# DeepGOMIMO: Deep Learning-Aided Generalized Optical MIMO with CSI-Free Blind
Detection

Xin Zhong, Chen Chen, Shu Fu, Zhihong Zeng, and Min Liu This work was
supported in part by the National Natural Science Foundation of China under
Grant 61901065, in part by the Fundamental Research Funds for the Central
Universities under Grant 2021CDJQY-013 and Grant 2020CDJQY-A001, and in part
by the China Postdoctoral Science Foundation under Grant 2021M693744.X. Zhong,
C. Chen, S. Fu, and M. Liu are with the School of Microelectronics and
Communication Engineering, Chongqing University, Chongqing 400044, China
(e-mail: {c.chen, 201912131098, shufu, liumin}@cqu.edu.cn).Z. Zeng is with the
LiFi Research and Development Centre, Institute for Digital Communications,
The University of Edinburgh, EH9 3JL, UK (e-mail: zhihong.zeng@ed.ac.uk).

###### Abstract

Generalized optical multiple-input multiple-output (GOMIMO) techniques have
been recently shown to be promising for high-speed optical wireless
communication (OWC) systems. In this paper, we propose a novel deep learning-
aided GOMIMO (DeepGOMIMO) framework for GOMIMO systems, where channel state
information (CSI)-free blind detection can be enabled by employing a specially
designed deep neural network (DNN)-based MIMO detector. The CSI-free blind DNN
detector mainly consists of two modules: one is the pre-processing module
which is designed to address both the path loss and channel crosstalk issues
caused by MIMO transmission, and the other is the feed-forward DNN module
which is used for joint detection of spatial and constellation information by
learning the statistics of both the input signal and the additive noise. Our
simulation results clearly verify that, in a typical indoor 4 $\times$ 4 MIMO-
OWC system using both generalized optical spatial modulation (GOSM) and
generalized optical spatial multiplexing (GOSMP) with unipolar non-zero 4-ary
pulse amplitude modulation (4-PAM) modulation, the proposed CSI-free blind DNN
detector achieves near the same bit error rate (BER) performance as the
optimal joint maximum-likelihood (ML) detector, but with much reduced
computational complexity. Moreover, since the CSI-free blind DNN detector does
not require instantaneous channel estimation to obtain accurate CSI, it enjoys
the unique advantages of improved achievable data rate and reduced
communication time delay in comparison to the CSI-based zero-forcing DNN (ZF-
DNN) detector.

###### Index Terms:

Optical wireless communication, multiple-input multiple-output, deep learning,
blind detection.

##  I Introduction

Due to the exhaustion of radio frequency (RF) spectrum resources, optical
wireless communication (OWC) which explores the infrared, visible light or
ultra-violet spectrum has been envisioned as a promising candidate to satisfy
the ever-increasing data demand in future indoor environments [1]. In recent
years, bidirectional OWC, which is also named light fidelity (LiFi), has been
widely considered as one of the key enabling technologies for 5G/6G and
Internet of Things (IoT) communications [2, 3, 4, 5]. Although OWC systems
have many inherent advantages such as abundant license-free spectrum
resources, no electromagnetic interference (EMI) and enhanced physical-layer
security, the practically achievable capacity of OWC systems is largely
limited by the small modulation bandwidth of commercial off-the-shelf (COTS)
optical elements, especially for illumination light-emitting diodes (LEDs)
[6].

As a very natural way to efficiently improve the achievable capacity of indoor
OWC systems using LEDs, multiple-input multiple-output (MIMO) transmission has
attracted great attention recently, which fully exploits the existing LED
fixtures in the ceiling of a typical room to harvest substantial diversity or
multiplexing gain [7, 8, 9]. So far, various optical MIMO techniques have been
introduced for OWC systems, among which optical spatial multiplexing (OSMP)
and optical spatial modulation (OSM) are two most popular ones. Specifically,
OSMP can achieve full multiplexing gain and hence a relative high spectral
efficiency, but suffers from severe inter-channel interference (ICI) [10]. In
contrast, OSM can remove ICI by activating a single LED to transmit signal at
each time slot. Although OSM can transmit additional index bits, only one
constellation symbol can be transmitted at each time slot, and hence it is
challenging for OSM systems to achieve high spectral efficiency [11]. Lately,
generalized optical MIMO (GOMIMO) techniques, including generalized OSM (GOSM)
and generalized OSMP (GOSMP), have been further proposed to boost the capacity
of MIMO-OWC systems [12, 13, 14, 15]. In GOSM systems, multiple LEDs are
activated to transmit the same signal, and therefore more index bits can be
transmitted and the diversity gain can also be increased. In GOSMP systems,
only a subset of LEDs are activated to transmit different signals, resulting
in reduced multiplexing gain. However, additional index bits can be
transmitted and the ICI can also been reduced in GOSMP systems. Our previous
work [15] has already clearly demonstrated the superiority of GOMIMO
techniques in comparison to conventional OSM and OSMP.

