# OFDM-Based Optical Spatial Modulation

Anil Yesilkaya, Rui Bian, Iman Tavakkolnia, and Harald Haas Manuscript
received December 31, 2018; revised April 06, 2019; accepted May 16, 2019.
Date of publication XXXX XX, 2019; date of current version XXXX XX, 2019. The
guest editor coordinating the review of this paper and approving it for
publication was Prof. Ananthanarayanan Chockalingam.Authors are with the LiFi
Research and Development Centre, Institute for Digital Communications, School
of Engineering, University of Edinburgh, Edinburgh, EH9 3FD, UK. Anil
Yesilkaya, Rui Bian and Iman Tavakkolnia contributed equally to this work.

###### Abstract

SM has proven to be a promising multiple-input-multiple-output (MIMO)
technique which provides high energy efficiency and reduces system complexity.
In SM, only one transmitter is active at any given time while the rest of them
remain silent. The index of the active transmitter carries information. This
spatial information is in addition to the data carried by the constellation
symbols in the signal domain. Therefore, SM increases the transmission rate of
the communication system compared to single-input-single-output and space-time
block coding (STBC)-MIMO. For signal domain data encoding, OFDM has been
widely adopted. The key benefits in multi-carrier intensity-modulation and
direct-detection (IM/DD) systems are: i) the capability to achieve high
spectral efficiency and ii) the ability to effectively mitigate direct-current
(DC) wander effects and the impact of ambient light. However, current off-the-
shelf light emitting diodes (LEDs) which are used as transmit entities are
primarily bandwidth limited. Thus, there are benefits of combining SM and OFDM
to enhance transmission speeds while maintaining low complexity. In this
paper, the two most common OFDM-based SM types, namely frequency domain SM
(FD-SM) and time domain SM (TD-SM), are investigated for optical wireless
communications (OWC). Moreover, proof-of-concept experimental results are
presented to showcase practical feasibility of both techniques. The obtained
results are also compared with Monte Carlo simulations. The results show that
TD-SM with an optimal maximum-a-posteriori-probability (MAP) detector
significantly outperforms FD-SM. It can be inferred from the results that TD-
SM is a strong candidate among OFDM-based optical SM systems for future
optical IM/DD wireless communication systems.

###### Index Terms:

Spatial modulation (SM), Orthogonal frequency division multiplexing (OFDM),
Light fidelity (LiFi), Optical wireless communications (OWC)

††publicationid: pubid: © 2019 IEEE. Digital Object Identifier
10.1109/JSTSP.2019.2920577

##  I Introduction

Advanced capabilities of mobile devices such as virtual/augmented reality
(VR/AR), high definition video/audio streaming, low-latency gaming and
internet-of-things (IoT) create a data-greedy ecosystem. The recent forecasts
show that the global mobile data traffic will reach 49 exabytes
($\textrm{10}^{\textrm{18}}$ bytes) by 2021[1]. It is also reported in [2]
that if the mobile data demand keeps increasing at the same rate as that of
the last 10 years, the radio frequency (RF) spectrum will be completely
saturated by 2035. Thus, the higher frequency portion of the spectrum is
starting to be considered as a viable solution for the potential congestion.
Similar to this trend, optical wireless communications (OWC) offers the
utilization of the optical portion of the electromagnetic spectrum e.g.,
visible light (VL), infra-red (IR) and ultra-violet (UV) to create a wireless
broadband medium between the ends. Similarly, the light-fidelity (LiFi) which
uses VL in the downlink and IR in the uplink forms a bi-directional optical
attocellular network that supports multi-user connectivity and seamless
handover[3]. In LiFi, ordinary off-the-shelf light emitting diodes (LEDs) and
photo-diodes (PDs) will serve as the transmit and receive units, respectively.
As the VL and IR rays are naturally blocked by opaque walls, LiFi could
achieve very high area spectral efficiency due to significantly reduced co-
channel interference in dense network deployments. Moreover, similar features
help enhance security and reduce latency in LiFi [4, 5]. The information is
carried by fluctuations in light intensity which is detected by the front-end
optics and decoded at the receiver (RX). This incoherent transmission
technique is referred to as intensity modulation and direct detection (IM/DD).
Since the optical power of the LEDs are intrinsically limited to be real and
positive valued, the information bearing signal is also limited by the same
fundamental constraints. Furthermore, eye safety and linearity considerations
as well as the dynamic range of the front-end opto-electronic elements bring
additional limitations to the transmitted signal. Therefore, techniques
developed and optimized for RF systems cannot be applied straightforwardly to
IM/DD systems without significant changes.

The limited electrical bandwidth of the optical elements motivates the use of
spatial multiplexing (SMX) [6, 7, 8, 9, 10, 11, 12] in OWC systems. For a
$N_{\textrm{t}}$ LED and $N_{\textrm{r}}$ PD multiple-input-multiple-output
(MIMO)-OWC system, the data rate could be enhanced
$\min\\{N_{\textrm{t}},N_{\textrm{r}}\\}$-fold by harnessing SMX. However, the
inter-channel interference (ICI) caused by channel coupling and detection
complexity at the RX side are the major drawbacks of SMX. Spatial modulation
(SM) is an alternative MIMO technique which avoids ICI and simplifies the RX
complexity in exchange for reduced data rates [13, 14, 15, 16, 17, 18]
compared to ordinary SMX systems. In basic SM systems, only a single
transmitter out of an array of transmitters is active at any given time.
Therefore, the ICI at the receiver side is completely eliminated along with
reduced search space at the detector. Furthermore, the channel matrix
effectively becomes a column vector which ensures the invertibility of the
channel coupling as long as there are no null elements. The active transmitter
is random and is selected based on a subset of information bits. The index of
the active transmitter carries the spatial symbol while the emitted signal
from the active transmitter conveys the constellation symbol simultaneously.
The adoption of orthogonal frequency division multiplexing (OFDM) in SM is
proposed for the first time in [16]. OFDM in conjunction with SM applied to
optical systems is proposed in [19]. In the proposed system, each symbol
conveyed by a subcarrier is also accompanied by a spatial symbol such that for
each subcarrier there is only one active transmitter. This technique will be
referred to as frequency domain SM (FD-SM) throughout the paper. It should be
noted that FD-SM cannot fully benefit from the advantages of initial SM
systems. As each LED is associated with an OFDM stream, $N_{\textrm{t}}$
modulators and $N_{\textrm{r}}$ demodulators are required at the transmitter
(TX) and RX sides, respectively. Moreover, ICI in FD-SM becomes inevitable due
to the non-zero time domain samples. Hence, the system performance could
become severely degraded if the channel matrix is ill conditioned.

In order to address the issues associated with FD-SM, an alternative method is
proposed in [20] for the optical systems which will be referred to as time
domain SM (TD-SM) throughout the paper. In TD-SM, after parallel-to-serial
conversion at the OFDM modulator, each time domain sample is accompanied by a
spatial symbol. Accordingly, each time domain sample carries a spatial symbol
along with a constellation symbol. Thus, a single OFDM modulator and
demodulator pair suffice. Moreover, ICI is also completely avoided since only
a single transmitter is active per time instant as the SM principle suggests.
The TD-SM technique has been advanced and combined with an optimal maximum-a-
posteriori-probability (MAP) detector in [21].

