83 | 43.95
TABLE IX: Optimum settings of the proposed HCRNN and HCDRNN. Structure |  HCRNN |  HCRDNN 1 |  HCRDNN 2
---|---|---|---
# Filters |  3 |  2 |  3
Filter size |  12$\times$1$\times$1 |  12$\times$1$\times$1 |  12$\times$1$\times$1
# Rec. neurons |  9 |  7 |  5
# Dense neurons |  N/A |  11 |  12
Conv. activation |  $ReLU$ |  $ReLU$ |  $ReLU$
Rec. activation |  $ReLU$ |  $ReLU$ |  $ReLU$
Dense activation |  N/A |  $ReLU$ |  $ReLU$
Learning rate |  5$\times$1$0^{-3}$ |  5$\times$1$0^{-3}$ |  5$\times$1$0^{-3}$
Batch size |  62 |  158 |  158
Optimizer |  Adam |  Adam |  Adam
TABLE X: Optimum settings of the state-of-the-art NN architectures. Structure | # Hidden layers neurons | Activation | Learning rate | Batch size | Opti-mizer
---|---|---|---|---|---
RV-TDNN [References] | 18 | $ReLU$ | 5$\times$1$0^{-3}$ | 22 | Adam
RNN [References] | 20 | $tanh$ | 2.5$\times$1$0^{-3}$ | 158 | Adam
CV-TDNN [References] | 7 | $\mathbb{C}ReLU$ | 4.5$\times$1$0^{-3}$ | 62 | Adam
LWGS [References] | 9 | $\mathbb{C}ReLU$ | 4.5$\times$1$0^{-3}$ | 62 | Adam
MWGS [References] | 12 | $\mathbb{C}ReLU$ | 4.5$\times$1$0^{-3}$ | 62 | Adam
Deep RV-TDNN [References] | (10-10-10) | $ReLU$ | 5$\times$1$0^{-3}$ | 22 | Adam
Deep RNN [References] | (16-16-16) | $tanh$ | 2.5$\times$1$0^{-3}$ | 158 | Adam
Deep CV-TDNN [References] | (4-4-4) | $\mathbb{C}ReLU$ | 4.5$\times$1$0^{-3}$ | 62 | Adam

##  VI Results and Discussions

In this section, the proposed and the state-of-the-art NN-based cancelers are
assessed for canceling the SI in a realistic FD system and compared with the
polynomial-based canceler’s performance. The performance assessment includes
studying the proposed NN architectures’ prediction capabilities, MSE
performance, achieved SI cancellation, power spectral density (PSD) of the
modeled SI signal, computational complexity, and memory requirements. For the
sake of comprehensive evaluation, we compare the performance of the proposed
NNs with the shallow and deep RV-TDNN, RNN, and CV-TDNN introduced in
[References] and [References]; further, we consider the performance of the
LWGS and MWGS investigated in [References]. In this work, all NN architectures
are implemented using 3.5.7 Python software with 2.0.0 TensorFlow and 2.3.1
Keras versions. Moreover, the NNs are trained using the dataset described in
Section V-A over 15 random network initializations. For the proposed NN
architectures, we employ the optimum settings summarized in Table IX, whereas
for the state-of-the-art NNs, we use the optimum settings listed in Table X.
Besides, we employ a memory length $M=13$ for the polynomial and all NN-based
cancelers. It is worth mentioning that the optimum settings for training the
proposed and the state-of-the-art NN-based cancelers, summarized in Tables IX
and X respectively, are chosen by applying trial and error methodology such
they achieve a similar cancellation performance to the polynomial canceler at
$P=5$ (i.e., 44.45 dB cancellation).

###  VI-A Prediction Capabilities of the Proposed NN Architectures

In this subsection, the prediction capabilities of the proposed HCRNN, HCRDNN
1, and HCRDNN 2 architectures are assessed in Figs. 5(a), (b), and (c),
respectively. Herein, we show the time-domain waveforms for 200 input-output
sample pairs of the SI signal predicted by the proposed NN architectures and
their corresponding ground-truth (actual) values. As can be seen from the
figures, there is a consistency between the actual and predicted values of the
SI signal modeled by the proposed NNs. This consistency substantiates the
ability of the proposed HCRNN and HCRDNN-based cancelers to model the SI
correctly. In addition, it can be inferred from the figures that the proposed
NNs have similar modeling capabilities since their network settings are
intentionally set such that they achieve a comparable cancellation performance
to the polynomial-based canceler with $P=5$.