In order to successfully implement GOMIMO systems, an efficient MIMO detection
scheme should be adopted. Generally, the joint maximum-likelihood (ML)
detector serves as the optimal detector for GOMIMO systems [16]. Nevertheless,
the ML detector usually has high computational complexity, making it
infeasible in practical applications. Instead, the combination of zero-forcing
(ZF) equalization and ML detection can be a practical low-complexity detection
scheme for GOMIMO systems [16]. However, ZF equalization inevitably leads to
noise amplification due to high channel correlation in typical indoor MIMO-OWC
systems. Moreover, the ZF-ML detector also suffers from the adverse effect of
error propagation, since the detection error of spatial symbols might
propagate to the estimation of constellation symbols.

With the rapid development of machine learning technology, deep learning has
revealed its great potential in wireless communication systems [17, 18]. Most
recently, deep learning techniques have also been applied in OWC systems for
binary signaling design [19], mitigation of both linear and nonlinear
impairments [20], energy-efficient resource management [21], and so on. More
specifically, a ZF-based deep neural network (DNN) detection scheme has been
proposed for MIMO detection in GOMIMO systems [22]. The obtained results in
[22] show that the ZF-DNN detector can achieve comparable bit error rate (BER)
performance as the optimal joint ML detector with greatly reduced
computational complexity. Nevertheless, the ZF-DNN detector takes the ZF
equalized signal as its input, which requires accurate channel state
information (CSI), i.e., the MIMO channel matrix, to successfully perform ZF
equalization. Although CSI can be estimated by using training symbols [23],
training-based instantaneous channel estimation inevitably causes both the
loss of achievable data rate and the increase of communication time delay.

Figure 1: Block diagram of a general $N_{r}\times N_{t}$ GOMIMO system. Insets
(a) and (b) show the mapping tables of GOSM and GOSMP, respectively.

In this paper, to address the disadvantages of CSI-based ZF-DNN detection due
to the requirement of instantaneous CSI for ZF equalization, we for the first
time propose a DeepGOMIMO framework for GOMIMO systems where CSI-free MIMO
detection is achieved by a novel blind DNN detection scheme. By adding a
specially designed pre-processing module before the feed-forward DNN module,
CSI-free blind detection can be successfully enabled for GOMIMO systems.
Numerical simulations are extensively conducted to evaluate the performance of
the proposed CSI-free blind DNN detector, which is also compared with other
three benchmark schemes including the joint ML detector, the ZF-ML detector
and the ZF-DNN detector. Our simulation results verify the advantages of the
proposed CSI-free blind DNN detector in comparison to other benchmark schemes
in GOMIMO systems.

The rest of this paper is organized as follows. In Section II, we describe the
mathematical model of a general GOMIMO system. In Section III, we introduce
four detection schemes for GOMIMO systems. Detailed simulation setup and
results are presented in Section IV. Finally, Section V concludes the paper.

##  II System Model

In this section, we introduce the mathematical model of a general GOMIMO
system equipped with $N_{t}$ LEDs and $N_{r}$ photo-detectors (PDs). The
channel model is first described and then the basic principle of GOMIMO is
further reviewed.

###  II-A Channel Model

Letting $\bf{x}$ $=[x_{1},x_{2},\cdots,x_{N_{t}}]^{T}$ be the transmitted
signal vector, $\bf{H}$ represent the $N_{r}\times N_{t}$ MIMO channel matrix
and $\bf{n}$ $=[n_{1},n_{2},\cdots,n_{N_{r}}]^{T}$ denote the additive noise
vector, the received signal vector $\bf{y}$
$=[y_{1},y_{2},\cdots,y_{N_{r}}]^{T}$ is obtained by

$\bf{y}=Hx+n,$ (1)

and the corresponding channel matrix $\bf{H}$ can be expressed by

$\bf{H}=\left[\begin{array}[]{ccc}h_{11}&\cdots&h_{1N_{t}}\\\
\vdots&\ddots&\vdots\\\ h_{N_{r}1}&\cdots&h_{N_{r}N_{t}}\end{array}\right],$
(2)

where $h_{rt}$ ($r=1,2,\cdots,N_{r};t=1,2,\cdots,N_{t}$) denotes the direct
current (DC) channel gain between the $r$-th PD and the $t$-th LED. Assuming
that each LED follows the general Lambertian radiation pattern and only the
line-of-sight (LOS) transmission is considered, $h_{rt}$ is calculated by [24]

$h_{rt}=\frac{(l+1)\rho A}{2\pi
d_{rt}^{2}}\mathrm{cos}^{m}(\varphi_{rt})T_{s}(\theta_{rt})g(\theta_{rt})\mathrm{cos}(\theta_{rt}).$
(3)

In (3), $l=-\mathrm{ln}2/\mathrm{ln}(\mathrm{cos}(\Psi))$ denotes the
Lambertian emission order, with $\Psi$ being the semi-angle at half power of
the LED; $\rho$ and $A$ represent the responsivity and the physical area of
the PD, respectively; $d_{rt}$ is the distance between the $r$-th PD and the
$t$-th LED; $\varphi_{rt}$ and $\theta_{rt}$ are the emission angle and the
incident angle, respectively; $T_{s}(\theta_{rt})$ is the gain of optical
filter; $g(\theta_{rt})=\frac{n^{2}}{\mathrm{sin}^{2}\Phi}$ is the gain of
optical lens, where $n$ and $\Phi$ are the refractive index and the half-angle
field-of-view (FOV) of the optical lens, respectively.

