# IMRecoNet: Learn to Detect in Index Modulation Aided MIMO Systems with
Complex Valued Neural Networks

Chenwu Zhang, Hancheng Lu, and Jinxue Liu  This work was supported in part by
the National Science Foundation of China (No.61771445, No.61631017,
No.91538203). Chenwu Zhang, Hancheng Lu and Jinxue Liu are with School of
Information Science and Technology, University of Science and Technology of
China, Hefei 230027, China (email<EMAIL_ADDRESS><EMAIL_ADDRESS>jxliu18@mail.ustc.edu.cn).

###### Abstract

Index modulation (IM) reduces the power consumption and hardware cost of the
multiple-input multiple-output (MIMO) system by activating part of the
antennas for data transmission. However, IM significantly increases the
complexity of the receiver and needs accurate channel estimation to guarantee
its performance. To tackle these challenges, in this paper, we design a deep
learning (DL) based detector for the IM aided MIMO (IM-MIMO) systems. We first
formulate the detection process as a sparse reconstruction problem by
utilizing the inherent attributes of IM. Then, based on greedy strategy, we
design a DL based detector, called IMRecoNet, to realize this sparse
reconstruction process. Different from the general neural networks, we
introduce complex value operations to adapt the complex signals in
communication systems. To the best of our knowledge, this is the first attempt
that introduce complex valued neural network to the design of detector for the
IM-MIMO systems. Finally, to verify the adaptability and robustness of the
proposed detector, simulations are carried out with consideration of
inaccurate channel state information (CSI) and correlated MIMO channels. The
simulation results demonstrate that the proposed detector outperforms existing
algorithms in terms of antenna recognition accuracy and bit error rate under
various scenarios.

###### Index Terms:

Index modulation, multiple-input multiple-output, signal detection, sparse
reconstruction, complex-valued neural network.

##  I Introduction

With the rise of new mobile services, such as virtual reality, augmented
reality and 4K video transmission, wireless air interface is facing tremendous
pressure and challenges [1, 2]. In order to further improve the throughput of
the wireless network, the massive multiple-input multiple-output (MIMO) [3, 4]
as well as millimeter wave (mmWave) technology[5] have become the key
technologies in the fifth generation (5G) wireless communication systems,
which has attracted extensive and enthusiastic research in recent years. It is
undeniable that the massive MIMO does bring significant throughput and
reliability improvements by operating the large-scale transmit antenna (TA)
array at the base station in diversity or multiplexing mode. However, energy
consumption in MIMO and mmWave technology have become a big concern in 5G
wireless communication[5]. As the number of TA grows, the huge power
consumption [6] and expensive hardware cost caused by radio frequency (RF)
chains bound to each TA also appear, which make the actual deployment of
massive MIMO face severe challenges especially when the mmWave
communication[5] is used. Hence, how to balance the performance of wireless
communication systems with the power consumption and hardware cost is still an
issue worthy of attention when the massive MIMO and mmWave communication
technology are practically applied to current wireless communication systems.

As an assistive technology that can reduce the power consumption and hardware
cost, index modulation (IM) [7] has attracted extensive research in recent
years. IM is a general term for a series of innovative digital modulation
technologies, which conveys information bits not only on the modulated symbols
but also on the on-off status of some resource blocks (such as antennas,
subcarriers and time slots) [8, 9, 10]. In this paper, we focus on the IM with
MIMO, i.e., only part of the TAs are seletected to be active for data
transmission based on the part of the information bits. Due to this unique
property, it is obvious that IM can bring the remarkable reduction of power
consumption and hardware cost of transmitter in 5G communication. Although IM
is excellent in power consumption and hardware cost, the significant increase
in receiver complexity caused by it cannot be ignored. In the IM aided MIMO
(IM-MIMO) systems, not only the transmitted symbol but also the activation
status of each TA need to be detected at the receiver, which greatly increases
the complexity compared with the traditional MIMO detectors. More
specifically, the increase in receiver complexity mainly comes from the
detection of antenna activation status. Once the activation status of the TAs
is detected, some linear MIMO detectors, such as least square (LS) and linear
minimum mean square error (LMMSE), can be used to quickly detect the signal.
Therefore, the detection of the activation status of TAs directly determines
the complexity of the entire receiver.

For the detection in the IM-MIMO systems, several algorithms have been
proposed over the past decades. In [11], Yang _et al._ applied the deep
learning technique to the signal detection of SM-MIMO system and problems were
solved by data-driven prediction approaches. Jeganathan _et al._ proposed an
exhaustive search based algorithm [12], which has the optimal performance but
with high complexity. To further reduce the complexity, several algorithms
based on convex optimization [13] or probability model [14] have been
investigated. However, performance, convergence and complexity issues hinder
the practicality of these algorithms. In particular, imperfect channel state
information (CSI) should be considered, for the reason that it is difficult to
do perfect channel estimation when the number of TA becomes larger.

This work addresses the detection of the IM-MIMO system from a different
perspective. Instead of tackling the detection problem analytically, we
leverage the recent advances in deep learning (DL) to design a data-driven
algorithm to achieve a better performance with low complexity. In particular,
the proposed approach establishes a connection between sparse reconstruction
problem and minimizing a loss function in training a convolution neural
network (CNN), rather than directly solve the detection problem of the IM-MIMO
systems.

DL has gotten great success in computer vision, natural language processing
and some other applications. Recent research has shown that DL has powerful
potential in solving difficult wireless communication problems, such as
channel estimation [15], signal detection [16] and power control [17].
However, current DL libraries, such as Tensorflow and PyTorch, only support
real value operations, which does not match the processing of complex signals
in communication systems. A question thus arises: if complex value operations
are adopted to the design of CNN, can we achieve better performance than the
real value based CNN with the same structure?

In this paper, we answer this question by proposing a DL based detector,
called IMRecoNet, which has been embedded complex value operations, to solve
the detection problem of the IM-MIMO systems efficiently and robustly. This
work is a further advancement of our previous work [16] that introduces the
complex value operations into the design of neural networks. The proposed
IMRecoNet has two unique features: one is that it is insensitive to the CSI,
which means it can still maintain excellent performance under imperfect CSI.
And the other is that it supports complex value operations to adapt the
complex signals in wireless communication systems. The main contributions of
this paper are summarized as follows.

  * •

By utilizing the unique attribute of IM, we model the transmission process of
the IM-MIMO system as a compressed sensing (CS) process. Correspondingly, the
detection process at the receiver can be modeled as a sparse reconstruction
problem, i.e., the inverse process of CS. Then, we divide this sparse
reconstruction problem into two sub-problems. One is the support111Set of
indices corresponding to nonzero elements in vector $\mathbf{s}$ is called the
support $\mathbf{\Omega}_{\mathbf{s}}$ of $\mathbf{s}$. If
$\mathbf{s}=\left[0,2,0,0,3\right]$, then
$\mathbf{\Omega}_{\mathbf{s}}=supp_{{\color[rgb]{0,0,1}1}}(\mathbf{s})=\left[1,4\right]$.
For matrix,
$\mathbf{X}=\left[\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{n}\right]$,
its support is
$\mathbf{\Omega}_{\mathbf{X}}=supp_{{\color[rgb]{0,0,1}1}}(\mathbf{x}_{1})\cup
supp_{{\color[rgb]{0,0,1}1}}(\mathbf{x}_{2})\cup\cdots\cup
supp_{{\color[rgb]{0,0,1}1}}(\mathbf{x}_{n})$. identification (i.e., the
detection of active TAs), and the other one is sparse estimation (i.e., the
detection of transmitted symbols on the active TAs).

  * •

To solve the sparse reconstruction problem (i.e., support identification and
sparse estimation) with low complexity and good robustness, we design a DL
based detector, called IMRecoNet, which combines two CNNs with linear MIMO
detector (i.e., LS). IMRecoNet can directly predict the activation probability
of each TA without iteration, which guarantees its low complexity. In
addition, IMRecoNet does not need CSI as its input, which means it can perform
well with inaccurate channel estimation, even without channel estimation. This
feature guarantees its robustness in various scenarios. With the aid of
IMRecoNet, linear MIMO detector can directly detect the signal with low
complexity.

  * •

Considering that signal processing in wireless communication systems is based
on complex numbers, complex-valued operations (i.e., complex convolution and
complex activation) are adopted to the design of IMRecoNet, which can extract
amplitude and phase characteristics of complex signal.

  * •

In order to verify the adaptability and robustness of the proposed IMRecoNet,
we carry out simulations with the consideration of inaccurate channel
estimation at the receiver and the correlated MIMO channel. Simulation results
demonstrate that the proposed IMRecoNet performs as well as traditional
algorithms under perfect CSI, while the performance of the proposed IMRecoNet
is far superior to traditional algorithms under imperfect CSI and correlated
MIMO channel. Apart from this, the IMRecoNet that introduces complex-valued
operations also has shown better performance than the real number network with
the same structure.

