A Novel OFDM-Based Time Domain Quadrature GSM for Visible Light Communication System

Abstract

In order to improve the spectral efficiency (SE) as well as the receiver performance of band-limited visible light communications (VLCs), two orthogonal frequency division multiplexing (OFDM)-based quadrature generalized multiple-input multiple-output (QG-MIMO) transmission schemes, including time domain (TD) quadrature generalized spatial modulation (TD-QGSM) and TD quadrature generalized spatial multiplexing (TD-QGSMP), are proposed in this paper. Firstly, the constellation symbols in the frequency domain are split into in-phase and quadrature components to perform the OFDM modulation separately. Then, the corresponding time domain signal is spatially mapped on different light emitting diodes (LEDs) for achieving the diversity or multiplexing. In addition, we also propose an illegal vector correction (IVC)-based orthogonal matching pursuit (OMP) detection algorithm to deal with the error propagation and noise amplification effect, where a novel correction criterion is involved for assisting the index vectors estimation and thus for improving the demodulation performance. The simulation results demonstrate that the SE can be significantly improved by the proposed schemes as compared with the existing OFDM-based generalized MIMO schemes, with the TD-QGSM increasing by at least 56.5% and the TD-QGSMP increasing by at least 72.3%. Moreover, the bit error rate (BER) performance can be further improved when applying the proposed IVC-OMP detection method, which outperforms the traditional maximum-likelihood and maximum ratio combining (ML-MRC) detection by at least 62.5%.

1. Introduction

The upcoming 6G communications will have 100–1000 times higher transmission rates than 5G networks, which are expected to fulfill the emerging requirements in challenging applications, including virtual presence, industry 4.0, and personalized health services [1]. Nevertheless, the current frequency bands of radio frequency (RF) communication are limited and may not be sufficient to meet the growing demands. Therefore, exploring new frequency sources to address the spectrum scarcity issue has become very important [2]. Visible light communication (VLC) technologies have been widely considered as a promising RF-alternative option for future indoor communications due to their abundant license-free spectrum, no electromagnetic interference radiation, and simultaneous illumination and communication [3,4]. Due to the low cost and simple modulation, light emitting diodes (LEDs) are usually employed as the optical transmitters in the VLC systems [5]. However, the bandwidth of LEDs is limited; thus, the achievable spectral efficiency (SE) of the VLC systems will be affected to a certain extent as the broadband signal is injected [6]. Various methods have been proposed to address the above issue. Advanced modulation formats that can effectively utilize the available bandwidth have been extensively investigated in VLC systems [7,8]. In addition, the advanced transmission schemes, such as the multiple-input multiple-output (MIMO) architecture, can also be deployed in VLC to enhance the system capacity under the fixed modulation bandwidth [9].

Spatial modulation (SM) and spatial multiplexing (SMP) are types of MIMO schemes that can be used in VLC systems, which have undergone a great deal of research attention due to their negligible inter-channel interference and low implementation complexity [10,11]. However, achieving the higher SE was still challenging since the signal transmission is only implemented through a single activated LED while the remaining are just utilized for illumination. To address this issue, researchers proposed generalized spatial modulation (GSM), which can selectively activate multiple LEDs for efficient data transmission in the spatial domain [12,13]. In addition, as a multicarrier modulation technology, orthogonal frequency division multiplexing (OFDM) is capable of making full use of the limited bandwidth, and can be employed to combat the inter symbol-interference introduced by multiple reflections from the walls of the indoor environment. By involving the OFDM in the GSM scheme, the capacity of the VLC system can be increased consequently. In [14], two OFDM-based VLC-GSM schemes were proposed where one is a frequency domain GSM (FD-GSM) and the other is a time domain GSM (TD-GSM). However, the realizable SE contributed by the constellation part remains unchanged from that of the traditional SM, which constrains the total achievable SE. In [15], the OFDM-based FD-generalized SMP (FD-GSMP) and TD-generalized SMP (TD-GSMP) were proposed to improve the achievable SE. However, the system complexity is very high and it is also difficult to achieve a good trade-off between the SE and the BER performance, which is the critical challenge in practical applications.

