# Detect to Learn: Structure Learning with Attention and Decision Feedback for
MIMO-OFDM Receive Processing

Jiarui Xu, Lianjun Li, Lizhong Zheng, and Lingjia Liu J. Xu, L. Li and L. Liu
are with Wireless@Virginia Tech, Department of Electrical & Computer
Engineering, Virginia Tech, Blacksburg, Virginia 24061. L. Zheng is with the
Department of Electrical Engineering and Computer Science, Massachusetts
Institute of Technology, Cambridge, MA 02139. The work is supported by US
National Science Foundation (NSF) and Intel under grants CNS-2003059,
CNS-2002908 and CCF-1937487.

###### Abstract

The limited over-the-air (OTA) pilot symbols in multiple-input-multiple-output
orthogonal-frequency-division-multiplexing (MIMO-OFDM) systems presents a
major challenge for detecting transmitted data symbols at the receiver,
especially for machine learning-based approaches. While it is crucial to
explore effective ways to exploit pilots, one can also take advantage of the
data symbols to improve detection performance. Thus, this paper introduces an
online attention-based approach, namely RC-AttStructNet-DF, that can
efficiently utilize pilot symbols and be dynamically updated with the detected
payload data using the decision feedback (DF) mechanism. Reservoir computing
(RC) is employed in the time domain network to facilitate efficient online
training. The frequency domain network adopts the novel 2D multi-head
attention (MHA) module to capture the time and frequency correlations, and the
structural-based StructNet to facilitate the DF mechanism. The attention loss
is designed to learn the frequency domain network. The DF mechanism further
enhances detection performance by dynamically tracking the channel changes
through detected data symbols. The effectiveness of the RC-AttStructNet-DF
approach is demonstrated through extensive experiments in MIMO-OFDM and
massive MIMO-OFDM systems with different modulation orders and under various
scenarios.

###### Index Terms:

Symbol Detection, Decision Feedback, Attention mechanism, Multi-head
attention, MIMO-OFDM, massive MIMO-OFDM, Online Training, Structural Knowledge

##  I Introduction

Deep learning (DL) has attracted significant attention due to its overwhelming
privilege in computer vision (CV), natural language processing (NLP), and
robotics. DL has also demonstrated great potential in MIMO-OFDM symbol
detection tasks. Efforts have been devoted to utilizing multi-layer perceptron
(MLP) [1], convolution neural network (CNN) [2, 3, 4], long short-term memory
(LSTM) network [5, 6], or generative adversarial network (GAN) [7] to jointly
estimate the channel and detect symbols. These approaches do not rely on
explicit modeling of the system and the estimation of the channel state
information (CSI), but instead, they treat the model as a black box and use
neural networks (NNs) to deal with more complicated wireless systems with non-
linear device components. Moreover, DL-based approaches offer a new avenue for
symbol detection tasks with a large number of OFDM subcarriers and high
modulation orders, thanks to their lower computational complexity compared to
conventional optimal detection algorithms [8].

While DL-based approaches are promising, such approaches have their own
challenges to be applied in real-world MIMO-OFDM systems. One of the main
challenges is the data requirement. Training NNs requires a huge volume of
labeled data along with a long training time. In modern cellular networks, OTA
labeled data is scarce and extremely expensive to obtain, resulting in limited
training data for symbol detection tasks. Attempts have been made to address
such issues by training with extensive channel realizations offline and
initializing the network with offline weights for online adaptation [1].
However, such approaches are hard to be adopted to the dynamically changing
environments in practice, where the systems have different transmission modes
with link adaptation, rank adaptation, and schedules operating on a subframe
(1 millisecond) basis [8, 9]. The discrepancy between modes for offline and
online training can result in model mismatch and prohibit the offline weights
from being utilized online.

Another challenge is how to design an interpretable network with domain
knowledge. Learning both the known and unknown features requires a large
amount of training data. The desired network should embed what we have known
into the architecture instead of learning everything from scratch. Recent
advances employ deep-unfolding NNs to improve network interoperability for
detection [10, 11, 12, 13], transceiver design [14, 15], and channel
estimation [16]. Specifically, for symbol detection, such approaches [10, 13,
11, 12] build the network by unfolding conventional iterative algorithms and
thus reduce the number of trainable parameters in the network. For example,
MMNet [12] converts each iteration of the iterative soft-thresholding
algorithms into network layers and only learns a few parameters of the
algorithm with NN in each layer. It has shown superior performance in
independent and identically distributed (i.i.d.) Gaussian channels and 3D MIMO
channels released by 3rd Generation Partnership Project (3GPP) [17]. However,
due to the stack of multiple layers, the number of parameters grows, leading
to a demand for a large-size training dataset. Furthermore, the assumption of
the perfect CSI knowledge also limits its utilization in practice.

Our previous RC based approaches [18, 19, 20] adopts the RC in the time domain
to conduct symbol detection and thus can be learned with only a limited amount
of online training data. More recently, RC-Struct [21] extends the idea of the
time domain RC-based approaches by introducing an extra frequency domain
structure-based network. The network consists of a time domain RC and a
frequency domain classifier. By processing in both domains, the time domain
convolution and superposition operation of the wireless channel, the time-
frequency structure of OFDM, and the frequency domain structure of the
repetitive modulation constellation pattern, can all be embedded into the
underlying NNs to take advantage of the available structural knowledge of the
MIMO-OFDM system. The incorporation of such structural knowledge helps to
significantly reduce the needed training overhead and improve the system
performance in a realistic environment. However, these RC-based methods only
learn from the pilot OFDM symbols and then test on the rest data OFDM symbols.
While they still work in the scenario where the channel is gradually evolving
over different OFDM symbols within a subframe, it does not learn the dynamic
features that are provided by the data symbols.

In this paper, we introduce an online attention-based approach, RC-
AttStructNet-DF, to tackle the above challenges. We follow our previous work,
RC-Struct, to embed structural knowledge of the MIMO-OFDM system into the
network architecture. To further improve detection performance, we adopt an
additional attention mechanism in our approach, which is achieved by a 2D MHA
module and attention loss. Inspired by the MHA module in the Transformer [22],
we develop the 2D MHA to capture time and frequency correlations in a two-
dimensional manner. The attention loss is designed for the frequency domain
network to assign different weights to the training loss of different training
samples based on their confidence levels. Moreover, unlike RC-Struct which
only learns from the pilot symbols, RC-AttStructNet-DF also takes advantage of
the data symbols. Specifically, a DF mechanism, which takes the detected data
symbols as the training data, is used to dynamically track the channel
variations within a subframe. To mitigate the error propagation of the DF
procedure, the frequency domain StructNet is designed on top of the RC-Struct,
where a parameter estimation (PE) layer is additionally adopted. The
customized StructNet is shown to be robust to incorrect labels owing to the
embedded structural information, making it a suitable network to be applied
with the DF process. The introduced PE layer further facilitates the DF
mechanism by learning the network parameters with the underlying NN.
Evaluation results demonstrate the effectiveness of adopting the attention
mechanism to improve detection performance, and the benefits of employing the
StructNet along with the DF mechanism to mitigate the issue of error
propagation and dynamically update network parameters. Meanwhile, extensive
experiments have been conducted to show the considerable performance gain of
RC-AttStructNet-DF over the conventional model-based methods and the state-of-
the-art learning-based approaches in both the MIMO-OFDM system and the massive
MIMO-OFDM system under the 3GPP 3D channel [17]. Evaluation results also
demonstrate that RC-AttStructNet-DF can perform better than the conventional
approaches with a practical pilot pattern specified by the 3GPP 5G NR. Our
contributions are summarized as the following:

  * •

We develop an attention-based symbol detection approach, where attention is
achieved by adopting a novel 2D MHA module and attention loss. The 2D MHA is
constructed based on the MHA module in the Transformer, which captures time
and frequency correlations in a two-dimensional manner. The attention loss is
designed to assign different weights to the training loss of different
training samples according to their confidence levels.

  * •

We exploit the DF mechanism to take advantage of the data symbols to improve
detection performance. Specifically, the DF algorithm updates the network
weights by utilizing the detected data symbols as the training data so that
the channel changes across different OFDM data symbols can be dynamically
tracked.

  * •

We design a learning-efficient network, StructNet, in the frequency domain on
top of the existing RC-Struct. The customized StructNet is shown to be robust
to incorrect labels owing to the embedded structural information, which
effectively mitigates the issue of error propagation typically encountered in
the DF procedure. Moreover, the introduced PE layer in StructNet enables the
network to dynamically update its parameters, further facilitating the
underlying DF mechanism.

  * •

Extensive performance evaluations have been conducted to show that RC-
AttStructNet-DF outperforms RC-Struct in the MIMO-OFDM system under various
scenarios with relatively high user mobility. Furthermore, RC-AttStructNet-DF
demonstrates its effectiveness in large-scale antenna array settings,
extending the application scenario from the MIMO system to the massive MIMO
system. Experimental results also reveal the potential of employing RC-
AttStructNet-DF in a practical pilot pattern specified by the 3GPP 5G NR.

The rest of the paper is organized as follows. Sec. II presents the MIMO-OFDM
system and problem statement. Sec. III briefly discusses RC, the MHA module,
and the atomic decision neuron network (ADNN) exploited in RC-Struct. Sec. IV
introduces the frequency-domain network, StructNet, in a simple MIMO system.
Then Sec. V discusses how to apply StructNet in the MIMO-OFDM system and
introduces the RC-AttStructNet-DF. Complexity analysis is provided in Sec. VI.
A toy experiment for analyzing StructNet is shown in Sec. VII. Sec. VIII
compares the performance of RC-AttStructNet-DF with state-of-the-art methods
in both the MIMO-OFDM system and the massive MIMO-OFDM system. The paper is
concluded in Sec. IX.

Notations: Scalar, vector, and matrix are denoted by non-bold letter, bold
lowercase letter, and bold uppercase letter. The superscripts $(\cdot)^{t}$
and $(\cdot)^{f}$ are used to differentiate the time domain and the frequency
domain signals, respectively. $\mathbb{C}(\mathbb{R})$ represents the complex
(real) number set. Other sets are denoted by calligraphic letters.

##  II Problem Formulation

We consider a MIMO-OFDM system with $N_{t}$ transmit antennas, $N_{r}$ receive
antennas, and $N_{\mathrm{sc}}$ subcarriers. The information is modulated in
the frequency domain on a subframe basis, where each subframe consists of $N$
OFDM symbols. The $n$-th OFDM symbol in the frequency domain can be
represented as $\boldsymbol{X}_{n}^{f}\in{\mathbb{C}}^{N_{t}\times
N_{\mathrm{sc}}}$. The frequency domain OFDM symbols are converted to the time
domain through an inverse fast Fourier transform (IFFT) and cyclic prefix (CP)
addition with length $N_{\mathrm{cp}}$. Then all the time domain OFDM symbols
are concatenated together to form the transmitted time domain signal
$\boldsymbol{X}^{t}=\left[\boldsymbol{X}_{0}^{t},\boldsymbol{X}_{1}^{t},\dots,\boldsymbol{X}_{N-1}^{t}\right]\in\mathbb{C}^{N_{t}\times
N(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$, where
$\boldsymbol{X}_{n}^{t}\in\mathbb{C}^{N_{t}\times(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$
is the $n$-th OFDM symbol in the time domain.