The rest of this paper is organized as follows. Related work are discussed in
Section II. The system model and problem formulation are given in Section III.
Section IV details the architecture of the proposed IMRecoNet and the
implementation of adopting complex value operations to the design of
IMRecoNet. The simulation results are provided in Section V. Finally,
conclusions are drawn in Section VI.

_Notation:_ Vectors and matrices are denoted by lowercase and uppercase bold
letters, respectively. $\left(\cdot\right)^{T}$, $\left(\cdot\right)^{H}$, and
$\left(\cdot\right)^{-1}$ stand for transpose, Hermitian transpose and matrix
inversion operations, respectively. $\|\cdot\|_{0}$ denotes the Zero norm,
$\|\cdot\|_{1}$ denotes the Taxicab norm, $\|\cdot\|_{2}$ denotes the
Euclidean norm, while $\|\cdot\|_{F}$ is the Frobenius norm.
$\langle\mathbf{x},\mathbf{y}\rangle$ denotes the inner product between vector
$\mathbf{x}$ and vector $\mathbf{y}$. $\mathcal{CN}(0,\sigma^{2})$ represents
the complex Gaussian distribution with zero mean and $\sigma^{2}$ variance.
$C_{N}^{K}$ and $\lfloor\cdot\rfloor$ denote the binomial coefficient and
floor function, respectively. $supp_{1}(\cdot)$ represents the support of
vectors(i.e. the indices of non-zero element) and $supp_{2}(\cdot)$ represents
the support of matrices(i.e. the detection of transmitted symbols on the
active TAs). $\varnothing$ represents the empty set. $\circledast$ is the
convolution operator. $\Re\left\\{\cdot\right\\}$ and
$\Im\left\\{\cdot\right\\}$ denote the real part and the imaginary part of a
vector or matrix, respectively. $Cov(\cdot)$ represents the covariance
operation.

##  II Related Work

###  II-A Index Modulation Aided Wireless Communication Systems

IM is an assistive technology aimed at reducing the power consumption and
hardware cost, which has been widely applied to existing wireless
communication systems in different ways. Research in [9] and [18] applied IM
to MIMO systems and orthogonal frequency division multiplexing (OFDM) systems,
respectively. To further improve the spectrum efficiency, literature [14]
combined IM with OFDM (OFDM-IM) and apply OFDM-IM to MIMO systems. Besides,
applying IM across multiple domains (such as space, frequency and time
domains) also attracts some attention and research [19][20] including
integrating IM with non-orthogonal multiple access [21][22]. IM reduces power
consumption and hardware cost while significantly increasing the complexity of
the receiver. It is worth noting that the traditional detectors in classical
MIMO systems cannot be directly applied to the IM aided MIMO wireless
communication systems. Since part of the information bits are conveyed on the
indices of the active resource blocks, not only the modulated symbols but also
the indices of the active resource blocks need to be detected. Therefore, how
to design an efficient detector for IM aided wireless communication systems is
still a key issue.

Literature [12] and [23] analyzed the maximum likelihood (ML) based detector,
which has the optimal performance. Although the ML based detector has the
optimal performance, it should not be ignored that its complexity increases
exponentially with increase of the number of TAs and the signal constellation
size. Attentive to the high computational complexity of the ML detection,
researchers proposed other methods for the detection of the IM-MIMO systems.
Mesleh _et al._ proposed the matched filtering detection [24]. This method
decouples the active antenna detection and the modulated symbol detection.
Specifically, the active antennas are first estimated, and then, the modulated
symbols are estimated. Literature [13] and [25] proposed a detection scheme
based on the angle between its corresponding channel vector and the received
signal. In [14], authors investigated two types of detection algorithms based
on the sequential Monte Carlo theory, which achieve near-optimal error
performance while maintaining low computational complexity.

The detection becomes more difficult if imperfect CSI is considered at the
receiver. All the methods mentioned before assume that the receiver has known
perfect CSI. However, perfect channel estimation is difficult and takes up
more system resources. From a more practical perspective, imperfect CSI should
be considered in the design of the detector for IM-MIMO systems.

###  II-B Deep Learning Meets Wireless Communication Systems

In view of the great success of DL in various aspects, researchers began to
apply DL to wireless communication systems in [26]. From physical layer to MAC
layer, many works have appeared.

Channel estimation is a typical case of successful application of DL in
wireless communication systems. Authors in [15] introduced DL to the channel
estimation in OFDM systems. To tackle the challenge that traditional DL method
is not reliable enough when the conditions of online deployment of the common
neural networks are not consistent with the channel models used in the
training stage, they proposed a novel meta learning based channel estimation.
Considering the multiple channels as multiple tasks, a meta-learner which
could learn the general characteristics of these tasks was trained off-line.
When deploying the approach online, the meta-learner will be used to guide the
adjustment of a new network by using a few pilots for fine-tuning. For
exploiting the expert knowledge in wireless communications, Gao _et al._
proposed a model-driven DL approach [27]. This model-driven approach can
overcome the heavy demand of a huge amount of training data and use expert
knowledge to facilitate further performance improvement. Besides, DL has the
potential to improve performance in the areas of wireless communications, such
as channel decoding [28], CSI feedback [29], precoding [30] and end-to-end
communication [31].

Narrow down to the detection of wireless communication systems, authors in
[32] investigated a model-driven DL detector for MIMO systems. In particular,
this DL based detector is designed by unfolding an iterative algorithm and
adding some trainable parameters. Compared with the data-driven DL based
detector, this model-driven DL based detector can be rapidly trained with a
much smaller dataset, while the channel estimation taken into consideration.
Similar to [32], literature [33] also proposed a model-driven DL based
detector by unfolding the iteration of a projected gradient descent algorithm,
which has comparable performance with approximate message passing based
detector and is more robust to ill-conditioned channels. For the IM aided
wireless communication systems, there are some research on the design of DL
based detector. Authors in [34] proposed a CNN based detection framework for
OFDM-IM systems, which achieved near ML detector performance with much lower
complexity. Specifically, the received symbols are transformed to polar
coordinates to help CNN detect the indices of active subcarriers and
2-dimensional convolution layers are utilized to fully exploit the inherent
information in the received symbols. Besides, authors in [35] proposed a deep
learning based detection scheme for the visible light communication system
using generalized spatial modulation, where a deep neural network consisting
of several neural layers is applied to detect the received signals.

There is no doubt that the research of DL in wireless communication system is
currently enthusiastic. However, existing DL libraries, such TensorFlow and
PyTorch, only support real value operations, while signals in wireless
communication systems are often represented as complex values. When training
the network, we have to split the complex number into two parts, i.e., the
real part and the imaginary part. In recent years, research on complex valued
CNN has appeared. Authors in [36] proposed several key components in complex
valued CNN, such as complex convolution, complex activation and complex batch
normalization. The CNN embedded with complex value operation has shown great
potential on audio-related tasks. Besides, literature [37] and [38] applied
complex valued CNN to magnetic resonance imaging domain and speech enhancement
domain, respectively.

Figure 1: The block diagram of the transmitter of the IM-MIMO system, and the
structure of transmitted signal in spatial domain and time domain. For the
structure of transmitted signal, the solid colored squares represent the
transmitted complex signal, while hollow colorless squares represent zero. And
the $i$-th row represents the transmitted signal through the $i$-th TA in time
domain, and the $j$-th column represents the transmitted signal vector for all
TAs at the $j$-th transmit slot of a frame. Thus, the blank rows indicate that
their corresponding TAs are off and do not send anything during a frame.

##  III System Model and Problem Formulation

In this section, the system model of IM-MIMO is first given, where IM is
applied to the spatial domain. Then, the joint sparsity of the transmitted
signal in IM-MIMO systems, which is introduced by IM, is analyzed. And the
detection process at the receiver is formulated as a sparsity reconstruction
problem.

###  III-A System Model

We consider an IM-MIMO system that equipped with $N_{t}$ transmit antennas
(TAs), of which block diagram is depicted in Fig.1. Different from the
traditional MIMO systems, only $N_{u}\ (0<N_{u}<N_{t})$ TAs are active for
data transmission in IM-MIMO systems, and $N_{u}$ active TAs are selected by
part of the information bits. Obviously, there exist $C_{N_{t}}^{N_{u}}$ TA
combinations (TAC) for selecting $N_{u}$ TAs from $N_{t}$ TAs. In order to
facilitate data mapping, only
$N_{L}=2^{\lfloor\log_{2}(C_{N_{t}}^{N_{u}})\rfloor}$ TACs are legal and
generated according to [7] based on $N_{t}$ and $N_{u}$ in advance. We denote
the $N_{L}$ legal TACs as
$\mathbb{J}=\left\\{J_{1},J_{2},\cdots,J_{N_{L}}\right\\}$.