Furthermore, to enhance the reception performance, numerous research studies have been devoted to the design of favorable detection methods. A maximum-likelihood (ML) detection algorithm that jointly demodulates the transmitter serial number and the transmit symbols was proposed in [16], which required many iterations over all template information, therefore yielding an excellent BER performance. However, the ML detection algorithm possesses extremely high complexity. To solve this problem, the maximum ratio combining (MRC) method, which combines the signals by multiplying different coefficients to different signals of the diversity, is employed in MIMO systems. Compared to selective combining and equal gain combining, the MRC can both achieve better BER performance and have much lower complexity than the ML method [17,18]. Moreover, the ML-MRC detection can also be deployed in OFDM-based generalized MIMO systems, which have less complexity than the traditional ML algorithm [15,19]. Nevertheless, the detection of the transmitter sequence number will seriously affect the symbol correctness in the following demodulation steps, leading to an unexpected BER performance reduction. Hence, developing an efficient detection algorithm with low complexity and satisfactory BER performance remains an urgent issue. In [20], a sparse signal reconstruction detection method combining greedy algorithm and ML was proposed for generalized space shift keying (GSSK) in a VLC system with low complexity and good BER performance. However, it cannot be directly used in VLC-GSM-based multicarrier transmission schemes.

In this paper, two novel OFDM-based quadrature generalized MIMO (QG-MIMO) transmission schemes, in terms of TD quadrature GSM (TD-QGSM) and TD quadrature GSMP (TD-QGSMP), are proposed in a VLC system, which can significantly improve the SE performance while achieving an excellent BER performance compared to the existing OFDM-based generalized MIMO schemes. In addition, by utilizing the spatial sparsity, the compressed sensing (CS) theory is involved and an illegal vector correction (IVC)-based orthogonal matching pursuit (OMP) detection algorithm is proposed to reconstruct the sparse received signal from its compressed observations with high probability by employing a correction criterion to adjust the estimated index vector when it fell into the illegal detection region. The simulation results show that the proposed QG-MIMO schemes can provide excellent SE and BER performance, which becomes more apparent with high modulation order. Moreover, the BER performance can be further improved by the proposed IVC-based OMP detection algorithm as compared with the traditional detection methods.

The remainder of the paper is organized as follows. Section 2 presents the system model of a general indoor VLC-MIMO system. In Section 3, two novel OFDM-based QG-MIMO schemes are proposed, and the analytical SE is also calculated. In Section 4, the CS-based detection algorithm is proposed to improve the demodulation performance and the computational complexity is also analyzed. In Section 5, the simulation results and discussions are demonstrated. In Section 6, the conclusion of this paper is provided.

Notation: Let ${\left(·\right)}^T$ denote the transpose of a vector or matrix and $⌊·⌋$ represent the floor operator, respectively. $\mathcal{C}\left(\mathcal{W},\mathcal{Z}\right)$ is used to denote the binomial coefficient as $\mathcal{Z}$ out of $\mathcal{W}$ is chosen. In addition, ${\left(·\right)}^{-1}$ and ${∥·∥}_2$ are used to denote the inverse of a matrix and the $L_2$-norm operation, respectively.

2. Channel Model

Figure 1 shows the geometrical setup of a general indoor VLC-MIMO system, which consists of $N_t$ LEDs and $N_r$ photodiodes (PDs). The LEDs are mounted on the ceiling for stable illumination and the user is equipped with multiple PDs, which are all oriented vertically downwards and upwards, respectively. In addition, we assume that the PDs are located at 0.85 m above the receiving plane. Since the total received optical power of the line-of-sight (LOS) link exceeds at least 95% of the total amount, only the LOS link transmissions are considered in this work.