TABLE I: Notation in the system

Symbol | Definition
---|---
$N_{t}$ | Number of transmit antennas
$N_{r}$ | Number of receive antennas
$N_{\mathrm{sc}}$ | Number of OFDM subcarriers
$N_{\mathrm{cp}}$ | Length of cyclic prefix
$N_{p}$ | Number of pilot symbols in one OFDM frame (training set)
$N_{d}$ | Number of data symbols in one OFDM frame (testing set)
$N$ | Total number of symbols in one OFDM frames
$L$ | Total number of channel delays
$\boldsymbol{Y}^{t}\in{\mathbb{C}}^{N_{r}\times N(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$ | The received time domain signal
$\boldsymbol{X}^{t}\in{\mathbb{C}}^{N_{t}\times N(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$ | The transmitted time domain signal
$\boldsymbol{h}_{n_{r},n_{t}}^{t}\in{\mathbb{C}}^{L}$ | The channel impulse response between receiver $n_{r}$ and transmitter $n_{t}$
$\boldsymbol{y}_{n_{r}}^{t}\in{\mathbb{C}}^{N(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$ | The received signal at $n_{r}$-th receiver in time domain
$\boldsymbol{x}_{n_{t}}^{t}\in{\mathbb{C}}^{N(N_{\mathrm{sc}}+N_{\mathrm{cp}}})$ | The transmitted signal at $n_{t}$-th transmitter in time domain
$\boldsymbol{Y}_{n}^{t}\in{\mathbb{C}}^{N_{r}\times(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$ | The received $n$-th OFDM symbols in time domain
$\boldsymbol{X}_{n}^{t}\in{\mathbb{C}}^{N_{t}\times(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$ | The transmitted $n$-th OFDM symbols in time domain
$\boldsymbol{Y}_{n}^{f}\in{\mathbb{C}}^{N_{r}\times N_{\mathrm{sc}}}$ | The received $n$-th OFDM symbols in frequency domain
$\boldsymbol{X}_{n}^{f}\in{\mathbb{C}}^{N_{t}\times N_{\mathrm{sc}}}$ | The transmitted $n$-th OFDM symbols in frequency domain

Denote the time domain signal at the $n_{t}$-th transmit antenna as
$\boldsymbol{x}_{n_{t}}^{t}\in\mathbb{C}^{N(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$,
and the $L$-tap channel between the receive antenna $n_{r}$ and transmit
antenna $n_{t}$ as $\boldsymbol{h}_{n_{r},n_{t}}^{t}\in{\mathbb{C}}^{L}$,
where the channel is gradually evolving across different OFDM symbols. Then at
the receiver side, the received signal can be written as

$\displaystyle\boldsymbol{y}_{n_{r}}^{t}=\sum_{n_{t}=0}^{N_{t}-1}\boldsymbol{h}_{n_{r},n_{t}}^{t}\circledast
g(\boldsymbol{x}_{n_{t}}^{t})+\boldsymbol{w}^{t},$ (1)

where
$\boldsymbol{y}_{n_{r}}^{t}\in\mathbb{C}^{N(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$
is the received signal at the $n_{r}$-th receive antenna; $\circledast$ stands
for the convolution operation; $g(\cdot)$ represents the non-linear distortion
caused by transmitter components, such as power amplifier (PA);
$\boldsymbol{w}^{t}$ denotes the additive white Gaussian noise (AWGN). Let
$\boldsymbol{Y}^{t}\in\mathbb{C}^{N_{r}\times
N(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$ represent the received time domain signal
for all the receive antennas, and
$\boldsymbol{Y}_{n}^{t}\in\mathbb{C}^{N_{r}\times(N_{\mathrm{sc}}+N_{\mathrm{cp}})}$
is the $n$-th received OFDM symbol in the time domain. The frequency domain
counterpart of the received signal
$\boldsymbol{Y}_{n}^{f}\in{\mathbb{C}}^{N_{r}\times N_{\mathrm{sc}}}$ is
acquired by removing CP and following with a fast Fourier transform (FFT). The
notations are summarized in Table I.

The MIMO-OFDM symbol detection task is to recover the transmitted symbol
$\boldsymbol{X}_{n}^{f}$ from the received time-domain observations
$\boldsymbol{Y}_{n}^{t}$ for each OFDM symbol. Pilot symbols, which are known
at both the transmitter and receiver sides, will be embedded in each subframe
to facilitate the detection of unknown data symbols. For ease of discussion,
we assume the first $N_{p}$ OFDM symbols are the pilots, and the rest of the
$N_{d}$ symbols are the unknown data. However, the introduced approach can
also be applied to other pilot patterns, which will be shown in Sec. VIII-E.
In this work, the training dataset for our NN-based approach consists of the
transmitted and received time domain pilot signals $\boldsymbol{X}_{n}^{t}$
and $\boldsymbol{Y}_{n}^{t}$ along with the frequency domain pilot symbols
$\boldsymbol{X}_{n}^{f}$.

##  III Preliminary

###  III-A Reservoir computing

RC is a recurrent neural network (RNN) based approach that can process
sequential data. It only conducts training on the output layer and fixes the
randomly initialized weights of the input layer and the RNN-based reservoir.
As the detailed discussion about RC has been provided in our previous work
[18, 19, 23, 20, 24, 21], we mainly focus on the processing procedures of RC.
Denote the input data as $\boldsymbol{U}\in\mathbb{C}^{N_{in}\times T}$, where
$\boldsymbol{u}(m)$ represents the $m$-th column of $\boldsymbol{U}$. The
processing of RC at the $m$-th time step can be represented by the following
state transition equation and the output equation:

$\displaystyle\boldsymbol{s}(m)=f(\boldsymbol{W}\boldsymbol{s}(m-1)+\boldsymbol{W}_{\mathrm{in}}\;\boldsymbol{u}(m)),\;\boldsymbol{\hat{o}}(m)=\boldsymbol{W}_{\mathrm{out}}\;\boldsymbol{z}(m),$
(2)

where $\boldsymbol{s}(m)\in\mathbb{C}^{N_{n}}$ is the state vector;
$\boldsymbol{W}_{\mathrm{in}}\in\mathbb{C}^{N_{n}\times N_{in}}$ and
$\boldsymbol{W}\in\mathbb{C}^{N_{n}\times N_{n}}$ are the fixed input weight
matrix and reservoir transition matrix, respectively; the $f(\cdot)$ is the
hyperbolic tangent function;
$\boldsymbol{\hat{o}}(m)\in\mathbb{C}^{N_{\mathrm{out}}}$ is the predicted
output at $m$-th time step;
$\boldsymbol{z}(m)=\left[\boldsymbol{s}(m)^{T},\boldsymbol{u}(m)^{T}\right]^{T}\in\mathbb{C}^{N_{n}+N_{in}}$
is the state and input concatenation;
$\boldsymbol{W}_{\mathrm{out}}\in\mathbb{C}^{N_{\mathrm{out}}\times(N_{n}+N_{in})}$
is the trainable output weight matrix. During training, the output weight
matrix is learned by the least square (LS) solution

$\displaystyle\boldsymbol{\hat{W}}_{\mathrm{out}}=\arg\min_{\boldsymbol{W}_{\mathrm{out}}}\|\boldsymbol{W}_{\mathrm{out}}\boldsymbol{Z}-\boldsymbol{O}\|_{F}^{2}=\boldsymbol{O}\boldsymbol{Z}^{\dagger},$
(3)

where
$\boldsymbol{Z}=[\boldsymbol{z}(0),\boldsymbol{z}(1),\dots,\boldsymbol{z}(T-1)]\in\mathbb{C}^{(N_{n}+N_{in})\times
T}$ is the concatenation of $\boldsymbol{z}(m)$,
$\boldsymbol{O}\in\mathbb{C}^{N_{\mathrm{out}}\times T}$ is the training
label, and $\boldsymbol{Z}^{\dagger}$ is the Moore-Penrose inverse of
$\boldsymbol{Z}$.

###  III-B Multi-head attention

Figure 1: The MHA module.

The MHA module is first introduced in the Transformer [22], which allows the
model to attend to information from different representation subspaces. As
shown in Fig. 1, the MHA module consists of multiple attention blocks. For
each block, the attention function maps the input into the query, key, and
value subspaces as the query, key, and value embeddings. The output of the
attention block, or the attention head, is obtained by the scaled dot product
of the three embeddings. Particularly, the attention block can be computed by
the following equations:

$\displaystyle\boldsymbol{Q}=\boldsymbol{U}\cdot\boldsymbol{W}_{q},\boldsymbol{K}=\boldsymbol{U}\cdot\boldsymbol{W}_{k},\boldsymbol{V}=\boldsymbol{U}\cdot\boldsymbol{W}_{v},$
(4)
$\displaystyle\mathrm{head}=f_{sm}(\frac{\boldsymbol{Q}\boldsymbol{K}^{T}}{\sqrt{N_{k}}})\cdot\boldsymbol{V},$
(5)

where $\boldsymbol{U}\in\mathbb{R}^{N_{in1}\times N_{in2}}$ is the input to
the attention block; $\boldsymbol{W}_{q}\in\mathbb{R}^{N_{in2}\times N_{k}}$,
$\boldsymbol{W}_{k}\in\mathbb{R}^{N_{in2}\times N_{k}}$, and
$\boldsymbol{W}_{v}\in\mathbb{R}^{N_{in2}\times N_{v}}$ are linear projections
that project the input to the query, key, and value embedding spaces,
respectively; $f_{sm}(\cdot)$ represents the softmax function; $N_{k}$ and
$N_{v}$ are the dimensions of key embedding and value embedding, respectively.
In the MHA module, the heads of each attention block are aggregated by
concatenation and a linear projection:

$\displaystyle\mathrm{MultiHead}=\mathrm{Concat}(\mathrm{head}_{0},\mathrm{head}_{1},\dots,\mathrm{head}_{N_{a}-1})\cdot\boldsymbol{W}_{o},$
(6)

where $\boldsymbol{W}_{o}\in\mathbb{R}^{N_{a}N_{v}\times N_{o}}$ is the output
linear project, $N_{a}$ is the number of attention blocks, and $N_{o}$ is the
output dimension.

###  III-C Atomic decision neuron network

The ADNN [25, 21] is a frequency domain symbol detection network that utilizes
a single binary classifier to conduct the multi-class detection task. The
design of the network exploits the repetitive structure of the modulation
constellation, which improves the training efficiency by reducing the network
size and utilizing training samples more efficiently.

