# Deep Autoencoder-based Z-Interference Channels with Perfect and Imperfect
CSI

Xinliang Zhang, , Mojtaba Vaezi  This paper has been presented in part at IEEE
Wireless Communications and Networking Conference (WCNC), 2023 [1] and IEEE
International Conference on Communications (ICC), 2023 [2]. This work was
supported by the U.S. National Science Foundation under Grants ECCS-2301778
and CNS-2239524. X. Zhang and M. Vaezi are with the Department of Electrical
and Computer Engineering, Villanova University, Villanova, PA 19085 USA
(e-mail<EMAIL_ADDRESS>mvaezi@villanova.edu).

###### Abstract

A deep autoencoder (DAE)-based structure for end-to-end communication over the
two-user Z-interference channel (ZIC) with finite-alphabet inputs is designed
in this paper. The proposed structure jointly optimizes the two
encoder/decoder pairs and generates interference-aware constellations that
dynamically adapt their shape based on interference intensity to minimize the
bit error rate (BER). An in-phase/quadrature-phase (I/Q) power allocation
layer is introduced in the DAE to guarantee an average power constraint and
enable the architecture to generate constellations with nonuniform shapes.
This brings further gain compared to standard uniform constellations such as
quadrature amplitude modulation. The proposed structure is then extended to
work with imperfect channel state information (CSI). The CSI imperfection due
to both the estimation and quantization errors are examined. The performance
of the DAE-ZIC is compared with two baseline methods, i.e., standard and
rotated constellations. The proposed structure significantly enhances the
performance of the ZIC both for the perfect and imperfect CSI. Simulation
results show that the improvement is achieved in all interference regimes
(weak, moderate, and strong) and consistently increases with the signal-to-
noise ratio (SNR). For instance, more than an order of magnitude BER reduction
is obtained with respect to the most competitive conventional method at weak
interference when $\rm SNR>15dB$ and two bits per symbol are transmitted. The
improvements reach about two orders of magnitude when quantization error
exists, indicating that the DAE-ZIC is more robust to the interference
compared to the conventional methods.

###### Index Terms:

Interference channel, Z-interference, imperfect CSI, autoencoder,
constellation design.

##  I Introduction

Interference is a central issue in today’s multi-cell networks. The
information-theoretic model for a multi-cell network is the interference
channel (IC). There have been many efforts to find the capacity of the IC
either with the same generality and accuracy used by Shannon for point-to-
point systems [3, 4, 5, 6] or by seeking approximate solutions with a
guaranteed gap to optimality at any signal-to-noise ratio (SNR) [7]. However,
the capacity region of the two-user IC is only known for strong interference
where decoding and canceling the interference is optimal [3]. Also, at very
weak interference, sum-capacity is achievable by treating interference as
noise [8, 9, 10], whereas, in general, decoding part of the interference and
treating the remaining as noise is the best achievable scheme to date [5].

The aforementioned Shannon-theoretic works are based on Gaussian inputs.
Despite being theoretically optimal, Gaussian alphabets are continuous and
unbounded, and thus, are rarely applied in real-world communication. In
practice, signals are generated using finite alphabet sets, such as phase-
shift keying (PSK) and quadrature amplitude modulations (QAM). The performance
gap between the finite alphabet input and the Gaussian input design is non-
negligible [11]. However, conventional finite-alphabet approaches are based on
predefined uniform constellations like QAM. These constellations are defined
for point-to-point systems [12, 13, 14, 15] and their constellation shaping is
oblivious to interference. Such an inability to respond to interference is an
obstacle to improving the bit-error rate and spectral efficiency of today’s
interference-limited communication systems.

In this paper, we consider the two-user single-input single-output (SISO) one-
sided IC, also known as the Z-interference channel (ZIC). With Gaussian
signaling, the capacity region of this channel is known only in the strong and
very strong interference regimes [6]. However, Gaussian signaling is
unsuitable for practical applications. Previous work has studied ZIC with
finite alphabet sets in specific regimes and predefined uniform
constellations. In [16], it is shown that rotating one input constellation
(alphabet) can improve the sum-rate of the two-user IC in strong/very strong
interference regimes. Later, an exhaustive search for finding the optimal
rotation of the signal constellation was presented in [17]. In addition, a
signaling design is proposed in [18] which applies a rotation to the channel
which resembles rotating the input. The main focus of the above papers is to
maximize the achievable rates, and they do not study bit-error rate (BER)
performance. However, BER is a key metric and interference can severely reduce
the BER by distorting the received constellation, when uniform constellations
like QAM are employed.

Recent research has proved end-to-end learning as a promising approach for
encoding, decoding, and signal representation to reduce BER [19].
Particularly, deep autoencoder (DAE) is a popular architecture for
implementing end-to-end learning. It consists of an encoder that transforms
input data into a low-dimensional representation to find its structure and a
decoder that reconstructs the original input from this representation. DAE-
based end-to-end communication is an emerging approach to finite-alphabet
communication in which BER is the main performance measure and constellation
design is inherent. DAE-based communication is introduces both for single- and
multi-user systems by various groups [19, 20, 21, 22, 23, 24]. These studies
indicate that the DAE surpasses current solutions and enhances performance
beyond conventional methods [24].

Specifically, utilizing two DAEs for the transmitter/receiver pairs enables
effective signal separation/decoding of original data even in the presence of
interference, thereby paving the way for enhancing communication performance
over the IC, as investigated in prior studies [19, 25, 26]. However, these
studies have limitations as they focus solely on symmetric interference
scenarios and compare their results against simple baselines like quadrature
phase shift keying (QPSK), despite the fact that QPSK performs much worse than
rotated QPSK in the context of the IC [16, 17, 18]. Further, the DAE designs
in [19, 25, 26] produce symbols with fixed power levels and lack the ability
to generalize to QAM-like constellations. Thus, they do not efficiently use
the in-phase and quadrature-phase (I/Q) plane. Additionally, these designs
assume perfect knowledge of channel state information (CSI), and the
transition from perfect CSI to imperfect CSI remains unexplored.

###  I-A Motivation and Contribution

The above limitations has motivated us to investigate DAEs potential for the
long-lasting problem of interference in more practical settings. We shed light
on DAE-based communication over asymmetric interference with both perfect and
imperfect CSI. Specifically, we design and train novel DAE-based architectures
for the ZIC with finite-alphabet inputs. In the ZIC, two DAEs should be
considered in two transmitter-receiver pairs. The two DAE pairs cooperate to
avoid interference and adapt their constellation to the interference
intensity. Our work is motivated by the following question: Can we design
interference-aware constellations using DAEs? Will the gains remain/vanish if
CSI is not perfect? We answer these questions by developing new structures and
explaining how the designed nonuniform constellations lend themselves to
interference mitigation in different regimes, and thus improving the BER.

The ZIC is characterized by local, short-range interference [27], where far-
away users are not affected by interference. This simple channel model is a
fundamental building block for more complex interference networks and its
understanding is crucial in the field of interference research. As such, it
has been widely studied in the literature of interference [6, 28, 29], as it
provides insights into the limits interference-limited scenarios. By
understanding the behavior of interference in the ZIC, we can develop more
effective strategies to mitigate interference in more complex networks.

The main contributions of the paper are as follows:

  * •

We design a DAE-based transmission structure for the ZIC and demonstrate its
effectiveness across weak, moderate, and strong interference levels. We
propose incorporating an average power constraint normalization layer that
enables nonuniform constellations, resulting in more efficient utilization of
the I/Q plane. The designed constellations are adaptive to the interference
intensity, and morph in a way that the ‘receivers’ see distinguishable
symbols, thereby improving BER even in the presences of interference. We also
conduct a neural network ablation study to demonstrate the impact and
necessity of each design element in our proposed model. This analysis provides
valuable insights into the significance of each component and also serves to
validate the overall architectural effectiveness.

  * •

We extend the proposed structure to the finite-alphabet ZIC with imperfect
CSI, where we consider both estimation and quantization errors. Particularly,
CSI estimation errors pose confuses for DAE training and testing performance.
Meanwhile, quantization errors, arising from limited feedback capacity,
introduce undesirable rotations to the constellations. In order to develop a
DAE that is robust against these errors, we introduce an equivalent system
model that reduces the CSI parameters and ultimately lowers the BER.