Moreover, the additive noise in typical OWC systems consists of both shot and
thermal noises, and it is reasonable to model the additive noise as a real-
valued zero-mean additive white Gaussian noise (AWGN) [8]. Letting $N_{0}$
denote the noise power spectral density (PSD) and $B$ be the signal bandwidth,
the power of the additive noise is given by $P_{n}=N_{0}B$.

###  II-B Principle of GOMIMO

The concept of GOMIMO was first proposed in [15], which aims to fully explore
the potential of MIMO transmission for spectral efficiency enhancement of
bandlimited OWC systems. Specifically, GOMIMO techniques can be generally
divided into two main categories: one is GOSM where all the activated LED
transmitters transmit the same signal, and the other is GOSMP where the
activated LED transmitters transmit different signals. For more details about
the GOMIMO techniques, please refer to our previous work [15].

Fig. 1 illustrates the block diagram of a general $N_{r}\times N_{t}$ GOMIMO
system, where $N_{a}$ ($1\leq N_{a}\leq N_{t}$) LEDs are activated for signal
transmission during GOMIMO mapping. As we can see, the input bits are first
divided into two streams: one is fed into the constellation mapper which
converts the binary bits into constellation symbols, and the other is sent
into the LED index selector which selects the desired LEDs to transmit the
generated constellation symbols accordingly. Based on the obtained
constellation symbol vector $\bf{c}$ and spatial index vector $\bf{v}$, GOMIMO
(GOSM or GOSMP) mapping is performed to generate the transmitted signal vector
$\bf{x}$. The mapping tables for GOSM and GOSMP with $N_{t}=4$ and $N_{a}=2$
are given in insets (a) and (b) of Fig. 1, respectively. At the receiver side,
the received signal vector $\bf{y}$ is fed into the GOMIMO detector which
finally yields the output bits. The detailed GOMIMO detection schemes will be
discussed in the following section.

In typical LED-based OWC systems, intensity modulation with direct detection
(IM/DD) is generally applied due to the non-coherence nature of LEDs. As a
result, only real-valued non-negative signals can be successfully transmitted
in the IM/DD OWC systems [24]. In this work, unipolar $M$-ary pulse amplitude
modulation ($M$-PAM) is adopted as the modulation format for GOMIMO systems.
In order to avoid the loss of spatial information when performing GOMIMO
mapping, the $M$-PAM symbols cannot have zero values [15]. Therefore, unipolar
non-zero $M$-PAM modulation is utilized here and the corresponding intensity
levels are given by

$I_{m}=\frac{2I_{\textrm{av}}}{M+1}m,~{}~{}~{}~{}m=1,\cdots,M,$ (4)

where $I_{\textrm{av}}$ denotes the average optical power emitted [8]. Using
$M$-PAM modulation, the spectral efficiencies (bits/s/Hz) of the $N_{r}\times
N_{t}$ GOMIMO system with $N_{a}$ activated LEDs applying GOSM and GOSMP
mappings are respectively given by

${\eta}_{\textrm{GOSM}}=\log_{2}(M)+\lfloor\log_{2}(C(N_{t},N_{a}))\rfloor,$
(5)
${\eta}_{\textrm{GOSMP}}=N_{a}\log_{2}(M)+\lfloor\log_{2}(C(N_{t},N_{a}))\rfloor,$
(6)

where $\lfloor\cdot\rfloor$ denotes the floor operator which outputs an
integer smaller or equal to its input value and $C(\cdot,\cdot)$ represents
the binomial coefficient.

##  III Detection Schemes for GOMIMO Systems

In this section, we first introduce two conventional detection schemes for
GOMIMO systems utilizing $M$-PAM modulation, including the optimal joint ML
detection and the ZF-ML detection. After that, we further present two deep
learning-aided detection schemes, including the CSI-based ZF-DNN detection and
our newly proposed CSI-free blind DNN detection.

###  III-A Joint ML Detection

Assuming perfect CSI, joint ML detection is the optimal detection scheme for
GOMIMO systems with $M$-PAM modulation. More specifically, the joint ML
detector estimates the transmitted constellation and spatial information
simultaneously in a joint manner. By applying the joint ML detector, the
transmitted signal vector $\mathbf{x}$ can be estimated by

$\hat{\mathbf{x}}_{\textrm{JML}}=\arg\min_{\mathbf{x}\in\mathbb{X}}{\left\|\mathbf{y}-\mathbf{H}\mathbf{x}\right\|}^{\textrm{2}},$
(7)

where ${\lVert\cdot\rVert}_{2}$ denotes the modulus operator and $\mathbb{X}$
represents the set of all the considered transmitted signal vectors.

Although the joint ML detection can achieve optimal performance, it suffers
from high computational complexity. Therefore, it is usually not feasible to
apply the joint ML detector in practical GOMIMO systems.