####  III-C1 Problem formulation

Consider a $N_{r}\times N_{t}$ MIMO system in the frequency domain,

$\displaystyle\boldsymbol{y}^{f}=\boldsymbol{H}^{f}\boldsymbol{x}^{f}+\boldsymbol{w}^{f},$
(7)

where $\boldsymbol{y}^{f}\in\mathbb{C}^{N_{r}}$ is the received signal,
$\boldsymbol{H}^{f}\in\mathbb{C}^{N_{r}\times N_{t}}$ represents the channel
in the frequency domain, $\boldsymbol{x}^{f}\in\mathbb{C}^{N_{t}}$ stands for
the transmitted $M$ quadrature amplitude modulation ($M$-QAM) symbol, and
$\boldsymbol{w}^{f}\in\mathbb{C}^{N_{r}}$ is the additive noise that may not
need to be Gaussian noise. The real-valued form of $\boldsymbol{y}^{f}$ and
$\boldsymbol{x}^{f}$, which are used as the input and output of the network,
are obtained by $\boldsymbol{\tilde{y}}^{f}=f_{R}(\boldsymbol{y}^{f})$ and
$\boldsymbol{\tilde{x}}^{f}=f_{R}(\boldsymbol{x}^{f})\in\mathcal{A}^{2N_{t}}$,
where $f_{R}(\cdot)$ is the complex to real value function
$f_{R}(\boldsymbol{x})=\left[\Re{\\{\boldsymbol{x}\\}}^{T},\Im{\\{\boldsymbol{x}}\\}^{T}\right]^{T}$
and $\mathcal{A}$ is the set $\\{-2K-1,-2K+1,\dots,2K-1,2K+1\\}$ with
$K=\frac{\sqrt{M}-2}{2}$. Note that the $\boldsymbol{\tilde{x}}^{f}$ are now
the $\sqrt{M}$ pulse amplitude modulation ($\sqrt{M}$-PAM) symbols. The real-
valued form of channel is acquired by:

$\displaystyle\boldsymbol{\tilde{H}}^{f}=\begin{bmatrix}\Re{\\{\boldsymbol{H}^{f}\\}},-\Im{\\{\boldsymbol{H}^{f}\\}}\\\
\Im{\\{\boldsymbol{H}^{f}\\}},\Re{\\{\boldsymbol{H}^{f}\\}}\end{bmatrix}.$ (8)

Then the problem becomes a classification task with labels in the set
$\mathcal{A}$, which can be approximated by the Naive Bayesian approximation

$\displaystyle\operatorname*{argmax}_{\boldsymbol{\tilde{x}}^{f}}P\\{\boldsymbol{\tilde{x}}^{f}|\boldsymbol{\tilde{y}}^{f}\\}\approx\operatorname*{argmax}_{\boldsymbol{\tilde{x}}^{f}}\prod_{n\in\\{0,1,\dots,2N_{t}-1\\}}P_{n}\\{{\tilde{x}}_{n}^{f}|\boldsymbol{\tilde{y}}^{f}\\},$
(9)

where the $P_{n}\\{\cdot|\boldsymbol{\tilde{y}}^{f}\\}$ represents the
marginal distribution of the $n$-th entry of $\boldsymbol{\tilde{x}}^{f}$. The
ADNN is designed to learn functions that approximate the
$P_{n}\\{\cdot|\boldsymbol{\tilde{y}}^{f}\\}$.

####  III-C2 Binary detection

For simplicity, we start with the QPSK detection, where the
${\tilde{x}}_{n}^{f}$ is in the class set $\\{+1,-1\\}$. Denote the function
approximated by the binary classifier as $\mathcal{B}_{n}$. We can directly
apply the binary classifier to estimate the marginal likelihood ratio with

$\displaystyle{P_{n}\\{{\hat{x}}_{n}^{f}=+1|\boldsymbol{\tilde{y}}^{f}\\}\over
P_{n}\\{{\hat{x}}_{n}^{f}=-1|\boldsymbol{\tilde{y}}^{f}\\}}\approx{\mathcal{B}_{n}(\hat{b}_{n}=+1;\boldsymbol{\tilde{y}}^{f})\over\mathcal{B}_{n}(\hat{b}_{n}=-1;\boldsymbol{\tilde{y}}^{f})}=:{\mathcal{L}}_{b}^{+-}(\boldsymbol{\tilde{y}}^{f}),$
(10)

where ${\hat{x}}_{n}^{f}$ is the estimated transmitted symbol, $\hat{b}_{n}$
is the predicted binary label, and ${\mathcal{L}}_{b}^{+-}$ denotes the
likelihood ratio of the binary classifier.

####  III-C3 Multi-class detection

Figure 2:  The architecture of ADNN. “Share” means that the weights of all the
binary classifiers are shared.

When transmitting the $M$-QAM symbol, we perform the multi-class
classification task with the class set $\mathcal{A}$ adopting the following
shifting principle [25]:

$\displaystyle{{P_{n}\\{{\hat{x}}_{n}^{f}=-2k+1|\boldsymbol{\tilde{y}}^{f}\\}}\over{P_{n}\\{{\hat{x}}_{n}^{f}=-2k-1|\boldsymbol{\tilde{y}}^{f}\\}}}={P_{n}\\{{\hat{x}}_{n}^{f}=+1|\boldsymbol{\tilde{y}}^{f}+2k\cdot\boldsymbol{\tilde{h}}_{n}^{f}\\}\over
P_{n}\\{{\hat{x}}_{n}^{f}=-1|\boldsymbol{\tilde{y}}^{f}+2k\cdot\boldsymbol{\tilde{h}}_{n}^{f}\\}},$
(11)

where $\boldsymbol{\tilde{h}}_{n}^{f}$ is the $n$-th column of the channel
$\boldsymbol{\tilde{H}}^{f}$ and $k=-K,-K+1,\dots,+K$. The principle is based
on the fact that the constellation points of the QAM modulation share the same
distance $2$ and have a repetitive structure. By shifting the received signal
$\boldsymbol{\tilde{y}}^{f}$ in the direction of
$\boldsymbol{\tilde{h}}_{n}^{f}$ with step size $2k$, which corresponds to
$\boldsymbol{\tilde{y}}^{f}+2k\cdot\boldsymbol{\tilde{h}}_{n}^{f}$, the
transmit symbol $-2k+1$ and $-2k-1$ are shifted to $+1$ and $-1$,
respectively. Note that the binary label $+1$ and $-1$ is determined by the
step size plus the transmit symbol. Then the likelihood ratio between class
$-2k+1$ and $-2k-1$ can be estimated by conducting a binary classification in
$\\{+1,-1\\}$ with input
$\boldsymbol{\tilde{y}}^{f}+2k\cdot\boldsymbol{\tilde{h}}_{n}^{f}$, which is
denoted as

$\displaystyle{{P_{n}\\{{\hat{x}}_{n}^{f}=-2k+1|\boldsymbol{\tilde{y}}^{f}\\}}\over{P_{n}\\{{\hat{x}}_{n}^{f}=-2k-1|\boldsymbol{\tilde{y}}^{f}\\}}}\approx{\mathcal{L}}_{b}^{+-}(\boldsymbol{\tilde{y}}^{f}+2k\cdot\boldsymbol{\tilde{h}}_{n}^{f}).$
(12)

In this way, the multi-class classification is transformed into multiple
binary detection processes. For ease of discussion, the operation
$\boldsymbol{\tilde{y}}^{f}+2k\cdot\boldsymbol{\tilde{h}}_{n}^{f}$ is referred
to as the “shifting process” for the remainder of this paper, where the step
size $2k$ is referred to as the “shifting parameter” and denoted as $s_{n}$.
Note that while the above transformation is designed for QAM modulation, the
same idea can be generalized to other modulation schemes with a customized
shifting process.

At the testing time, the received signal is tested along with all the possible
values of the shifting parameter $s_{n}$ in set
$\mathcal{S}=\\{-2K,-2K+2,\dots,2K-2,2K\\}$. The probability of each class can
be obtained by collecting all the likelihood ratios in eq.( 12) with

$P_{n}\\{{\hat{x}}_{n}^{f}=-2k+1|\boldsymbol{\tilde{y}}^{f}\\}\\\ \approx
P_{n}\\{{\hat{x}}_{n}^{f}\\!=\\!-2K-1|\boldsymbol{\tilde{y}}^{f}\\}\\!\prod_{k^{\prime}=k}^{K}\\!{\mathcal{L}}_{b}^{+-}(\boldsymbol{\tilde{y}}^{f}+2k^{\prime}\cdot\boldsymbol{\tilde{h}}_{n}^{f}),$
(13)

where each probability can be computed by solving the equation
$\sum_{a\in\mathcal{A}}P_{n}\\{{\hat{x}}_{n}^{f}=a|\boldsymbol{\tilde{y}}^{f}\\}=1$.
The estimated ${\hat{x}}_{n}^{f}$ is determined by the class with the maximum
probability. In Fig. 2, we show the architecture of ADNN and an example of
when it is tested in the $16$-QAM case. The ADNN is employed as the frequency
domain network in RC-Struct, where the channel
$\boldsymbol{\tilde{h}}_{n}^{f}$ is estimated by the linear minimum mean
square error (LMMSE) method.

####  III-C4 Construction of binary training samples

The complex-valued $M$-QAM symbols are converted to real-valued $\sqrt{M}$-PAM
symbols for training. For clarity, we define two types of training samples:
the PAM training samples and the binary training samples. The PAM training
samples refer to the real-valued transmit and receive signal pairs. The binary
training samples are generated to train the designed binary classifier using
the PAM training samples. For each PAM training sample, we construct two
binary training samples, which include a positive binary sample and a negative
binary sample. As mentioned in Sec. III-C3, the binary label is determined by
$b_{n}=s_{n}+{\tilde{x}}_{n}^{f}$. At the training time, the shifting
parameter is set as $s_{n}=-{\tilde{x}}_{n}^{f}+b_{n}$ to control if a binary
sample is positive or negative. Specifically, for each PAM training sample
$({\tilde{x}}_{n}^{f},\boldsymbol{\tilde{y}}^{f})$, positive and negative
binary training samples are generated as

$\displaystyle(b_{n}^{+}=+1,\boldsymbol{\tilde{y}}^{f},s_{n}^{+}=-{\tilde{x}}_{n}^{f}+1),$
$\displaystyle(b_{n}^{-}=-1,\boldsymbol{\tilde{y}}^{f},s_{n}^{-}=-{\tilde{x}}_{n}^{f}-1).$
(14)

In this way, each PAM training sample is augmented into two binary training
samples and thus can be utilized more efficiently. In Tab. II, we show
examples of positive and negative shifting parameters corresponding to $4$-PAM
symbols.

TABLE II: Construction of binary training sample Transmit symbol $\tilde{x}_{n}^{f}$ | $s_{n}^{+}=-\tilde{x}_{n}^{f}+1$ | $s_{n}^{-}=-\tilde{x}_{n}^{f}-1$
---|---|---
$-3$ | $4$ | $2$
$-1$ | $2$ | $0$
$+1$ | $0$ | $-2$
$+3$ | $-2$ | $-4$

##  IV Frequency domain network — StructNet

In this section, we introduce the frequency domain network, StructNet, in a
MIMO system and analyze the properties of the network. The discussion of how
to apply StructNet in the MIMO-OFDM symbol detection task is provided in Sec.
V.