Another unique application of SM to IM/DD systems is the spatial complex-
bipolar number encoding. Unlike conventional SM, the spatial domain is
exploited to realize the transmission of complex and bipolar valued signals in
IM/DD systems. In [22], a $2\times 2$ system is proposed in which the positive
valued real OFDM time domain samples are transmitted from the first LED.
Moreover, the absolute value of the negative samples are also sent
simultaneously from the second LED in the proposed method. Thus, the
constraint of having strictly positive valued transmission signal as imposed
by IM/DD is efficiently overcome. Note that in [22], the reality of the time
domain samples are ensured by applying Hermitian symmetry in the frequency
domain. Similarly, another $2\times 2$ technique is given in [23], suggesting
the transmission of the positive valued real and imaginary parts from the
first and second LEDs, respectively. Note that the Hermitian symmetry
requirement is removed in [23] by spatial encoding and the positive values are
ensured by a direct-current (DC) bias and zero clipping. In [24, 25, 26], both
limitations of IM/DD are jointly compromised by the proposed $4\times 4$
system. Accordingly, positive real and imaginary parts of OFDM time domain
samples are transmitted from the first and third LEDs while absolute values of
the negative real and imaginary from the second and fourth LEDs, respectively.
Therefore, neither Hermitian symmetry nor DC bias are required in [24, 25,
26]. These spatial real/complex and positive/negative encoding systems are
referred to as complex-bipolar encoded SM (CBE-SM) throughout the paper.
Similar to FD-SM, CBE-SM cannot also benefit from the advantages of SM with
increasing $N_{\textrm{t}}$. For large MIMO systems, CBE-SM introduces ICI and
complexity penalties. For convenience and as an overview, OFDM-based optical
SM systems are summarized in Fig. 1.

In this paper, the system models for both FD-SM and TD-SM are introduced. A
bit error ratio (BER) performance comparison for both systems is conducted
using Monte Carlo simulations. Lastly, the comparison is corroborated for the
first time by an experimental proof-of-concept study of both FD-SM and TD-SM
techniques.

The rest of the paper is organized as follows: a brief history of spatial
index modulation is provided in Section II. The channel characteristics for
the OWC systems are given in Section III. The details of the system models of
FD-SM and TD-SM systems are given in Section IV. In Section V, the
experimental setup and methodology are introduced. Computer simulations and
experimental results are provided in Section VI. Finally, conclusions are
drawn in Section VII.

_Notation_ : Throughout the paper, matrices and column vectors are in bold
uppercase and bold lowercase letters, respectively. The $m^{\textrm{th}}$ row
and $n^{\textrm{th}}$ column element of a matrix $\mathbf{A}$ is given by
$A(m,n)$. Similarly, the $k^{\textrm{th}}$ element of a vector $\mathbf{a}$ is
denoted by $a(k)$. The transpose of a matrix/vector and the Euclidean norm of
a vector are expressed by $(\cdot)^{\textrm{T}}$ and $||{\cdot}\rvert\rvert$,
respectively. A real normal distribution with mean $\mu$ and variance
$\sigma^{2}$ is represented by $\mathcal{N}(\mu,\sigma^{2})$. The argument
maximum, minimum, argument minimum, complex conjugate, modulus and statistical
expectation operators are denoted by $\arg\max\\{\cdot\\}$, $\min\\{\cdot\\}$,
$\arg\min\\{\cdot\\}$, $(\cdot)^{\ast}$, $|\cdot|$ and
$\textrm{E}\\{\cdot\\}$, respectively. The dot product of two vectors
$\mathbf{a}$ and $\mathbf{b}$ is given by $\mathbf{a}\cdot\mathbf{b}$. The
continuous time unit step, Dirac delta and Q functions are expressed by
$u(\cdot)$, $\delta(\cdot)$ and $\textrm{Q}(\cdot)$, respectively. The
Q-function is defined by
$\textrm{Q}(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-(u^{2}/2)}du$. Also,
the discrete time Dirac delta function is given by $\delta[\cdot]$. Lastly,
the $m\times 1$ zeros vector is denoted by $\mathbf{0}_{m\times 1}$.

ptOFDM-Based Optical SMptFrequency Domain[19, 22, 27, 28, 29]ptTime Domain[20,
21]ptCB Encoded[22, 24, 25, 26, 23, 30]

Figure 1: Taxonomy of the OFDM-based optical SM systems. 1965 Permutation
Modulation[31]1990 Antenna Hopping [32]2001 Space Shift Keying[33]2002 Spatial
Data Multiplexing(SDM)[13]2004 AH capacity analysis[34]2005 Benefits of SDM
[14]2006 Spatial Modulation[15]/with OFDM [16]2008 Generalized SSK[35]2010
Generalized SM[36, 37], FBE-SM[38], DSTSK[39]2012 MA-GSM[40], GSSK-OWC[41]2015
Quadrature SM[42], Enhanced SM[43] Figure 2: History of the SM-based systems.

##  II A brief history of spatial index modulation

A brief summary of spatial index modulation techniques applied to RF and OWC
systems is given in Fig. 2. The very first idea of encoding information to the
permutation of the transmit entities is shown by Slepian in 1965 as
permutation modulation (PM) [31]. PM paved the way for SM systems which later
became a part of the more generalized concept called index modulation. The
interested reader may be referred to the following overview paper on SM [44].
It took 25 years for the first application of PM to appear in the spatial
domain, referred to as antenna hopping (AH), which is proposed in [32]. The
capacity analysis for the channel/antenna hopping technique is carried out in
[34]. In 2001, a space modulation technique referred to as space shift keying
(SSK) which uses different channel conditions to encode and decode the
information is proposed [33]. Unlike SM, SSK only encodes information in
spatial symbols, and it does not encode information in the signal space. As a
consequence, SSK exhibits an additional complexity reduction, but the
achievable spectral efficiencies are limited. SSK is exhaustively analyzed for
RF communication systems under different channel conditions [45, 46]. A novel
data multiplexing technique is proposed in [13] for a $4\times 4$ system that
carries all the information of the four streams by activating a single antenna
per transmission step. However, the system required parity bits which
compromised spectrum efficiency. The work in [13] has been extended in [14]
where the limitations of parity bits is overcome completely. The new technique
was referred to as SM for the first time in [15]. In 2006, an OFDM-based SM is
also proposed[16]. The generalization of SSK, namely generalized SSK (GSSK) is
proposed in 2008[35]. The number of active antennas in GSSK could be
arbitrarily chosen which increases the spectral efficiency significantly. In a
similar manner, the generalization of SM, namely generalized SM (GSM) is
proposed in 2010 [36, 37]. It is worth noting that the active transmitters in
GSM transmit the same information in order to avoid ICI. In the same year, the
first application of SM to OWC systems is also presented[47]. Furthermore, a
novel technique to allow arbitrary number of transmitters in SM systems namely
fractional bit encoded SM (FBE-SM) is also proposed[38]. Another novel method
which constitutes the framework of the differential SM systems, namely
differential space-time shift keying (STSK) is proposed in [39]. In
differential STSK, the necessity for the receiver side channel state
information is removed. In [40] and [48], another generalization of SM is
proposed, such that each active antenna transmits different signal in exchange
for ICI and increased detection complexity. The generalization of SSK for OWC
by relaxation of the number of active LEDs is proposed in [41]. Accordingly,
the number of active LEDs could take any value per transmitted symbol which
increases the spectral efficiency significantly. In 2015, a technique, named
quadrature SM (QSM) to enhance the spectral efficiency of SM is proposed[42].
In QSM, the I (in-phase) and Q (quadrature) parts of a complex bipolar
constellation symbol are separated and transmitted from antennas chosen
independently. It is important to note that the chosen antenna for I and Q
could coincide, which will not introduce ICI as both parts are orthogonal. As
both I and Q parts carry different spatial symbols, the total data carried in
QSM becomes larger compared to conventional SM. Unlike conventional SM
methods, two antennas are active per given time instant in QSM unless I and Q
parts of the symbol concur. This basic idea is used to enhance the number of
transmit symbol possibilities in a technique referred to as enhanced SM (ESM)
[43]. In ESM, two different constellation sets, $M$-ary and $\frac{M}{2}$-ary,
are employed as primary and secondary modulations. Hence, $M$-ary
constellation is employed if the number of active antennas is one where
$\frac{M}{2}$-ary constellation is used when there are two active antennas. It
should also be noted that the number of the transmit combinations in both QSM
and ESM are the same, though ESM has more flexibility on the transmit
constellation design.