Assume L is the inverse fast Fourier transform (IFFT) length always employed in the OFDM module. Let $\mathbf{H}\in {\mathbb{R}}^{N_r\times N_t}$ denote the MIMO channel matrix, $\mathbf{s}\in {\mathbb{R}}^{N_t\times L}$ denote the transmitted signal matrix, and $\mathbf{n}\in {\mathbb{R}}^{N_r\times L}$ denote the additive white Gaussian noise (AWGN) with zero mean. Thus, the received signal $\mathbf{y}\in {\mathbb{R}}^{N_r\times L}$ can be expressed by

$$\mathbf{y}=\mathbf{Hs}+\mathbf{n}.$$

(1)

In addition, $\mathbf{H}$ can be determined by

$$H=\left[\begin{array}{ccc}{h}_{11}& \cdots & h_{1N_t}\\ ⋮& \ddots & ⋮\\ h_{N_{r}1}& \cdots & h_{N_r{N}_t}\end{array}\right],$$

(2)

where $h_{\lambda \mu }\left(\lambda =1,2,\dots ,N_r;\mu =1,2,\dots ,N_t\right)$ is the direct-current channel gain between the $\mu$-th LED and the $\lambda$-th PD. As the LEDs follow the Lambertian radiation pattern [21], $h_{\lambda \mu }$ can be calculated by

$$h_{\lambda \mu }=\frac{\left(w+1\right)\rho A}{2\pi d_o^2}{cos}^w\left({\phi }_o\right)T_s\left({\theta }_o\right)g\left({\theta }_o\right)cos\left({\theta }_o\right),$$

(3)

where $w=-ln2/ln\left(cos\left(\Psi \right)\right)$ represents the Lambertian emission order, $\Psi$ is the semi-angle at half power of the LED, $\rho$ is the responsivity, and A is the PD’s active area. In addition, $d_o$, ${\phi }_o$, and ${\theta }_o$ are used to denote the distance, the angle of emission, and the angle of incidence between the $\mu$-th LED and the $\lambda$-th PD, respectively. $T_s\left({\theta }_o\right)$ is the gain of optical filter, $g\left({\theta }_o\right)=\frac{n^2}{{sin}^2\Phi }$ is the gain of optical lens, where n is the refractive index and $\Phi$ is the half-angle field of view (FOV) of the optical lens.

3. Two Proposed OFDM-Based QG-MIMO Schemes

In this section, the fundamental principles of two novel OFDM-based QG-MIMO transmission schemes, including OFDM-based TD-QGSM and TD-QGSMP, are proposed first. Then, the theoretical SE performance of the two proposed schemes is analyzed accordingly.

3.1. TD-QGSM

The schematic diagram of the OFDM-based TD-QGSM system is shown in Figure 2, where M-ary quadrature amplitude modulation (M-QAM) is adopted in OFDM modulation. As shown in Figure 2, first, the input bits are divided into two parts for mapping in two different dimensions. In each time slot, we choose N LEDs for activation. Therefore, the first $⌊lo{g}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋$ bits are encoded in the spatial dimension based on gray coding while the remaining $2\times {log}_{2}M$ bits are used to map the constellation symbols. We split all the constellation symbols into real and imaginary parts; thus, the total constellation symbols vector $\mathbf{c}$ can be described as $\mathbf{c}={\mathbf{c}}_{Re}+j\times {\mathbf{c}}_{Im}$, where ${\mathbf{c}}_{Re}$ and ${\mathbf{c}}_{Im}$ denote the real part and the imaginary part of $\mathbf{c}$, respectively. Then, two groups of OFDM modulators are employed to perform the OFDM modulation according to ${\mathbf{c}}_{Re}$ and ${\mathbf{c}}_{Im}$ so as to obtain the in-phase and quadrature modulated signals. Since the intensity modulation is usually deployed in VLC system to generate the real-valued intensity waveform to drive the LED transmitters, a DC bias is required to be added to the final modulated signal. The modulated signal can be expressed as

$$s_l=\left\{\begin{array}{c}{x}_l+b,x_l⩾-b\hfill \\ 0,x_l<-b\hfill \end{array}\right.,$$

(4)

where $l\in \left\{0,1,\dots ,L-1\right\}$ denotes the index of the time slot. $s_l$ denotes the TD sample in the l-th time slot, $x_l$ denotes the real-valued bipolar TD signal generated at the output of IFFT, and b signifies the added DC bias. Finally, the in-phase-part-modulated signal is added to the quadrature-part-modulated signal to generate the signal vector $\mathbf{u}$.