###  III-B ZF-ML Detection

In order to avoid the high computational complexity of joint ML detection, a
low-complexity ZF-ML detection scheme can be applied in GOMIMO systems, which
is basically a three-step detection scheme [25, 15]. In the first step, ZF
equalization is performed for MIMO demultiplexing. The estimate of the
transmitted signal vector $\mathbf{x}$ after ZF equalization can be obtained
by

$\bf{\hat{x}}_{\textrm{ZF}}=H^{{\dagger}}y=x+H^{{\dagger}}n,$ (8)

where $\bf{H^{{\dagger}}}$ denotes the pseudo inverse of $\bf{H}$ [10].

In the second step, ML detection is executed to obtain the estimate of the
spatial index vector according to $\bf{\hat{x}}_{\textrm{ZF}}$. Finally, in
the third step, the estimate of the constellation symbol vector can be
obtained accordingly by using $\bf{\hat{x}}_{\textrm{ZF}}$ and the estimate of
the spatial index vector. For more details about the principle of ZF-ML
detection for GOMIMO systems, please refer to our previous work [15].

Compared with joint ML detection, the computational complexity of ZF-ML
detection is significantly reduced. Nevertheless, the performance of ZF-ML
detection is also largely degraded in comparison to that of joint ML
detection, which can be explained as follows. On the one hand, ZF equalization
inevitably causes severe noise amplification due to the high channel
correlation in typical MIMO-OWC systems [8], which might greatly degrade the
performance of GOMIMO systems. On the other hand, the detection error of
spatial symbols might propagate to the estimation of the constellation symbols
[26], which leads to further substantial performance degradation of GOMIMO
systems.

Figure 2: Schematic diagram of the proposed CSI-free blind DNN detector
consisting of a pre-processing module and a feed-forward DNN module.

###  III-C CSI-Based ZF-DNN Detection

To efficiently address both the high computational complexity issue of joint
ML detection and the noise amplification and error propagation issues of ZF-ML
detection, a ZF-DNN detection has been proposed for GOSMP systems in [22]. The
key idea of the ZF-DNN detection scheme is to employ a feed-forward DNN module
to directly and simultaneously estimate the transmitted spatial and
constellation bits by taking the ZF equalized signal vector
$\bf{\hat{x}}_{\textrm{ZF}}$ as input. For more details about the
implementation of the ZF-DNN detector, please refer to [22, 26]. In a word,
the feed-forward DNN module can fulfill the tasks of spatial index vector
estimation, constellation symbol vector estimation, spatial symbol
demodulation and constellation symbol demodulation at the same time.
Simulation results in [22] clearly show that, by selecting a proper training
signal-to-noise ratio (SNR), the ZF-DNN detector can achieve nearly the same
BER performance as the optimal joint ML detector, but with a significantly
reduced computational complexity.

Despite the near-optimal BER performance and low computational complexity of
the ZF-DNN detector, it takes the ZF equalized signal vector
$\bf{\hat{x}}_{\textrm{ZF}}$ as the input of the feed-forward DNN module. As
per (8), $\bf{\hat{x}}_{\textrm{ZF}}$ is obtained by multiplying the received
signal vector $\bf{y}$ with $\bf{H^{{\dagger}}}$, i.e., the pseudo inverse of
the channel matrix $\bf{H}$. Generally, the CSI (i.e., the channel matrix) can
be efficiently estimated by transmitting training symbols [23]. Nevertheless,
the use of training symbols for accurate CSI estimation inevitably reduces the
achievable data rate of GOMIMO systems, especially for low SNR scenarios.
Furthermore, since the channel matrix is highly related to the specific
location of the MIMO receiver, i.e., the PD array, channel estimation needs to
be executed instantaneously with the change of receiver location. In
consequence, instantaneous channel estimation inevitably introduces additional
communication time delay and computational complexity in practical GOMIMO
systems.

###  III-D Proposed CSI-Free Blind DNN Detection

Considering the many disadvantages of CSI-based ZF-DNN detection due to the
requirement of instantaneous CSI for ZF equalization, in this work, we for the
first time propose a novel CSI-free blind DNN detection scheme for GOMIMO
systems. Fig. 2 depicts the schematic diagram of the proposed CSI-free blind
DNN detector, which consists of a pre-processing module and a feed-forward DNN
module. It can be seen that a pre-processing module is placed in front of the
feed-forward DNN module in the proposed CSI-free blind DNN detector, which is
the key to deal with the impact of MIMO transmission through free-space
channels and hence realize blind detection without the need of CSI.
Specifically, as can be found from (1), the impact of MIMO transmission on the
transmitted signal vector $\bf{x}$ can be characterized from the following two
aspects. Firstly, since the channel coefficients in typical MIMO-OWC systems
are within the region from $10^{-6}$ to $10^{-4}$ [8, 27], the electrical path
loss caused by MIMO transmission is about 80 to 120 dB. Secondly, MIMO
transmission also inevitably leads to channel crosstalk, which might cause
severe ICI, especially for GOSMP systems. As a result, the designed pre-
processing module should be able to address both the path loss issue and the
channel crosstalk issue caused by MIMO transmission.