##  III OWC Channel Model

Figure 3: Geometrical model of the point-to-point optical wireless link Figure
4: The block diagram of the FD-SM system.

In this section, the properties of the optical wireless channel are presented.
Throughout the paper, a $N_{\textrm{t}}\times N_{\textrm{r}}$ non-imaging
receiver based point-to-point MIMO-OWC link will be considered. It is reported
in [49] that the multipath effects arising from the side walls, ceiling and
floor reflections are negligible as long as a dominant line-of-sight (LoS)
link exists. Thus, only the LoS DC channel gains are considered in this work.
It is worth noting that all the LEDs are assumed to be operating within their
linear dynamic range. The geometric parameters of the OWC link are depicted in
Fig. 3. Accordingly, the position vectors of the $i^{\textrm{th}}$ LED and
$j^{\textrm{th}}$ PD are given by $\mathbf{l}_{i}$ and $\mathbf{p}_{j}$,
respectively. Moreover, the normal vectors of the $i^{\textrm{th}}$ LED and
$j^{\textrm{th}}$ PD are denoted by $\mathbf{n}_{i}$ and $\mathbf{o}_{j}$,
respectively. The LoS DC optical wireless channel gains between
$i^{\textrm{th}}$ LED and $j^{\textrm{th}}$ PD are given as follows[50]:

$H(j,i)=\frac{(m+1)A_{\textrm{PD}}}{2\pi||{\mathbf{d}_{j,i}}\rvert\rvert^{2}}\cos^{m}(\theta_{j,i})\cos(\psi_{j,i})\mathds{1}_{\kappa_{1/2}}\left(\psi_{j,i}\right),$
(1)

where $i\in\\{1,2,\cdots,N_{\textrm{t}}\\}$ and
$j\in\\{1,2,\cdots,N_{\textrm{r}}\\}$. The parameter $m$ is the Lambertian
mode number which is defined by $m=-1/\log_{2}(\cos(\Phi_{1/2}))$. The semi-
angle of the half power of the LEDs is denoted by $\Phi_{1/2}$. Note that all
the LEDs and PDs are assumed to be identical. The detection area of the PDs is
given by $A_{\textrm{PD}}$. The radiance and incidence angles between
$i^{\textrm{th}}$ LED and $j^{\textrm{th}}$ PD are denoted by $\theta_{j,i}$
and $\psi_{j,i}$, respectively. The Euclidean distance vector from
$j^{\textrm{th}}$ PD to $i^{\textrm{th}}$ LED is given by

$\mathbf{d}_{j,i}=\mathbf{l}_{i}-\mathbf{p}_{j}.$

The radiance and incidence angles could alternatively be represented as
follows:

$\theta_{j,i}=\arccos\frac{-\mathbf{n}_{i}\cdot\mathbf{d}_{j,i}}{||{\mathbf{d}_{j,i}}\rvert\rvert}~{}~{}\textrm{and}~{}~{}\psi_{j,i}=\arccos\frac{\mathbf{o}_{j}\cdot\mathbf{d}_{j,i}}{||{\mathbf{d}_{j,i}}\rvert\rvert}.$
(2)

By combining (1) and (2), we can express the DC channel gains by

$H(j,i)=\frac{A_{\textrm{PD}}(m+1)(-\mathbf{n}_{i}\cdot\mathbf{d}_{j,i})^{m}(\mathbf{o}_{j}\cdot\mathbf{d}_{j,i})}{2\pi||{\mathbf{d}_{j,i}}\rvert\rvert^{m+3}}\mathds{1}_{\kappa_{1/2}}\left(\psi_{j,i}\right).$
(3)

The indicator function $\mathds{1}_{\kappa_{1/2}}\left(\cdot\right)$,
determines whether the incident rays are in the field-of-view (FoV) of the PD
or not

$\mathds{1}_{\kappa_{1/2}}\left(x\right)=\left\\{\begin{array}[]{lr}1,&\text{if
}|x|\leq\kappa_{1/2}\\\ 0,&\text{otherwise}.\end{array}\right.$

The FoV semiangle of the PD is denoted by $\kappa_{1/2}$. Consequently, the
$N_{\textrm{r}}\times N_{\textrm{t}}$ channel matrix could be obtained as
$\mathbf{H}=H(j,i),~{}\forall i,j$.

Like in any communication system, the performance of OWC systems is heavily
affected by noise. However, in OWC, there are two major types of noise
sources, namely i) shot noise and ii) thermal noise. The photo-generated shot
noise occurs due to the discrete excitement levels in the photo-detector
devices. For an OWC link, the shot noise, which generally follows a Poisson
distribution, arises due to the transmitted signal and ambient light. Unlike
shot noise, thermal noise is due to the front-end circuitry and thus signal
independent. Thermal noise could be modeled by a Gaussian distribution[51]. In
practical LiFi systems, the DC forward current of LEDs are relatively high
compared to the information carrying signal in order to meet the illumination
requirements. Hence, the overall noise is dominated by the photo-induced shot
noise. Moreover, the high intensity shot noise consists of many independent
filtered Poisson processes. Under these circumstances the central limit
theorem can be applied and the total noise typically follows a Gaussian
distribution[52]. Consequently, the effective noise in OWC systems shows
additive white Gaussian noise (AWGN) characteristics.

##  IV OFDM-Based Optical Spatial Modulation

In this section, we provide a deeper insight into OFDM-based SM for MIMO-OWC
systems. Then, the FD-SM approach and the recently proposed TD-SM technique
will be explained in greater detail.

Figure 5: The block diagram of the TD-SM system.

###  IV-A Conventional Frequency Domain SM

FD-SM is a widely adopted OFDM-based SM technique used in both RF and optical
domains. The block diagram of the $N_{\textrm{t}}\times N_{\textrm{r}}$ FD-SM
system is given in Fig. 4. The incoming user bits are partitioned into
$(k_{\textrm{c}}+k_{\textrm{s}})$-bits where the number of constellation and
spatial bits are given by $k_{\textrm{c}}=\log_{2}(M)$ and
$k_{\textrm{s}}=\log_{2}(N_{\textrm{t}})$, respectively. The constellation
modulation order is denoted by $M$. The following one-to-one mapping encodes
locations and values of active subcarriers to a
$\left(\frac{N}{2}-1\right)\times N_{\textrm{t}}$ subcarriers matrix
$\mathbf{Q}$. The parameter $N$ denotes the fast Fourier transform (FFT) size.
Accordingly, the columns of $\mathbf{Q}$ represent independent OFDM streams,
which will each be associated with an LED. The rows of $\mathbf{Q}$ indicate
the subcarrier level spatial information. In other words, the index and value
of an active subcarrier convey the spatial and constellation symbols,
respectively. The active subcarriers are modulated by $M$-ary quadrature
amplitude modulation (QAM) ($M$-QAM). Following the SM principle, only a
single subcarrier has a non-zero value per row of $\mathbf{Q}$. The
$i^{\textrm{th}}$ column vector of $\mathbf{Q}$ is denoted by
$\mathbf{q}_{i}$. The probability mass function (PMF) of the independently
identically distributed (i.i.d.) elements of $\mathbf{q}_{i}$ is given by