As shown in Fig.1, each IM-MIMO frame is comprised of a total number of
$b=b_{1}+b_{2}$ bits. The first $b_{1}=\log_{2}(N_{L})$ bits are called
spatial bits and used to select a TAC
$J=\left\\{j_{1},j_{2},\cdots,j_{N_{u}}\right\\}$ from the TAC set
$\mathbb{J}$, where
$j_{1},j_{2},\cdots,j_{N_{u}}\in\left\\{1,2,\cdots,N_{t}\right\\}$ are the
index of active TAs. Baesed on TAC $J$, antenna activation pattern (AAP)
$\mathbf{g}\triangleq\left[g_{1},g_{2},\cdots,g_{N_{t}}\right]$ is defined as

$g_{\color[rgb]{0,0,1}k}=\begin{cases}0,&{\color[rgb]{0,0,1}k}\notin J\\\
1,&{\color[rgb]{0,0,1}k}\in J\end{cases}$ (1)

which $``1"$ means the corresponding TA is active and $``0"$ means the
corresponding TA is inactive. For example, when $N_{t}=4$ and
$N_{u}={\color[rgb]{0,0,1}2}$, the mapping relationship between the input
binary bits, the TAC and the AAP is shown in Table. I.

TABLE I: The mapping relationship between the input binary bits, the TAC and the AAP with $N_{t}=4,N_{u}={\color[rgb]{0,0,1}2}$. $\mathbf{b}=[b_{0},b_{1}]$ | TAC Index | TAC | AAP
---|---|---|---
$[0\ 0]$ | 0 | $\left\\{1,3\right\\}$ | $[1,0,1,0]$
$[0\ 1]$ | 1 | $\left\\{1,4\right\\}$ | $[1,0,0,1]$
$[1\ 0]$ | 2 | $\left\\{2,4\right\\}$ | $[0,1,0,1]$
$[1\ 1]$ | 3 | $\left\\{2,3\right\\}$ | $[0,1,1,0]$

Since only $N_{u}$ TAs are activated for actual information transmission, the
IM-MIMO system only needs to be configured with $N_{u}$ signal processing
links. After selecting the TAC, the left $b_{2}=N_{u}d$ bits are equally split
to $N_{u}$ blocks and processed on $N_{u}$ signal links, which means each link
processes $d$ bits. After quadrature amplitude modulation (QAM) with
constellation $\mathcal{S}\left(|\mathcal{S}|=M\right)$ and OFDM, the signals
on $N_{u}$ links are transmitted through $N_{u}$ active TAs, respectively. We
assume that an IM-MIMO frame is transmitted through $T$ time slots. After
passing through the MIMO channel, the received signal vector at the $j$-th
$\left(j\in\left[1,T\right]\right)$ time slot of the $i$-th IM-MIMO frame can
be represented as

$\mathbf{y}_{j,i}=\mathbf{H}_{j,i}\left(J_{i}\right)\mathbf{s}_{j,i}+\mathbf{n}_{j,i},$
(2)

where
$\mathbf{H}_{j,i}=\left[\mathbf{h}_{1},\mathbf{h}_{2},\cdots,\mathbf{h}_{N_{t}}\right]\in\mathbb{C}^{N_{r}\times
N_{t}}$ and $\mathbf{s}_{j,i}\in\mathcal{S}^{N_{u}\times 1}$ denote the MIMO
channel matrix and actually transmitted signal vector at the $j$-th time slot
of the $i$-th IM-MIMO frame, repectively.
$\mathbf{n}_{j,i}\in\mathbb{C}^{N_{r}\times 1}$ represents the additive white
Gaussian noise (AWGN) vector with zero mean and unit variance at the $j$-th
time slot in the $i$-th IM-MIMO frame. Particularly, $J_{i}$ is the selected
TAC for the $i$-th IM-MIMO frame, and
$\mathbf{H}_{j,i}\left(J_{i}\right)=[\mathbf{h}_{j_{1}},\mathbf{h}_{j_{2}},\cdots,\mathbf{h}_{j_{N_{u}}}]$
consists of the columns of $\mathbf{H}_{j,i}$ indexed by $J_{i}$.

Since the information bits are conveyed on the TAC $J_{i}$ and actually
transmitted signal vector $\mathbf{s}_{j,i}$, both $J_{i}$ and
$\mathbf{s}_{j,i}$ should be detected at the receiver. Based on Eq.(2), the
maximum likelihood detector (MLD) [12] can be formulated as

$<\hat{J}_{i},\mathbf{\hat{s}}_{j,i}>=\mathop{\arg\min}_{J_{i}\in\mathbb{J},\mathbf{s}_{j,i}\in\mathcal{S}^{N_{u}}}{\|\mathbf{y}_{j,i}-\mathbf{H}_{j,i}\left(J_{i}\right)\mathbf{s}_{j,i}\|}_{2}^{2}.$
(3)

It should be noted that the MLD jointly detects $J_{i}$ and
$\mathbf{s}_{j,i}$, which entails an exhaustive search space. Therefore, there
will be a prohibitive complexity with large $N_{t}$, $N_{u}$ and $M$.

###  III-B Problem Formulation

Let define the following mapping relationship

$\mathbf{s}_{j,i}\in\mathbb{C}^{N_{u}\times
1}\xrightarrow{J_{i}}\mathbf{x}_{j,i}\in\mathbb{C}^{N_{t}\times 1},$ (4)

which means that inserting $N_{t}-N_{u}$ zeros to $\mathbf{s}_{j,i}$ according
to the TAC $J_{i}$. By doing this mapping, all the TAs transmit information
according to the following rules: the $N_{u}$ active TAs transmit the
modulated symbols generated by $N_{u}$ signal processing links, while
$N_{t}-N_{u}$ inactive TAs transmit zeros.

Based on the mapping relationship expressed as Eq.(4), the structure of
transmitted signal in an IM-MIMO frame can be depicted as the right of Fig.1.
Due to the fact that there are only $N_{u}$ TAs are activated during the data
transmission of an IM-MIMO frame, it is clear that there are only $N_{u}$ non-
zero element in each transmitted signal vector (i.e.,
$\mathbf{x}_{j,i},j\in\left[t,t+T-1\right]$), which has been shown in Fig.1.
When $N_{u}$ is far less than $N_{t}$ (i.e., $N_{u}\ll N_{t}$),
$\mathbf{x}_{j,i}$ can be considered as a sparse vector and the sparsity is
$N_{u}$. For each IM-MIMO frame, the TAC is only selected once, which means
the position of $N_{u}$ non-zero elements in each transmitted signal vector of
an IM-MIMO frame are same. Based on the mapping relationship in Eq.(4) and the
structure of the transmitter shown in Fig.1, we have

$supp_{{\color[rgb]{0,0,1}1}}(\mathbf{x}_{j,i})=supp_{{\color[rgb]{0,0,1}1}}(\mathbf{g}_{i})=J_{i},\forall
j\in\left[1,T\right],$ (5)
${\|supp_{{\color[rgb]{0,0,1}1}}(\mathbf{x}_{j,i})\|}_{0}={\|\mathbf{g}_{i}\|}_{0}=N_{u},\forall
j\in\left[1,T\right],$ (6)

where $supp_{{\color[rgb]{0,0,1}1}}(\cdot)$ denotes the support (i.e., the
indices of non-zero element) of a vector. Eq.(5) means all the transmitted
signal vectors in the same IM-MIMO frame share the same support, and Eq.(6)
means the sparsity of these vectors are all $N_{u}$. These two properties
determine that the all transmitted signal vectors in the same IM-MIMO frame
are jointly sparse and each transmitted signal matrix of an IM-MIMO frame
$\mathbf{X}_{i}=\left[\mathbf{x}_{1,i},\mathbf{x}_{2,i},\cdots,\mathbf{x}_{T,i}\right]\in\mathbb{C}^{N_{t}\times
T}$ has the structural sparsity.

Assuming that the period to send an IM-MIMO frame is less than the coherence
time of the channel, the channel impulse response does not change for
transmitting all signal vectors in an IM-MIMO frame. Therefore, we have

$\mathbf{H}_{1,i}=\mathbf{H}_{2,i}=\cdots=\mathbf{H}_{T,i}=\mathbf{H}_{i},$
(7)

where $\mathbf{H}_{i}\in\mathbb{C}^{N_{r}\times N_{t}}$ is the channel impulse
response during the transmission of the $i$-th IM-MIMO frame. Based on Eq.(4)
and Eq.(7), Eq.(2) can be rewritten as

$\mathbf{Y}_{i}=\mathbf{H}_{i}\mathbf{X}_{i}+\mathbf{N}_{i},$ (8)

where
$\mathbf{Y}_{i}=\left[\mathbf{y}_{1,i},\mathbf{y}_{2,i},\cdots,\mathbf{y}_{T,i}\right]\in\mathbb{C}^{N_{r}\times
T}$ represents the $i$-th received IM-MIMO frame and
$\mathbf{N}_{i}=\left[\mathbf{n}_{1,i},\mathbf{n}_{2,i},\cdots,\mathbf{n}_{T,i}\right]\in\mathbb{C}^{N_{r}\times
T}$ represents the AWGN for the $i$-th IM-MIMO-OFDM frame. Since the
transmitted signal vectors in an IM-MIMO frame are jointly sparse and
$\mathbf{X}_{i}$ has the property of structural sparsity, the transmission
process expressed as Eq.(8) can be considered as a compress sensing process.
Accordingly, the detection process at the receiver to recover the transmitted
signal can be formulated as a joint sparse reconstruction model (JSRM). In
JSRM, $\mathbf{X}_{i}$ is the transmitted signal matrix composed of the
jointly sparse signal vectors that needs to be recovered, $\mathbf{H}_{i}$ is
the sensing matrix and $\mathbf{Y}_{i}$ is the received signal matrix that
contains the measurement vectors. Based on the structure sparsity of
$\mathbf{X}_{i}$, and each signal vector in $\mathbf{X}_{i}$ shares the same
sensing matrix (i.e., $\mathbf{H}_{i}$), $\mathbf{X}_{i}$ can be jointly
recovered through $\mathbf{Y}_{i}$, which can be formulated as a optimization
problem with the consideration of AWGN.