In the spatial domain, the spatial information will be encoded by selecting the activated LED transmitters. Assume that N out of $N_t$ total LEDs are chosen to convey the same TD sample in each time slot while the rest LEDs are set to be zeros. Let ${\mathbf{v}}_l={\left[v_l^1,v_l^2,\dots ,v_l^N\right]}^T$ denote spatial index vector at the l-th time slot, ${\beta }_l$ denote the corresponding TD samples for a given time slot, and the elements of the vector ${\mathbf{s}}_l={\left[s_l^1,s_l^2,\dots s_l^{N_t}\right]}^T$ that jointly mapped by ${\mathbf{v}}_l$ and $\mathbf{u}$ can be provided by

$$s_l^k=\left\{\begin{array}{c}{\beta }_l,ifk=v_l^i\hfill \\ 0,other\hfill \end{array}\right.,$$

(5)

where $k\in \left\{1,2,\dots N_t\right\}$ and $i\in \left\{1,2,\dots N\right\}$ denote the index of the elements of the signal vector and the index vector, respectively.

Table 1. TD-QGSM mapping table in the case of $N_t=6$ and $N=4$.

Spatial Bits	LED 1	LED 2	LED 3	LED 4	LED 5	LED 6
000	${\beta }_l$	${\beta }_l$	${\beta }_l$	${\beta }_l$	0	0
001	${\beta }_l$	${\beta }_l$	${\beta }_l$	0	${\beta }_l$	0
010	${\beta }_l$	${\beta }_l$	${\beta }_l$	0	0	${\beta }_l$
011	${\beta }_l$	${\beta }_l$	0	${\beta }_l$	${\beta }_l$	0
100	0	${\beta }_l$	${\beta }_l$	${\beta }_l$	0	${\beta }_l$
101	0	${\beta }_l$	${\beta }_l$	0	${\beta }_l$	${\beta }_l$
110	0	${\beta }_l$	0	${\beta }_l$	${\beta }_l$	${\beta }_l$
111	0	0	${\beta }_l$	${\beta }_l$	${\beta }_l$	${\beta }_l$

demonstrates the mapping table in the case of $N_t=6$ and $N=4$. Take a set of spatial bits such as “110” from the mapping table as an example; the spatial index vector can be expressed by ${\mathbf{v}}_l={\left[2,4,5,6\right]}^T$, which means that the joint vector is transmitted by the 2nd, 4th, 5th, and 6th LED transmitters simultaneously, while the input bits of the 1st and 3rd LED transmitters will be set to zeros. Therefore, the transmitted vector can be obtained as ${\mathbf{s}}_l={\left[0,{\beta }_l,0,{\beta }_l,{\beta }_l,{\beta }_l\right]}^T$. Finally, the vector ${\mathbf{s}}_l$ is used to modulate those N active LEDs.

At the receiver side, the TD signal vector $\widehat{\mathbf{u}}$ and the spatial index vector ${\widehat{\mathbf{v}}}_l$ are first estimated using a suitable detection algorithm, where the details are presented in Section 4. We split $\widehat{\mathbf{u}}$ into the in-phase and quadrature parts by using the separation operation of real and imaginary and then perform the OFDM demodulation on these two parts, respectively. After constellation demapping and parallel-to-serial (P/S) conversion, the information bits can be recovered accordingly. As for the spatial part bits, they can be produced by demapping the index vector ${\widehat{\mathbf{v}}}_l$ directly.