$p_{\mathbf{q}_{i}}[w]=\frac{1+\left(M(N_{\textrm{t}}-1)-1\right)\delta[|w|]}{MN_{\textrm{t}}},$
(4)

for $w\in\\{0,\mathcal{C}\\}$. The $M$-QAM constellation alphabet is denoted
by $\mathcal{C}$. Thus, the electrical power of $\mathbf{q}_{i}$ becomes
$\textrm{E}\\{\mathbf{q}_{i}^{2}\\}=1/N_{\textrm{t}}$. Note that the average
electrical power of the $M$-QAM constellation symbols are set to unity. In
order to ensure that the resultant signal is real after the inverse FFT (IFFT)
operation, Hermitian symmetry is imposed on $\mathbf{Q}$ as follows:

$\mathbf{S}=[\mathbf{0}_{N_{\textrm{t}}\times
1}~{}~{}\mathbf{Q}^{\textrm{T}}~{}~{}\mathbf{0}_{N_{\textrm{t}}\times
1}~{}~{}\tilde{\mathbf{Q}}^{\textrm{T}}]^{\textrm{T}}.$ (5)

The conjugate symmetric matrix of $\mathbf{Q}$ is given by
$\tilde{\mathbf{Q}}$. Therefore, the elements of $\mathbf{S}$ become,
$S(k,i)=S(N-k,i)^{\ast},~{}\forall i,$ where $k\in\\{0,1,\cdots,N-1\\}$ and
$S(0,i)=S(N/2,i)=0,~{}\forall i$. Then, the real and bipolar time domain
sample vectors are obtained by applying an $N$-point IFFT to the
$i^{\textrm{th}}$ column vector of $\mathbf{S}$,

$\mathbf{x}_{i}=\mathbf{F}^{-1}\mathbf{s}_{i},\quad\textrm{where
}i\in\\{1,2,\cdots,N_{\textrm{t}}\\}.$ (6)

The $N\times N$ IFFT matrix is given by $\mathbf{F}^{-1}$. Moreover, the
$i^{\textrm{th}}$ column vector of the resultant $N\times N_{\textrm{t}}$ time
domain samples matrix $\mathbf{X}$ is denoted by $\mathbf{x}_{i}$. For a
relatively large IFFT size ($N>64$), the real and bipolar i.i.d.
$\mathbf{x}_{i}$ vectors are approximated with a Gaussian distribution,
$\mathbf{x}_{i}\sim\mathcal{N}(0,\sigma^{2})$ where
$\sigma^{2}=\frac{1}{NN_{\textrm{t}}}$. In order to satisfy the unipolarity
restriction of IM/DD systems, a DC bias is introduced to $\mathbf{x}_{i}$ and
the remaining negative samples are clipped to zero. The element-wise biasing
and clipping effects are given by
$\bar{\mathbf{x}}_{i}=\left[\mathbf{x}_{i}+B\right]^{U}_{L}$, where
$[x]^{U}_{L}\coloneqq L+\left(\min\\{x,U\\}-L\right)u(x-L)$. The lower and
upper clipping values are denoted by $L$ and $U$, respectively. The DC bias
value is defined by $B=r\sigma$ where $r$ is the bias proportionality
constant. The DC bias level in decibels is defined by
$B_{\textrm{dB}}=10\log_{10}(r^{2}+1)$. The probability density function (PDF)
of the biased and clipped elements of the $\bar{\mathbf{x}}_{i}$ is given by

$\begin{split}f_{\bar{\mathbf{x}}_{i}}(v)&=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-(v-B)^{2}/2\sigma^{2}}\Pi\left(v\right)\\\
&+\textrm{Q}\left(\frac{B-L}{\sigma}\right)\delta(v-L)+\textrm{Q}\left(\frac{U-B}{\sigma}\right)\delta(v-U),\end{split}$
(7)

where $\Pi(v)=u(v-L)-u(v-U)$. Note from (7) that the locations of the lower
clipped samples are completely random in
$\bar{\mathbf{X}}=[\bar{\mathbf{x}}_{1}~{}~{}\bar{\mathbf{x}}_{2}~{}~{}\cdots~{}~{}\bar{\mathbf{x}}_{N_{\textrm{t}}}]$.
Even for the best case scenario, where $L=0$, the likelihood that a single
element in $\bar{\mathbf{X}}$ per row will become non-zero is extremely low.
Therefore, ICI is unavoidable in FD-SM. The average transmitted electrical
power per LED for the FD-SM is calculated by using (7) as follows:

$\begin{split}\textrm{E}\\{\bar{\mathbf{x}}_{i}^{2}\\}&=\frac{\sigma}{\sqrt{2\pi}}\left((L+B)e^{\frac{-(L-B)^{2}}{2\sigma^{2}}}-(U+B)e^{\frac{-(U-B)^{2}}{2\sigma^{2}}}\right)\\\
&+(B^{2}-L^{2}+\sigma^{2})\textrm{Q}\left(\frac{L-B}{\sigma}\right)\\\
&+(U^{2}-B^{2}-\sigma^{2})\textrm{Q}\left(\frac{U-B}{\sigma}\right)+L^{2}.\end{split}$
(8)

Therefore, the total transmit electrical power of FD-SM becomes,
$P_{\textrm{e,FD-SM}}=N_{\textrm{t}}\textrm{E}\\{\bar{\mathbf{x}}_{i}^{2}\\}$.
After the IFFT, a cyclic prefix (CP) sequence of length $N_{\textrm{CP}}$ is
added to each $\bar{\mathbf{x}}_{i}$. Lastly, digital-to-analog converted
$(N+N_{\textrm{CP}})$-length column vectors are fed to the $N_{\textrm{t}}$
LEDs in a serial fashion. The spectral efficiency of FD-SM could be determined
by using time ($G_{\textrm{T}}$) and frequency domain ($G_{\textrm{F}}$)
utilization factors as follows:

$\begin{split}\eta_{\textrm{FD-
SM}}&=\frac{1}{2}\left(k_{\textrm{c}}+k_{\textrm{s}}\right)G_{\textrm{F}}G_{\textrm{T}}\\\
&=\frac{1}{2}\left(k_{\textrm{c}}+k_{\textrm{s}}\right)\left(\frac{N-2}{N}\right)\left(\frac{N}{N+N_{\textrm{CP}}}\right)\\\
&\approx\frac{1}{2}\left(k_{\textrm{c}}+k_{\textrm{s}}\right)\quad\textrm{bits
per channel use (bpcu)}.\end{split}$ (9)

The factor $1/2$ in (9) stems from the Hermitian symmetry imposed on
$\mathbf{S}$.

At the receiver side, the optical signals are converted back to the electrical
domain by the PDs. The $N_{\textrm{r}}\times N$ real and positive valued
baseband received signal matrix after the CP removal and analog-to-digital
conversion is given by

$\mathbf{Y}=\mathbf{H}\bar{\mathbf{X}}^{\textrm{T}}+\mathbf{N},$ (10)

where $\mathbf{N}$ denotes the $N_{\textrm{r}}\times N$ AWGN matrix. The
$j^{\textrm{th}}$ row vector of $\mathbf{N}$ follows an i.i.d Gaussian
distribution, $\mathbf{n}_{j}\sim\mathcal{N}(0,\sigma_{\textrm{N}}^{2})$ for
$j\in\\{1,2,\cdots,N_{\textrm{r}}\\}$. The $N_{\textrm{r}}\times
N_{\textrm{t}}$ optical channel impulse response matrix, detailed in the
previous section, is denoted by $\mathbf{H}$. For simplicity, only the flat
frequency response channels are considered in both simulations and experiments
throughout the paper. The process of obtaining flat channels in experimental
set-up will be detailed in the Section V. Note that OFDM would still
outperform the single carrier systems due to its inherent resilience to DC
wander effects and slow ambient light fluctuations (below the minimum
modulation frequency which is typically around 1 MHz). The digital-to-analog,
electrical-to-optical, optical-to-electrical and analog-to-digital conversion
coefficients are taken as unity without loss of generality.