  * •

Our design directly accepts the transmission bits as its input rather than
converting them to symbols and using one-hot vectors for the DAE input. This
has two advantages. First, our design can directly minimize the BER and we do
not need to worry about optimal bit-to-symbol mapping. Second, it reduces the
complexity as to transmit $b$ bits, the input and output layers require only
$b$ neurons whereas the one-hot vector method needs $2^{b}$ neurons.

It worth mentioning that, for benchmarking purposes, we use rotated uniform
constellations, which have been proven to be more competitive than their
unrotated counterparts. Our proposed DAE-ZIC shows significantly better BER
performance for all interference regimes (weak, moderate, and strong
interference). For perfect CSI, when averaged over SNRs from 0 to 20dB, 44%
reduction in BER is achieved. At certain SNRs and interference regimes, the
improvement is over an order of magnitude. For imperfect CSI, the overall
reduction is about 40%. The gap between the DAE-ZIC and conventional methods
is even larger when quantization error is applied.

###  I-B Organization

The remainder of this paper is organized as follows. We elaborate on the ZIC
system model in Section II. The DAE design and the training approach are
introduced in Section III. The system model with an imperfect CSI is derived
in Section IV. The modifications for the DAE with an imperfect CSI is
introduced in Section V. Numerical results are presented in Section VI, and
the paper is concluded in Section VII.

Notation: $(\cdot)^{T}$ denotes transpose, $\mathbb{E}\\{\cdot\\}$ denotes
expectation, and ${\rm{diag}}(\lambda_{1},\dots,\lambda_{n})$ represents the
diagonal matrix with elements $\lambda_{1},\dots,\lambda_{n}$.
$\mathcal{N}(\mu,\sigma^{2})$ and $\mathcal{CN}(\mu,\sigma^{2})$ are real and
complex Gaussian distributions where $\mu$ and $\sigma^{2}$ are the mean and
variance. $|\cdot|$ is the amplitude of a complex number.

##  II System Model of ZIC with Perfect CSI

Figure 1 shows the system model of a two-user complex SISO ZIC with perfect
CSI. The two transmitter-receiver pairs wish to reliably transmit their
messages while the transmission of the first pair interferes with the
transmission of the second. The four nodes are named Tx1, Tx2, Rx1, and Rx2,
as shown in Fig. 1. $h_{ij}$ refers to the channel gain from the $i$th
transmitter to the $j$th receiver and $i,j\in\\{1,2\\}$. For ZIC, $h_{21}=0$.
The received signals at the receivers can be written as

$\displaystyle y_{1}=h_{11}x_{1}+h_{21}x_{2}+n_{1},$ (1a) $\displaystyle
y_{2}=h_{22}x_{2}+n_{2},$ (1b)

in which $x_{1}$ and $x_{2}$ denote the transmitted symbols of Tx1 and Tx2.
The transmitted signals are complex-valued with finite-alphabets and variances
$\mathbb{E}\\{|x_{1}|^{2}\\}=P_{1}$ and $\mathbb{E}\\{|x_{2}|^{2}\\}=P_{2}$ in
which $P_{1}$ and $P_{2}$ are the power budgets of the two transmitters. The
channel coefficients are complex random variables

$\displaystyle h_{ij}\triangleq
r_{ij}e^{j\theta_{ij}}\sim\mathcal{CN}(\mu_{h},\sigma_{h}^{2}),$ (2)

where $\mu_{h}$ and $\sigma_{h}^{2}$ are the mean and variance of the channel
distribution, and $r_{ij}$ and $\theta_{ij}$ represent the magnitude and phase
of $h_{ij}$. Also, $n_{1}$ and $n_{2}$ are the complex-valued independent and
identically distributed (i.i.d.) additive white Gaussian noise with zero means
and variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$. Without loss of
generality, we assume the noise powers at the two receivers’ sides are the
same, i.e., $\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}$ [28].

Figure 1: System model of the ZIC.

###  II-A The Equivalent System Model of the ZIC

It is known that, without loss of generality, the channel gains of the direct
transmission links can be modeled as one, shown in Fig. 2(b), [29, 30, 28].
The interference gain is also real-valued for both real- and complex-valued
systems. When CSI is available and if we apply pre- and post-processing
illustrated in Fig. 2(a), such a system model in Fig. 1 is equivalent to that
of Fig. 2(b).111We describe this process here as we will need later in Section
IV where CSI is not perfect.  The Tx2 applies $e^{j(\theta_{11}-\theta_{21})}$
to cancel the phase of $h_{21}$ and align its phase with that of $h_{11}$. The
Rx1 and Rx2 applies $h_{11}^{-1}$ and
$h_{22}^{-1}e^{j(\theta_{21}-\theta_{11})}$ to normalize the channel gain to
one. Then, the received post-processed signals are

$\displaystyle\bar{y}_{1}=h_{11}^{-1}y_{1}=x_{1}+r_{21}r_{11}^{-1}x_{2}+n_{1}h_{11}^{-1},$
(3a)
$\displaystyle\bar{y}_{2}=h_{22}^{-1}e^{j\theta_{21}}y_{2}=x_{2}+e^{j\theta_{21}}{n_{1}}h_{22}^{-1}.$
(3b)

By defining

$\displaystyle\sqrt{\alpha}\triangleq r_{21}r_{11}^{-1},$ (4a)
$\displaystyle\bar{n}_{1}\triangleq
n_{1}h_{11}^{-1},\;\textmd{and},\;\bar{n}_{2}\triangleq
e^{j\theta_{21}}{n_{1}}h_{22}^{-1},$ (4b)

we have the system model in Fig. 2(b) as

$\displaystyle\bar{y}_{1}=\bar{h}_{11}x_{1}+\sqrt{\alpha}x_{2}+\bar{n}_{1},$
(5a) $\displaystyle\bar{y}_{2}=\bar{h}_{22}x_{2}+\bar{n}_{2}.$ (5b)

where $\bar{h}_{11}=\bar{h}_{22}=1$ and $\bar{h}_{21}=\sqrt{\alpha}$ are the
equivalent channel gains.

(a) Original system model (b) Equivalent system model

Figure 2: The equivalent system model of the ZIC.

Thus, the two system models in Fig. 1 and Fig. 2 are equivalent. Hence, we
follow the existing studies and use the system model in Fig. 2(b), and
consider a fixed $\sqrt{\alpha}$ at each time. It is worth mentioning that
both actual channel gains and noise ($h_{ij}$ and $n_{i}$, $i,j\in\\{1,2\\}$)
are Gaussian. In this paper, we assume a slow fading scenario with

$\displaystyle\bar{n}_{i}\sim\mathcal{CN}(0,\sigma_{i}^{2}r_{ii}^{-2}).$ (6)

##  III Deep Autoencoder with Perfect CSI

Existing studies [16, 17, 31] use standard QAM constellations at each
transmitter. Such constellations are fixed and are not adjustable according to
the interference intensity. To further improve the transmission performance,
we propose a DAE-based transmission for the two-user ZIC, named DAE-ZIC. The
architecture is shown in Fig. 3.

Figure 3: The architecture of the two-user DAE-ZIC implemented by two pairs of
deep autoencoders. Each transmitter of the ZIC contains two sub-networks. Sub-
network 1 mainly generates the constellation and sub-network 2 is used to
implement the average power constraint. The receivers decode their bits from
the received signal. This architecture is based on the system model in Fig.
2(b). $\eta$ is a power control parameter defined in (15).

###  III-A The Architecture of DAE-ZIC

The DAE-ZIC consists of two pairs of DAEs. Each pair performs an end-to-end
transmission, which includes input bits, autoencoder at the transmitter,
channel and noise layers, autoencoder at the receiver, and final output bits.

####  III-A1 Network Input

Each transmitter sends $N_{s}$ bits to the corresponding receiver. The
interference channel coefficient $\sqrt{\alpha}$ is known at the transmitter
and receiver and is appended to the input bit vector. Then, both the
transmitters and receivers have the knowledge of the CSI. The two transmitters
are expected to jointly design their constellations and the receivers will
decode correspondingly.