As shown in Fig. 2, our specially designed pre-processing module mainly
contains two parts: one is the amplitude scaling part and the other is the
feature extraction part. Specifically, the amplitude scaling part is adopted
to address the path loss issue by multiplying the received signal vector
$\bf{y}$ with a scaling factor $\alpha$. Note that a proper $\alpha$ value is
determined in advance for each receiver location in the GOMIMO system, and
hence no instantaneous CSI is needed to achieve amplitude scaling. Moreover,
the feature extraction part is used to address the channel crosstalk issue,
which multiplies the scaled received signal vector $\alpha\bf{y}$ by a feature
matrix $\bf{F}$. Hence, the output signal vector of the pre-processing module
in the CSI-free blind DNN detector, i.e., $\bf\hat{y}$
$=[\hat{y}_{1},\hat{y}_{2},\cdots,\hat{y}_{N_{r}}]^{T}$, can be obtained by

$\bf\hat{y}=\alpha\bf{F}\bf{y}.$ (9)

In order to provide enough information for the following feed-forward DNN
module to efficiently learn and remove the impact of channel crosstalk caused
by MIMO transmission, the feature matrix $\bf{F}$ should be able to reflect
all the potential signal superposition cases at the receiver side.
Consequently, according to the mapping tables of both GOSM and GOSMP in Fig.
1, we adopt the corresponding unified mapping matrix as the feature matrix,
i.e.,

$\bf{F}=\left[\begin{array}[]{llll}1&1&0&0\\\ 1&0&1&0\\\ 0&1&0&1\\\
0&0&1&1\end{array}\right].$ (10)

TABLE I: Simulation Parameters Parameter | Value
---|---
Room dimension | $5\ \textrm{m}\times 5\ \textrm{m}\times 3\ \textrm{m}$
Height of receiving plane | 0.85 m
Number of LEDs | 4
Semi-angle at half power of LED | $60^{\circ}$
LED spacing | 2.5 m
Gain of optical filter | 0.9
Refractive index of optical lens | 1.5
Half-angle FOV of optical lens | $72^{\circ}$
Number of PDs | 4
Responsivity of PD | 1 $\mathrm{A/W}$
Active area of PD | 1 $\textrm{cm}^{2}$
PD spacing | 10 cm
Number of activated LEDs, $N_{a}$ | 2
PAM levels, $M$ | 4

Subsequently, the pre-processed signal vector $\bf\hat{y}$ is fed into a feed-
forward DNN module, which mainly consists of an input layer, multiple hidden
layers, an output layer and a decision layer. Since $\bf\hat{y}$ is a vector
with $N_{r}$ elements, the input layer contains $N_{r}$ neurons accordingly.
Moreover, we set totally four fully-connected hidden layers in the feed-
forward DNN module, which are used to learn the statistical characteristics of
both the input signal and the additive noise. The number of neurons in the
$i$-th ($1\leq i\leq 4$) hidden layer is denoted by $L_{i}$, and the rectified
linear unit (ReLU) function, i.e., $f_{\textrm{ReLU}}(\alpha)=\max(0,\alpha)$,
is adopted as the activation function of the hidden layers. For the output
layer, it adopts the Sigmoid function, i.e., $f_{\textrm{Sigmoid
}}(\alpha)=1/\left(1+\exp^{-\alpha}\right)$, as the activation function to
generate a fuzzy bit information, so as to map the output of each neuron
within the range [0, 1]. Since the DNN detector takes the input binary bits
corresponding to a transmitted signal vector as the output, both the output
layer and the decision layer have the same number of neurons, which is equal
to the spectral efficiency of the GOMIMO system, i.e.,
$S=\eta_{\textrm{GOMIMO}}$. Therefore, letting $\mathbf{z}_{k}$ denote the
output of the $k$-th ($1\leq k\leq 6$) layer of the feed-forward DNN module,
the corresponding input-output relationship can be described by

${\mathbf{z}}_{k}=\left\\{\begin{array}[]{ll}\alpha\bf{F}\bf{y},&k=1\\\
f_{\textrm{ReLU}}\left(\mathbf{W}_{k-1}{\mathbf{z}}_{k-1}+\mathbf{b}_{k-1}\right),&2\leq
k\leq 5\\\
f_{\textrm{Sigmoid}}\left(\mathbf{W}_{k-1}{\mathbf{z}}_{k-1}+\mathbf{b}_{k-1}\right),&k=6\end{array},\right.$
(11)

where $\mathbf{W}_{p}$ and $\mathbf{b}_{p}$ with $1\leq p\leq 5$ represent the
corresponding weight matrix and the bias vector, respectively.

Finally, the decision layer is utilized to determine the fuzzy output of each
neuron in the output layer to be 0 or 1. Letting
$\mathbf{z}_{6}=[z_{1},z_{2},\dots z_{S}]^{T}$ and $\bf\hat{b}$
$=[\hat{b}_{1},\hat{b}_{2},\cdots,\hat{b}_{S}]^{T}$ respectively denote the
fuzzy output vector of the output layer and the final output binary bit
vector, the $q$-th ($q=1,2,\cdots,S$) binary bit in $\bf\hat{b}$ can be
estimated by

$\hat{b}_{q}=\left\\{\begin{array}[]{ll}0,&z_{q}<0.5\\\ 1,&z_{q}\geq
0.5\end{array}.\right.$ (12)

In the proposed CSI-free blind DNN detector, we adopt the mean-square error
(MSE) loss function to measure the difference between the transmitted bit
vector $\bf{b}$ and the corresponding estimated bit vector $\bf\hat{b}$, which
is given by

$e_{\textrm{MSE}}=\frac{1}{S}\left\|\bf\hat{b}-\bf{b}\right\|^{2}.$ (13)

##  IV Simulation Results

In this section, we evaluate and compare the performance of four different
detection schemes in a typical indoor GOMIMO system through numerical
simulations.