(a) HCRNN. (b) HCRDNN 1. (c) HCRDNN 2.

Fig. 5: Prediction capabilities of the proposed NN architectures.

###  VI-B MSE Performance

The MSE performance in the training and testing phases of the proposed NN
architectures is depicted in Figs. 6 and 7, respectively, compared to the
state-of-the-art NNs. The MSE is utilized to evaluate the average squared
difference between the ground-truth and predicted SI signal modeled by the
various NN architectures. As seen from the figures, the MSE values of all NNs
are comparable in both training and testing phases (this can be clearly
observed from the inset graphs in Figs. 6 and 7). As previously stated, the
reason behind this is that all NNs are set to attain a comparable cancellation
performance, and that is why they achieve a similar MSE performance. Moreover,
it can be observed from Figs. 6 and 7 that there are no overfitting signs for
the proposed and the state-of-the-art NNs as they perform well in both
training and testing phases. Additionally, it can be seen from the figures
that the proposed NNs converge to similar, low MSE values in both training and
testing phases, which indicates the goodness of the solution provided by the
proposed NN architectures.

###  VI-C Achieved SI Cancellation

The boxplots quantifying the total SI cancellation achieved by the proposed
and the state-of-the-art NN-based canceler is depicted in Fig. 8. Here, we
calculate the mean cancellation provided by the NN-based cancelers over the
employed 15 network initializations. As seen from the figure, all NNs attain a
comparable cancellation performance to the polynomial canceler at $P=5$ (red
dashed-line in Fig. 8). Specifically, the mean cancellation achieved by all
NN-based cancelers varies from 44.40 to 45.27 dB, which is very close to the
polynomial target cancellation. It is worth noting that we configure the NN
settings to achieve a comparable cancellation performance to the polynomial
canceler as it is not easy to adjust the setting to exactly achieve the target
cancellation.

Fig. 6: MSE in the training phase for the proposed and the state-of-the-art NN
architectures. Fig. 7: MSE in the testing phase for the proposed and the
state-of-the-art NN architectures.

###  VI-D PSD Performance

In this subsection, we evaluate the PSD of the SI signal before applying any
SI cancellation techniques (blue curve in Fig. 9). Further, the PSD of the
residual SI signal after the linear cancellation process is assessed (red
curve). Similarly, we depict the PSD of the SI signal after performing non-
linear cancellation using the polynomial and NN-based cancelers. Finally, we
show the PSD of the receiver background noise when there is no transmission,
i.e., the receiver’s noise floor (black curve). As can be inferred from Fig.
9, the linear canceler provides 37.90 dB SI cancellation by bringing the SI
signal’s power down from -42.74 dBm to -80.60 dBm. Moreover, the polynomial-
based canceler attains an additional 6.6 dB cancellation by bringing the
residual SI signal’s power down from -80.60 dBm to -87.19 dBm, making it very
close to the receiver noise floor (approximately 3 dB above the receiver’s
noise floor). A similar task is performed by the proposed and the state-of-
the-art NN-based cancelers, in which they cancel the SI signal after the
linear canceler by 6.55-7.40 dB, making it very akin to the receiver’s
background noise level as illustrated in the inset graph of Fig. 9.

###  VI-E Computational Complexity and Memory Requirements

In this subsection, we assess the computational complexity of the proposed and
the state-of-the-art NN-based cancelers in terms of the number of FLOPs and
calculate the complexity reduction provided by each canceler compared to the
polynomial canceler. Similarly, we quantify the memory requirements of each
NN-based canceler in terms of the number of parameters and calculate the
amount of reduction compared to the baseline canceler. The results of the
comparison are graphically shown in Fig. 10 and numerically summarized in
Table XI.

Fig. 8: Total SI cancellation of the proposed and the state-of-the-art NN-
based cancelers. Fig. 9: PSD of the modeled SI signal using the proposed and
the state-of-the-art NN-based cancelers.