$\displaystyle\hat{\mathbf{X}}_{i}$
$\displaystyle=\mathop{\arg\min}_{\mathbf{X}_{i}}\ {\parallel
supp_{{\color[rgb]{0,0,1}2}}(\mathbf{X}_{i})\parallel}_{1}$ (9) $\displaystyle
s.t.\ $ $\displaystyle Eq.(\ref{formula-supp1}),Eq.(\ref{formula-supp2})$ (9a)
$\displaystyle\frac{1}{T}\sum_{j=1}^{T}{\|\mathbf{y}_{j,i}-\mathbf{H}_{i}{\color[rgb]{0,0,1}\mathbf{x}}_{j,i}\|}_{2}^{2}<\epsilon$
(9b)

where
$supp_{{\color[rgb]{0,0,1}2}}(\mathbf{X}_{i})=supp_{{\color[rgb]{0,0,1}1}}(\mathbf{x}_{1,i})\cup
supp_{{\color[rgb]{0,0,1}1}}(\mathbf{x}_{2,i})\cup\cdots\cup
supp_{{\color[rgb]{0,0,1}1}}(\mathbf{x}_{T,i})$ and $\epsilon$ is a
predetermined noise level of the system.

There exist many methods to realize the sparse reconstruction, such as greedy
based methods like orthogonal matching pursuit (OMP) [39] and simultaneous
orthogonal matching pursuit (SOMP) [40], relaxed mixed norm minimization
methods and Bayesian methods. However, these methods need the prior knowledge
of sensing matrix, which is the channel matrix (i.e., $\mathbf{H}_{i}$) in our
system model. Considering the fact that perfect channel estimation leads to
greater overhead and may be unrealistic in actual wireless communication
systems, we introduce DL to realize the signal detection with the
consideration of imperfect channel estimation, which will be detailed in the
next section.

##  IV Deep Learning Based Detection Framework

To be more practical, the impact brought by imperfect channel estimation and
the correlation between antennas should be considered at the receiver. Aiming
at solving the joint sparse reconstruction problem formulated in Section III-B
with the consideration of imperfect channel estimation and the correlation
between antennas, a DL base detection framework, called IMRecoNet, is designed
in this section. Besides, for adapting to the signal represented by complex
values in wireless communication systems, complex-valued operations are
introduced to the design of IMRecoNet.

###  IV-A The architecture of IMRecoNet

According to the system model in Section III-A, IM-MIMO systems transmit
information through TAC $J_{i}$ and actually transmitted signal vector
$\mathbf{s}_{j,i}$. Therefore, we can decompose the joint sparse
reconstruction problem formulated in Section III-B into two steps. The first
step is to estimate the position of non-zero elements (i.e., the TAC $J_{i}$),
and the second step is to estimate the specific value of the non-zero elements
(i.e., the actually transmitted signal matrix
$\mathbf{S}_{i}=\left[\mathbf{s}_{1,i},\mathbf{s}_{2,i},\cdots,\mathbf{s}_{T,i}\right]\in\mathbb{C}^{N_{u}\times
T}$). Based on $J_{i}$ and $\mathbf{S}_{i}$, the transmitted signal matrix
$\mathbf{X}_{i}$ for the $i$-th IM-MIMO frame can be reconstructed easily. We
design a DL based signal detection framework, called IMRecoNet, to realize the
two steps mentioned before, whose architecture is shown in Fig.2. IMRecoNet
consists of two CNNs and a traditional zero-forcing (ZF) detector. At the
first stage, the AAP detection (AAPD) subnet is utilized to detect the
activation status of each TA. At the second stage, ZF detector combined with
signal enhancement (SE) subnet realize to estimate the actually transmitted
signal based on the activation information of TAs predicted by AAPD subnet. It
is worth noting that both AAPD subnet and SE subnet are running in an end-to-
end manner, which means they do not need CSI at the training and detection
period.

Figure 2: The architecture of proposed IMRecoNet.

In fact, the signals transmitted by different TAs pass through different
wireless channels to reach the receiver. Therefore, the “fingerprint”
information of the wireless channel is hidden in the received signal, and it
also contains the activation state information of each TA. Based on this
feature hidden in the received signal, the data-driven AAPD subnet can mine
this internal feature well and predict the activation probability of all TAs
at once. The architecture of AAPD subnet is shown in Fig.3(a).

(a) AAPD

(b) SE

Figure 3: The architecture of AAPD and SE subnet.

As shown in Fig.3(a), the input of AAPD subnet is the received signal matrix
for an IM-MIMO frame and then two convolutional layers are used for feature
extraction. In order to speed up the convergence of the network, two
convolutional layers use ReLU as the activation function. The prediction of
the activation state of each TA is actually a two-class problem (i.e., active
or inactive). Therefore, three full-connection layers are used to map features
extracted by the convolution layers, and Sigmoid function is adopted as the
activation function of the output layer to give the activation probability of
each TA. Since AAPD subnet solves a classification problem, we use binary
cross-entropy function to train it, which is expressed as

$\displaystyle L_{AAPD}(\bm{\theta}_{1})$
$\displaystyle=\frac{1}{N_{t}}\sum_{m=1}^{N_{t}}\left[\mathbf{g}_{i}(m)\log(\mathbf{p}_{i}(m))\right]$
(10)
$\displaystyle+\sum_{m=1}^{N_{t}}\left[(1-\mathbf{g}_{i}(m))\log(1-\mathbf{p}_{i}(m))\right],$

where $\bm{\theta}_{1}$ denotes the parameters of AAPD subnet,
$\mathbf{g}_{i}$ is the AAP for the $i$-th IM-MIMO frame defined in Eq.(1),
and $\mathbf{p}_{i}=\left[p_{1,i},p_{2,i},\cdots,p_{N_{t},i}\right]$
represents the activation probability vector of TAs for the $i$-th IM-MIMO
frame. The entire workflow of AAPD subnet can be expressed as

$\displaystyle\mathbf{p}_{i}=f_{AAPD}(\mathbf{Y}_{i};\bm{\theta}_{1}),$ (11)
$\displaystyle\hat{J}_{i}=\left\\{n\ having\ the\ top\ N_{u}\ largest\
\mathbf{p}_{i}(n)\right\\},$ (12)

where $f_{AAPD}(\cdot)$ represents the function of AAPD subnet, $\hat{J}_{i}$
is the estimated TAC for the $i$-th IM-MIMO frame and
$n\in\left\\{1,2,\cdots,N_{t}\right\\}$ is the index of TAs.

Since the inactive TAs actually do not make any contribution to the received
signal, when $J_{i}$ is obtained through the AAPD subnet, Eq.(8) can be
rewritten as

$\mathbf{Y}_{i}=\mathbf{H}_{i}(\hat{J}_{i})\mathbf{S}_{i}+\mathbf{N}_{i},$
(13)

where $\mathbf{H}_{i}(\hat{J}_{i})\in\mathbb{C}^{N_{r}\times N_{u}}$ consists
of the columns of $\mathbf{H}_{i}$ according to the elements of $J_{i}$, and
$\mathbf{S}_{i}=\left[\mathbf{s}_{1,i},\mathbf{s}_{2,i},\cdots,\mathbf{s}_{T,i}\right]\in\mathbb{C}^{N_{u}\times
T}$ is the actually transmitted signal matrix. For Eq.(13), we can obtain the
initial estimation through ZF detector, which can be expressed as

$\mathbf{\hat{S}}_{i}^{ZF}=\mathbf{W}_{i}^{ZF}\mathbf{Y}_{i}={\left[{\mathbf{H}_{i}(\hat{\mathbf{J}}_{i})}^{H}{\mathbf{H}_{i}(\hat{\mathbf{J}}_{i})}\right]}^{-1}{\mathbf{H}_{i}(\hat{\mathbf{J}}_{i})}^{H}{\mathbf{Y}_{i}},$
(14)

where
$\mathbf{W}_{i}^{ZF}={\left[{\mathbf{H}_{i}(\hat{\mathbf{J}}_{i})}^{H}{\mathbf{H}_{i}(\hat{\mathbf{J}}_{i})}\right]}^{-1}{\mathbf{H}_{i}(\hat{\mathbf{J}}_{i})}^{H}$
is the interference cancellation matrix. Considering the noise enhancement
introduced by ZF detector and channel estimation error, we further introduce
SE subnet in the second stage to enhance $\mathbf{\hat{S}}_{i}^{ZF}$ estimated
by ZF detector through Eq.(14), whose architecture is shown in Fig.3(b).