####  III-A2 Transmitter DAE

As shown in Fig. 3, the DAE of the transmitter contains two sub-networks: Sub-
network 1 and Sub-network 2. Sub-network 1 converts the input bit-vector to
symbols with unit power in the I/Q components. Sub-network 2 performs power
allocation, which controls the power of the I/Q components. The two sub-
networks combines in parallel to yield a nonuniform constellation. The batch
normalization in sub-network 1 together with sub-network 2 realize the average
power constraint at the transmit antenna. Having an average power constraint
is necessary especially for SISO systems. In this way, the I/Q plane is used
efficiently, like QAM. Otherwise, the DAE can only produce constant-power
constellations, like PSK, where constellation points are on a circle which is
not efficient in terms of BER.

The components of sub-network 1 are fully connected layers (FCNN), residual
connections, or shortcuts, to alleviate the vanishing gradient effect, and the
output batch normalization layer. The activation function of the FCNN layers
is tanh, except for the last layer which has two hidden nodes and no
activation function.

Let the batch size be $N_{B}$. The output of the last FCNN is

$\displaystyle\mathbf{X}_{\textmd{fcnn}}\triangleq[\mathbf{x}_{\textmd{fcnn}}^{\textmd{I}},\;\mathbf{x}_{\textmd{fcnn}}^{\textmd{Q}}],$
(7)

where $\mathbf{x}_{\textmd{fcnn}}^{\textmd{I}}$ and
$\mathbf{x}_{\textmd{fcnn}}^{\textmd{Q}}\in\mathbb{R}^{N_{B}\times 1}$ are the
outputs of the two hidden nodes and represent I/Q of the complex-valued
signal. Since the FCNN has unbounded outputs, it cannot guarantee a power
constraint at the transmitter. We propose the transmitter design shown in Fig.
3 to achieve an average power constraint at each antenna. First, we use batch
normalization in sub-network 1 to unify the average power of I/Q
independently. The batch normalization layer linearly normalizes
${\mathbf{x}}_{\textmd{fcnn}}^{\textmd{I}}$ and
${\mathbf{x}}_{\textmd{fcnn}}^{\textmd{Q}}$, in which the normalized vectors
$\mathbf{x}_{B}^{\textmd{I}}$ and $\mathbf{x}_{B}^{\textmd{Q}}$ are given by

$\displaystyle\mathbf{x}_{B}^{\textmd{I}}\triangleq\beta^{\textmd{I}}\cdot{\mathbf{x}}_{\textmd{fcnn}}^{\textmd{I}},\;\;\mathbf{x}_{B}^{\textmd{Q}}\triangleq\beta^{\textmd{Q}}\cdot{\mathbf{x}}_{\textmd{fcnn}}^{\textmd{Q}},$
(8)

where $\bm{\beta}\triangleq[\beta^{\textmd{I}},\;\beta^{\textmd{Q}}]^{T}$
contains two factors for normalization.

In sub-network 2, the power allocation of the I/Q components is determined by
the FCNN layers, which take the input value $\sqrt{\alpha}$ into account. The
FCNN layers calculates the powers of I/Q components, and the power
normalization block limits the total power to $P_{t}$, thus achieving the
intended power control. The powers of $\mathbf{x}_{B}^{\textmd{I}}$ and
$\mathbf{x}_{B}^{\textmd{Q}}$ (i.e., each batch of I/Q signals) are multiplied
by $\gamma^{\textmd{I}}$ and $\gamma^{\textmd{Q}}$, which are the outputs of
Sub-network 2. Defining the input and output of the power normalization layer
as
${\bm{\gamma}}_{0}\triangleq[\gamma^{\textmd{I}}_{0},\;\gamma^{\textmd{Q}}_{0}]^{T}\in\mathbb{R}^{2\times
1}$ and
${\bm{\gamma}}\triangleq[\gamma^{\textmd{I}},\;\gamma^{\textmd{Q}}]^{T}\in\mathbb{R}^{2\times
1}$ respectively, the power normalization layer normalizes its input and
scales its power, i.e.,
${\bm{\gamma}}=\sqrt{P_{t}}\frac{{\bm{\gamma}}_{0}}{|{\bm{\gamma}}_{0}|}$.
Thus, ${\bm{\gamma}}^{T}{\bm{\gamma}}=P_{t}$. Such an operation can be done
via the Lambda layer in Keras [32]. Finally, the outputs of the batch
normalization and power normalization are multiplied together, i.e.,

$\displaystyle\mathbf{x}_{\textmd{out}}^{\textmd{I}}\triangleq\gamma^{\textmd{I}}\cdot\mathbf{x}_{B}^{\textmd{I}},\;\;\mathbf{x}_{\textmd{out}}^{\textmd{Q}}\triangleq\gamma^{\textmd{Q}}\cdot\mathbf{x}_{B}^{\textmd{Q}}.$
(9)

The powers of $\mathbf{x}_{\textmd{out}}^{\textmd{I}}$ and
$\mathbf{x}_{\textmd{out}}^{\textmd{Q}}$ are $\gamma^{\textmd{I}}$ and
$\gamma^{\textmd{Q}}$, respectively. In short, batch normalization is applied
to I and Q components separately and along the time axis (batch by batch)
while power normalization scales the power of I/Q components at each time to
reach the average power constraint. That is, the two normalization operations
are implemented in different dimensions. The final output of the transmitter
is

$\displaystyle\mathbf{X}$
$\displaystyle\triangleq[\mathbf{x}_{\textmd{out}}^{\textmd{I}},\;\mathbf{x}_{\textmd{out}}^{\textmd{Q}}]$
$\displaystyle=[\mathbf{x}_{B}^{\textmd{I}},\;\mathbf{x}_{B}^{\textmd{Q}}]\cdot\textmd{diag}({\bm{\gamma}})$
$\displaystyle=[\mathbf{x}_{\textmd{fcnn}}^{\textmd{I}},\;\mathbf{x}_{\textmd{fcnn}}^{\textmd{Q}}]\cdot\textmd{diag}({\bm{\beta}})\cdot\textmd{diag}({\bm{\gamma}}).$
(10)

where $\mathbf{x}_{\textmd{fcnn}}^{\textmd{I}}$ and
$\mathbf{x}_{\textmd{fcnn}}^{\textmd{Q}}$ are the output of the FCNN layer in
sub-network 1 and represent the preliminary I/Q signals, $\bm{\beta}$
normalizes the power of each batch, and $\bm{\gamma}$ controls the power of
the I/Q signals such that average power is $P_{t}$.

Figure 4: Form a SISO complex-valued channel by a 2x2 MIMO real-valued
channel.

####  III-A3 Channel Implementation

The channel is formed following (1). The complex-valued SISO channel is
achieved by real values intuitively shown in Fig. 4. At Rx1, the received
signal is $\bar{y}_{1}=\bar{y}_{1}^{\textmd{I}}+j\bar{y}_{1}^{\textmd{Q}}$,

$\displaystyle\left[\begin{matrix}\bar{y}_{1}^{\textmd{I}}\\\
\bar{y}_{1}^{\textmd{Q}}\end{matrix}\right]=\left[\begin{matrix}x_{1}^{\textmd{I}}\\\
x_{1}^{\textmd{Q}}\end{matrix}\right]+\bar{h}_{21}\cdot\left[\begin{matrix}x_{2}^{\textmd{I}}\\\
x_{2}^{\textmd{Q}}\end{matrix}\right]+\left[\begin{matrix}\bar{n}_{1}^{\textmd{I}}\\\
\bar{n}_{1}^{\textmd{Q}}\end{matrix}\right],$ (11)

where $\bar{y}_{1}^{\textmd{I}}$ and $\bar{y}_{1}^{\textmd{Q}}$ are the I/Q
components of the received signal, $x_{i}^{\textmd{I}}$ and
$x_{i}^{\textmd{Q}}$ are the I/Q components of the $i$the transmitter, and
$\bar{n}_{i}^{\textmd{I}},\bar{n}_{i}^{\textmd{Q}}\sim\mathcal{N}(0,\frac{1}{2}\sigma_{N}^{2}r_{ii}^{-2})$
are the I/Q components of the complex-valued. Similarly, the received signal
for Rx2 is $\bar{y}_{2}=\bar{y}_{2}^{\textmd{I}}+j\bar{y}_{2}^{\textmd{Q}}$,

$\displaystyle\left[\begin{matrix}\bar{y}_{2}^{\textmd{I}}\\\
\bar{y}_{2}^{\textmd{Q}}\end{matrix}\right]=\left[\begin{matrix}x_{2}^{\textmd{I}}\\\
x_{2}^{\textmd{Q}}\end{matrix}\right]+\left[\begin{matrix}\bar{n}_{2}^{\textmd{I}}\\\
\bar{n}_{2}^{\textmd{Q}}\end{matrix}\right].$ (12)

The additive Gaussian white noise (AWGN) is implemented by a Gaussian noise
layer in Keras. The noise power is set according to SNR in the training and
testing stages.