###  IV-A Design of StructNet

StructNet is a frequency domain network that builds upon ADNN, where it also
embeds the repetitive structure of the modulation constellation into the
network. In ADNN, the shifting process is performed using the ground truth
channel. When adopted in RC-Struct, the LMMSE estimated channel is utilized
due to the difficulty in obtaining perfect channel knowledge in practice.
While the estimated channel can serve as an alternative to conduct the
shifting process, it lacks the ability to adapt to changes in the environment.
To address this issue, an additional PE layer is introduced in StructNet to
estimate channel and perform the shifting process. The weights of the PE layer
are dynamically updated according to channel variations without relying on
perfect channel knowledge.

The architecture of StructNet is depicted in Fig. 3, comprising a PE layer and
a binary classifier. The weights of the PE layer, denoted by
$\boldsymbol{\hat{h}}_{n}^{f}$, also represent an estimate of the ground truth
channel $\boldsymbol{\tilde{h}}_{n}^{f}$. The PE layer takes the shifting
parameter $s_{n}\in\mathcal{S}$ as input and is implemented using a linear NN
layer. The binary classifier is an MLP. The PE layer is initialized with the
LMMSE estimated channel and updated along with the binary classifier through
backpropagation. The network is trained using the cross-entropy loss.

Figure 3: The architecture of StructNet.

One notable aspect of StructNet is that its PE layer for channel estimation is
learned without assuming knowledge of the ground truth channel. Unlike other
NN-based channel estimation approaches, which require ground truth channels to
calculate the network loss, StructNet adopts a more realistic setting and does
not assume perfect CSI. Instead, the PE layer is trained based on the loss of
the binary classification. Such an update of the PE layer is achieved by
embedding the repetitive modulation constellation structure into the network
design. In addition, the embedded structural information also enables
StructNet to utilize training samples more efficiently, as discussed in
Sec.III-C4, and to be more robust to incorrect labels, as analyzed in
Sec.IV-B. The robustness to incorrect labels makes StructNet less susceptible
to error propagation when applied in the DF mechanism. Furthermore, the
introduced PE layer further facilitates the DF approach due to the dynamic
updating of the network parameters.

Figure 4: Architecture of RC-AttStructNet-DF network.

###  IV-B Training with incorrect labels

Due to the unique training sample construction process discussed in Sec.
III-C4, given any percentage of the incorrect PAM labels, there are at most
$50\%$ incorrect binary labels during the training stage. We first explain the
reason for this conclusion with a concrete example and then discuss it in a
more general case.

For ease of discussion, we use $4$-PAM as an example, but the conclusion
generalizes to $\sqrt{M}$-PAM labels. Note that there are two kinds of labels
in our setting: the PAM labels and the binary labels. The PAM label is the
transmitted $\sqrt{M}$-PAM symbol in the set $\mathcal{A}$. The binary label
is in the set $\\{+1,-1\\}$, indicating whether a sample is positive or
negative. Assume we have a correct PAM label with ${\tilde{x}}_{n}^{f}=-1$ and
the corresponding incorrect PAM label is ${\bar{x}}_{n}^{f}=+1$. Then the
training samples created by the incorrect label ${\bar{x}}_{n}^{f}=+1$ can be
written as

$\displaystyle(\bar{b}_{n}^{+}=+1,\boldsymbol{\tilde{y}}^{f},{s}_{n}^{+}=-{\bar{x}}_{n}^{f}+1=0),$
$\displaystyle(\bar{b}_{n}^{-}=-1,\boldsymbol{\tilde{y}}^{f},{s}_{n}^{-}=-{\bar{x}}_{n}^{f}-1=-2).$
(15)

Note that the actual binary label is determined by
${b}_{n}=s_{n}+{\tilde{x}}_{n}^{f}$. As the actual PAM label is
${\tilde{x}}_{n}^{f}=-1$, the actual binary label for the constructed positive
sample, however, is ${b}_{n}^{+}=s_{n}^{+}+{\tilde{x}}_{n}^{f}=-1<0$,
resulting in an incorrect binary positive sample. For the negative sample, the
actual binary label is ${b}_{n}^{-}=s_{n}^{-}+{\tilde{x}}_{n}^{f}=-3<0$, which
is a correct binary sample.

In general, suppose ${\bar{x}}_{n}^{f}$ is the incorrect PAM label
corresponding to the actual PAM label ${\tilde{x}}_{n}^{f}$. The positive
sample is created by setting $s_{n}^{+}=-{\bar{x}}_{n}^{f}+1$ and the negative
sample is constructed by setting $s_{n}^{-}=-{\bar{x}}_{n}^{f}-1$. As the
actual PAM label is ${\tilde{x}}_{n}^{f}$, the actual binary label for the
positive sample is
${b}_{n}^{+}={\tilde{x}}_{n}^{f}+s_{n}^{+}={\tilde{x}}_{n}^{f}-{\bar{x}}_{n}^{f}+1$
and the actual label for the negative sample is
${b}_{n}^{-}={\tilde{x}}_{n}^{f}+s_{n}^{-}={\tilde{x}}_{n}^{f}-{\bar{x}}_{n}^{f}-1$.
The positive sample is considered to be incorrect if
${b}_{n}^{+}={\tilde{x}}_{n}^{f}-{\bar{x}}_{n}^{f}+1<0$, i.e.,
${\tilde{x}}_{n}^{f}-{\bar{x}}_{n}^{f}<-1$. The negative sample is incorrect
when ${b}_{n}^{-}={\tilde{x}}_{n}^{f}-{\bar{x}}_{n}^{f}-1>0$, i.e.,
${\tilde{x}}_{n}^{f}-{\bar{x}}_{n}^{f}>1$. Since
${\tilde{x}}_{n}^{f},{\bar{x}}_{n}^{f}\in\mathcal{A}$ and
${\tilde{x}}_{n}^{f}\neq{\bar{x}}_{n}^{f}$, the distance
${\tilde{x}}_{n}^{f}-{\bar{x}}_{n}^{f}$ can at most satisfy one of the
inequalities between ${\tilde{x}}_{n}^{f}-{\bar{x}}_{n}^{f}<-1$ and
${\tilde{x}}_{n}^{f}-{\bar{x}}_{n}^{f}>1$ for any value of
${\tilde{x}}_{n}^{f}$ and ${\hat{x}}_{n}^{f}$. Thus, there is at least one
correct positive sample or correct negative sample for any incorrect PAM
sample. The percentage of incorrect binary samples is at most $50\%$. The
incorrect percentage only equals $50\%$ when all the training PAM labels are
incorrect. This unique property of StructNet makes it less affected by
detection errors. As one of the major issues of DF-based approaches is error
propagation, StructNet can mitigate the issue by showing robustness to the
detection errors and therefore work well with the DF approach.

##  V Introduced approach — RC-AttStructNet-DF

In this section, we introduce the RC-AttStructNet-DF approach. As shown in
Fig. 4, the architecture of RC-AttStructNet-DF consists of three essential
components: the time domain RC network, the 2D MHA module, and the frequency
domain StructNet network. The time domain RC network is exploited to decouple
the transmitted data streams and deconvolve the channel for equalization [19,
20]. Furthermore, we design a DF mechanism to dynamically update the network
with the detected data symbols, especially in high mobility scenarios. The RLS
algorithm is used in the time domain as part of the DF mechanism to adaptively
update the weights on an OFDM symbol basis. In the frequency domain, the 2D
MHA module is designed to capture the time and frequency correlation of the
signal. Subsequently, the frequency domain network, StructNet, is utilized to
conduct the multi-class classification. The residual connection [26] is
employed to connect the 2D MHA and the StructNet. Additionally, we develop an
attention loss for the DF mechanism in the frequency domain, which weights the
training loss of different samples based on their confidence level. In the
following subsections, we provide detailed training and testing procedures for
our approach.

###  V-A RC with DF

In the time domain, the convolution and the superposition operation of the
wireless channel are conducted on the transmit signal. The RC reverses such an
operation by jointly decoupling the different transmitted data streams and
deconvolving the channel for equalization [19, 20].

####  V-A1 Learning from pilot symbols

The input to the RC network is the time domain received signal
$\boldsymbol{Y}^{t}$ and the target output is the time domain transmitted
signal $\boldsymbol{X}^{t}$. The training dataset for learning from pilot
symbols can be represented as

$\displaystyle\mathcal{D}_{rc}\triangleq\\{\boldsymbol{Y}_{n}^{t},\boldsymbol{X}_{n}^{t}\\}_{n=0}^{N_{p}-1}.$
(16)

The initial output weights of RC are learned with the pilot symbols through
the LS solution shown in eq. (3).

####  V-A2 DF with data symbols

The DF mechanism is utilized to dynamically update the output weights of RC
with the detected data symbols in a symbol-by-symbol manner. During the DF
procedure, the RLS [27, 28] method is adopted to recursively update the output
weights. The initial weights for the RLS algorithm are the weights learned
from the pilot symbols.

Specifically, we use RC trained by the pilot symbols to generate the
estimation for the first data symbol
$\boldsymbol{\hat{X}}_{N_{p}}^{t}\in{\mathbb{C}}^{N_{t}\times(N_{\mathrm{cp}}+N_{\mathrm{sc}})}$.
For the $n$th data symbol, the estimation $\boldsymbol{\hat{X}}_{n}^{t}$
($n=N_{p}+1,\dots,N-1$) is detected by the RC learned with the $(n-1)$-th data
symbol. The estimation is then converted into frequency domain
$\boldsymbol{\hat{X}}_{n}^{f}$ and then each symbol is mapped to the nearest
constellation points, which generates $\boldsymbol{\bar{X}}_{n}^{f}$. The
frequency domain $\boldsymbol{\bar{X}}_{n}^{f}$ is transformed back to the
time domain $\boldsymbol{\bar{X}}_{n}^{t}$ through the IFFT operation. Then
the output weights of RC are recursively updated by minimizing the objective:

$\displaystyle\arg\min_{\boldsymbol{W}_{\mathrm{out}}^{(n)}(m)}\sum_{m^{\prime}=0}^{m}\alpha^{m-m^{\prime}}\|\boldsymbol{W}_{\mathrm{out}}^{(n)}(m^{\prime})\boldsymbol{z}_{n}(m^{\prime})-\boldsymbol{\bar{x}}_{n}^{t}(m^{\prime})\|_{2}^{2},$
(17)

where $\boldsymbol{W}_{\mathrm{out}}^{(n)}(m)$ is the weight learned by the
$n$-th data symbol at step $m$;
$\boldsymbol{z}_{n}(m)\in\mathbb{C}^{N_{n}+N_{t}}$ is the concatenation of the
RC state and input for the $n$-th data symbol at step $m$;
$\boldsymbol{\bar{x}}_{n}^{t}(m)\in{\mathbb{C}}^{N_{t}}$ is the $m$-th column
of $\boldsymbol{\bar{X}}_{n}^{t}$; $\alpha\in(0,1]$ is the forgetting factor;
and $m\\!=\\!0,1,\\!\dots\\!,N_{\mathrm{sc}}\\!+\\!N_{\mathrm{cp}}\\!-\\!1$.
The forgetting factor $\alpha$ indicates how much we trust the previous
samples. When $\alpha<1$, the smaller the $\alpha$, the fewer weights we put
on the old samples than the recent ones.