As shown in Fig.3(b), SE subnet use residual learning [41] to enhance
$\mathbf{\hat{S}}_{i}^{ZF}$. SE subnet is composed of input layer, feature
extraction layer and output layer. The input of the SE subnet is the rough
estimation result of the ZF detector, and the output is the final detection
result after enhancement, which is essentially a fitting problem. Therefore,
the SE subnet uses the mean square error (MSE) as its loss function:

$L_{SE}(\bm{\theta}_{2})=\frac{1}{N}\sum_{i=1}^{N}\|f_{SE}(\hat{S}_{i}^{ZF};\bm{\theta}_{2})-\mathbf{S}_{i}\|_{2}^{2},$
(15)

where $\bm{\theta}_{2}$ denotes the parameter set of SE subnet, and
$f_{SE}(\cdot)$ represents the function of SE subnet.

###  IV-B Complex-valued IMRecoNet

The mainstream DL libraries (such as Tensorflow [42] and PyTorch [43]) are
designed for processing real-valued data, which is inconsistent with the
processing of complex-valued signals in wireless communication systems. In
order to improve the ability of IMRecoNet to process complex-valued signals,
we adopt complex-valued operations into AAPD subnet and SE subnet.

_1) Complex-Valued Differentiation:_ When training neural networks, the
backpropagation algorithm needs to calculate gradient to update weight
parameters. However, the introduction of complex values limit the gradient
calculation of the activation function and loss function during
backpropagation. Wirtinger calculus [44] provides an alternative method for
calculating the differentiation of a complex function, which is defined as

$\small\frac{\partial f}{\partial z}\triangleq\frac{1}{2}(\frac{\partial
f}{\partial x}-j\frac{\partial f}{\partial y}),$ (16)

$\frac{\partial f}{\partial\bar{z}}\triangleq\frac{1}{2}(\frac{\partial
f}{\partial x}+j\frac{\partial f}{\partial y}).$ (17)

Eq.(16) and Eq.(17) are called as $\mathbb{R}$-derivative and conjugate
$\mathbb{R}$-derivative, respectively. Based on Eq.(16) and Eq.(17),
researchers in [45] and [46] define the complex gradient as

$\nabla_{z}f=2\frac{\partial f}{\partial\bar{z}}.$ (18)

Although Tensorflow does not directly support the implementation of complex
values, its automatic differentiation algorithm supports the use of complex
values [47]. For $f:\mathbb{C}\rightarrow\mathbb{C}$, Tensorflow calculates
the complex gradient as

$\nabla_{z}f=\overline{\frac{\partial f}{\partial
z}+\frac{\partial\bar{f}}{\partial z}}.$ (19)

It is straightforward to prove that Eq.(18) and Eq.(19) are equivalent based
on the characteristics in [44].

For any real-valued loss function
$\mathcal{L}:\mathbb{C}\rightarrow\mathbb{R}$ and any complex-valued function
$g:\mathbb{C}\rightarrow\mathbb{C}$, the complex gradient can be obtained
through chain rule as

$\frac{\partial\mathcal{L}\circ
g}{\partial\bar{z}}=\frac{\partial\mathcal{L}}{\partial g}\frac{\partial
g}{\partial\bar{z}}+\frac{\partial\mathcal{L}}{\partial\bar{g}}\frac{\bar{g}}{\partial\bar{z}}.$
(20)

Eq.(18) and Eq.(20) make it possible to train complex-valued neural networks.

_2) Complex-Valued Convolution Layer:_ Assuming that the $l$-th complex-valued
convolution layer outputs $n$ complex feature maps (denoted as
$\mathbf{M}^{l}_{1\sim n}$), we can expand the $n$ complex feature maps to
$2n$ real feature maps as: $\mathbf{\widetilde{M}}^{l}_{1\sim
n}=\Re\left\\{\mathbf{M}^{l}_{1\sim n}\right\\}$ and
$\mathbf{\widetilde{M}}^{l}_{n+1\sim 2n}=\Im\left\\{\mathbf{M}^{l}_{1\sim
n}\right\\}$. Expanding the $m$ complex convolution kernels
$\mathbf{W}^{l+1}_{1\sim m}$ of the $(l+1)$-th convolution layer into $2m$
real convolution layers (i.e., ${\mathbf{\widetilde{W}}}^{l}_{1\sim m}$ and
${\mathbf{\widetilde{W}}}^{l}_{m+1\sim 2m}$ ) in the same way, the complex
valued convolution layer can be defined as

$\displaystyle\mathbf{\widetilde{M}}^{l+1}_{1\sim m}$
$\displaystyle=\Re\left\\{\mathbf{W}^{l+1}_{1\sim
m}\circledast\mathbf{M}^{l}_{1\sim n}\right\\}$ (21)
$\displaystyle=\mathbf{\widetilde{W}}^{l+1}_{1\sim
m}\mathbf{\widetilde{M}}^{l}_{1\sim n}-\mathbf{\widetilde{W}}^{l+1}_{m+1\sim
2m}\mathbf{\widetilde{M}}^{l}_{n+1\sim 2n},$

$\displaystyle\mathbf{\widetilde{M}}^{l+1}_{m+1\sim 2m}$
$\displaystyle=\Im\left\\{\mathbf{W}^{l+1}_{1\sim
m}\circledast\mathbf{M}^{l}_{1\sim n}\right\\}$ (22)
$\displaystyle=\mathbf{\widetilde{W}}^{l+1}_{m+1\sim
2m}\mathbf{\widetilde{M}}^{l}_{1\sim n}+\mathbf{\widetilde{W}}^{l+1}_{1\sim
m}\mathbf{\widetilde{M}}^{l}_{n+1\sim 2n},$

where $\circledast$ represents convolution operation. Combining
$\mathbf{\widetilde{M}}^{l+1}_{1\sim m}$ and
$\mathbf{\widetilde{M}}^{l+1}_{m+1\sim 2m}$ can get the final output of the
$(l+1)$-th complex convolution layer (i.e., $\mathbf{{M}}^{l+1}_{1\sim m}$).

_3) Complex-Valued Activation Function:_ For the complex activation function,
we extend ReLU and Sigmoid function through following expression to make them
support complex values.

$\displaystyle f_{1}(z)$
$\displaystyle=Sigmoid(\Re\left\\{z\right\\})+jSigmoid(\Im\left\\{z\right\\}),$
(23) $\displaystyle f_{2}(z)$
$\displaystyle=ReLU(\Re\left\\{z\right\\})+jReLU(\Im\left\\{z\right\\}).$ (24)

###  IV-C Batch Normalization of Complex Value

In order to avoid the potential over-fitting hazards, we use the batch
normalization for the complex number to accelerate the convergence speed of
the neural network, and prevent the gradient disappearance as well as control
explosion. Different from the real number, complex number has real and
imaginary parts. If the complex numbers are only scaled and shifted to make
the mean to $0$ and the variance to $1$ as real numbers, there is no guarantee
that real part and imaginary part have the same variance, and the distribution
of the normalized value may be offset. For normalized complex domain, it can
be regarded as a two-dimensional matrix whiten problem, by using two main
components of the square root of the variance of each component to scale the
value. This can be done by a covariance matrix, calculated as follows:

$\displaystyle\widetilde{\mathbf{x}}=(\mathbf{V})^{-\frac{1}{2}}(\mathbf{x}-\mathbb{E}\\{\mathbf{x}\\})$
(25)

where $\mathbf{V}$ is the covariance matrix with the definition

$\displaystyle\mathbf{V}$
$\displaystyle=\left(\begin{array}[]{cc}V_{rr}&V_{ri}\\\ V_{ir}&V_{ii}\\\
\end{array}\right)=\left(\begin{array}[]{cc}Cov(\Re(\mathbf{x}),\Re(\mathbf{x}))&Cov(\Re(\mathbf{x}),\Im(\mathbf{x}))\\\
Cov(\Im(\mathbf{x}),\Re(\mathbf{x}))&Cov(\Im(\mathbf{x}),\Im(\mathbf{x}))\\\
\end{array}\right)$ (29)

The above normalization procedure eliminates the relevance of the imaginary
part and real part in complex number and avoids the real part and imaginary
part in adjusting coupling, which reduces over-fitting risk in the complex
neural network[48]. Similar to the batch normalization of real number, the
complex batch normalization also uses two parameters $\gamma$ and $\beta$ to
scale and translate the normalized complex number, which is shown in the
following type:

$\displaystyle\mathbb{C}BN=\mathbf{\gamma}\mathbf{\widetilde{x}}+\mathbb{\beta}$
(30)

$\gamma$ is a matrix with four parameters, which is given as

$\gamma=\left(\begin{array}[]{cc}\gamma_{rr}&\gamma_{ri}\\\
\gamma_{ir}&\gamma_{ii}\\\ \end{array}\right)$

$\beta$ is a complex vector with the same dimension as
$\mathbf{\widetilde{x}}$. In order to make the output of complex batch
normalization obey the standard Gaussian distribution, both $\gamma_{rr}$ and
$\gamma_{ir}$ need to be initialized to $\frac{1}{\sqrt{2}}$. Besides, the
real part and the imaginary part of the complex offset $\beta$ need to be
initialized to 0.