####  III-A4 Receiver DAE

The received signals are $\bar{y}_{1}$ and $\bar{y}_{2}$. To ensure the
receiver networks have a finite input range, we use batch normalization layers
in Keras unifying the power of the received signals, i.e.,

$\displaystyle{y}_{\textmd{B},i}=\xi\cdot\bar{y}_{i},\;\;E\\{|{y}_{\textmd{B},i}|^{2}\\}=1,\;\;\forall
i\in\\{1,2\\},$ (13)

where $\xi$ is a coefficient to reach the unit power. The process details and
settings are the same as those in the transmitter.

We further define the desired signal for Rx1 as

$\displaystyle x_{\textmd{D},1}\triangleq x_{1}+\sqrt{\alpha}x_{2}.$ (14)

$x_{\textmd{D},1}$ contains the true desired signal $x_{1}$ and the
interference $\sqrt{\alpha}x_{2}$. The goal of the receiver is to decode
$x_{1}$ for an arbitrary $x_{2}$ in its constellation. The desired signal of
Rx2 is $x_{\textmd{D},2}\triangleq x_{2}$. However, the normalization of the
received signal (13) causes the power of the desired signal to vary with the
SNR. Hence, the autoencoder should adjust the decoding boundary according to
the SNR, which is an extra burden. So, we turn to normalize the desired signal
using linear factor, $\eta$, multiplied on the batch normalization output,
i.e.,

$\displaystyle
y_{\textmd{D},i}=\eta\cdot{y}_{\textmd{B},i},\;\eta\triangleq\sqrt{1+\frac{P_{\textmd{D,i}}}{\sigma_{N}^{2}}},\quad\forall
i\in\\{1,2\\},$ (15)

where $P_{\textmd{D},i}$ is the power of the desired signal $x_{\textmd{D},i}$
and $\sigma_{N}^{2}$ is the noise power. The batch normalization normalizes
the desired signals using pre-processing $\eta$. Then, the normalized signal,
$y_{\textmd{D},i}$, together with the feature of the ZIC,
$h_{21}=\sqrt{\alpha}$, are sent to the rest of the FCNN layers. The final
output of the DAE is an estimation of the transmitted bit-vectors,
$\hat{\mathbf{s}}_{1}$ and $\hat{\mathbf{s}}_{2}$, as shown in Fig. 3.

The activation function of the output layer is sigmoid function, i.e.,
$f(s)=\frac{1}{1+e^{-s}}$. Thus, assuming the input of the final layer, marked
as Rx Output in Fig. 3, of $i$th receiver is $\mathbf{y}_{F,i}$, the output
equations can be written as $\hat{\mathbf{s}}_{i}=f(\mathbf{y}_{F,i})$. The
sigmoid function is commonly used because it outputs a value between 0 and 1,
which can be interpreted as a probability of the input belonging to the
positive class. This function is also differentiable, which is important for
backpropagation during training of neural networks.

####  III-A5 Loss Function

In our DAE-ZIC, each receiver has its own estimation of the transmitted bits.
Then, the overall loss function of the DAE-ZIC is
$\mathcal{L}=\mathcal{L}_{1}+\mathcal{L}_{2}$, where $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$ are the loss at Rx1 and Rx2. Each output vector in the two
receivers represents binary messages, then the network can be trained using
binary cross-entropy loss:

$\displaystyle\mathcal{L}_{i}=\frac{1}{N_{B}}\sum_{n=1}^{N_{B}}(\mathbf{s}_{i,n})^{T}\log\hat{\mathbf{s}}_{i,n}+(1-\mathbf{s}_{i,n})^{T}\log(1-\hat{\mathbf{s}}_{i,n}),$
(16)

where $i\in\\{1,2\\}$ distinguishes the users, $N_{B}$ is the batch size,
$\mathbf{s}_{i,n}$ is the $n$th input bit-vector in the batch, and
$\hat{\mathbf{s}}_{i,n}$ is the corresponding output. The loss function treats
each element of the DAE output as a 0/1 classification task. The binary cross-
entropy loss function is commonly used for multi-label classification
problems, where each example can have multiple binary labels. The loss
function measures the difference between the predicted probability of each
label being present in the example and the true probability of the label being
present. Finally, the loss is the summation of the loss of $N_{s}$ tasks,
where $N_{s}$ is the number of bits in the transmission. In the training
process, the backpropagation algorithm passes $\mathcal{L}_{1}$ to Rx1 and it
will further go to Tx1 and Tx2, whereas $\mathcal{L}_{2}$ affects Rx2 and Tx2.

###  III-B Training Procedure of the DAE-ZIC

Due to the difficulty of training a single network across all values of the
interference gain $\alpha$, distinct instances of DAEs are employed for a few
different ranges of $\alpha$. For each training session, a value for $N_{s}$
is selected and the desired range for $\alpha\in[\alpha_{\min},\alpha_{\max}]$
is specified. During the training process, all four sub-networks are trained
simultaneously. The DAE is trained iteratively using random values of $\alpha$
within this interval. For each $\alpha$, the DAE undergoes training for
$E_{p}$ epochs with a mini-batch size of $N_{B}$ and a constant learning rate
of $l_{r}$. After training the DAE for $N_{d}$ different values of $\alpha$,
the learning rate is reduced to $d_{r}l_{r}$. A comprehensive description of
the training procedure, including the simultaneous training of all sub-
networks to adapt to the ZIC interference patterns, can be found in Algorithm
1.

Algorithm 1 Training Procedure for the DAE-ZIC

1:Fix $N_{s}$, $\alpha_{\min}$, and $\alpha_{\max}$.

2:Set $P_{t}=1$W and SNR$=10$dB.

3:Set $N_{\alpha}=30,000$, the number of channels.

4:Set $E_{p}=10$, the number of the epochs.

5:Set $N_{B}=10^{4}$, the batch size.

6:Set $l_{r}=10^{-2}$, the initial learning rate, which will drop to
$d_{r}l_{r}=0.95l_{r}$ after every $N_{d}=200$ trained channels.

7:Initialize the DAE-ZIC network.

8:for index $i_{\alpha}$ from $1$ to $N_{\alpha}$ do

9: Uniformly and randomly select one $\alpha\in[\alpha_{\min},\alpha_{\max}]$.

10: Randomly generate $h_{11}$ and $h_{22}$ using (2).

11: Normalize the channel using (5).

12: Update $\bar{h}_{21}=\sqrt{\alpha}$ in the DAE-ZIC in (11).

13: Set the variance of the noise layer according to (6).

14: for index $i_{e}$ from $1$ to $E_{p}$ do

15: Randomly generate $N_{B}$ bit vectors.

16: Update the weights of the DAE-ZIC using Adam.

17: end for

18: Set learning rate $l_{r}=d_{r}l_{r}$ if $i_{\alpha}/N_{d}$ is an integer.

19:end for

##  IV System Model with Imperfect CSI

In this section, we consider the ZIC with imperfect CSI and find its
equivalent channel. The CSI imperfectness comes from two sources: the error in
the estimation of the CSI at each receiver and the error due to the
quantization of the CSI before feeding it back to the transmitter. The system
model is depicted in Fig. 5(a), in which both the estimation error and
quantization error are considered. In general, the notation is similar to
Section II-A. One main difference is that the estimation errors
$\varepsilon_{ij}$ occur at the receivers when we estimate the channel
coefficients $\hat{h}_{ij}$. The imperfectness of $\hat{h}_{ij}$ affects the
decoding process. Besides, the quantization error occurs when the parameters
need to be fed back to another node. For example, the Tx2 applies

$\displaystyle\theta_{q}\triangleq Q(\hat{\theta}_{11}-\hat{\theta}_{21}),$
(17)

to cancel the phase of $h_{21}$ where $\theta_{11}$ and $\hat{\theta}_{21}$
are estimated by Rx1 and $Q(\cdot)$ is a quantization function. Ideally,
$\theta_{q}$ is expected to be $\hat{\theta}_{11}-\hat{\theta}_{21}$, however,
the quantization error exists which is defined as

$\displaystyle\theta_{\delta}\triangleq\theta_{q}-(\hat{\theta}_{11}-\hat{\theta}_{21}),$
(18)

The quantization error is included in $\bar{h}_{21}$ in Fig. 5(b). The details
of the system model and the two types of errors are given as follows.