####  VI-E1 Computational Complexity

Table XI illustrates the reduction in the number of FLOPs provided by the
proposed HCRNN and HCRDNN-based cancelers compared to the polynomial and the
state-of-the-art NN-based cancelers. As seen from Table XI, the proposed NN-
based cancelers achieve a superior enhancement in the computational complexity
compared to the polynomial-based canceler. For instance, the HCRNN provides a
52.18% reduction in the number of FLOPs while maintaining the same
cancellation performance of the polynomial canceler. In addition, the HCRDNN 1
and HCRDNN 2 achieve 55.07% and 53.47% reduction compared to the polynomial
canceler, respectively. However, the shallow and deep RV-TDNN, RNN, and CV-
TDNN barely attain one half of the complexity reduction provided by the
proposed NN architectures.

On the other hand, the proposed HCRNN, HCRDNN 1, and HCRDNN 2 architectures
outperform the state-of-the-art MWGS by providing 18%, 20.92%, and 19.32% more
reduction in FLOPs, respectively. Further, they attain 2.4%, 5.3%, and 3.7%
more saving in FLOPs compared to the LWGS-based canceler, respectively. The
above-mentioned results substantiate the proposed cancelers’ superiority in
modeling the SI with low computational complexity compared to the polynomial
and the state-of-the-art NN-based cancelers.

Fig. 10: Computational complexity and memory requirements of the proposed and
the state-of-the-art NN-based cancelers.

####  VI-E2 Memory Requirements

Similarly, the reduction in the number of parameters of the proposed HCRNN and
HCRDNN-based cancelers compared to the polynomial and the state-of-the-art NN-
based cancelers is illustrated in Table XI. As can be observed, the shallow
and deep structures of the RV-TDNN and RNN significantly increase the required
parameters compared to the polynomial canceler. The former comes from dealing
with the real and imaginary parts separately, while the latter results from
employing a large number of feedback connections to efficiently detect the
data sequences. On the other hand, the proposed HCRNN, HCRDNN 1, and HCRDNN 2
reduce the number of parameters by 26.60%, 20.51%, and 28.53% compared to the
polynomial canceler, respectively. This reduction is due to the application of
a convolutional layer before the recurrent layer in the proposed NN
architectures, which mitigates the huge memory requirements of the recurrent
layer. It can also be seen from Table XI that the reduction in parameters
provided by the proposed NNs is comparable to that attained by the shallow and
deep CV-TDNN. Further, the LWGS and MWGS provide the most reduction in the
number of parameters.

However, the proposed NNs outperform the LWGS and MWGS in the number of FLOPs.
More importantly, it can be inferred from Table XI that among all NN
architectures, the proposed HCRDNN 1 attains the best reduction in FLOPs.
Furthermore, the proposed HCRNN achieves the best compromise among the
cancellation performance, number of FLOPs, and parameters. As such, the
proposed NN-based cancelers offer high design flexibility for hardware
implementation, depending on the system demands.

TABLE XI: Reduction in the number of FLOPs and parameters for the proposed and the state-of-the-art NN-based cancelers compared to the polynomial canceler with $P=5$. Network |  SI Canc. (dB) | Complexity | Complexity Reduction
---|---|---|---
|  |  # Par. |  # FLOPs |  # Par. |  # FLOPs
Polynomial ($P\negthinspace=$ 5) |  44.45 |  312 |  1558 |  - |  -
RV-TDNN |  44.76 |  550 |  1156 |  +76.28% |  -25.80%
RNN |  44.94 |  528 |  1210 |  +69.23% |  -22.34%
CV-TDNN |  44.50 |  238 |  1166 |  -23.72% |  -25.16%
LWGS |  44.48 |  162 |  782 |  -48.08% |  -49.81%
MWGS |  44.40 |  212 |  1026 |  -32.05% |  -34.15%
Deep RV-TDNN |  44.73 |  538 |  1120 |  +72.44% |  -28.11%
Deep RNN |  45.27 |  1420 |  3106 |  +355.13% |  +99.36%
Deep CV-TDNN |  44.63 |  228 |  1106 |  -26.92% |  -29.01%
HCRNN |  44.50 |  229 |  745 |  -26.60% |  -52.18%
HCRDNN 1 |  44.44 |  248 |  700 |  -20.51% |  -55.07%
HCRDNN 2 |  44.41 |  223 |  725 |  -28.53% |  -53.47%

##  VII Future Research Directions

In this work, the proposed NN architectures have been verified using a dataset
that is captured by a single-input single-output FD test-bed. The joint design
of multiple-input multiple-output (MIMO) and FD should be considered for B5G
wireless networks [References]-[References], and the prospects of the proposed
NNs could be generalized and verified using datasets captured by massive MIMO
FD test-beds. Nevertheless, several challenges need to be solved prior to the
deployment and design of such a system, which include but are not limited to
the self- and cross-talks among the transceiver’s antennas [References]. Such
challenges deserve a full investigation and will be studied in future work.