In order to detect the transmitted symbols, the channel coupling effects in
the received signal needs to be removed first. Therefore, the channel
decoupled received signal matrix which is obtained after the feed-forward
equalization is given by $\hat{\mathbf{Y}}=\mathbf{W}\mathbf{Y}$. For the sake
of simplicity, the zero-forcing (ZF) equalizer, which is widely adopted in the
literature, is chosen as the feed-forward equalizer,
$\mathbf{W}=\mathbf{H}^{-1}$. It should be noted that the channel matrix
$\mathbf{H}$ becomes singular if the OWC link geometry has perfect symmetry.
In other words, either rows and/or columns of $\mathbf{H}$ become linearly
dependent. In practice, the perfect symmetry of the link is impossible to
maintain due to the non-ideal geometry and characteristics of the front-end
opto-electronics. However, the similarity between the rows and/or columns of
the channel matrix would bring significant performance degradation. Hence,
non-singular channel matrices with various condition numbers are considered
throughout this paper without loss of generality. The ZF equalized and serial-
to-parallel converted $N\times N_{\textrm{r}}$ received time domain samples
matrix is given by
$\hat{\mathbf{Y}}=\bar{\mathbf{X}}+\mathbf{N}^{\textrm{T}}\left(\mathbf{H}^{\textrm{T}}\right)^{-1}$.
The $N$-point FFT of the $i^{\textrm{th}}$ column vector of $\hat{\mathbf{Y}}$
is obtained by $\hat{\mathbf{s}}_{i}=\mathbf{F}\hat{\mathbf{y}}_{i}$ for
$i\in\\{1,2,\cdots,N_{\textrm{r}}\\}$. After the FFT, the ZF equalized
subcarriers matrix becomes
$\hat{\mathbf{S}}=[\hat{\mathbf{s}}_{1}~{}~{}\hat{\mathbf{s}}_{2}~{}~{}\cdots~{}~{}\hat{\mathbf{s}}_{N_{\textrm{r}}}]$.
Then, the data carrying active subcarrier indexes (spatial symbols) and their
values (constellation symbols) are extracted from the $\hat{\mathbf{S}}$. For
the optimal detection, both spatial and constellation symbols must be decoded
simultaneously[35]. The joint maximum-likelihood (ML) detector for the spatial
and constellation symbols is given as follows:

$\begin{split}\left(\hat{j}(i),\hat{c}(i)\right)&=\arg\max\limits_{\begin{subarray}{c}1\leq
j\leq N_{\textrm{r}}\\\ \forall
c\in\mathcal{C}\end{subarray}}p\left(\hat{S}(i,j)~{}|~{}c\right)\\\
&=\arg\min\limits_{\begin{subarray}{c}1\leq j\leq N_{\textrm{r}}\\\ \forall
c\in\mathcal{C}\end{subarray}}||\hat{S}(i,j)-c||,\\\ &\quad\forall
i\in\\{0,1,\cdots,N-1\\}.\end{split}$ (11)

In (11), the constellation symbol drawn from the $M$-QAM set is denoted by
$c\in\mathcal{C}$. Moreover, the estimated spatial and constellation symbols
are stored for each subcarrier in
$\hat{\mathbf{j}}=[\hat{j}(0),\hat{j}(1),\cdots,\hat{j}(N-1)]^{\textrm{T}}$
and
$\hat{\mathbf{c}}=[\hat{c}(0),\hat{c}(1),\cdots,\hat{c}(N-1)]^{\textrm{T}}$,
respectively. Finally, the transmitted user bits are simply obtained back by
inputting $\hat{\mathbf{j}}$ and $\hat{\mathbf{c}}$ into the inverse mapping
function.

(a) $M=4$

(b) $M=256$

Figure 6: Spectral efficiency comparison for TD-SM, FD-SM and RC techniques.

###  IV-B Time Domain SM

The fundamental difference between FD-SM and TD-SM is that FD-SM dismisses the
main advantages of SM while TD-SM retains all of them at the cost of stringent
time synchronization requirement. Furthermore, the achievable spectral
efficiency in TD-SM is significantly higher compared to FD-SM due to the time
domain spatial symbol mapping. The block diagram of TD-SM is given in Fig. 5.
First, the incoming user bits are partitioned into
$(k_{\textrm{c}}+k_{\textrm{s}})$-bits similar to the FD-SM. Then,
$k_{\textrm{c}}$ bits are used to modulate the $(\frac{N}{2}-1)\times 1$
subcarriers vector $\mathbf{q}$ by $M$-QAM. It should be noted that in TD-SM
technique, no spatial information exists in the frequency domain. Again, the
Hermitian symmetry is imposed on the frequency domain frame as follows:

$\mathbf{s}=[0~{}~{}\mathbf{q}^{\textrm{T}}~{}~{}0~{}~{}\tilde{\mathbf{q}}^{\textrm{T}}]^{\textrm{T}},$
(12)

where $\tilde{\mathbf{q}}$ is the conjugate symmetric vector of $\mathbf{q}$.
Therefore, $s(k)=s(N-k)^{\ast}$ for $k\in\\{0,1,\cdots,N-1\\}$. Due to the
time domain spatial symbol encoding, a single OFDM modulator/demodulator pair
is always suffice in TD-SM. Hence, the $N\times 1$ real and bipolar time
domain samples vector $\mathbf{x}$ is obtained after the IFFT operation by
$\mathbf{x}=\mathbf{F}^{-1}\mathbf{s}$. In order to make the resultant signal
unipolar, the vector $\mathbf{x}$ is DC-biased and clipped similar to the FD-
SM. The PDF for the elements of the biased and clipped vector
$\bar{\mathbf{x}}$ before the spatial symbol encoding is the same as in (7).
In TD-SM, only a single LED is transmitting a non-zero value per time instant.
The transmitting LED is determined by the spatial mapping of $k_{\textrm{s}}$
bits as suggested by the SM principle. To this end, in TD-SM method, $N$ time
domain samples are available to encode spatial symbols in contrast to FD-SM
which encodes $\frac{N}{2}-1$ subcarriers for that purpose. The PDF for the
elements in column vectors of the transmit matrix after the spatial symbol
encoding is expressed by using (7) as follows:

$f_{\bar{\mathbf{z}}_{i}}(v)=\frac{f_{\bar{\mathbf{x}}}(v)(1-\delta(v))+(N_{\textrm{t}}-1)\delta(v)}{N_{\textrm{t}}}.$
(13)

The $i^{\textrm{th}}$ column vector for the TD-SM transmission matrix
$\bar{\mathbf{Z}}$ is denoted by $\bar{\mathbf{z}}_{i}$. The PDF of the biased
and clipped transmit signal vector, $f_{\bar{\mathbf{x}}}(v)$, follows (7). By
using (13), the average transmitted electrical power per LED in TD-SM becomes