3.2. TD-QGSMP

The diagram of the OFDM-based TD-QGSMP scheme is shown in Figure 3. As shown in the figure, the input bits are first divided into two parts for mapping in two different dimensions. The first $⌊lo{g}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋$ bits are encoded in the spatial dimension by using the gray code in each time slot. Different from the previous scheme, the proposed TD-QGSMP allocates $2\times N{log}_{2}M$ bits for constellation mapping, and then a serial-to-parallel (S/P) conversion is performed to convert the serial $2\times N{log}_{2}M$ bits to N parallel bit streams for constellation mapping. After that, the N groups constellation symbols are split into real and imaginary parts, and then the constellation symbols vector ${\mathbf{c}}_{\mathbf{z}}$ can be expressed as ${\mathbf{c}}_{\mathbf{z}}={\mathbf{c}}_{Re,z}+j\times {\mathbf{c}}_{Im,z}$ with $z\in \left\{1,2,\dots N\right\}$ denoting the index of the modulated signal group. Subsequently, the two parts of the constellation symbols are separately performed regarding the OFDM modulation to generate the TD signal, which will be added by an appropriate DC bias to drive the LED. Finally, the z-th group in-phase-part-modulated signal is added to the z-th group quadrature-part-modulated signal to generate the signal vector ${\mathbf{u}}_{\mathbf{z}}$.

In the spatial domain, N out of total $N_t$ LEDs will be chosen to transit different TD samples and the remaining LEDs are silent. After GSMP mapping, the elements of the resultant transmitted vector ${\mathbf{s}}_l={\left[s_l^1,s_l^2,\dots s_l^{N_t}\right]}^T$ can be obtained by

$$s_l^k=\left\{\begin{array}{c}{\chi }_l^i,ifk=v_l^i\hfill \\ 0,other\hfill \end{array}\right.,$$

(6)

where $k\in \left\{1,2,\dots N_t\right\}$ and $i\in \left\{1,2,\dots N\right\}$.

Table 2. TD-QGSMP mapping table in the case of $N_t=6$ and $N=4$.

Spatial Bits	LED 1	LED 2	LED 3	LED 4	LED 5	LED 6
000	${\chi }_l^1$	${\chi }_l^2$	${\chi }_l^3$	${\chi }_l^4$	0	0
001	${\chi }_l^1$	${\chi }_l^2$	${\chi }_l^3$	0	${\chi }_l^4$	0
010	${\chi }_l^1$	${\chi }_l^2$	${\chi }_l^3$	0	0	${\chi }_l^4$
011	${\chi }_l^1$	${\chi }_l^2$	0	${\chi }_l^3$	${\chi }_l^4$	0
100	0	${\chi }_l^1$	${\chi }_l^2$	${\chi }_l^3$	0	${\chi }_l^4$
101	0	${\chi }_l^1$	${\chi }_l^2$	0	${\chi }_l^3$	${\chi }_l^4$
110	0	${\chi }_l^1$	0	${\chi }_l^2$	${\chi }_l^3$	${\chi }_l^4$
111	0	0	${\chi }_l^1$	${\chi }_l^2$	${\chi }_l^3$	${\chi }_l^4$

shows the mapping table that corresponds to $N_t=6$ and $N=4$. Take a set of spatial bits such as “011” from the mapping table as an example and the corresponding spatial index vector is ${\mathbf{v}}_l={\left[1,2,4,5\right]}^T$, which means that the 1st, 2nd, 4th, 5th LED transmitters will be selected to convey the sample ${\chi }_l^1$, ${\chi }_l^2$, ${\chi }_l^3$, ${\chi }_l^4$, respectively. Therefore, the transmitted vector can be obtained as ${\mathbf{s}}_l={\left[{\chi }_l^1,{\chi }_l^2,0,{\chi }_l^3,{\chi }_l^4,0\right]}^T$.