The output weight $\boldsymbol{\hat{W}}_{\mathrm{out}}^{(n)}(m)$ is
recursively updated by

$\displaystyle\boldsymbol{\hat{W}}_{\mathrm{out}}^{(n)}(m)=\boldsymbol{\hat{W}}_{\mathrm{out}}^{(n)}(m-1)+\boldsymbol{e}_{n}(m)\boldsymbol{v}_{n}^{T}(m),$
(18)

where
$\boldsymbol{e}_{n}(m)=\boldsymbol{\bar{x}}_{n}^{t}(m)-\boldsymbol{\hat{W}}_{\mathrm{out}}^{(n)}(m-1)\boldsymbol{z}_{n}(m)$
is the error on the current sample $m$ when estimated with weight matrix at
the $(m-1)$-th step, and $\boldsymbol{v}_{n}(m)$ is the gain vector computed
by the following equation [28]:

$\displaystyle\boldsymbol{v}_{n}(m)=\frac{\boldsymbol{\Phi}_{n}^{-1}(m-1)\boldsymbol{z}_{n}(m)}{\alpha+\boldsymbol{z}_{n}^{T}(m)\boldsymbol{\Phi}_{n}^{-1}(m-1)\boldsymbol{z}_{n}(m)}.$
(19)

The
$\boldsymbol{\Phi}_{n}^{-1}(m)=(\sum_{m^{\prime}=0}^{m}\alpha^{m-m^{\prime}}\boldsymbol{z}_{n}(m^{\prime})\boldsymbol{z}_{n}^{T}(m^{\prime}))^{-1}$
is the inverse of the weighted correlation of $\boldsymbol{z}_{n}(m)$ and is
recursively updated with

$\displaystyle\boldsymbol{\Phi}_{n}^{-1}\\!(m)\\!=\\!\alpha^{-1}(\boldsymbol{\Phi}_{n}^{-1}(m-1)-\boldsymbol{v}_{n}(m)\left[\boldsymbol{z}_{n}^{T}(m)\boldsymbol{\Phi}_{n}^{-1}(m-1)\right]).$
(20)

The output weight $\boldsymbol{\hat{W}}_{\mathrm{out}}^{(n)}$ is determined by
the weight learned at $(N_{\mathrm{sc}}\\!+\\!N_{\mathrm{cp}}\\!-\\!1)$-th
step.

###  V-B 2D MHA and StructNet with DF

In the frequency domain, the output of the RC $\boldsymbol{\hat{X}}_{n}^{t}$
is transformed to the frequency domain
$\boldsymbol{\hat{X}}_{n}^{f}\in\mathbb{C}^{N_{t}\times N_{\mathrm{sc}}}$,
where the corresponding target output is $\boldsymbol{X}_{n}^{f}$. Denote
${\hat{X}}_{n}^{f}(n_{t},n_{\mathrm{sc}})$ and
${X}_{n}^{f}(n_{t},n_{\mathrm{sc}})$ as the $(n_{t},n_{\mathrm{sc}})$-th entry
of the $\boldsymbol{\hat{X}}_{n}^{f}$ and $\boldsymbol{X}_{n}^{f}$. The
complex values are mapped to real values by
$\boldsymbol{i}_{n}^{f}(n_{t},n_{\mathrm{sc}})=f_{R}({\hat{X}}_{n}(n_{t},n_{\mathrm{sc}}))$
and
$\boldsymbol{o}_{n}^{f}(n_{t},n_{\mathrm{sc}})=f_{R}({{X}}_{n}(n_{t},n_{\mathrm{sc}}))$.

####  V-B1 Learning from pilot symbols

Figure 5: 2D MHA module.

The input first passes through the 2D MHA module. Unlike the MHA module in the
Transformer that only captures feature correlations along the same dimension,
the 2D MHA works in a two-dimensional manner, incorporating both time and
frequency dimensions. As shown in Fig. 5, two attention blocks, including a
time attention block and a frequency attention block, are utilized in the
module to capture the time correlation and frequency correlation,
respectively. The input to the time attention block is the real-valued
frequency domain signal $\boldsymbol{I}^{f}(n_{t})\in\mathbb{R}^{N_{p}\times
2N_{\mathrm{sc}}}$, which is a concatenation of
$\boldsymbol{i}_{n}^{f}(n_{t},n_{\mathrm{sc}})$ along the time and frequency
dimensions. The input to the frequency attention block is the frequency domain
signal $\boldsymbol{\tilde{I}}^{f}(n_{t})\in\mathbb{R}^{N_{\mathrm{sc}}\times
2N_{p}}$ obtained by a permutation of $\boldsymbol{I}^{f}(n_{t})$. To
facilitate the feature aggregation of each block, the value embedding
dimension for each attention block is set to be equal to the input size. The
output of the frequency attention block is permutated and then aggregated with
the time attention block output through concatenation and linear projection.
The final output of 2D MHA is added to the input through the residual
connection and then passed through StructNet. Note that the 2D MHA is only
employed when learning from pilot symbols to help obtain a better estimate of
the data symbols. During the DF procedure, the network is updated on a symbol
basis, which does not allow the module to capture the two-dimensional feature
along the time dimension. Therefore, the 2D MHA is skipped during the DF
procedure.

####  V-B2 Attention loss

The attention loss is designed for the frequency domain network to re-weight
the training loss of different samples according to their confidence level.
The idea behind it is to force the network to pay more attention to confident
samples. The attention loss for each sample can be written as
$\boldsymbol{a}_{n}^{f}(n_{t},n_{\mathrm{sc}})\odot\ell(f_{s}(\boldsymbol{i}_{n}^{f}(n_{t},n_{\mathrm{sc}})),\boldsymbol{{o}}_{n}^{f}(n_{t},n_{\mathrm{sc}}))$,
where $\boldsymbol{a}_{n}^{f}(n_{t},n_{\mathrm{sc}})$ is the corresponding
weights of this training sample, $\odot$ is the Hadamard product,
$\ell(\cdot)$ stands for the cross-entropy loss, and $f_{s}(\cdot)$ represents
the function approximated by the frequency domain network. When training with
the pilot symbols, the confidence level for all the training samples is the
same, and thus the weights for all the training samples are set to be 1, i.e.,
$\boldsymbol{a}_{n}^{f}(n_{t},n_{\mathrm{sc}})=[1,1]^{T}$. Note that the
output of the frequency domain network is the detected data symbol along with
its predicted probability. The predicted probability indicates how confident
the network is in the detected data symbol. During the DF procedure, the
training labels are the detected symbols. The attention weights are set as the
predicted probability of detected symbols. It is true that the detected data
symbol with a high predicted probability is not ensured to be correct.
However, if the network predicts a high probability for the detected data
symbol, it is more likely to be correct than the detected data symbol with a
low predicted probability. When the detected data symbol has a high predicted
probability, we put more weights on it than the detected symbol with a low
predicted probability. In addition, only symbols with a predicted probability
larger than $\eta$ are used during the DF process. Therefore, the attention
weights can be written as
$\boldsymbol{a}_{n}^{f}(n_{t},n_{\mathrm{sc}})=q(P(\boldsymbol{\tilde{o}}_{n}^{f}(n_{t},n_{\mathrm{sc}})))$,
where $P(\boldsymbol{\tilde{o}}_{n}^{f}(n_{t},n_{\mathrm{sc}}))$ is the
probability of the predicted symbol
$\boldsymbol{\tilde{o}}_{n}^{f}(n_{t},n_{\mathrm{sc}})$ provided by the
frequency domain network. The $q(\cdot)$ denotes the following non-linear
function:

$\displaystyle q(x)=\begin{cases}x,&\text{if }x\geq\eta\\\
0,&\text{otherwise}\end{cases},$ (21)

where $\eta$ is the probability threshold. With the attention loss, the error
propagation issue of the DF process can be alleviated by assigning weights to
the loss based on the predicted probability and selecting samples with high
confidence.

####  V-B3 DF with data symbols

After training the 2D MHA and the StructNet with the pilot symbols, we adopt
the DF mechanism to learn from the detected data symbols on an OFDM symbol
basis. As the two-dimensional features do not exist when updating in a symbol-
by-symbol fashion, the 2D MHA module is not used and only the StructNet is
fine-tuned with the data symbols. Specifically, we obtain the detected data
signals $\boldsymbol{\hat{X}}_{n}^{t}$ in the time domain utilizing the
updated RC and transform it to frequency domain
$\boldsymbol{\hat{X}}_{n}^{f}$. Then we test the StructNet updated by the
$n-1$th data symbol with $\boldsymbol{\hat{X}}_{n}^{f}$ to get the estimated
symbol $\boldsymbol{\tilde{X}}_{n}^{f}$ for the $n$th data symbol. The pair
$(\boldsymbol{\hat{X}}_{n}^{f},\boldsymbol{\tilde{X}}_{n}^{f})$ is exploited
as the training data to fine-tune the StructNet. Note that for $n=N_{p}$,
i.e., the first data symbol, the training label is obtained by testing with
the pilot-trained 2D MHA and the StructNet. For the rest data symbols, the
training labels for fine-tuning are obtained only by the fine-tuned StructNet.

###  V-C Summary of symbol detection procedure

The symbol detection procedure of RC-AttStructNet-DF includes two parts:
offline initialization and online training. We explain the offline
initialization and summarize the online training in this subsection. The
procedure is also shown in Algorithm 1.

Algorithm 1 Symbol Detection Procedure of RC-AttStructNet-DF

1:# Offline initialization

2:Prepare artificial training data

3:Train a binary classifier with artificial training data

4:Initialize binary classifier in StructNet with learned weights

5:# Online training

6:for Each OFDM subframe do

7: # Learning from pilot symbols

8: Prepare pilot training data $D_{rc}$ with time domain received and
transmitted pilot signals as defined in eq. (16)
---

9: Train output weights of RC with $D_{rc}$ using one-shot LS solution in eq.
(3)
---

10: Generate RC output (time domain RC pilot output) when tested on received
pilot signals
---

11: Transform time domain RC pilot output to frequency domain (frequency
domain RC pilot output)

12: Utilize frequency domain RC pilot output and transmitted pilot symbols to
estimate effective channel using LMMSE

13: Initialize PE layer of StructNet with estimated effective channel

14: Prepare pilot training data for 2D MHA and StructNet with frequency domain
RC pilot output and transmitted pilot symbols

15: Train 2D MHA and StuctNet with pilot training data using attention loss

16: # DF with data symbols

17: for OFDM symbol $n=N_{p}:N-1$ (data symbols) do

18: Generate RC output ($n$-th time domain RC data output) by testing on
$n$-th time domain received data signal

19: Prepare training dataset for RC with $n$-th time domain received data
signal and $n$-th time domain RC data output

20: Update RC weights by RLS algorithm following eq. (17)

21: Transform $n$-th time domain RC data output to frequency domain ($n$-th
frequency domain RC data output)

22: if $n=N_{p}$ then

23: Generate $n$-th detected data symbols with 2D MHA and StructNet by testing
on $n$-th frequency domain RC data output
---

24: else

25: Generate $n$-th detected data symbols with StructNet by testing on $n$-th
frequency domain RC data output
---

26: end if

27: Prepare training dataset for StructNet with $n$-th frequency domain RC
data output and $n$-th detected data symbols
---

28: Fine-tune StructNet weights with attention loss

29: end for

30:end for

####  V-C1 Offline initialization

The binary classifier in StructNet is initialized with offline weights that
are trained by artificially generated data. Note that this offline training
does not require any prior knowledge of the channel, which differentiates our
method from other work with offline initialization and online adaptation.
Specifically, the training labels are randomly generated symbols
$\boldsymbol{E}\in\mathcal{A}^{N_{t}\times N_{sp}}$, where $\mathcal{A}$
represents the set $\\{-1,+1\\}$ and $N_{sp}$ is the number of offline
training samples. The training inputs are the noise-contaminated symbols or
the received signal $\boldsymbol{E}+\boldsymbol{G}$, where
$\boldsymbol{G}\in\mathbb{R}^{N_{t}\times N_{sp}}$ is the Gaussian noise. In
this way, the initial weights of the binary classifier are learned to conduct
nearest neighbor mapping and facilitate the online training of the full
StructNet, i.e., the PE layer and the binary classifier. Note that these
offline weights, once trained, are fixed. The same weights are used for
initialization when conducting online detection on different subframes.