(a) Original system model with imperfect CSI. The angle $\theta_{q}$ is the
pre-processing angle defined in (17). (b) Equivalent system model with
imperfect CSI. The angle $\theta_{\delta}$ is the quantization error defined
in (18).

Figure 5: System model of the ZIC with imperfect CSI.

####  IV-1 Estimation Error

The estimated channel gain is modeled as in [33]

$\displaystyle\hat{h}_{ij}\triangleq\hat{r}_{ij}e^{j\hat{\theta}_{ij}}={h}_{ij}-\varepsilon_{ij},$
(19)

where

$\displaystyle h_{ij}\sim\mathcal{CN}(\mu_{h},\sigma_{h}^{2})$ (20)

is the actual channel with mean $\mu_{h}$ and variance $\sigma_{h}^{2}$, and

$\displaystyle\varepsilon_{ij}\sim\mathcal{CN}(0,\sigma_{E}^{2})$ (21)

is the estimation error in which $\sigma_{E}^{2}$ is the variance of the
error, and $ij\in\\{11,21,22\\}$. $\hat{h}_{11}$ and $\hat{h}_{21}$ are
estimated by Rx1 while $\hat{h}_{22}$ is estimated by Rx2. The estimated
channel coefficients are determined by the actual channel and noise, hence,
$\hat{h}_{ij}\sim\mathcal{CN}(\mu_{h},\sigma_{h}^{2}+\sigma_{E}^{2})$. Once
the actual channel and noise are determined, we have
$h_{ij}=\hat{h}_{ij}+\varepsilon_{ij}$ as the channels in Fig. 5(a).

Tx2 keeps the pre-processing based on a feedback angle, $\theta_{q}$. Rx1
applies post-processing in Fig. 2 based on the $\hat{h}_{ij}$. Then,
$\bar{y}_{i}$ in (3a) becomes

$\displaystyle\bar{y}_{1}=$
$\displaystyle\left(1+\frac{\varepsilon_{11}}{\hat{h}_{11}}\right)x_{1}+\frac{n_{1}}{\hat{h}_{11}}+e^{j\theta_{q}}\left(\frac{\hat{h}_{21}}{\hat{h}_{11}}+\frac{\varepsilon_{21}}{\hat{h}_{11}}\right)x_{2},$
(22a) $\displaystyle\bar{y}_{2}=$
$\displaystyle\left(1+\frac{\varepsilon_{22}}{\hat{h}_{22}}\right)x_{2}+\frac{n_{2}}{\hat{h}_{22}},$
(22b)

which can be rewritten as

$\displaystyle\bar{y}_{1}=$
$\displaystyle\bar{h}_{11}x_{1}+\bar{h}_{21}x_{2}+\bar{n}_{1},$ (23a)
$\displaystyle\bar{y}_{2}=$ $\displaystyle\bar{h}_{22}x_{2}+\bar{n}_{2},$
(23b)

by defining

$\displaystyle\bar{h}_{ii}\triangleq(1+\varepsilon_{ii}\hat{h}_{ii}^{-1}),$
(24a)
$\displaystyle\bar{h}_{21}\triangleq(\hat{r}_{21}\hat{r}_{11}^{-1}e^{j\theta_{\delta}}+\varepsilon_{21}\hat{h}_{11}^{-1}),$
(24b) $\displaystyle\bar{n}_{i}\triangleq
n_{i}\hat{h}_{ii}^{-1}\sim\mathcal{CN}(0,\sigma_{N}^{2}\hat{r}_{ii}^{-2}).$
(24c)

In (24b), $\theta_{\delta}$ is the residual angle caused by the quantized
feedback, which is defined in (18) where ${\theta}_{q}$ is the feedback angle
as (25b).

####  IV-2 Quantization Error

Next, the transmitters require the knowledge of CSI to perform pre-processing
and modification on constellation for enhanced performance. However, due to
the limited feedback resources, the feedback information is quantized. This
generates another source of imperfection. The estimated and quantized
parameters owned by each end are shown in Fig. 6. Rx1 has the estimated
$r_{11}$, $r_{21}$, $\theta_{11}$, and $\theta_{21}$. Rx2 has the estimated
$r_{22}$, and $\theta_{22}$. To make all the four nodes access to the channel
knowledge, both

$\displaystyle\alpha_{q}=Q(\hat{\alpha})=Q({\hat{r}_{21}^{2}}\cdot{\hat{r}_{11}^{-2}}),$
(25a)
$\displaystyle\color[rgb]{0,0,0}\theta_{q}=Q(\hat{\theta})=Q(\hat{\theta}_{11}-\hat{\theta}_{21})=\hat{\theta}_{11}-\hat{\theta}_{21}+\theta_{\delta},$
(25b)

are sent to the transmitter and Rx2, where $Q(\cdot)$ is a uniform quantizer
with accuracy $N_{q}$, $\hat{\alpha}$ is the estimation of the interference
intensity, $\hat{\theta}=\hat{\theta}_{11}-\hat{\theta}_{21}$, and
$\theta_{\delta}$ is the same quantization error in (18). The quantizer
$Q(\cdot)$ uniformly divides the region of the input variable into $2^{N_{q}}$
segments. The region for $\alpha$ and angle are $[0,3]$ and $[-\pi,\pi]$,
respectively. The middle value of the segment is the quantization result if
the input value of $Q(\cdot)$ is within this segment.

Figure 6: Feedback process for the ZIC transmission. $\alpha_{q}$ and
$\theta_{q}$ are the only feedback parameters.

####  IV-3 Implementation of the imperfect ZIC

To implement the imperfect channel model, we randomly generate actual channels
$h_{11}$ and $h_{22}$ and estimation errors $\varepsilon_{11}$,
$\varepsilon_{22},$ and $\varepsilon_{21}$. Then, $\hat{h}_{11}$ and
$\hat{h}_{22}$ are determined by (19). We give interference gain $\alpha$ and
then $h_{21}=\sqrt{\alpha}e^{i\theta_{21}}$, where $\theta_{21}$ is a random
uniformly distributed angle on $[0,2\pi)$. After actual channels are
generated, the receivers will have the estimated CSI and can normalize the
channel as Fig. 5(b). The receivers will then assume the channel gains are
unity and the Rx1 assumes the interference gain is
${\hat{\alpha}}=|\hat{r}_{21}\hat{r}_{11}^{-1}|^{2}$.

It is worth mentioning that, $\bar{y}_{1}$ or $\bar{y}_{2}$ will go to
infinity when the estimated channel gains ($r_{11}$ or $r_{22}$) are close to
zero. In this case, the transmission is not reliable due to the wrong
information obtained from the channel estimation no matter if the equalization
in (23a) is applied or not. The receivers will suffer from a mismatch between
the received symbol and the constellation. In the channel generation for the
imperfect model, we keep the channel only if

$\displaystyle\max(\left|\frac{\varepsilon_{11}}{\hat{h}_{11}}\right|,\left|\frac{\varepsilon_{22}}{\hat{h}_{22}}\right|,\left|\frac{\varepsilon_{21}}{\hat{h}_{11}}\right|)<T.$
(26)

where $T$ is a threshold. We use $T=1$ in this paper so that the estimation
errors is not dominating in $\bar{h}_{ii}$ in (24a).

To summarize, for $i,j\in\\{1,2\\}$,

  * •

The actual channels coefficients are $h_{ij}$ in (20).

  * •

The estimated channels coefficients are $\hat{h}_{ij}$, as in (19);

  * •

After the equalization, the equivalent channel coefficients are
$\bar{h}_{ij}$, as in (24a)-(24b);

  * •

After normalization, Rx1 and Rx2 will assume the direct channel gains are one
and the interference gain is $\hat{\alpha}$.

  * •

Rx1 knows both the estimated and quantized parameters. Rx1 sends feedback
parameters ($\hat{\alpha}_{q}$ and $\hat{\theta}_{q}$) to Tx1, Tx2, and Rx2.