###  IV-D Training and Deployment IMRecoNet

Since IMRecoNet contains two independent CNNs, we use a step-by-step training
method to train IMRecoNet. Based on the generated training dataset, we first
train the AAPD subnet. Based on the trained AAPD subnet, we generate the rough
estimation result $\mathbf{S}^{ZF}_{i}$ of the transmitted signal matrix
$\mathbf{S}_{i}$ according to Eq.(14). Finally, we use the rough estimation
result of the transmitted signal matrix $\mathbf{S}^{ZF}_{i}$ and the actual
transmitted signal matrix $\mathbf{S}_{i}$ to train the SE subnet. The entire
training process is shown in Algorithm 1.

1 Input: $\left\\{\mathbf{g}_{i}\right\\}_{i=1}^{N}$: AAP set;
$\left\\{\mathbf{S}_{i}\right\\}_{i=1}^{N}$: actually transmitted signal
matrix; $\left\\{\mathbf{Y}_{i}\right\\}_{i=1}^{N}$: received signal matrix;
$\left\\{\mathbf{H}_{i}\right\\}_{i=1}^{N}$: channel matrix;
$\gamma_{1},\gamma_{2}$: error threshold

2 Output: $\bm{\Theta}=\left\\{\bm{\theta}_{1},\bm{\theta}_{2}\right\\}$:
parameter set of IMRecoNet;

3

4initialize parameters $\bm{\theta}_{1}$;

5 while  _$error\geqslant\gamma_{1}$_ do

6 use dataset $\left\\{\mathbf{Y}_{i},\mathbf{g}_{i}\right\\}_{i=1}^{N}$ to
train AAPD subnet;

7 minimize Eq.(10), update paremeters $\bm{\theta}_{1}$;

8

9 end while

10get $\mathbf{p}_{i}$ through Eq.(11);

11 get $\hat{J}_{i}$ through Eq.(12);

12 get $\mathbf{S}^{ZF}_{i}$ through Eq.(14);

13 initialize paremeters $\bm{\theta}_{2}$;

14 while  _$error\geqslant\gamma_{2}$_ do

15 use dataset
$\left\\{\mathbf{S}^{ZF}_{i},\mathbf{S}_{i}\right\\}_{i=1}^{i=N}$ to train SE
subnet;

16 minimize Eq.(15), update parameters $\theta_{2}$;

17

18 end while

19Return: $\bm{\Theta}=\left\\{\bm{\theta}_{1},\bm{\theta}_{2}\right\\}$;

Algorithm 1 Training IMRecoNet

When the optimal parameters set
$\bm{\Theta}=\left\\{\bm{\theta}_{1},\bm{\theta}_{2}\right\\}$ is obtained,
the signal detection process of IMRecoNet can be expressed as

$\displaystyle<\hat{\mathbf{J}}_{i},\hat{\mathbf{S}}_{i}>$
$\displaystyle=f_{IMRecoNet}(\mathbf{Y}_{i};\bm{\Theta})$ (31)
$\displaystyle=f_{SE}(f_{ZF}(f_{AAPD}(\mathbf{Y}_{i};\bm{\theta}_{1}),\mathbf{H}_{i},\mathbf{Y}_{i});\bm{\theta}_{2})$

where $f_{IMRecoNet}(\cdot)$ and $f_{SE}(\cdot)$ are the function represented
by the IMRecoNet and ZF detector, respectively. First, the AAPD subnet
predicts the activation probability of each TA based on the received signal
matrix $\mathbf{Y}_{i}$. The antenna activation vector $\mathbf{p}_{i}$
predicted by the AAPD subnet will be mapped to the TAC $\hat{J}_{i}$. After
that, according to Eq.(14), using the the received signal matrix
$\mathbf{Y}_{i}$, the channel matrix $\hat{\mathbf{H}}_{i}$ obtained by the
receiving end and the TAC $\hat{J}_{i}$ predicted in the previous step, the
rough estimation result of the ZF detector $\mathbf{S}^{ZF}_{i}$ can be
calculated. Finally, the SE subnet is used to enhance the rough estimation
result of the ZF detector to obtain the final signal detection result
$\hat{\mathbf{S}}_{i}$. The signal detection process of IMRecoNet is shown in
Algorithm 2.

1 Input: $\mathbf{Y}_{i}$: the received signal matrix; $\mathbf{H}_{i}$:
channel matrix;

2 Output: $\hat{J}_{i}$: the predicted TAC; $\hat{\mathbf{S}}_{i}$: the
actually transmitted signal matrix;

3

4load IMRecoNet according to
$\bm{\Theta}=\left\\{\bm{\theta}_{1},\bm{\theta}_{2}\right\\}$;

5 while  _receive $\mathbf{Y}_{i}$_ do

6 predict activation probability of each TA:
$\mathbf{p}_{i}=f_{AAPD}(\mathbf{Y}_{i};\bm{\theta}_{1})$;

7 map $\mathbf{p}_{i}$ to $\hat{J}_{i}$
$\mathbf{p}_{i}\rightarrow\hat{J}_{i}$;

8 calculate $\hat{\mathbf{S}}_{i}^{ZF}$ according to Eq.(14);

9 enhance $\hat{\mathbf{S}}_{i}^{ZF}$:
$\hat{\mathbf{S}}_{i}=f_{SE}(\hat{\mathbf{S}}_{i}^{ZF};\bm{\theta}_{2})$;

10

11 end while

12Return: $\hat{J}_{i},\hat{\mathbf{S}}_{i}$ ;

Algorithm 2 Signal Detection Process of IMRecoNet

###  IV-E Complexity Analysis

The number of parameters of a deep learning model is one of the important
indicators to measure the complexity of the model, and it also affects the
convergence speed of the model. For IMRecoNet, whether using real-valued or
complex-valued operations, the main modules are convolution layer, full-
connection layer and batch normalization layer. Among them, the batch
normalization layer has fewer parameters and accounts for a relatively small
proportion of the weight parameters of the entire IMRecoNet. Therefore, the
number of weight parameters of IMRecoNet mainly comes from the convolution
layer and the full-connection layer. The theoretical calculation results of
the number of weight parameters of the convolution layer and the full-
connection layer are shown in Table II and Table III, respectively.

For a real-valued convolution layer with $N_{in}$ feature maps and $N_{out}$
feature maps, the kernel size is $k\times k\times N_{in}$, therefore, one
convolution layer contains $k\times k\times N_{in}\times N_{out}$ parameters.
When using complex-valued convolution, every two real-valued feature maps
correspond to a complex-valued feature map, i.e., $\frac{N_{in}}{2}$
($\frac{N_{out}}{2}$) complex-valued feature maps consist of
$\frac{N_{in}}{2}$ ($\frac{N_{out}}{2}$) real parts and $\frac{N_{in}}{2}$
($\frac{N_{out}}{2}$) imaginary parts. Complex-valued convolution kernels also
have real parts and imaginary parts. According to the rules of complex-valued
convolution operation expressed in Eq.(21) and Eq.(22), it can be known that
the dimensions of the real and imaginary convolution kernels are both $k\times
k\times\frac{N_{in}}{2}$. Since the real part feature maps and the imaginary
part feature maps share the same real part convolution kernels and the
imaginary part convolution kernels, according to Eq.(21) and Eq.(22), there
exist $k\times k\times\frac{N_{in}}{2}\times\frac{N_{out}}{2}$ real part
parameters and $k\times k\times\frac{N_{in}}{2}\times\frac{N_{out}}{2}$
imaginary part parameters, i.e., a total of $\frac{k\times k\times
N_{in}\times N_{out}}{2}$ parameters, for connecting $\frac{N_{in}}{2}$ input
complex-valued feature maps to $\frac{N_{out}}{2}$ output complex complex-
valued feature maps. Compared with real-valued convolution, complex-valued
convolution cut the number of parameters by half, greatly reduces the total
number of IMRecoNet’s parameters.