TABLE III: Training complexity

Method | RC | Frequency domain network
---|---|---
RC-Struct | $\mathcal{O}(V(N_{n}+N_{\mathrm{train}}+N_{t})(N_{n}+N_{r})N_{\mathrm{train}})$ | $\mathcal{O}(8N_{t}N_{h}N_{\mathrm{ep}}N_{\mathrm{sc}}N_{p})$
RC-AttStructNet-DF | $\mathcal{O}(V(N_{n}+N_{\mathrm{train}}+N_{t})(N_{n}+N_{r})N_{\mathrm{train}}+((N_{n}+N_{r})N_{t}+4(N_{n}+N_{r})^{2})N_{\mathrm{test}})$ | $\mathcal{O}(4N_{p}N_{\mathrm{sc}}N_{t}(2N_{k}+N_{\mathrm{sc}})+8N_{t}(N_{\mathrm{ep}}N_{p}+N_{\mathrm{epdf}}N_{d})N_{h}N_{\mathrm{sc}})$

TABLE IV: Testing complexity Method | Complexity
---|---
LMMSE with LMMSE-CSI | $\mathcal{O}(N_{p}N_{\mathrm{sc}}^{2}N_{a}^{2}+N_{d}N_{\mathrm{sc}}(N_{a}^{3}+N_{a}^{2}+N_{a}))$
LMMSE with LMMSE-Interpolation | $\mathcal{O}((N_{p}N_{\mathrm{sc}}^{2}+7N_{d}N_{\mathrm{sc}})N_{a}^{2}+N_{d}N_{\mathrm{sc}}(N_{a}^{3}+N_{a}^{2}+N_{a}))$
SD with LMMSE-Interpolation | $\mathcal{O}((N_{p}N_{\mathrm{sc}}^{2}+7N_{d}N_{\mathrm{sc}})N_{a}^{2}+N_{d}N_{\mathrm{sc}}M^{N_{a}}(2N_{a}^{2}+2N_{a}-1))$
RC-Struct | $\mathcal{O}(V(N_{n}+N_{r})(N_{n}N_{\mathrm{test}}+N_{t})+4N_{t}N_{h}N_{\mathrm{sc}}N_{d})$
RC-AttStructNet-DF | $\mathcal{O}(V(N_{n}+N_{r})(N_{n}N_{\mathrm{test}}+N_{t})+4N_{\mathrm{sc}}N_{t}(2N_{k}+N_{\mathrm{sc}}+N_{h}N_{d}))$

####  V-C2 Online training

The time domain RC network and the frequency domain network are learned
separately. When learning from pilot symbols, RC is first trained with the
time domain received and transmitted pilot signals. Then after the training of
RC, the output of RC is transformed into the frequency domain. The frequency
domain network is learned by taking frequency domain RC output as the input
and the transmitted pilot symbols in the frequency domain as the training
label. When learning from detected data symbols with the DF mechanism, the
network weights are updated in a symbol-by-symbol manner. For each data
symbol, we generate the RC output of the data symbol by testing on the time
domain received data signal. This RC output is employed as the training label
to re-train RC with the RLS algorithm. The time domain RC output is then
converted to frequency domain. The detected data symbol is obtained by testing
the frequency domain network on frequency domain RC output. Similarly, this
detected data symbol is utilized as the training label for fine-tuning
frequency domain network.

It is noteworthy that the combination of RC in the time domain and the
frequency domain network is critical in our design. This is because even
though StructNet is designed to learn from the training samples efficiently,
training StructNet along with 2D MHA still requires a relatively large amount
of training data. Since RC can efficiently decouple different data streams and
equalize the channel in time domain, as shown in our previous work [19, 20],
the classification task in frequency domain becomes much more accessible to
tackle after the processing of RC.

##  VI Complexity Analysis

This section analyzes the computational complexity of RC-AttStructNet-DF. The
complexity is compared with RC-Struct [21], LMMSE detector, and sphere
decoding (SD) approach. In the analysis, we mainly consider the computation
cost of matrix multiplication and pseudo-inverse, as the costs for matrix
addition and element-wise operation are negligible compared to these main
factors. For ease of discussion, we denote the number of training and testing
samples in the time domain as
$N_{\mathrm{train}}=(N_{\mathrm{cp}}+N_{\mathrm{sc}})N_{p}$ and
$N_{\mathrm{test}}=(N_{\mathrm{cp}}+N_{\mathrm{sc}})N_{d}$. As the
complexities of RC-Struct have been provided in [21], we summarize the
conclusions in Tab. III and Tab. IV and mainly focus on the complexity
analysis of RC-AttStructNet-DF. $V$ denotes the number of cascaded RCs.

In the time domain, RC is first learned with the pilot symbols and then
updated with the detected data symbols. As the DF is utilized, the training
complexity will be larger than RC-Struct due to the adoption of the RLS update
procedure with extra training on the data symbols. The complexity for training
RC with pilot symbols is the same as RC-Struct, which is
$\mathcal{O}(V(N_{n}+N_{\mathrm{train}}+N_{t})(N_{n}+N_{r})N_{\mathrm{train}})$.
The RLS procedure on the data symbols has three steps to update the output
weights. The complexity for updating
$\boldsymbol{\hat{W}}_{\mathrm{out}}^{(n)}(m)$ in eq. (18) for each sample is
$\mathcal{O}((N_{n}+N_{r})N_{t})$. The update of $\boldsymbol{v}_{n}(m)$ in
eq. (19) has a complexity of
$\mathcal{O}((N_{n}+N_{r})^{2}+N_{n}+N_{r})\approx\mathcal{O}((N_{n}+N_{r})^{2})$.
The complexity for updating $\boldsymbol{\Phi}^{-1}_{n}(m)$ in eq. (20) is
$\mathcal{O}(3(N_{n}+N_{r})^{2})$. Then the complexity for all the samples
with the RLS approach is
$\mathcal{O}(((N_{n}+N_{r})N_{t}+4(N_{n}+N_{r})^{2})N_{\mathrm{test}})$. Thus,
the training complexity in time domain is
$\mathcal{O}(V(N_{n}+N_{\mathrm{train}}+N_{t})(N_{n}+N_{r})N_{\mathrm{train}}+((N_{n}+N_{r})N_{t}+4(N_{n}+N_{r})^{2})N_{\mathrm{test}})$.
At the testing stage, time domain RC in RC-AttStructNet-DF has the same
complexity as RC-Struct, which is
$\mathcal{O}(V(N_{n}+N_{r})(N_{n}N_{\mathrm{test}}+N_{t}))$.

The frequency domain network is composed of the 2D MHA module and the
StructNet. For simplicity, we assume the time and frequency attention block in
the 2D MHA adopts the same key embedding dimension $N_{k}$. Then the
complexity for the time attention block is
$\mathcal{O}((4N_{k}N_{p}N_{\mathrm{sc}}+4N_{p}N_{\mathrm{sc}}^{2})N_{t})$.
The complexity for frequency attention block is
$\mathcal{O}((4N_{k}N_{p}N_{\mathrm{sc}}+4N_{p}N_{\mathrm{sc}})N_{t})$. The
output linear project has a complexity of
$\mathcal{O}(8N_{p}N_{\mathrm{sc}}N_{t})$. Thus, the total complexity for
training 2D MHA with pilot symbols is
$\mathcal{O}(4N_{p}N_{\mathrm{sc}}N_{t}(2N_{k}+N_{\mathrm{sc}}+3))\approx\mathcal{O}(4N_{p}N_{\mathrm{sc}}N_{t}(2N_{k}+N_{\mathrm{sc}}))$.
During the DF procedure, 2D MHA is only adopted for testing the first data
symbol, resulting in a complexity of
$\mathcal{O}(4N_{\mathrm{sc}}N_{t}(2N_{k}+N_{\mathrm{sc}}))$.

For StructNet, it consists of a PE layer and a binary classifier. The PE layer
conducts an element-wise multiplication and thus the complexity is ignored
here. Denote $N_{h}$ as the number of neurons in the input layer and
$N_{\mathrm{ep}}$ as the number of training epochs. When training with pilot
symbols, the training complexity of StructNet is the same as the frequency
domain network of RC-Struct, which is
$\mathcal{O}(8N_{t}N_{h}N_{\mathrm{ep}}N_{\mathrm{sc}}N_{p})$. When DF is
adopted, the extra training complexity is
$\mathcal{O}(8N_{t}N_{h}N_{\mathrm{epdf}}N_{\mathrm{sc}}N_{d})$, where
$N_{\mathrm{epdf}}$ is the number of fine-tuning epochs for DF. Thus, the
total complexity is
$\mathcal{O}(8N_{t}(N_{\mathrm{ep}}N_{p}+N_{\mathrm{epdf}}N_{d})N_{h}N_{\mathrm{sc}})$.
As the number of testing samples in the frequency domain is
$N_{t}N_{\mathrm{sc}}N_{d}$, the testing complexity is
$\mathcal{O}(4N_{t}N_{h}N_{\mathrm{sc}}N_{d})$.

(a)

(b)

Figure 6: Comparison of SER. (a) with different $Eb/No$’s in dB. (b) with
different percentages of incorrect PAM labels.

The LMMSE approach is a low-complexity linear detector that is widely used in
communication systems. The SD detector [29] is a non-convex solver that
approaches the optimal maximum likelihood (ML) detection, which has high
detection complexity and thus is rarely used in practice. As channel
estimation is needed as the input to both methods, we adopt LMMSE for the
underlying channel estimation. As our previous work [19] has analyzed the
complexities of both methods with the LMMSE channel estimation in details, we
summarize the conclusions in Tab. IV. The “LMMSE-CSI” indicates that the
channel estimates are obtained by only utilizing the pilot symbols. Meanwhile,
the “LMMSE-Interpolation” means that the channel estimates are interpolated
over the data symbols using the pilot-estimated CSI. To simplify the
expression, we assume that the number of antennas satisfies
$N_{a}=N_{r}=N_{t}$.