$\textrm{E}\\{\bar{\mathbf{z}}^{2}_{i}\\}=\frac{\textrm{E}\\{\bar{\mathbf{x}}^{2}\\}}{N_{\textrm{t}}}.$
(14)

where $\textrm{E}\\{\bar{\mathbf{x}}^{2}\\}$ can be calculated by using (8).
Thus, the total electrical transmit power of TD-SM is also given by
$P_{\textrm{e,TD-SM}}=\textrm{E}\\{\bar{\mathbf{x}}^{2}\\}$. It can be deduced
from (13) that even under the worst case scenario, where $L\neq 0$, only a
single LED will be active per time in TD-SM which implies ICI-free
transmission. As a result, noise and complexity enhancement emerging from the
ZF equalization at the receiver side are avoided in TD-SM. It should also be
noted that OWC systems must satisfy the minimum illumination requirements
along with communication functionality. Accordingly, the average illumination
provided by the TD-SM technique will be lower compared to FD-SM due to
$N_{\textrm{t}}-1$ idle LEDs in each time instant. However, this problem can
easily be overcome by introducing a proper DC bias to the idle LEDs without
causing any performance degradation. Note that the illumination bias is
intrinsically different than $B$ where $B$ is essential for signal
transmission in IM/DD. Moreover, the illumination bias will not influence the
effective signal-to-noise-ratio (SNR) of the system as it does not carry any
information. Thus, the illumination bias is omitted in both simulations and
experiments without loss of generality. Moving on, CP added and digital-to-
analog converted vectors are fed to the LEDs in a serial manner. Thus, the
spectral efficiency of the TD-SM becomes

$\begin{split}\eta_{\textrm{TD-
SM}}&=\frac{1}{2}k_{\textrm{c}}G_{\textrm{F}}G_{\textrm{T}}+k_{\textrm{s}}\\\
&\approx\frac{1}{2}k_{\textrm{c}}+k_{\textrm{s}}\quad\textrm{bpcu}.\end{split}$
(15)

It is obvious from (9) and (15) that the amount of data carried by the
constellation symbols are the same in both FD-SM and TD-SM. However, due to
the time-domain encoding of the spatial symbols, the amount of data conveyed
in the spatial domain is doubled in the TD-SM technique. The spectral
efficiency comparisons between TD-SM, FD-SM and repetition coding (RC) are
given for various numbers of transmitters, $N_{\textrm{t}}$, and various
signal constellation sizes, $M$, in Fig. 6. In RC, all the LEDs transmit the
same signal [11]. It is observed from Fig. 6 that TD-SM achieves the highest
spectral efficiency among the considered techniques for any number of LEDs and
constellation orders.

At the RX side, the optical signal is captured and translated into the
electrical domain by the PDs. After analog-to-digital conversion and CP
removal, the real and positive valued baseband received electrical signal
vector per time instant becomes

Input : $\mathbf{y}$, $\mathbf{H}$, $U$, $L$, $B$, $\sigma$ and
$\sigma_{\textrm{n}}$.

Output : Estimated time domain sample and spatial symbol vectors
$\hat{\bar{\mathbf{x}}}=[\hat{\bar{x}}(0),\hat{\bar{x}}(1),\cdots,\hat{\bar{x}}(N-1)]^{\textrm{T}}$
and
$\hat{\mathbf{i}}=[\hat{i}(0),\hat{i}(1),\cdots,\hat{i}(N-1)]^{\textrm{T}}$,
respectively.

Initialize;

for  _t=0:(N-1)_ do

$\hat{i}(t)=\arg\min\limits_{1\leq i\leq
N_{\textrm{t}}}\left\\{||\mathbf{y}-\mathbf{h}_{i}\hat{\bar{x}}^{i}(t)||^{2}+\frac{\sigma_{\textrm{n}}^{2}\hat{\bar{x}}^{i}(t)}{\sigma^{2}}\left(\hat{\bar{x}}^{i}(t)-2B\right)\right\\}$;

$\hat{\bar{x}}(t)=\hat{\bar{x}}^{\hat{i}(t)}(t)$;

end for

Algorithm 1 The algorithm for the joint MAP estimator.

$\mathbf{y}=\mathbf{h}_{i}\bar{x}(t)+\mathbf{n},$ (16)

where $\mathbf{h}_{i}$ denotes the $i^{\textrm{th}}$ column vector of
$\mathbf{H}$. The $t^{\textrm{th}}$ captured OFDM time sample is given by
$\bar{x}(t)$ for $t\in\\{0,1\cdots,N-1\\}$. The $N_{\textrm{r}}\times 1$ AWGN
vector is denoted by $\mathbf{n}$, which follows an i.i.d. Gaussian
distribution, $\mathbf{n}\sim\mathcal{N}(0,\sigma_{\textrm{n}}^{2})$. The
spatial symbols in TD-SM must be decoded in the time domain. According to
[20], the channel coupling effect on the spatial symbols is simply reversed by
the ZF equalizer. Please note that the performance degradation that emerges
from the (near) rank deficient matrices in ZF equalization of FD-SM is
completely avoided in TD-SM. The effective channel is always rank one in TD-SM
as seen from (16). Therefore, the channel coupling can simply be reversed by
the multiplication of reciprocals of the corresponding column vector elements.
In other words, unlike FD-SM, there is always a solution for ZF equalization
in TD-SM as long as $H(j,i)\neq 0,~{}\forall i,j$. Due to the truncated
Gaussian distributed characteristics of the time domain samples, the ZF
detector proposed in [20] becomes sub-optimal. In [21], the optimal MAP
estimator which jointly detects the transmitted samples and the LED indexes is
proposed. Note that the optimal MAP estimator utilizes the PDF of the time
domain samples as well as the variance of the effective noise in the
estimation process. The MAP estimator used to detect the time domain samples
is formulated by

(a)

(b)

Figure 7: The block diagram (a) and photo (b) of the experimental set-up.

$\hat{\bar{x}}(t)=\arg\max\limits_{\bar{x}(t)}p\left(\mathbf{y}~{}|~{}\bar{x}(t)\right)p\left(\bar{x}(t)\right).$
(17)

If we plug (7) into (17), and after simple manipulations, the MAP estimation
of the transmitted sample becomes

$\begin{split}\hat{\bar{x}}(t)&=\arg\min\limits_{\bar{x}(t)}\left\\{\mathcal{M}^{\textrm{MAP}}\right\\}\\\
&=\arg\min\limits_{\bar{x}(t)}\left\\{\sigma^{2}||{\mathbf{y}-\mathbf{h}_{i}\bar{x}(t)}\rvert\rvert^{2}+\sigma_{\textrm{n}}^{2}\left(\bar{x}(t)-B\right)^{2}\right\\}.\end{split}$
(18)

By taking the derivative of (18) and equating it to zero, we obtain the joint
estimation of the $t^{\textrm{th}}$ sample value and $i^{\textrm{th}}$ active
LED index as follows:

$\hat{\bar{x}}^{i}(t)=\left[\frac{\sigma^{2}(\mathbf{y}^{\textrm{T}}\mathbf{h}_{i}+\mathbf{h}_{i}^{\textrm{T}}\mathbf{y})+2B\sigma_{\textrm{n}}^{2}}{2(\sigma^{2}\mathbf{h}_{i}^{\textrm{T}}\mathbf{h}_{i}+\sigma_{\textrm{n}}^{2})}\right]^{U}_{L}.$
(19)

Note that (19) is conditioned on $i$. Hence, the active LED index is estimated
by the minimization of the MAP metric, $\mathcal{M}^{\textrm{MAP}}$, given in
(18). After substituting $\bar{x}(t)$ in $\mathcal{M}^{\textrm{MAP}}$ by (19),
the estimated active LED index becomes

$\begin{split}&\hat{i}(t)=\arg\min\limits_{1\leq i\leq
N_{\textrm{t}}}\left\\{\mathcal{M}^{\textrm{MAP}}\left(\hat{\bar{x}}^{i}(t)\right)\right\\}\\\
&=\arg\min\limits_{1\leq i\leq
N_{\textrm{t}}}\left\\{||\mathbf{y}-\mathbf{h}_{i}\hat{\bar{x}}^{i}(t)||^{2}+\frac{\sigma_{\textrm{n}}^{2}\hat{\bar{x}}^{i}(t)}{\sigma^{2}}\left(\hat{\bar{x}}^{i}(t)-2B\right)\right\\}.\end{split}$
(20)

Then, the transmitted OFDM sample value is determined by plugging (20) back
into (19). For further clarity, the joint MAP estimation procedure is
summarized in Algorithm 1. Then, the FFT of the estimated samples vector is
taken by using $\hat{\mathbf{s}}=\mathbf{F}\hat{\bar{\mathbf{x}}}$. Finally,
the transmitted user bits are recovered by demapping $\hat{\mathbf{s}}$ and
$\hat{\mathbf{i}}$ jointly.