###### Remark 1

If $\sigma_{E}^{2}=0$ and $N_{q}\to\infty$ (i.e., the CSI is perfect), the
system in Fig. 5(b) reduces to that of Fig. 2(b).

##  V Deep Autoencoder for ZIC with Imperfect CSI

The channel implementation should follow the imperfect ZIC model in Section
IV. The equivalent channels, $\bar{h}_{ij}$s, are set into the channel layers
inside the DAE-ZIC. Different from the perfect channel case in (11)-(12) and
Fig. 4, the I/Q components become

$\displaystyle\left[\begin{matrix}\bar{y}_{1}^{\textmd{I}}\\\
\bar{y}_{1}^{\textmd{Q}}\end{matrix}\right]=\bar{\mathbf{H}}_{11}\left[\begin{matrix}x_{1}^{\textmd{I}}\\\
x_{1}^{\textmd{Q}}\end{matrix}\right]+\bar{\mathbf{H}}_{21}\left[\begin{matrix}x_{2}^{\textmd{I}}\\\
x_{2}^{\textmd{Q}}\end{matrix}\right]+\left[\begin{matrix}\bar{n}_{1}^{\textmd{I}}\\\
\bar{n}_{1}^{\textmd{Q}}\end{matrix}\right],$ (27a)
$\displaystyle\left[\begin{matrix}\bar{y}_{2}^{\textmd{I}}\\\
\bar{y}_{2}^{\textmd{Q}}\end{matrix}\right]=\bar{\mathbf{H}}_{22}\left[\begin{matrix}x_{2}^{\textmd{I}}\\\
x_{2}^{\textmd{Q}}\end{matrix}\right]+\left[\begin{matrix}\bar{n}_{2}^{\textmd{I}}\\\
\bar{n}_{2}^{\textmd{Q}}\end{matrix}\right],$ (27b)

in which $x_{i}^{\textmd{I}}$ and $x_{i}^{\textmd{Q}}$ are I/Q components of
the transmitted signals, where $i\in\\{1,2\\}$ denotes the users; similarly,
$\bar{y}_{i}^{\textmd{I}}$ and $\bar{y}_{i}^{\textmd{Q}}$ are I/Q components
of the received signals; and, $\bar{n}_{i}^{\textmd{I}}$ and
$\bar{n}_{i}^{\textmd{Q}}$ are the noises. $\bar{\mathbf{H}}_{ij}$ is the real
form of the complex-valued channel $\bar{h}_{ij}$ in (24a)-(24b),

$\displaystyle\bar{\mathbf{H}}_{ij}=\left[\begin{matrix}\bar{h}_{ij}^{\textmd{I}}&-\bar{h}_{ij}^{\textmd{Q}}\\\
\bar{h}_{ij}^{\textmd{Q}}&\quad\bar{h}_{ij}^{\textmd{I}}\end{matrix}\right].$
(28)

where $\bar{h}_{ij}=\bar{h}_{ij}^{\textmd{I}}+j\bar{h}_{ij}^{\textmd{Q}}$.
Therefore, we add an FCNN layer to implement each of the channels. The weight
of the channel layer is $\bar{\mathbf{H}}_{ij}$ in (28). These channel layers
have zero bias, no activation function, and are non-trainable. The weight will
be replaced only if the channel is updated during the training and testing.

Figure 7: Implementation of the channels inside the proposed DAE network.

In contrast to the scenario with perfect CSI, illustrated in Fig. 3, the
presence of imperfections in the CSI introduces distinct considerations to our
approach.

  * •

Differences in parameters. In the case of imperfect CSI, the parameter
$\alpha$ is not directly available for utilization. The imperfectness at Rx1
side comes from the estimation of the interference gain $\hat{\alpha}$ and
angle $\hat{\theta}$. Rx1 transmits $\alpha_{q}$ and $\theta_{q}$ (quantized
values of $\hat{\alpha}$ and $\hat{\theta}$) to Tx1, Tx2, and Rx2, as shown in
Fig. 6. Thus, the information in three nodes suffer from both sources of
imperfectness, i.e., estimation and quantization errors.

  * •

Differences in network input configurations. For Rx1, the input comprises the
non-quantized $\sqrt{\hat{\alpha}}$. Furthermore, Rx1 is uniquely positioned
to incorporate the quantization error of the feedback angle,
$\theta_{\delta}$, as an additional input. This is because Rx1 oversees the
quantization process, and thus is aware of the quantization discrepancy. As a
result, $\theta_{\delta}$ serves as an input to the network at Rx1.

The training process with imperfect channels is similar to Algorithm 1. We
still use Algorithm 1 but lines 10–13 of that algorithm will be replaced by
Algorithm 2.

Algorithm 2 Channel Layer Preparation for DAE-ZIC

1:Inputs: $\mu_{h}$, $\sigma_{h}^{2}$, $\sigma_{E}^{2}$, $T$, and $N_{q}$.

2:while 1 do

3: Randomly generate $h_{11}$ and $h_{22}$ using (20).

4: Randomly generate $\varepsilon_{11}$, $\varepsilon_{22}$, and
$\varepsilon_{21}$ using (21).

5: if (26) satisfied then

6: Break.

7: end if

8:end while

9:Uniformly generate $\Delta\theta_{q}$ in $\frac{1}{2^{N_{q}}}[-\pi,\pi]$.

10:Update $\theta_{\delta}$ using (18).

11:Normalize the channels using (24a) and (24b).

12:Set the channel layers using (27) and according to Fig. 7.

13:Set the variance of the noise layer according to (24c).

14:Update $\alpha_{q}$ and $\theta_{q}$ using (25a) and (25b).

15:Complete the feedback process according to Fig. 6.

##  VI Performance Analysis

The performance is evaluated and compared for the three methods below:

  * •

DAE-ZIC: The proposed method which designs new constellations based on the
interference intensity.

  * •

Baseline-1: The transmitters directly use standard QAM.

  * •

Baseline-2: Tx1 uses standard QAM, while Tx2 rotates the standard QAM symbols
based on the interference intensity [16, 17, 31].

First, we illustrate and analyze the constellations given by the proposed DAE-
ZIC methods with perfect CSI. Then, the BER performance of the three methods
is analyzed under perfect CSI assumption. Finally, we compare the performance
of the three methods under imperfect channels, including imperfect estimation
and quantization in the feedback.

(a) Baseline-1, $\alpha=0.5$

(b) Baseline-2, $\alpha=0.5$

(c) DAE-ZIC, $\alpha=0.5$

(d) Baseline-1, $\alpha=1$

(e) Baseline-2, $\alpha=1$

(f) DAE-ZIC, $\alpha=1$

(g) Baseline-1, $\alpha=2$

(h) Baseline-2, $\alpha=2$

(i) DAE-ZIC, $\alpha=2$

Figure 8: Constellation of th DAE-ZIC and baselines at Rx1 for three values of
$\alpha$. The colors distinguish different symbols. Each of the four clusters
of the same color are caused by one of the four symbols from the interference
with different noise.

###  VI-A Network Ablation Study

In this section, we conduct an ablation study to analyze the impact of network
design and training parameters. Specifically, we investigate the effects of
incorporating residual connections and selecting various training parameters.
To this end, we design and perform six ablation experiments, which are
detailed in Table I. The first experiment (Exp. 1) examines the presence or
absence of shortcuts in the proposed method. Through Exp. 2 to Exp. 5, we
investigate the impact of feeding or not feeding the training parameter
$\sqrt{\alpha}$ into different sub-networks at the transmitters and receives.
Exp. 6 explores the importance of Sub-network 2 which is a new design to
perform power allocation.