The analysis shows that RC-AttStructNet-DF has higher training and testing
complexity than RC-Struct due to the additional 2D MHA module and DF
procedure. However, the training and testing complexities of these two
approaches are still in the same order of magnitude. Compared with the
conventional approaches, RC-AttStructNet-DF has higher complexity than the
LMMSE approach due to the extra training stage, and a lower complexity than
the SD detector.

##  VII Toy experiment: MIMO system with Gaussian channel

In this section, we provide a toy experiment in a MIMO system with the
Gaussian channel to analyze the properties of StructNet. We start with
analyzing the effectiveness of the PE layer and then empirically show the
robustness of StructNet to incorrect labels.

###  VII-A Experimental setting

In the toy experiment, we assume the classifier has sufficient data and time
to be trained, and the PE layer is initialized with an inaccurate LMMSE
estimated channel. To satisfy such an assumption, the number of training
samples is set to be $1000$, among which $4$ samples are utilized for LMMSE
channel estimation, and the rest $996$ samples are used for training the
classifier. The trained network is tested with $3000$ samples. For simplicity,
we test in a $2\times 2$ MIMO with $4$-PAM modulation. $100$ Gaussian channel
realizations are tested. The channels are selected to have condition numbers
smaller than $1.5$ to mimic the setting with dynamic transmission modes, where
data is transmitted in channels with small condition numbers. In Sec. VIII, we
remove such assumptions and evaluate our method in more realistic 3GPP 3D
channels.

###  VII-B Effectiveness of PE layer

We compare three approaches: 1) ADNN-GT: The ADNN with perfect channel
knowledge; 2) ADNN-LMMSE: The ADNN with LMMSE estimated channel; 3) StructNet:
our introduced approach. The classifiers in all these three networks are
comprised of two linear layers connected with the hyperbolic tangent non-
linear function, and trained with the cross-entropy loss. In Fig. 6 (a), we
plot the symbol error rate (SER) as a function of bit energy to noise ratio
($E_{b}/N_{o}$). Compared with ADNN-LMMSE, StructNet can achieve better
performance. The performance gain is more significant with a relatively high
$E_{b}/N_{o}$. In addition, StructNet is shown to have comparable SER
performance with ADNN-GT, where the perfect channel knowledge is used. The
results indicate that even if the PE layer starts from an inaccurate
initialization, StructNet can achieve comparable performance with the approach
exploiting the ground truth CSI.

###  VII-C Experiments of training with incorrect labels

In Sec. IV-B, we analyzed the reason why StructNet is robust to incorrect PAM
labels. In this section, we conduct the experiment to show how the performance
of StructNet is affected by different percentages of incorrect training
labels. In the experiment, we randomly select a certain percentage of PAM
samples to be incorrect. The performance is evaluated under $5$ dB
$E_{b}/N_{o}$. Besides the two methods mentioned above, we also compare the
performance with a four-class classifier. For a fair comparison, the four-
class classifier also adopts two linear layers and the hyperbolic tangent non-
linear function, except that the output layer is of size $4$. The results of
SER versus the incorrect PAM label percentage are shown in Fig. 6 (b). As
opposed to the general four-class classifier that is significantly affected by
incorrect labels, StructNet performs reasonably well with even $70\%$
incorrect PAM labels. This is because the $70\%$ incorrect PAM labels only
contribute to $35\%$ incorrect binary labels, making it easier to handle by
the network. The results are consistent with our analysis and demonstrate the
ability of StructNet to combat the incorrect training samples. Such a property
allows StructNet to mitigate the error propagation issue inherent in DF-based
approaches, making it suitable to be applied with the DF mechanism.

##  VIII Evaluation with 3GPP-3D channel

In this section, we show the performance of RC-AttStructNet-DF both in the
MIMO-OFDM system and the massive MIMO-OFDM system. Unlike the setting in RC-
Struct [21], the experiments are conducted at a user speed of $30$ km/h. We
compare the introduced approach with the conventional model-based methods and
state-of-the-art learning-based strategies.

(a)

(b)

(c)

Figure 7: BER comparison in the MIMO-OFDM system. (a) QPSK (b) $16$ QAM (c)
$64$ QAM

###  VIII-A Experimental setting

In the experiments, the number of subcarriers is set to be
$N_{\mathrm{sc}}=512$ and the CP length is $N_{\mathrm{cp}}=32$. Each subframe
has a total of $N=20$ OFDM symbols, among which $N_{p}=4$ OFDM symbols are the
training pilot and $N_{d}=16$ OFDM symbols are the data symbols. Note that
only $4$ pilot symbols are used as the training data for each subframe, which
is different from other learning-based approaches that exploit a large amount
of training data. The wireless channels are generated following the 3GPP 3D
MIMO model [17] with the QuaDRiGa simulator [30]. The user speed is set as
$30$ km/h, which is different from the setting in RC-Struct with a speed of
$5$ km/h. In addition, gray coding is adopted in the experiments.

In terms of the setting of RC, the number of neurons is $N_{n}=16$, and the
number of layers is $V=1$. In [19], it is shown that the utilization of a
sliding window for the input to RC can increase the short-term memory capacity
of RC. Following this work, a sliding window of size $32$ is utilized to the
input. Note that all RC-based approaches compared in this work adopt the same
sliding window for a fair comparison. In the 2D MHA, the key embedding
dimension for the time and frequency attention block is set as $216$ and $8$,
respectively. For the StructNet in the frequency domain, the input layer has
$128$ neurons, and the output layer has $2$ neurons. In StructNet, the offline
weights of the binary classifier are trained with $2000$ randomly generated
symbols for $1000$ epochs. As choosing $E_{b}/N_{o}$ for training the offline
weights does not provide any significant performance gain and is also not
practical, the $E_{b}/N_{o}$ for each training sample is randomly chosen from
$0$ dB to $15$dB to obtain relatively generic offline weights. During online
training, we adopt an alternative training strategy, which updates the PE
layer and the binary classifier separately. Specifically, we first update the
binary classifier and fix the PE layer. Then the PE layer is updated with the
weights of the binary classifier fixed. The total number of training epochs is
set to be $20$. In addition, to reduce the computation complexity, nine
resource block groups (RBGs) are combined to train a single network. The
probability threshold for attention loss is set as $\eta=0.5$. It means that
we select the detected symbols that the network predicts to have over $50\%$
chance to be correct. In other words, if the network thinks the detected data
symbol has less than or equal to a $50\%$ of the chance being correct, the
detected symbol is less likely to be correct and thus we do not use it in the
DF procedure.

###  VIII-B BER Comparison in the MIMO-OFDM system and the massive MIMO-OFDM
system

We compare the following approaches: (1) _LMMSE+LMMSE-CSI_ : The LMMSE-based
symbol detector with LMMSE estimated CSI; (2) _LMMSE+LMMSE-Interpolation_ :
The LMMSE-based symbol detector using interpolated LMMSE CSI; (3) _SD+LMMSE-
Interpolation_ : The non-convex symbol detector that performs ML detection
with SD approach utilizing interpolated LMMSE CSI [29]; (4) _MMNet_ : The
MMNet network proposed in [12], where the network for each subcarrier has been
trained for $500$ iterations; (5) _RC-Struct_ : The RC-based approach using
LMMSE estimated shifting parameter in the frequency domain [21]; (6) _RC-
AttStructNet-DF_ : The introduced method in this paper. Note that the “LMMSE-
CSI” refers to that the LMMSE channel estimates only use the pilot symbols.
The “LMMSE-Interpolation” means that the channel estimates are interpolated
over data symbols using the pilot estimated CSI.

(a)

(b)

(c)

Figure 8: BER comparison in the massive MIMO-OFDM system. (a) QPSK (b) $16$
QAM (c) $64$ QAM

We conduct experiments in two system settings: the MIMO-OFDM system and the
massive MIMO-OFDM system. In the MIMO-OFDM system, the number of transmit
antennas is set as $N_{t}=4$ and the number of receive antennas is $N_{r}=4$.
In Fig. 7, we show the BER plot for QPSK, $16$ QAM, $64$ QAM,
respectively.111Note that the BER of RC-AttStructNet-DF (without channel
coding) ranges from $3\%$ to $14\%$, which lies within the typical BER range
specified by the 3GPP 5G NR [31, 32]. For instance, the user equipment (UE)
channel quality indicator (CQI) calculation is based on a target block error
rate (BLER) of 10% (after channel coding) [31], and radio link monitoring out-
of-sync BLER is set to be 10% [32]. The results show that all the learning-
based approaches and the SD method outperform LMMSE when the CSI is obtained
by using only the pilot symbols for all the tested modulation orders. With the
interpolated CSI, the BER of the LMMSE detection scheme decreases, as the
channel estimates become more accurate when interpolated over the data
symbols. As exhibited in Fig. 7 (b) and Fig. 7 (c), the SD approach has better
performance than the RC-Struct and MMNet in the high $Eb/No$ regime when $16$
QAM and $64$ QAM modulation orders are used. However, in the low $Eb/No$
regime, the performance of SD becomes worse than RC-Struct and MMNet due to
inaccurate channel estimates. The reason is that the channel estimates in the
low $Eb/No$ regime are more precise than in the high $Eb/No$ regime, leading
to performance degradation. These observations indicate that the performance
of the conventional approaches LMMSE and SD heavily relies on the accuracy of
the CSI estimation. The inherent error of the CSI estimation can jeopardize
the detection performance. On the other hand, as a learning-based approach
that does not rely on explicit channel estimation, RC-AttStructNet-DF is not
affected by such impairments and is shown to have outstanding performance gain
over conventional methods for all the tested modulation orders.

Regarding the learning-based approaches, in Fig. 7, we can see that the RC-
AttStructNet-DF consistently outperforms the MMNet algorithm and the RC-Struct
method under different scenarios. The reason is that for MMNet, it requires a
larger amount of training data than the setup in this paper to learn the
network weights. As an online over-the-air scenario is adopted in our
evaluation, the MMNet learned by the limited amount of training data suffers
from the model overfitting problem, resulting in performance degradation.
Different from MMNet, by embedding the structural knowledge of the MIMO-OFDM
system, RC-AttStructNet-DF can be learned with limited training data in an
online fashion. Furthermore, both the RC-Struct and the MMNet only learn from
the pilot symbols. Due to the relatively high user mobility, the neural
network weights only trained by the pilot symbols are not sufficient to track
the changes of the channel over data symbols, and thus have an unsatisfactory
detection performance when testing on the data symbols. Instead, RC-
AttStructNet-DF dynamically updates the network weights according to the
channel variation with the data symbols. With the specially designed
architecture and the dynamic adaptation, better performance is achieved by the
RC-AttStructNet-DF.