##  V Experimental Set-up and Methodology

In this section, the $4\times 4$ OFDM-based optical SM experimental set-up is
explained along with the corresponding methodology for evaluating the
performance of the system.

The block diagram and the photo of the experimental set-up are shown in Figs.
7(a) and 7(b), respectively. In the setup, $N_{\textrm{t}}=4$, VLMS1500-GS08
red LEDs are used where a Thorlabs ACL4532 aspheric condenser lens is placed
in front of each LED to collimate the output light of the LED. Then, four
digital signals are created in MATLAB using a laptop according to methods
described in previous sections (i.e., TD-SM or FD-SM). In order to comply with
the flat frequency response assumption, only 20 MHz of modulation bandwidth is
considered. Four channels of a high speed arbitrary waveform generator (AWG),
Keysight M8195A, is used to generate analog signals from the incoming digital
samples. The sampling rate of the AWG is 16 GSa/s and the resolution of the
built-in digital-to-analog converter (DAC) unit is 8 bits. Signals generated
by the AWG are amplified by Mini-Circuits ZHL-1A-S+ amplifier modules (AMP).
The DC bias voltage and the forward current of LEDs are fixed at 2.3 V and 100
mA, respectively, which are controlled by a DC power supply. These values are
chosen to minimize the nonlinear effect of LEDs. The DC bias is superimposed
with the analog information carrying signal through a bias-tee module, Mini-
Circuits ZFBT-4R2GW and the resulting signal is fed to the LEDs. The link
distance between the TX and RX is 80 cm.

At the receiver, $N_{\textrm{r}}=4$, LEC-RP0508 high sensitivity positive-
intrinsic-negative (PIN) PDs are used in four receiver circuits as shown in
Fig. 8. An operational amplifier chip, Texas Instruments LMH6629, is
incorporated into the circuit. Subsequently, signals are captured by four
channels of the Agilent DSA90804A high-speed oscilloscope (OSC). Lastly, the
captured signals are sent to the laptop for post-processing in MATLAB.

Figure 8: Schematic presentation of PD circuit. Figure 9: Channel frequency
responses of each LED for their dominant links. Figure 10: An example frame of
transmitted signals from different LEDs and the original digital signal for
the TD-SM system.

The channel matrix is estimated by training sequences, e.g., known OFDM
symbols, prior to data transmission. Note that any error in the channel
estimation may degrade the system performance. It is shown in the next section
that simulation results match the experimental results, and thus, channel
estimation in the assumed system model does not produce large errors.
Furthermore, the likelihood of significant channel estimation errors is
limited due to the quasi-static characteristics of the indoor optical wireless
communications channel. The channel states remain stationary since the
experimental setup does not include any moving components or other factors
that may alter the channel condition. Therefore, an accurate initial channel
estimation is valid throughout the signal transmission. The strongest front-
end channel frequency responses, including the optical and electrical
components, are shown in Fig. 9 for each pair of transmitters and receivers.
It is observed that frequency responses of the front-end system are almost
flat within a 20 MHz modulation bandwidth, i.e., the frequency response stays
within the 3 dB margin. The noise variance is estimated by comparing the
oversampled received signal and the known transmitted signal. The estimated
value is used for TD-SM signal detection (see (19)). Part of the oversampled
time domain signals and original OFDM samples for TD-SM are depicted in Fig.
10. It can be seen in Fig. 10 that at specific time samples, which are
designated for data transmission and detection, only one LED transmits a non-
zero value. Obviously, synchronization of signals at the TX and RX is of vital
importance. In this work, synchronization at the transmitter is ensured by
using the single high accuracy AWG with a built-in synchronization system.
Moreover, at the beginning of each generated signal, a unique sequence of
binary data is placed. The binary sequence is used to synchronize the received
signals after they are captured by the oscilloscope.

As described in section IV-B and observed in Fig. 10, an LED is active (i.e.,
“ON”) only when it is selected according to the spatial information. Thus, if
an LED is inactive (i.e., “OFF”), it must be completely turned off, meaning
that the DC bias is also zero. However, as seen in Fig. 7(a), the DC bias is
provided by an external DC power supply which cannot be controlled by the
digital input signal. Therefore, the DC bias is constantly applied to LEDs.
This is different from the theoretical system model and the TD-SM scheme. In
order to solve this problem, the bipolar signal is modified before adding the
DC bias. For each LED, zero values at the “OFF” time instants are replaced by
a low negative value. Thus, the summation of the constant DC bias and the
bipolar signal yields a very small value at the corresponding time instants.
Note that this modification is only necessary due to the hardware limitation
in our experimental setup, and it could be rectified in future practical
implementations of the TD-SM system, where the DC bias and the signal are
generated and controlled at the same time.

Figure 11: BER for diagonal channel $\mathbf{H}_{1}$.

##  VI Results and discussion

In this section, the BER performance of the described methods are compared. In
order to ensure a fair comparison, the spectral efficiency of all methods is
fixed at $\eta_{\textrm{FD-SM}}=\eta_{\textrm{TD-SM}}=4$ bpcu. Thus, 16-QAM
and 64-QAM modulation formats are considered for TD-SM and FD-SM,
respectively. The SNR is calculated as the ratio between the total received
electrical power and the noise at the RX. Monte-Carlo simulation results are
also presented. The optical channel gains and the noise variance obtained from
the experimental results are used for Monte-Carlo simulations. The number of
subcarriers and the DC bias are chosen as $N=256$ and $B_{\mathrm{dB}}=10$ dB,
respectively, for both the experimental system and the computer simulations.

Firstly, the optical lenses in front of the LEDs are adjusted such that a
diagonal channel matrix is realized. In other words, there is no overlap
between optical links. The condition number of the channel matrix, $\rho$
defined as the ratio of the maximum eigenvalue to the minimum eigenvalue of
the matrix, is equal to 1 in this case. The corresponding channel matrix is
referred to as $\mathbf{H}_{1}$. Secondly, optical lenses are moved slightly
in order to make the light spots larger at the receiver side so that they
overlap with each other. Consequently, another channel matrix, namely
$\mathbf{H}_{2}$, is obtained with a condition number greater than one, and in
this case $\rho=3.5$. In that scenario, almost 5% of the transmitted optical
power from each LED is detected by two other PDs in addition to the main PD.

Figure 12: BER for non-diagonal channel $\mathbf{H}_{2}$ with condition number
$\rho=3.5$.