TABLE II: Parameters of the Blind DNN Detector for GOSM Parameter | Value
---|---
Receiver locations | (2.5 m, 2.5 m, 0.85) $\mid$ (0 m, 0 m, 0.85)
Number of input nodes | 4
Number of hidden layers | 4
Number of neurons |  $L_{1}$ = 128, $L_{2}$ = 64, $L_{3}$ = 32, $L_{4}$ = 16
Number of output nodes | 4
Hidden layer activation | ReLU
Output layer activation | Sigmoid
Loss function | MSE
Optimizer | Adamax
Learning rate | 0.01 $\mid$ 0.001
Length of training set | 150000
Length of validation set | 50000
Scaling factor |  $1\times 10^{5}$ $\mid$ $2\times 10^{5}$
TABLE III: Parameters of the Blind DNN Detector for GOSMP Parameter | Value
---|---
Receiver locations | (2.5 m, 2.5 m, 0.85) $\mid$ (0 m, 0 m, 0.85)
Number of input nodes | 4
Number of hidden layers | 4
Number of neurons |  $L_{1}$ = 64, $L_{2}$ = 64, $L_{3}$ = 64, $L_{4}$ = 64
Number of output nodes | 6
Hidden layer activation | ReLU
Output layer activation | Sigmoid
Loss function | MSE
Optimizer | Adamax
Learning rate | 0.01 $\mid$ 0.005
Length of training set | 150000
Length of validation set | 50000
Scaling factor |  $1\times 10^{5}$ $\mid$ $1\times 10^{6}$

###  IV-A Simulation Setup

In our simulations, we consider a 4 $\times$ 4 ($N_{r}$ = $N_{t}$ = 4) GOMIMO
system configured in a typical 5 m $\times$ 5 m $\times$ 3 m room. The 2
$\times$ 2 square LED array is placed at the center of the ceiling and the
spacing between two adjacent LEDs is 2 m. The height of the receiving plane is
0.85 m, and two receiver locations over the receiving plane, i.e., the center
(2.5 m, 2.5 m, 0.85 m) and the corner (0 m, 0 m, 0.85 m), are considered for
performance evaluation. The receiver consists of a 2 $\times$ 2 square PD
array, where the spacing between two adjacent PDs is 10 cm. For both GOSM and
GOSMP mappings, two out of four LEDs are activated for signal transmission,
i.e., $N_{a}$ = 2. Moreover, unipolar non-zero 4-PAM modulation is adopted in
the GOMIMO system, and hence the corresponding spectral efficiencies for GOSM
and GOSMP mappings are 4 and 6 bits/s/Hz, respectively. In addition, we adopt
transmitted SNR as the measure to evaluate the BER performance of the GOMIMO
system [8, 15]. The other simulation parameters of the GOMIMO system can be
found in Table I.

(a) GOSM

(b) GOSMP

Figure 3: MSE loss of the proposed CSI-free blind DNN detector with receiver
located at the center of the receiving plane for (a) GSM and (b) GSMP.

The detailed parameters of the CSI-free blind DNN detectors for GOSM and GOSMP
are given in Tables II and III, respectively. For GOSM, the number of neurons
of four hidden layers is 128, 64, 32 and 16, respectively. The learning rate
is 0.01 when the receiver is located at the center of the receiving plane,
while it is reduced to 0.001 when the receiver is moved to the corner.
Moreover, the scaling factors are set to $1\times 10^{5}$ and $2\times 10^{5}$
when the receiver is located at the center and the corner, respectively. For
GOSMP, every hidden layer contains 64 neurons, and the learning rates are 0.01
and 0.005 when the receiver is located at the center and the corner of the
receiving plane, respectively. In addition, the scaling factors of $1\times
10^{5}$ and $1\times 10^{6}$ are used for center and corner received
locations, respectively. For both GOSM and GOSMP, the lengths of training set
and validation set are assumed to be 150000 and 50000, respectively. In order
to accelerate the convergence speed, we use the mini-batch technique in
training and each mini-batch contains 100 transmitted signal vectors.

###  IV-B MSE Loss

We first analyze the MSE loss of the proposed CSI-free blind DNN detector in
the 4 $\times$ 4 GOMIMO system. Figs. 3(a) and (b) show the MSE losses versus
the number of epochs for GOSM and GOSMP, respectively, where the receiver is
located at the center of the receiving plane. As we can see, the MSE loss
decreases rapidly with the increase of training epochs for both GOSM and
GOSMP. Moreover, the MSE loss is much reduced when a higher training SNR is
used, especially for GOSMP. It can be seen from Fig. 3(a) that the MSE loss
for GOSM fast converges with only a few epochs. For GOSMP, as shown in Fig.
3(b), about 20 epochs are required for the MSE loss to converge. Hence, owing
to the use of the mini-batch technique, the CSI-free blind DNN detector only
requires a very limited number of epochs for efficient training, indicating
that it can be deployed rapidly in practical applications.