TABLE I: Ablation settings and their performance for different values of $\alpha$. Settings | Proposed | Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | Exp. 5 | Exp. 6
---|---|---|---|---|---|---|---
Use shortcuts | ✓ | - | ✓ | ✓ | ✓ | ✓ | ✓
$\sqrt{\alpha}$ to sub-net 1 | ✓ | ✓ | - | ✓ | - | ✓ | ✓
$\sqrt{\alpha}$ to sub-net 2 | ✓ | ✓ | ✓ | - | - | ✓ | -
$\sqrt{\alpha}$ to the Rx | ✓ | ✓ | ✓ | ✓ | ✓ | - | ✓
Use sub-net 2 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | -
Performance | Proposed | Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | Exp. 5 | Exp. 6
$\alpha=0.5$ | 0.0252 | 0.0424 | 0.0324 | 0.0318 | 0.0340 | 0.0513 | 0.0934
$\alpha=1$ | 0.0236 | 0.1207 | 0.0294 | 0.0288 | 0.0281 | 0.0488 | 0.0391
$\alpha=1.5$ | 0.0212 | 0.1288 | 0.0253 | 0.0237 | 0.0308 | 0.0463 | 0.0328

####  VI-A1 The impact of residual connections

The incorporation of residual connections into the proposed DAE has a
significant impact on its performance, especially for bit inputs. The addition
of these shortcuts enables the network to increase its learning capacity and
improve its performance without the need for additional parameters or a wider
network. Moreover, these skip connections help preserve gradients, allowing
the network to learn representations at varying depths. As can be seen in the
first two columns of Table I, excluding residual connections significantly
increases the BER, indicating that residual connections play a crucial role in
the overall performance of the proposed DAE-ZIC.

####  VI-A2 The impact of $\sqrt{\alpha}$ at the Txs

We develop three experiments (Exp. 2, Exp. 3, and Exp. 4) in which
$\sqrt{\alpha}$ is fed into either sub-networks or non and compare their BER
with the proposed network in which $\sqrt{\alpha}$ is fed into bot sub-
networks. While there is not big performance difference between Exp. 2, Exp.
3, and Exp. 4, for all $\alpha$ settings, there is a significant improvement
when $\sqrt{\alpha}$ is fed into both sub-networks (proposed method). Sub-
network 1 primarily focuses on constellation design, whereas Sub-network 2
regulates the power of I/Q dimension. Since $\sqrt{\alpha}$ influences both
the constellation layout and the power distribution, providing $\sqrt{\alpha}$
to each sub-network is essential for optimizing the overall performance of the
system.

####  VI-A3 The impact of $\sqrt{\alpha}$ at the Rx

Exp. 5 examines the performance when $\sqrt{\alpha}$ is not fed to the
decoder. This experiment emphasizes the importance of incorporating
interference information at the decoder side, as otherwise the BER roughly
doubles (compared to the first column).

####  VI-A4 The impact of Sub-network 2

Exp. 6 demonstrates the importance of Sub-network 2, which is a new design to
ensure different powers at I/Q dimensions. Our simulations reveal that the
DAE-designed constellations downgraded to PSK in the absence of Sub-network 2.
This explains why neglecting Sub-network 2 yields results similar to the
Baseline-2 which uses rotated QPSK. On the contrary, as shown by the
simulations, the inclusion of Sub-network 2 allows the proposed structure to
design other shapes such as pulse-amplitude keying (PAM) at each transmitter.
The combination of these PAMs ultimately yields a QAM-like constellation at
the receiver affected by interference. These findings further emphasize the
importance of incorporating Sub-network2, as it enables effective power
control across the I/Q dimensions, crucial for ensuring the proper functioning
of the constellations.

###  VI-B Scenario I: Perfect CSI

(a) ${M}_{1}={M}_{2}=4$, $\alpha=0.5$

(b) ${M}_{1}={M}_{2}=4$, $\alpha=1$

(c) ${M}_{1}={M}_{2}=4$, $\alpha=1.5$

(d) ${M}_{1}={M}_{2}=8$, $\alpha=0.5$

(e) ${M}_{1}={M}_{2}=8$, $\alpha=1$

(f) ${M}_{1}={M}_{2}=8$, $\alpha=1.5$

Figure 9: The maximum (worst case) BER performance among the two users of DAE-
ZIC versus SNR. The BER is averaged over different interference gains which is
uniformly distributed in the given interval.

(a) ${M}_{1}={M}_{2}=4$

(b) ${M}_{1}={M}_{2}=8$

Figure 10: The maximum (worst case) BER performance versus interference gains.
The SNR is fixed as 10dB.

(a) ${M}_{1}={M}_{2}=4$, $\alpha=0.5$

(b) ${M}_{1}={M}_{2}=4$, $\alpha=1$

(c) ${M}_{1}={M}_{2}=4$, $\alpha=1.5$

Figure 11: The maximum (worst) BER performance among the two users versus SNR
when estimation error is considered.

####  VI-B1 Constellation Analysis

The received constellations at Rx1 for all three methods (the two baselines
and the proposed DAE-ZIC) are shown in Fig. 8. In this simulation, we set
$N_{s}=2$ so that each user has $2^{N_{s}}=4$ information symbols. The
channels are perfectly known, the transmit power is unity, and the SNR is 8dB.

Each sub-figure of Fig. 8 contains four four-symbol clusters differentiated by
different colors. Each cluster refers to a symbol transmitted to Rx1. Within
each cluster, there are four symbols, which are due to the symbols of Rx2. For
example, the blue colors denote symbol $1$ of Rx1, each contaminated by one of
the symbols of Rx2 (interference) and AWGN noise. It can be seen that the
location and distribution of symbols are different in each method. The
constellations of Baseline-1 (left column) are very crowded and even
overlapped when $\alpha=1$ in Fig 8(d). This is because Tx2 strongly
interferes with the transmission of Tx1 to Rx1 by directly applying 4-QAM.
Baseline-2 (middle column) rotates the constellation of Tx2, which enlarges
the space between symbols and thus makes the decoding easier. The proposed
DAE-ZIC (right-column) creates the most separable constellations. It can
better make use of the I/Q plane in constellation design based on the
interference intensity. The two cooperating DAEs can intelligently choose and
adjust various scaled constellation types to avoid constellation overlapping.
When $\alpha=0.5$, the DAE-ZIC designs parallelogram-shape constellations
compared with the square-shape constellations in the baselines. For
$\alpha=1$, both the constellation of both Tx1 and Tx2 morph to PAM. The two
PAM constellations are perpendicular to each other hence the overlapping is
eliminated. When $\alpha=2$, Tx1 uses a parallelogram-shape QAM and Tx2 uses a
PAM. By adapting their constellations to the interference intensity, the two
DAEs cooperate to avoid constellation overlap as much as possible. This is the
main reason that the DAE-ZIC outperforms the baselines.

In addition, compared with [26], where the symbols exhibit nearly equal power
in one stream, our design allows the I/Q domains to have different power
levels by incorporates sub-network 2. With this, the transmitter can generate
QAM-like constellations, further mitigating the impact of interference. It is
also worth noting that imperfect CSI is not considered in [26], we will show
in the next section.

####  VI-B2 BER Performance of the DAE-ZIC

To evaluate the effectiveness of the DAE-ZIC, we compare the BER of the three
methods over SNR in $[0,20]$dB and $0\leq\alpha\leq 3$. For the DAE-ZIC, we
divide interval $\alpha\in[0,3]$ in to six sub-intervals, i.e, $[0,0.5)$,
$[0.5,1)$, $\ldots$, $[2.5,3]$. For each sub-interval of $\alpha$, we train a
DAE-ZIC through Algorithm 1.

The BER versus the SNR is shown in Fig. 9. In each sub-figure, $\alpha$ is a
fixed value. In general, the DAE-ZIC outperforms the two baseline models,
especially in moderate and strong interference regimes. With $N_{s}=3$,
performance of DAE-ZIC drops at 0dB. The reason is we have trained the network
at $\textmd{SNR}=10$dB but have tested it for a range of SNRs from 0 to 20dB.
A potential way to improve is to train the DAE-ZIC with a variety of SNRs.