The BER performance results are presented in Figs. 11 and 12. Different SNR
values are obtained by scaling the digital signal at the AWG prior to
transmission. It is observed that in both scenarios, TD-SM with either MAP
(TD-SM-MAP) or ZF (TD-SM-ZF) detection outperforms FD-SM by about 7 dB. It is
also shown that the simulation results closely follow the experimental BER
measurement for both FD-SM and TD-SM. Note that the same measured channel
matrix from the experiment is used for simulations. In both Figs. 11 and 12,
the BER of TD-SM-MAP is slightly lower than that of TD-SM-ZF. It is worth
noting that higher SNR values would not be achievable in the experimental
system due to the limited output optical power of the LEDs. In order to
achieve high SNR regime, the peak-to-peak amplitude of the transmit signal has
to be increased which in turn worsens the effect of nonlinearity. The
nonlinearity introduced by the front-end opto-electronic devices can simply be
overcome by pre-distortion of the transmit signal[53]. For the sake of
brevity, both FD-SM and TD-SM systems are presented with their simplest form
without loss of generality.

It is important to emphasize that the difference between the FD-SM and TD-SM
systems would be more significant in scenarios where the channel is ill-
conditioned. As mentioned in the Section IV-B and presented in [21], TD-SM
exhibits inherent resilience against ill conditioned channels compared to FD-
SM due to a single active LED per time instant. Thus, the effective channel in
TD-SM is a column vector which is a rank one system. However, due to physical
limitations in the experimental setup, we were not able to realize channel
matrices with condition numbers higher than 3.5. Moreover, a larger overlap
between different links (i.e., a larger light spot at the RX) significantly
decreased the received optical power, and consequently, all system structures
could only be realized in their low SNR regime. Hence, the experimental
demonstration of such scenarios is the subject of future work. In [53], it is
shown that a larger array of transmitters and receivers may also change the
channel condition number. In order to investigate this case, simulation
results for a 16$\times$16 system are presented in Fig. 13 with
$\eta_{\textrm{FD-SM}}=\eta_{\textrm{TD-SM}}=5$ bpcu and two different channel
condition numbers of $\rho=1$ and $\rho=400$. To achieve the same spectral
efficiency, 64-QAM and 4-QAM modulation formats are employed in FD-SM and TD-
SM, respectively. It is observed that TD-SM-MAP achieves the lowest BER in
both channel conditions. For the $\rho=1$ case, TD-SM with either MAP or ZF
results in significantly lower BER compared to FD-SM due to the lower
modulation order $M$. A different behavior is observed for the $\rho=400$
case. While both FD-SM and TD-SM-ZF fail, TD-SM-MAP demonstrates better
performance because it benefits from MAP detection.

In future work, the effect of temporal dispersion and intersymbol interference
(i.e., effectively a frequency selective channel) on both FD-SM and TD-SM will
be investigated. As the spatial symbols only exist in the time domain in TD-
SM, a joint time-frequency domain detector will also be investigated to
resolve the issue.

Figure 13: BER results for a 16$\times$16 system with $\eta_{\textrm{FD-
SM}}=\eta_{\textrm{TD-SM}}=5$ bpcu and channel condition numbers $\rho=1$ and
$\rho=400$.

##  VII Conclusion

In this paper, two of the major OFDM-based optical SM techniques, namely FD-SM
and TD-SM were investigated. The spatial symbols are encoded in the subcarrier
level in FD-SM which brings ICI, TX/RX complexity and spectral efficiency
loss. Conversely, in TD-SM technique, the spatial symbols are encoded in the
time domain as suggested by conventional SM. It has been shown that TD-SM
inherits all the merits of conventional SM. Moreover, TD-SM techniques achieve
significantly higher spectral efficiency compared to FD-SM. Both systems have
been validated in terms of BER performance by Monte Carlo computer simulations
as well as, for the first time ever, experimental results. It has also been
shown by using extensive simulations and demonstrating the proof-of-concept
experimental results that the TD-SM with an optimal MAP detector outperforms
FD-SM in terms of the BER performance. Consequently, the experimental results
verify and suggest that TD-SM is a viable candidate for next-generation OFDM-
based SM.

## Acknowledgment

Professor Haas gratefully acknowledge the support of this research by the
Engineering and Physical Sciences Research Council (EPSRC) under an
Established Career Fellowship grant, EP/R007101/1. He also acknowledges the
financial support of his research by the Wolfson Foundation and the Royal
Society. Anil Yesilkaya acknowledges financial support for his PhD studies
from Zodiac Inflight Innovations (TriaGnoSys GmbH). The authors would like to
thank Dr. Tezcan Cogalan and Dr. Hossein Kazemi for their insightful comments
and suggestions.

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|  Anil Yesilkaya (S’11) received the B.Sc (1st Hons) and M.Sc degrees in
electronics engineering from the Kadir Has University, Istanbul, Turkey in
2014 and 2016, respectively. He was the recipient of the Best Paper Award at
the IEEE International Conference on Communications, in 2018. He is currently
pursuing the Ph.D. degree in digital communications with the University of
Edinburgh. His research interests include, PHY layer security, multiple-input
multiple-output optical wireless communications and LiFi-based in-flight
connectivity.
---|---
|  Rui Bian received the B.Eng. degree in information engineering from the
Wuhan University of Technology, Wuhan, China, in 2011, and the M.Sc. degree in
electronics from the University of Edinburgh, Edinburgh, U.K., in 2013, where
he is currently pursuing the Ph.D. degree in digital communications. His main
research interest is in visible light communication.
---|---
|  Iman Tavakkolnia (S’15–M’18) received the B.Sc. degree in telecommunication
engineering from the University of Tehran, Tehran, Iran, in 2006, the M.Sc.
degree in communication systems from the Sharif University of Technology,
Tehran, Iran, in 2011, and the Ph.D. degree in electrical engineering from the
University of Edinburgh, Edinburgh, U.K., in 2018. He is currently a Research
Associate with LiFi Research and Development Centre, University of Edinburgh.
His research interests include communication theory, optical fiber
communication, and visible light communication.
---|---
|  Harald Haas (S’98-AM’00-M’03-SM’16-F’17) received the Ph.D. degree from The
University of Edinburgh in 2001. He is currently the Chair of Mobile
Communications at The University of Edinburgh, and he is the Initiator, Co-
Founder, and Chief Scientific Officer of pureLiFi Ltd., and the Director of
the LiFi Research and Development Centre, The University of Edinburgh. He has
authored 430 conference and journal papers, including a paper in Science and
co-authored a book Principles of LED Light Communications Towards Networked
Li-Fi (Cambridge University Press, 2015). His main research interests are in
optical wireless communications, hybrid optical wireless and RF
communications, spatial modulation, and interference coordination in wireless
networks. He first introduced and coined spatial modulation and LiFi. LiFi was
listed among the 50 best inventions in TIME Magazine in 2011. He was an
invited speaker at TED Global 2011, and his talk on Wireless Data from Every
Light Bulb has been watched online over 2.4 million times. He gave a second
TED Global lecture in 2015 on the use of solar cells as LiFi data detectors
and energy harvesters. This has been viewed online over 2 million times. He
was elected as a fellow of the Royal Society of Edinburgh and a Fellow of the
IEEE in 2017. In 2012 and 2017, he was a recipient of the prestigious
Established Career Fellowship from the Engineering and Physical Sciences
Research Council (EPSRC) within Information and Communications Technology in
the U.K. In 2014, he was selected by EPSRC as one of ten Recognizing
Inspirational Scientists and Engineers Leaders in the U.K. He was a
corecipient of the EURASIP Best Paper Award for the Journal on Wireless
Communications and Networking in 2015 and the Jack Neubauer Memorial Award of
the IEEE Vehicular Technology Society. In 2016, he received the Outstanding
Achievement Award from the International Solid State Lighting Alliance. He was
a co-recipient of recent best paper awards at VTC-Fall, 2013, VTC-Spring 2015,
ICC 2016, ICC 2017 and ICC 2018. He is an Editor of the IEEE TRANSACTIONS ON
COMMUNICATIONS and the IEEE JOURNAL OF LIGHTWAVE TECHNOLOGIES.
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