###  IV-C BER Performance

(a) GOSM, center

(b) GOSM, corner

Figure 4: BER comparison of the proposed CSI-free blind DNN detector and three
benchmark detectors for GOSM at (a) the center and (b) the corner.

(a) GOSMP, center

(b) GOSMP, corner

Figure 5: BER comparison of the proposed CSI-free blind DNN detector and three
benchmark detectors for GOSMP at (a) the center and (b) the corner.

We further evaluate and compare the BER performance of the proposed CSI-free
blind DNN detector with the other three benchmark detectors in the 4 $\times$
4 GOMIMO system. Figs. 4(a) and (b) compare the BER performance of four
detectors for GOSM with the receiver located at the center and the corner of
the receiving plane, respectively. When the receiver is located at the center
of the receiving plane, as shown in Figs 4(a), the ZF-ML detector requires a
high transmitted SNR of 163.4 dB to achieve the target BER of $10^{-3}$.
However, the required SNR to reach BER = $10^{-3}$ is reduced to 138.9 dB for
the joint ML detector. As a result, a substantial 24.5-dB SNR gain can be
obtained by the joint ML detector in comparison to the ZF-ML detector, which
is mainly because that the ZF-ML detector suffers from severe noise
amplification and error propagation. Moreover, it can be further seen that the
ZF-DNN detector with an optimal 140-dB training SNR can achieve comparable BER
performance as the joint ML detector in the high SNR region, suggesting the
excellent error performance of the ZF-DNN detector under the condition of
accurate CSI for ZF equalization. Finally, for our proposed CSI-free blind DNN
detector, we investigate the impact of training SNR on its error performance
and three different training SNRs of 130, 140 and 150 dB are considered. It is
clearly shown that the CSI-free blind DNN detector with 140-dB training SNR
achieves nearly the same BER performance as the joint ML detector across the
whole SNR region, which slightly outperforms the ZF-DNN detector in the low
SNR region. However, the joint ML detector outperforms the CSI-free blind DNN
detector when a lower training SNR of 130 dB or a higher training SNR of 150
dB is adopted, and the reasons can be explained as follows. The DNN module can
better learn the statistics of the noise with a relatively small training SNR,
while the statistics of the data symbols can be more accurately learned when
the training SNR is relatively large. As a result, there exists an optimal
training SNR which can make a trade-off for the DNN module to learn the
statistics of both the noise and the data symbols and hence lead to a minimum
overall BER. When the receiver is moved to the corner of the receiving plane,
as shown in Figs 4(b), we can observe that the joint ML detector outperforms
the ZF-ML detector by an SNR gain of more than 40 dB at BER = $10^{-3}$, while
the ZF-DNN detector with an optimal training SNR of 160 dB obtains near-
optimal BER performance as the joint ML detector only for relatively low BERs.
Furthermore, the CSI-free blind DNN detector achieves comparable BER
performance as the joint ML detector in the high SNR region, which outperforms
the ZF-DNN detector in the low SNR region. It should be noted that an error
floor occurs for the CSI-free blind DNN detector with a lower training SNR of
140 dB, which is mainly due to the insufficient learning of the statistics of
the data symbols under a very noisy environment.

The BER versus transmitted SNR for GOSMP is plotted in Fig. 5. As we can see,
the ZF-DNN detector with an optimal training SNR can achieve very close
performance as the joint ML detector when the receiver is located at the
center of the receiving plane, but it performs worse than the joint ML
detector when the receiver is moved to the corner, especially in the low SNR
region. In contrast, the proposed CSI-free blind DNN detector can achieve
comparable BER performance as the joint ML detector for both center and corner
receiver locations. Moreover, error floors occur for the CSI-free blind DNN
detector when the adopted training SNR is too small or too large. It can be
further observed from Figs. 4 and 5 that the optimal training SNRs for the ZF-
DNN detector and the CSI-free blind DNN detector at the same receiver location
are generally the same in GOMIMO systems.

###  IV-D Impact of Input Pre-processing

Figure 6: BER comparison of the proposed CSI-free blind DNN detector where the
feed-forward DNN module having different inputs for both GOSM and GOSMP at the
center of the receiving plane.

(a) GOSM

(b) GOSMP

Figure 7: BER vs. $\textrm{log}_{10}\alpha$ of the proposed CSI-free blind DNN
detector for (a) GOSM and (b) GOSMP.