In the massive MIMO-OFDM system, we test in an uplink scenario with 4 transmit
antennas and 64 receive antennas. In particular, at the transmitter side, the
number of scheduled UE is 2, where each UE has 2 transmit antennas. At the
receiver side, the base station (BS) is equipped with a rectangular planar
array that has 8 azimuth antennas and 8 elevation antennas. Fig. 8 shows the
BER performance in the massive MIMO-OFDM system with QPSK, $16$ QAM, and $64$
QAM. The same trend holds as in the MIMO-OFDM system, where RC-AttStructNet-DF
achieves the lowest BER. The results further demonstrate the advantages of RC-
AttStructNet-DF over the other methods under different scenarios.

###  VIII-C BER comparison with nonlinear distortion

Figure 9: BER comparison in the nonlinear region.

(a)

(b)

Figure 10: The MIMO scattered pilot pattern in one resource block (RB). (a)
Conventional approaches. (b) RC-AttStructNet-DF.

In this section, we perform the experiment when PA distortion is applied to
the input signal. The PA model
$g(x)=\frac{x}{\left[1+\left(\frac{|x|}{x_{\mathrm{sat}}}\right)^{2\rho}\right]^{0.5\rho}}$
is adopted to introduce channel distortion [33], where $x$ is the input
transmitted signal, $x_{\mathrm{sat}}$ is the PA saturation level, and $\rho$
measures the smoothing parameter. We define the input back-off (IBO) as the
ratio between PA’s saturation power to the input power. The input signal is
distorted when the peak-to-average-power ratio (PAPR) of the input signal is
higher than IBO. We adopt $x_{\mathrm{sat}}=1$ and $\rho=3$ and set the non-
linear region as the case when IBO is smaller than $6.5$ dB. The experiments
are conducted in the $4\times 4$ MIMO-OFDM system with $16$ QAM modulation.
Fig. 9 shows the performance when the non-linear distortion exists. The
performance of the conventional schemes, such as the SD and LMMSE, is highly
affected by the system’s nonlinearity. Specifically, the BER of the SD method
increases quickly when the IBO reduces, as the estimated CSI becomes less
accurate with stronger signal distortion. As the distortion level increases,
the performance of the MMNet approach also degrades due to the linear
assumption for its system model. Furthermore, RC-based approaches perform
better than the conventional approaches when IBO is low, indicating that the
RC-based approaches are better at combating the nonlinearity. More
importantly, RC-AttStructNet-DF achieves the best performance among all the
schemes, demonstrating its generalization ability in different cases.

###  VIII-D Effectiveness of attention mechanism, PE layer, and DF

In this section, we demonstrate the effectiveness of incorporating the
attention mechanism, PE layer, and DF. The experiments are conducted in the
MIMO-OFDM system with $4$ transmit antennas and $4$ receive antennas. The
modulation order is set as $64$ QAM. In Fig. 11, we compare the BER
performance of four methods: (1) RC-Struct; (2) RC-Struct-DF: RC-Struct with
DF; (3) RC-StructNet-DF: the approach with PE layer and DF; (5) RC-
AttStructNet-DF: our introduced approach with the attention mechanism, PE
layer, and DF. As shown in Fig. 11, RC-Struct-DF outperforms RC-Struct, which
indicates the effectiveness of using DF when structural information is
incorporated in the network. By comparing the performance of RC-StructNet-DF
with RC-Struct-DF, we can see that the BER performance is further boosted when
the PE layer is adopted in the frequency domain. The results suggest that the
introduced PE layer in StructNet can further facilitate the DF mechanism and
improve the detection performance by allowing the network to dynamically
update the network parameters according to channel variations. Moreover, RC-
AttStructNet-DF achieves an additional performance gain over RC-StructNet-DF,
demonstrating that the attention mechanism is a valuable addition to our
network.

Figure 11: BER for testing effectiveness of different modules.

###  VIII-E BER comparison with practical pilot pattern

While our previous discussions are all based on the pilot structure where the
first $N_{p}$ OFDM symbols are all pilots, in this section, we show that our
introduced approach can also be applied to a scattered pilot pattern specified
by the 3GPP standard. We consider the MIMO pilot pattern shown in Fig. 10
following the 3GPP 5G NR [34, 35]. The pilot resource elements (REs) are
colored in yellow and the data REs are colored in blue. The white RE
represents the empty pilot symbols. As illustrated in Fig. 10 (a), for
conventional approaches, empty pilot symbols are transmitted. In addition,
pilots are set to be orthogonal among different antenna ports. The empty and
orthogonal settings of the pilot are utilized to eliminate pilot interference
and ensure a more accurate MIMO channel estimation. For our introduced
learning-based method, we adopt the pilot structure in Fig. 10 (b), where all
the pilots are randomly generated. It is noteworthy that the pilot pattern in
Fig. 10 (b) has the same training overhead as the pilot structure in Fig. 10
(a). The difference is that we try to avoid pilot interference using the empty
and orthogonal pilots for conventional approaches, while we keep the pilot
interference with the pilot pattern for learning-based methods. The reason is
that, for conventional detectors that rely on channel estimation, the received
pilots should be free of interference to obtain a more accurate estimated CSI.
However, for learning-based methods, the neural network needs to learn the
situation when interference exists to avoid the mismatch between the training
stage and the testing stage. More detailed discussions about the pilot pattern
design have been provided in our previous work [19].

The experiment is conducted in the $4\times 4$ MIMO-OFDM system with 16 QAM
modulation. The total number of OFDM symbols is set as 14 following the 3GPP
5G NR standard [34, 35]. The training overhead of this scattered pilot pattern
is approximately $19\%$. Note that the block pilot pattern, where there are
$20$ OFDM symbols in total and the first $4$ OFDM symbols are pilots, has a
training overhead of $20\%$, which also satisfies the pilot occupancy
requirement specified in [34, 35]. With the scattered pilot pattern, the RC-
AttStructNet-DF network is first trained with the pilot REs to obtain the
initial weights. Then the DF is conducted for the full subframe, including the
OFDM symbol with pilot REs, in a symbol-by-symbol manner. During the DF
procedure, when re-training with the detected OFDM symbol that has pilot REs,
the pilot positions are filled up with the ground truth pilots instead of
detected pilots to generate training labels. As displayed in Fig. 12, our
introduced RC-AttStructNet-DF approach outperforms both the conventional LMMSE
and SD methods with this practical scattered pilot pattern. More importantly,
the results further demonstrate the generalization ability of RC-AttStructNet-
DF and its potential to be adopted in a realistic setting.

Figure 12: BER comparison with the scattered pilot pattern.

###  VIII-F BER comparison with conventional methods using decision-directed
channel estimation

In this section, we conduct the performance comparison with conventional
approaches when the RLS-based decision-directed (DD) channel estimation [36]
is adopted. Specifically, the first four pilot OFDM symbols are utilized to
obtain the initial channel estimation using the RLS scheme. Then the
subsequent CSI corresponding to the data symbols is obtained by treating the
detected symbols as training data and estimated iteratively using the RLS
approach. The BER performance is provided in Fig. 14. The LMMSE detector and
SD detector with this DD channel estimation are referred to as “LMMSE+DD-CSI”
and “SD+DD-CSI”, respectively. The experiment is performed under the $4\times
4$ MIMO-OFDM system with $16$ QAM modulation. As shown in Fig. 14, by
dynamically estimating the channel of data symbols, “LMMSE+DD-CSI” has better
performance than the LMMSE detector with interpolated LMMSE CSI in the high
$E_{b}/N_{o}$ regime. However, due to the error propagation caused by the
inaccurate data symbol detection, the performance of “LMMSE+DD-CSI” is
degraded in mid to low $E_{b}/N_{o}$ regimes. The performance of the SD
detector is boosted by the DD channel estimates when compared with the SD
detector utilizing the interpolated LMMSE CSI. Compared with conventional
approaches, our introduced RC-AttStructNet-DF achieves better performance than
“LMMSE+DD-CSI” across different $E_{b}/N_{o}$’s and outperforms “SD+DD-CSI” in
the mid to low $E_{b}/N_{o}$ regimes. We further compare the performance when
the PA distortion is applied to the input signal. As indicated in Fig. 14, the
performance of the “LMMSE+DD-CSI” and “SD+DD-CSI” deteriorate when signal
distortion becomes stronger, especially for the “SD+DD-CSI”. On the other
hand, RC-AttStructNet-DF is less affected by the nonlinearity within the
system and exhibits better performance than the conventional approaches.

Figure 13: BER comparison with conventional schemes using DD-CSI.

Figure 14: BER comparison with adopting DD-CSI in nonlinear region.

###  VIII-G Empirical complexity of symbol detection approaches

To see the performance-complexity trade-off more straightforwardly, we compare
the empirical complexity of different methods under the same setting for Fig.
7 (c). Specifically, we show CPU run time for different methods in the setting
of the MIMO-OFDM system with 4 transmit antennas and 4 receive antennas and 64
QAM modulation. In Tab. V, we present both the training time and the total
processing time for a subframe of different methods, where the total
processing time includes both the training and testing time for a subframe.
Tab. V indicates that RC-AttStructNet-DF has a longer total processing time
than LMMSE-based approaches. However, RC-AttStructNet-DF can achieve a 65% and
57% BER reduction over the LMMSE with LMMSE-CSI and LMMSE with LMMSE-
Interpolation under 24 dB $E_{b}/N_{o}$, respectively, as shown in Fig. 7 (c).
While RC-AttStructNet-DF takes a longer total processing time than RC-Struct,
the CPU run time of these two approaches are still on the same scale. With the
same scale of the total processing time, RC-AttStructNet-DF performs
significantly better than RC-Struct. On the other hand, when compared with the
SD method, RC-AttStructNet-DF processes in a shorter amount of time and is
also demonstrated to have a better performance. More importantly, when
compared with the state-of-the-art learning-based approach MMNet, RC-
AttStructNet-DF processes over 100 times faster, which demonstrates the
potential of RC-AttStructNet-DF to be adopted for online detection.

TABLE V: CPU run time of symbol detection methods

Method | Training time (Sec.) | Total processing time (Sec.)
---|---|---
LMMSE+LMMSE-CSI | - | 0.12
LMMSE+LMMSE-Interpolation | - | 7.57
SD+LMMSE-Interpolation | - | 152.06
RC-Struct | 17.55 | 18.27
RC-AttStructNet-DF | 26.67 | 29.57
MMNet | 3014.52 | 3025.66

##  IX Conclusion

In this paper, we introduce an online attention-based approach, RC-
AttStructNet-DF, for conducting MIMO-OFDM symbol detection on a subframe
basis. The approach employs reservoir computing (RC) in the time domain. The
frequency domain network consists of the 2D MHA module along with a structure-
based network StructNet, which is learned with an attention loss. The 2D MHA
is exploited to capture the time and frequency correlations of the signal. The
StructNet is designed to mitigate the error propagation of the DF approach and
facilitate the DF mechanism. With the StructNet, the adopted DF mechanism
further enhances detection performance by learning from detected data symbols
and dynamically tracking channel changes within a subframe. Extensive
experiments in 3GPP 3D channels demonstrate the effectiveness of RC-
AttStructNet-DF in detection under different scenarios with the BER
performance and the effectiveness of different modules.

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