The BER performance versus $\alpha$ at $\textmd{SNR}=10$dB is shown in Fig.
10. In Fig. 10(a) and Fig. 10(b), we set $N_{s}=2$ and $N_{s}=3$, i.e.,
${M}_{1}={M}_{2}=4$ and ${M}_{1}={M}_{2}=8$. When interference is very weak,
i.e., $\alpha\in[0,0.25]$, the three methods have similar BERs. The proposed
DAE-ZIC noticeably reduces the BER in weak, moderate, and strong interference
cases, where $\alpha\in[0.5,2]$. From Fig. 10(a), using DAE-ZIC over
$\alpha\in[0,3]$, BER is reduced 75.7% and 44.29% with respect to Baseline-1
and Baseline-2, respectively. When $\alpha\in[0.5,2]$, the improvement becomes
80.3% and 51.5%. When the interference is very strong, e.g., $\alpha>2.5$,
Baseline-2 slightly outperforms the DAE-ZIC. The reason could be that 4-QAM
with rotation may achieve optimal performance [16]. For $N_{s}=3$ in Fig.
10(b), 44.4% and 31.5% BER reduction is reached by the DAE-ZIC over
$\alpha\in[0,3]$. For $\alpha\in[0.5,2]$, DAE-ZIC outperforms the other two
methods with 45.0% and 33.4%. Baseline-1 performs poorly for
$\alpha\in[0.5,2]$, because the two added QAM constellations may get crossed
and overlap. Thus, Rx1 cannot decode its message. Such a phenomenon is
alleviated in Baseline-2 which simply rotates one QAM constellation and the
added constellations still have a reasonable symbol distance. The
normalization layer allows the DAE-ZIC to design constellations without any
regular-shape restrictions. Thus, the minimum distance at receivers can be
enlarged which results in a lower BER.

###  VI-C Scenario II: Imperfect CSI

The influences of the imperfect CSI on the BER performance are shown as
follows. Two types of imperfection are analyzed independently: CSI estimation
error and quantization error. For imperfect CSI, both baselines use the system
model in Fig. 5(b). When decoding, Rx1 will take $\theta_{\delta}$ into
decoding. The performance of the two baselines and the proposed DAE-ZIC are
evaluated for $\alpha\in[0,3]$ and SNR in $[0,20]$dB. In the simulation, the
actual direct channel $h_{ii}\sim\mathcal{CN}(1,0.1)$ where $i\in\\{1,2\\}$.
The estimation error $\varepsilon_{ij}$s are in
$\mathcal{CN}(0,\sigma_{E}^{2})$. Then the estimated channels are obtained
from (19). To be fair to the users, we use the maximum (worst) BER between
them as the measurement. Each result is averaged over 500 random channels.

(a) ${M}_{1}={M}_{2}=4$

(b) ${M}_{1}={M}_{2}=8$

Figure 12: The maximum (worst) BER performance of the two users under the
estimation error. The SNR is fixed as 10dB.

(a) ${M}_{1}={M}_{2}=4$, $\alpha=0.5$

(b) ${M}_{1}={M}_{2}=4$, $\alpha=1$

(c) ${M}_{1}={M}_{2}=4$, $\alpha=1.5$

Figure 13: The maximum (worst) BER performance of the two users versus SNR.
The quantization is considered while free from estimation error.

####  VI-C1 CSI with Estimation Error

The BER versus interference intensity $\alpha$ is shown in Fig. 12. Two levels
of the estimation error are examined. Specifically,
$\sigma_{E}^{2}\in\\{0.05,0.1\\}$. The proposed DAE-ZIC outperforms the
baselines almost for any $\alpha$. The percentage of the BER reduction is
shown in Table II. With a fixed $\alpha$, the BER versus SNR is shown in Fig.
11 for weak, moderate, and strong interference. Compared to the baseline
methods, the DAE-ZIC has a remarkable improvement in BER performance in all
interference and estimation error levels. As mentioned earlier, the main
difference between the proposed DAE-ZIC and the baselines is that the network
is able to re-design the constellation based on the interference intensity,
and that is how the DAE-ZIC reduces the BER.

TABLE II: BER Reductions of DAE-ZIC Compared to the Baseline Methods (in Percentage and $\alpha\in[0,3]$).  Compared to | Baseline-1 | Baseline-2
---|---|---
$N_{s}$ | 2 | 3 | 2 | 3
$\sigma_{E}^{2}$ | $0$ | 75.77% | 44.29% | 44.43% | 31.50%
$0.05$ | 55.40% | 38.97% | 39.12% | 31.43%
$0.1$ | 48.83% | 35.81% | 41.41% | 29.24%

####  VI-C2 CSI with Feedback Quantization

When channel estimation is perfect, the BER performance with feedback
quantization (with ${N_{q}=3}$) and without quantization (${N_{q}=\infty}$) is
shown in Fig. 13. The DAE-ZIC outperforms the baselines with and without
feedback quantization. In addition, the quantization error increases the BER
of all methods. However, the performance degradation of the DAE-ZIC is much
less than the other two methods. Especially, for $\alpha=1$ and $\alpha=1.5$),
where the interference is strong, the DAE-ZIC outperform other methods by more
than two orders of magnitude when $N_{q}=3$ and $\rm SNR=20$dB. Interestingly,
Baseline-1 performs better for $N_{q}=3$ compared to $N_{q}=\infty$. The
reason is that quantization of the angles may introduce rotation angle on Tx2,
acting like Baseline-2 unintentionally. This then reduces the constellation
overlapping and thus a better BER is reached.

With high interference intensities, i.e, $\alpha=1$ and $\alpha=1.5$, the BER
of the DAE-ZIC with quantization ($N_{q}=3$) is only slightly degraded
compared to the un-quantized case ($N_{q}=\infty$). This is because, while Txs
receive the quantized CSI, Rx1 knows CSI before and after quantization. Then,
it can to some extent correct the imperfectness in transmitters. Therefore,
there is no dramatic degradation for the DAE-ZICs. On the other hand,
Baseline-2 is more sensitive to the quantization error and thus a big gap of
the BER happens between $N_{q}=\infty$ and $N_{q}=3$. The reason is that
Baseline-2 only rotates the constellation in Tx2, which highly depends on the
phase shifted by the channels.

##  VII Conclusion

A novel DAE architecture for interference mitigation in the two-user ZIC with
perfect and imperfect CSI has been proposed. The DAE-ZIC minimizes the BER by
jointly designing transmit and receive DAEs and optimizing them together. In
this architecture, the average power constraint is realized by designing a
normalization layer. This enables the proposed DAE-ZIC to design more
efficient symbols to achieve lower BERs. BER simulations verify the
effectiveness of the proposed structure. We have compared the proposed DAE-ZIC
with two baseline models, and the DAE-ZIC outperforms both. With quantized
CSI, the gain obtained by the DAE-ZIC compared to the best conventional method
is remarkable and can be as large as two orders of magnitude at $\rm
SNR=20dB$. Getting back to the questions asked in the introduction, we
conclude that autoencoder is a viable solution for interference channels and
it outperforms the conventional methods by designing new, nonuniform
constellations which make the symbols separable at the interfered receiver
side.

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|  Xinliang Zhang (S’19) received the B.Eng. and M.Eng. degrees in Electrical
Engineering from Xidian University, Xi’an, China, in 2015 and 2018,
respectively. He obtained the Ph.D. degree from the Department of Electrical
and Computer Engineering, Villanova University in 2022. His research interests
lie in physical layer security, machine learning for wireless communications,
MIMO networks, and signal processing.
---|---
|  Mojtaba Vaezi (S’09–M’14–SM’18) received the B.Sc. and M.Sc. degrees from
Amirkabir University of Technology (Tehran Polytechnic) and the Ph.D. degree
from McGill University, all in Electrical Engineering. From 2015 to 2018, he
was with Princeton University as a Postdoctoral Research Fellow and Associate
Research Scholar. He is currently an Associate Professor of ECE at Villanova
University. Before joining Princeton, he was a researcher at Ericsson Research
in Montreal, Canada. His research interests include the broad areas of signal
processing and machine learning for wireless communications with an emphasis
on physical layer security and sixth-generation (6G) radio access
technologies. Among his publications in these areas is the book Multiple
Access Techniques for 5G Wireless Networks and Beyond (Springer, 2019). Dr.
Vaezi has held editorial positions at various IEEE journals, currently serving
as an Editor for IEEE Transactions on Communications along with his role as a
Senior Editor for IEEE Communications Letters and Senior Area Editor for IEEE
Signal Processing Letters. He is a recipient of several academic, leadership,
and research awards, including McGill Engineering Doctoral Award, IEEE Larry
K. Wilson Regional Student Activities Award in 2013, the Natural Sciences and
Engineering Research Council of Canada (NSERC) Postdoctoral Fellowship in
2014, Ministry of Science and ICT of Korea’s best paper award in 2017, IEEE
Communications Letters Exemplary Editor Award in 2018, the 2020 IEEE
Communications Society Fred W. Ellersick Prize, the 2021 IEEE Philadelphia
Section Delaware Valley Engineer of the Year Award, and the National Science
Foundation (NSF) CAREER Award in 2023.
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