On the other hand, applying different machine learning techniques, such as
support vector machines (SVM) [References], for SI cancellation in FD systems
is identified as another future direction of investigation. Several challenges
related to the high computational complexity of the SVM models can be
considered in the future.

##  VIII Conclusion

In this paper, two hybrid-layers NN architectures, referred to as hybrid-
convolutional recurrent NN (HCRNN) and hybrid-convolutional recurrent dense NN
(HCRDNN), have been proposed for the first time to model the FD system SI with
reduced computational complexity. The former exploits the weight sharing
characteristics and dimensionality reduction potential of the convolutional
layer to extract the memory effect and non-linearity incorporated into the
input signal using a reduced network scale. Moreover, it employs the high
modeling capabilities of the recurrent layer to help learn the temporal
behavior of the input signal. The latter exploits an additional dense layer to
build a deeper NN model with low complexity. The complexity analysis of the
proposed NN architectures has been conducted, and the optimum settings for
their training have been selected. Our findings demonstrate that the proposed
HCRNN and HCRDNN-based cancelers attain a reduction in the computational
complexity with 52% and 55% over the polynomial-based canceler, respectively,
while maintaining the same cancellation performance. In addition, the proposed
HCRNN and HCRDNN offer astounding complexity reduction over the shallow and
deep NN-based cancelers in the literature.

## Representation of Non-linear Systems using Polynomial-based Models

In the following, we review the common methods for representing non-linear
systems using polynomial-based models (e.g., Hammerstein and parallel-
Hammerstein), which are considered the cornerstone of approximating the SI in
FD transceivers. Herein, it is assumed that all signals are of a narrow-band
nature [References]. In addition, only the odd order non-linearities are
considered since the even order non-linearities lay outside the passband
[References] and are typically filtered by the BPFs in the FD transceiver
[References].

###  -A Hammerstein Model

The Hammerstein model is one of the widely-known models for approximating the
non-linear behavior with $M$-tap memory. In the Hammerstein model, a static
non-linearity $N(.)$ is employed in series with a linear filter $L(.)$ in
order to model the non-linearity with memory as follows [References]:

$y(n)=L\left[N\left[x\left(n\right)\right]\right]=\sum_{m=0}^{M}h_{m}\sum_{\underset{p\,odd}{p=1,}}^{P}a_{p}\,x(n-m)\left|\,x(n-m)\right|^{p-1},$
(46)

where $P$ indicates the order of non-linearity, whereas $h_{m}$ and $a_{p}$
represent the coefficients of the linear filter $L(.)$ and the non-linearity
$N(.)$, respectively.

###  -B Parallel-Hammerstein (PH) Model

An extended version of the Hammerstein model is the PH model, which is
constructed by combining the outputs of several Hammerstein models and can be
identified as [References]:

$y(n)=\sum_{\underset{p\,odd}{p=1,}}^{P}\sum_{m=0}^{M}h_{m,p}\,x(n-m)\left|x(n-m)\right|^{p-1},$
(47)

where $h_{m,p}$ represent the coefficients of the linear filter corresponding
to that order of non-linearity.

The PH model (47) can be rewritten as [References] as follows:

$y(n)=\sum_{\underset{p\,odd}{p=1,}}^{P}\sum_{m=0}^{M}h_{m,p}\,x(n-m)^{\frac{p+1}{2}}x^{*}(n-m)^{\frac{p-1}{2}},$
(48)

where $\left(.\right)^{*}$ indicates the complex conjugate operator. The PH
model (48) provides a more general polynomial representation for approximating
the non-linearity with memory and is considered the cornerstone of modeling
the SI in FD transceivers.

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