TABLE II: The number of parameters of real-valued and complex-valued convolution layer. | Input Channels | Output Channels | Kernel Size |  Parameters(Complexity)
---|---|---|---|---
Real-Valued | $N_{in}$ | $N_{out}$ | $k\times k$ | $k\times k\times N_{in}\times N_{out}$
Complex-Valued | $N_{in}/2$ real parts | $N_{out}/2$ real parts | $k\times k$ | $\left(k\times k\times N_{in}\times N_{out}\right)\ /\ 2$
$N_{in}/2$ imaginary parts | $N_{out}/2$ imaginary parts
TABLE III: The number of parameters of real-valued and complex-valued full-connection layer. | Input Dimension | Output Dimension |  Parameters(Complexity)
---|---|---|---
Real-Valued | $N_{in}$ | $N_{out}$ | $N_{in}\times N_{out}$
Complex-Valued | $N_{in}/2$ real parts | $N_{out}/2$ real parts | $\left(N_{in}\times N_{out}\right)\ /\ 2$
$N_{in}/2$ imaginary parts | $N_{out}/2$ imaginary parts

For the real-valued full-connection layer with input dimension $N_{in}$ and
output dimension $N_{out}$, the number of parameters is $N_{in}\times
N_{out}$. In the complex-valued full-connection layer, every two real numbers
correspond to a complex number, so the input and output dimensions will be
halved, that is, the input dimension becomes $\frac{N_{i}n}{2}$, and the
output dimension becomes $\frac{N_{out}}{2}$. The operation process of the
complex-valued full-connected layer is similar to that of the complex-valued
convolutional layer. The number of real and imaginary parameters required to
connect the $N_{in}$-dimensional input complex-valued vector to the
$N_{out}$-dimensional output complex-valued vector is both
$\frac{N_{in}}{2}\times\frac{N_{out}}{2}$, the total number of parameters is
$\frac{N_{in}\times N_{out}}{2}$, which is only half of the number of
parameters of real-valued full-connection layer.

From the above theoretical analysis, it can be seen that whether it is a
convolutional layer or a full-connection layer, the number of parameters based
on the complex-valued operation is only half of that based on the real-valued
operation. IMRecoNet mainly consists of convolutional layers and full-
connection layers. Therefore, the number of parameters of complex-valued
IMRecoNet is only about half that of the real-valued IMRecoNet. The smaller
number of parameters, on the one hand, ensures the convergence speed of the
complex-valued IMRecoNet during training, and on the other hand, it ensures
that it requires less computing resources and computing resources at runtime.
A detailed comparison will be given in the subsequent experimental analysis
part about the specific number of weight parameters of the complex-valued
IMRecoNet and the real-valued IMRecoNet. Besides, the performance of the
number of floating point operations (FLOPs) in the signal detection of
IMRecoNet, which is widely used in deep learning process, will be provided in
the simulation results to help illustrate the complexity of proposed method.

##  V Performance Evaluation

In this section, we evaluate and discuss the performance of proposed
IMRecoNet. Section V-A gives the details of the implementation of the proposed
IMRecoNet and several configurations of simulation. Section V-B demonstrates
the simulation results under various scenarios to show the adaptability and
robustness of the proposed IMRecoNet.

###  V-A Implementation

Both two subnets (i.e., AAPD subnet and SE subnet) in IMRecoNet are developed
through Keras [49] with Tensorflow as backend. The two subnets are trained
using gradient based Adam method on the Nvidia GTX 1080ti GPU environment. The
learning rate and batch size are set to 0.001 and 100, respectively. When we
generate training data and test data, the SNR is defined as

$SNR(dB)=10\log\frac{\mathbb{E}\left\\{\|\mathbf{H}_{i}\mathbf{x}_{j,i}\|_{2}^{2}\right\\}}{\mathbb{E}\left\\{\|\mathbf{n}_{j,i}\|_{2}^{2}\right\\}},$
(32)

to measure the noise level. In order to fully verify the performance of
IMRecoNet, IMRecoNet is trained and tested at each SNR of 5dB, 10dB, 15dB,
20dB and 25dB. We generate 100,000 sets of data through MATLAB R2020b for
large number of antennas scenario, of which training, validation and testing
datasets accounted for $60\%$,$20\%$ and $20\%$, respectively. To verify the
adaptability of IMRecoNet, both Rayleigh fading MIMO channel and correlated
MIMO channel are considered in our simulations:

_1) Rayleigh fading MIMO channel:_ The elements of channle matrix $\bm{H}$
obeys complex Gaussian distribution, i.e.,
$h_{i,j}\sim\mathcal{CN}(0,\frac{1}{N_{r}})$.

_2) Correlated MIMO channel:_ Based on the Rayleigh fading MIMO channel, we
can generate the correlated MIMO channel according to Kronecker model, which
can be expressed as

$\mathbf{H}_{c}=\mathbf{R}^{\frac{1}{2}}_{r}\mathbf{H}\mathbf{R}^{\frac{1}{2}}_{t},$
(33)

where $\mathbf{H}$ is the independent identical distributed (i.i.d.) Rayleigh
fading MIMO channel, $\mathbf{R}_{r}$ and $\mathbf{R}_{t}$ are the spatial
correlation matrix of receive antennas and transmit antennas, respectively.
$\mathbf{R}_{r}$ and $\mathbf{R}_{t}$ are generated through expontial
correlation model [50] with the same correlation coefficient $\rho$.
Specifically, the elements of $\mathbf{R}_{r}$ and $\mathbf{R}_{t}$ satisfy
the following relationship:

$\lambda_{i,j}=\begin{cases}\rho^{j-i},&i\leqslant j\\\
\lambda_{j,i}^{*},&i>j\end{cases}$ (34)

In order to demonstrate the robustness of the IMRecoNet, imperfect channel
estimation at the receiver is also considered in our simulations. Considering
the ML channel estimation [51][52], the imperfect channel estimation can be
expressed as

$\bm{\hat{H}}=\bm{H}+\Delta\bm{H},$ (35)

where $\bm{\hat{H}}$ and $\Delta\bm{H}$ denote the estimated channel matrix
and estimation error, respectively. $\Delta\bm{H}$ obeys zero mean i.i.d.
complex Gaussian distribution with
$\mathbb{E}\left[{|\Delta{h}_{m,n}|}^{2}\right]=\frac{N_{t}{\sigma}^{2}_{z}}{{N_{p}}{E_{p}}}$
[51], where $N_{p}$ and $E_{p}$ represent the number and the power of pilot
symbols, respectively.

We compare IMRecoNet with two traditional algorithms:

(1) Maximum Likelihood (ML) [12][23]: An exhaustive search algorithm jointly
estimates the TAC and actually transmitted signal vector, which exhibits the
best BER performance.

(2) Simultaneous Orthogonal Matching Pursuit (SOMP) [53] : A classic algorithm
in the field of compressed sensing, is a fast convergence algorithm based on
greedy method.

For further verifying the effectiveness and robustness of the proposed
IMRecoNet, we carry out simulations in different scenarios, whose details are
shown in Table IV. From the table, we can find that, as the number of TAs
(i.e., $N_{t}$) and active TAs (i.e., $N_{u}$) increases, legal TAC size shows
exponential growth, which means that the difficulty and complexity of signal
detection increase exponentially.

TABLE IV: Configurations of IM in different scenarios. | $N_{t}$ | $N_{u}$ | TAC Table Size
---|---|---|---
Scenario 1 | 4 | 1 | 4
Scenario 2 | 16 | 4 | 1024

(a) 4QAM (b) 16QAM

Figure 4: BER performance under Rayleigh fading MIMO channel with perfect
channel estiamtion ($N_{t}=4,N_{u}=1$).

###  V-B Simulation Results

_1) Performance under Rayleigh fading MIMO channel with perfect CSI:_ Fig. 4
shows the BER performance of the proposed IMRecoNet and two reference
algorithms under Rayleigh fading MIMO channel with perfect channel estimation.
The TA configuration is $N_{t}=4,N_{u}=1$. From the figure, we can find that
the ML based detector has the optimal performance both in 4QAM and 16QAM,
because it traverses all possible combinations of the legal TACs and the
actually transmitted signal vectors. It is obvious that whether it is real-
valued IMRecoNet or complex-valued IMRecoNet, its BER performance is
significantly ahead of the SOMP signal detection algorithm. And the proposed
IMRecoNet has the performance close to the optimal ML based detector. Besides,
the complex-valued IMRecoNet has a better performance than the real-valued
IMRecoNet with the same network structure. In the perfect CSI case, the BER
performance of complex-valued IMRecoNet has an average of 0.77dB gain than the
real-valued IMRecoNet. When it comes to 16QAM, the BER performance gain
increases into 1.31dB.

Figure 5: The BER performance of the proposed IMRecoNet under various TA
configurations.

_2) Performance under various TA configurations:_ Fig. 5 shows the BER
performance of the proposed IMRecoNet under various TA configurations in
Rayleigh fading MIMO channel. The modulation type is 4QAM. It can be seen from
Fig. 5 that IMRecoNet can achieve good BER performance in the two TA
configurations, i.e., $N_{t}=4,N_{u}=1$ and $N_{t}=16,N_{u}=4$, which
indicates that IMRecoNet has good robustness to changes in the number of TAs.
On the whole, when the TA configuration changes from $N_{t}=4,N_{u}=1$ to
$N_{t}=16,N_{u}=4$, the BER performance of IMRecoNet has a certain drop. This
is because that when $N_{t}=4,N_{u}=1$, only one TA is active for data
transmission, which means no interference. When $N_{t}=16,N_{u}=4$, there are
four TAs transmitting data at the same time, which has interference between
different TAs. The BER performance of the complex-valued IMRecoNet is better
than that of the real-valued IMRecoNet in both two cases. When
$N_{t}=4,N_{u}=1$, the BER performance of the complex-valued IMRecoNet has an
average gain of 0.70dB than the real-valued IMRecoNet. When the TA
configuration becomes $N_{t}=16,N_{u}=4$, the performance gain is expanded to
1.30dB.