It can be seen from Fig. 2 that the pre-processing module, which pre-processes
the input of the feed-forward DNN module, plays a vital role to guarantee that
the proposed CSI-free blind DNN detector can successfully perform MIMO
detection blindly without the need of CSI. In the next, we evaluate the impact
of input pre-processing on the performance of the CSI-free blind DNN detector.
Here, two different inputs of the feed-forward DNN module are considered: one
is $\alpha\bf{y}$, i.e., the pre-processing module only performs amplitude
scaling, and the other is $\alpha\bf{Fy}$, i.e., the pre-processing module
performs both amplitude scaling and feature extraction. Fig. 6 compares the
BER performance of the proposed CSI-free blind DNN detector where the feed-
forward DNN module having different inputs for both GOSM and GOSMP with the
receiver located at the center of the receiving plane. As we can see, for
GOSM, the BER performance is only slightly improved when the input is changed
from $\alpha\bf{y}$ to $\alpha\bf{Fy}$, and the SNR gain at BER = $10^{-3}$ is
only 0.8 dB. In contrast, for GOSMP, a noticeable BER improvement can be
obtained by replacing the input $\alpha\bf{y}$ with $\alpha\bf{Fy}$, and the
corresponding SNR gain at BER = $10^{-3}$ is increased to 2.4 dB. The
difference between BER improvements for GOSM and GOSMP can be explained as
follows. As discussed in Section III.D, since the feature matrix $\bf{F}$
contains the spatial mapping information of GOMIMO systems, the feed-forward
DNN module can use the spatial mapping information to remove the channel
crosstalk. As a result, the feed-forward DNN module with input $\alpha\bf{Fy}$
can efficiently mitigate the adverse effect of error propagation. However, in
GOSM systems, the activated LEDs are used to transmit the same signal and
hence error propagation only leads to reduced diversity gain, which might not
significantly degrade the BER performance. In contrast, since the activated
LEDs transmit different signals in GOSMP systems, error propagation leads to
the missing of constellation information and hence results in significant BER
degradation.

(a) GOSM

(b) GOSMP

Figure 8: Computation time comparison of the proposed CSI-free blind DNN
detector and three benchmark detectors for (a) GOSM and (b) GOSMP at the
center of the receiving plane.

Due to the substantial path loss during MIMO transmission, the received signal
needs to be properly amplified before it can be fed into the feed-forward DNN
module. Figs. 7(a) and (b) show the BER versus $\textrm{Log}_{10}\alpha$ with
different transmitted SNRs for GOSM and GOSMP, respectively. For GOSM, as
shown in Fig. 7(a), we can observe that a feasible range of $\alpha$ is around
[$10^{5}$, $10^{7}$] when the receiver is located at the center of the
receiving plane. Moreover, the feasible range of $\alpha$ keeps the same for
different transmitted SNR values. When the receiver is moved to the corner,
the feasible range of $\alpha$ is [$10^{5}$, $10^{8}$]. For GOSMP, as shown in
Fig. 7(b), the same feasible range of $\alpha$ is obtained as that of GOSM
when the receiver is located at the center of the receiving plane. However,
the feasible range of $\alpha$ for GOSMP is only around $10^{6}$ when the
receiver is moved to the corner. To successfully implement the proposed CSI-
free blind DNN detector, the proper $\alpha$ value with respect to each
receiver location is determined in advance.

###  IV-E Computational Complexity

Finally, we evaluate the computational complexity of the proposed CSI-free
blind DNN detector and compare it with other benchmark detectors, in terms of
computation time [28]. For both the CSI-free blind DNN detector and the ZF-DNN
detector, once the detector has been successfully trained, it can be used for
MIMO detection for a long period of time without further retraining, unless
the system parameters such as receiver location have been changed [22]. Hence,
only the computational complexity of the online detection process is
considered for the CSI-free blind DNN detector and the ZF-DNN detector, while
the complexity of the offline training process is not taken into account.
Figs. 8(a) and (b) compare the computation time of the proposed blind DNN
detectors and the other three benchmark detectors for GOSM and GOSMP,
respectively. As we can see, for GOSM, the CSI-free blind DNN detector, the
ZF-DNN detector and the ZF-ML detector require nearly the same computation
time which is less than 3 seconds. However, the joint ML detector requires
totally 48.42 seconds to finish the computation, which is significantly longer
than that of the other three detectors. It is the same for GOSMP that the CSI-
free blind DNN detector, the ZF-DNN detector and the ZF-ML detector require
comparable computation time, which is much shorter than that required by the
joint ML detector. Therefore, the proposed CSI-free blind DNN detector
achieves near optimal BER performance as the joint ML detector, but with
substantially lower computational complexity.

##  V Conclusion

In this paper, we have for the first time proposed a novel DeepGOMIMO
framework for GOMIMO systems, where a DNN-based detector is specially designed
to realize CSI-free blind detection of the received MIMO signals. The proposed
CSI-free blind DNN detector contains a pre-processing module and a feed-
forward DNN module, which are used to address the adverse effects of MIMO
transmission and to perform joint detection of spatial and constellation
information, respectively. It is shown by our simulation results that, in a
typical indoor 4 $\times$ 4 MIMO-OWC system adopting both GOSM and GOSMP with
unipolar non-zero 4-PAM modulation, the CSI-free blind DNN detector achieves
comparable BER performance as the optimal joint ML detector, which greatly
outperforms the ZF-ML detector. Moreover, the CSI-free blind DNN detector, the
ZF-DNN detector and the ZF-ML detector require nearly the same computation
time to perform detection, which is significantly shorter than that required
by the joint ML detector. In addition, compared with the ZF-DNN detector, the
CSI-free blind DNN detector can achieve improved achievable data rate and
reduced communication time delay since it does not require instantaneous
channel estimation to obtain accurate CSI for ZF equalization. In conclusion,
our proposed DeepGOMIMO can be a potential candidate for the implementation of
practical high-speed and low-complexity OWC systems.

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