_3) Performance under correlated MIMO channel with perfect channel
estimation:_ To verify the robustness of the proposed IMRecoNet, we evaluate
its performance under correlated MIMO channel modeled by Eq. (33) with perfect
channel estimation. TA configuration is set to $N_{t}=16,N_{u}=4$, the
correlation cofficient $\rho$ of $\mathbf{R}_{r}$ and $\mathbf{R}_{t}$ are
both set to 0.5, and the modulation type is 16QAM. From Fig. 6, it is clear
that the proposed IMRecoNet still maintain the BER performance close to the
optimal ML based detection algorithm. Beside, the BER performance of the
proposed IMRecoNet under correlated MIMO channel is still significantly ahead
of the BER performance of the SOMP based signal detection algorithm. The
results fully show that the proposed IMRecoNet has good robustness to
correlated MIMO channel.

Figure 6: (a).BER performance under correlated MIMO channel with perfect
channel estimation ($N_{t}=16,N_{u}=4$). (b).BER performance with channel
estimation.

_4) Robustness to the channel estimation:_ In order to evaluate the robustness
of IMRecoNet to channel estimation errors, we consider the following imperfect
channel estimation: $\hat{\mathbf{H}}=\mathbf{H}+\Delta\mathbf{H}$, where
$\Delta\mathbf{H}$ is the channel estimation error and
$\Delta\mathbf{H}\sim\mathcal{CN}(0,\sigma_{c}^{2})$. We use
$\hat{\mathbf{H}}$ instead of $\mathbf{H}$ as the input of IMRecoNet for
signal detection. Fig. 6 shows the BER performance under 4QAM for different
$\sigma_{c}$. In this simulation, TA configuration is $N_{t}=4,N_{u}=1$, and
SNR is set to 15dB. It can be seen from the result that, as the channel
estimation error increases, the BER performance of each algorithm decreases.
However, the BER performance of IMRecoNet is far superior to the SOMP based
signal detection algorithm, and is very close to the performance of the ML
based signal detection algorithm. When channel estimation error is less than
-30dB, the BER performance of IMRecoNet is slightly worse than that of the
optimal signal detection algorithm based. When the channel estimate is greater
than -30dB, the BER performance of IMRecoNet will begin to outperform the
performance of the ML based signal detection algorithm. This is becuase the
mapping relationship between the received signal and the antenna activation
mode is directly learned in deep learning based IMRecoNet instead of obtaining
channel state information via channel estimation. This result shows that
IMRecoNet has a certain robustness to channel estimation errors, and also
shows that SE Subnet can reduce signal detection error caused by calculation
error.

TABLE V: Comparison of the parameter size and storage size between real-valued IMRecoNet and complex-valued IMRecoNet | IMNet-Real |
---|---|---
| AD subnet | SE subnet | Total |
Parameter size | 1.337M | 0.036M | 1.373M |
Storage size | 5.099MB | 0.139MB | 5.238MB |
| IMNet-Complex |
| AD subnet | SE subnet | Total |
Parameter size | 0.668M | 0.019M | 0.687M |
Storage size | 2.549MB | 0.073MB | 2.622MB |

(a) perfect CSI

(b) imperfect CSI

Figure 7: AAP prediction accuracy under Rayleigh fading MIMO channel with
perfect/imferfect CSI ($N_{t}=4,N_{u}=1$).

(a) AAPD Subnet

(b) SE Subnet

Figure 8: Convergence speed of real-valued IMRecoNet and complex-valued
IMRecoNet.

_5) AAP Prediction Accurancy comparison under perfect CSI and imperfect CSI:_
Fig.7(a) and Fig.7(b) present the AAP prediction accuracy of the proposed
IMRecoNet and two baseline algorithms under Rayleigh fading MIMO channel with
perfect CSI and imperfect CSI,respectively. TA configuration is
$N_{t}=4,N_{u}=1$ and the modulation mode is 16QAM. We can find that the
optimal ML based detector and SOMP detector have about $10\%,20\%$ performance
degradation. Compared Fig.7(b) with Fig.7(a), the IMRecoNet has little
performance degradation under imperfect CSI. The reason is that ML and SOMP
based detector need the CSI as input while our proposed IMRecoNet does not
need CSI. Besides, it can be seen that IMNet-Complex performs better than the
IMNet-Real due to the complex-valued operations.

_6) Performance comparison for real-valued IMRecoNet and complex-valued
IMRecoNet:_ Table V shows the parameter size and the stoarge size of real-
valued IMRecoNet and complex-valued IMRecoNet. The TA configuration is
$N_{t}=4,N_{u}=1$. It can be found from the table that the parameter size of
complex-valued IMRecoNet is basically half of that of real-valued IMRecoNet,
which is consistent with the previous theoretical analysis. Thanks to the
greatly reduced parameter size, the storage size of complex-valued IMRecoNet
is only half of the complex-valued IMRecoNet. The introduction of complex-
valued operations greatly reduces the parameter size and the storage size of
the neural network model, which further increases the possibility of the
deployment and use of the neural network in the actual wireless communication
system.

The parameter size determines the complexity of the model. In addition to
occupying less storage space, a smaller parameter size also has a certain
impact on the convergence speed of the network during training period. Fig. 8
illustrates the convergence speed, which shows the iterations in the AAPD and
the SE subnet when the TA is configured with $N_{t}=4,N_{u}=1$, the modulation
mode is 16QAM, and the SNR is set to 5dB. In each iteration, the parameters of
CNN were updated based on gradient descent. Fig. 8(a) shows that the complex-
valued AAPD subnet has the similar convergence speed to the real-valued AAPD
subnet. However, the complex-valued AAPD subnet can obtain a lower loss value
than the real-valued AAPD subnet when they converge, which also means the
complex-valued AAPD subnet will have better performance. It can be found from
Fig.8(b) that, although the loss value of the complex-valued SE subnet and the
real-valued SE subnet are basically the same when converging, the complex-
valued SE subnet has a faster convergence speed. The complex-valued SE subnet
has basically converged after 10,000 iterations, while the real-valued SE
subnet reaches the state of convergence after 15,000 iterations.

_7) Complexity analysis and comparison:_ In order to further compare the
complexity of each algorithm, Fig. 9 shows the number of floating point
operations (FLOPs) required for signal detection for each algorithm in
$N_{t}=4,N_{u}=1$ and $N_{t}=16,N_{u}=4$, respectively. It can be seen from
the figure that, when the number of TA is small (i.e., $N_{t}=4,N_{u}=1$), the
ML based signal detection algorithm and the SOMP based signal detection
algorithm have lower FLOPs. When the number of TAs increase to
$N_{t}=16,N_{u}=4$, the FLOPs of the ML based algorithm increased sharply,
much larger than other signal detection algorithms. The FLOPs of SOMP based
signal detection algorithms also shows a significant increase. When
$N_{t}=16,N_{u}=4$, the FLOPs of the complex-valued IMRecoNet are lower than
the FLOPs of the ML based signal detection algorithm and the SOMP based signal
detection algorithm, and also lower than the FLOPs of the real-valued
IMRecoNet, which again illustrates that the introduction of the complex-valued
operations has played a key role in reducing the complexity of the model.

Figure 9: FLOPs for each algorithm.

##  VI Conclusion

In this paper, we design a detector for IM-MIMO systems from a machine
learning perspective. Inspired by the huge advances in artificial intelligence
in recent years, we propose to integrate deep learning to the design of
detector for IM-MIMO systems. Different from assuming the neural network as a
black-box and directly using it to detect, we first model the transmission of
IM-MIMO systems as a compressed sensing process and formulate the detection
process at the receiver as a sparse reconstruction problem. Based on the
formulated sparse reconstruction problem, we integrate CNN and a traditional
linear MIMO detector (i.e., LS), called IMRecoNet, to realize the detection
process. In order to adapt the neural network to complex signal processing in
wireless communications, complex value operations (i.e., complex convolution
and complex activation) are further introduced to the design of IMRecoNet.
Extensive simulations have been carried out to verify the performance of real
value and complex value based IMRecoNet under various scenarios. The BER
performance shows that IMRecoNet achieves better performance than the
traditional detectors, and the introduction of complex value operations can
indeed further improve performance.

The results of this paper has shown the great potential of deep learning in
the physical layer of wireless communications. Compared with computer vision
and natural language processing, wireless communication has its own unique
characteristics, such as complex value signals and strict delay requirements.
Therefore, it is inefficient and unrealistic to directly introduce deep
learning into wireless communications. In this paper, we first introduce
complex-valued operations to CNN for adapting the characteristic that signal
in wireless communications is represented as complex values. Compared with CNN
with real value operations, the introduction of complex operations leads to
certain performance improvement. Last but not least, only complex convolution
and simple complex activation are implemented in this paper, how to design
other deep learning components that match the characteristics of wireless
communications is still open and challenging.

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