At the receiver side, the spatial index vector ${\widehat{\mathbf{v}}}_l$ and TD signal vector ${\widehat{\mathbf{u}}}_z$ are estimated by using a suitable detection algorithm, which will be demonstrated in Section 4. Split ${\widehat{\mathbf{u}}}_z$ into the in-phase and quadrature parts by using the separation operation of real and imaginary and then perform OFDM demodulation and constellation demapping on these two parts, respectively, to obtain the information bits. Furthermore, the spatial part bits are accessible by demapping the index vector ${\widehat{\mathbf{v}}}_l$ directly.

3.3. SE Analysis

As for the OFDM-based TD-GSM scheme, Hermitian symmetry is not a constraint on the spatial symbol since the GSM procedure is performed in the time domain; the SE contributed by the spatial symbols is $⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋$. However, Hermitian symmetry still affects the symbols in the constellation mapping part as only $\frac{L}{2}-1$ subcarriers out of total subcarriers can transmit valid data, meaning that the SE contributed by each symbol is $\frac{\frac{L}{2}-1}{L}$, and then the SE contributed by the symbols used for constellation mapping is $\frac{\frac{L}{2}-1}{L}{log}_2\left(M\right)$. Therefore, the total SE of TD-GSM can be expressed by

$$\begin{array}{cc}\hfill R_{TD-GSM}& =\frac{\frac{L}{2}-1}{L}{log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋\hfill \\ \hfill & \approx \frac{1}{2}{log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋.\hfill \end{array}$$

(7)

As for the OFDM-based TD-QGSM scheme, a total of $2\times {log}_{2}M$ bits will be assigned in each time slot to perform the constellation mapping, and two groups of OFDM modulators will be deployed to modulate the constellation symbols to obtain the information symbols. Thus, the SE contributed by constellation part for TD-QGSM will be double that of the TD-GSM, which can be presented as $\frac{\frac{2L}{2}-2}{L}{log}_2\left(M\right)$. Due to the constant number of bits used to select the activation LEDs, the SE contributed by the spatial part remains the same as TD-GSM. The total SE of the proposed TD-QGSM can be expressed by

$$\begin{array}{cc}\hfill R_{TD-QGSM}& =\frac{\frac{2L}{2}-2}{L}{log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋\hfill \\ \hfill & \approx {log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋.\hfill \end{array}$$

(8)

As for the OFDM-based TD-QGSMP scheme, the constellation symbols will be divided into N different segments for transmission; the SE contributed by the spatial part remains the same as TD-QGSM. However, the SE contributed by constellation part becomes N times that of TD-QGSM, which can be expressed as $\frac{\frac{2L}{2}-2}{L}N{log}_2\left(M\right)$; hence, the total SE of the proposed TD-QGSMP can be provided as

$$\begin{array}{cc}\hfill R_{TD-QGSMP}& =\frac{\frac{2L}{2}-2}{L}N{log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋\hfill \\ \hfill & \approx N{log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋.\hfill \end{array}$$

(9)

In summary, both the two proposed transmission schemes enhance the overall SE by improving the SE of the constellation part, which results in a significant improvement effect, and the specific comparison will be presented in Section 5.1. Finally,

Table 3. Comparison of SE for different OFDM-based generalized MIMO transmission schemes.

Transmission Scheme	Achievable Rate	Feature
FD-GSM	$\frac{1}{2}{log}_2\left(M\right)+\frac{1}{2}⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋$	GSM performed in frequency domain.
FD-GSMP	$\frac{N}{2}{log}_2\left(M\right)+\frac{1}{2}⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋$	1. Constellation mapping symbols divided into N groups; 2. GSM performed in frequency domain.
FD-GQSM	$\frac{1}{2}{log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋$	1. I/Q separation performed before OFDM modulation; 2. GSM performed in frequency domain.
TD-GSM	$\frac{1}{2}{log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋$	GSM performed in time domain.
TD-GSMP	$\frac{N}{2}{log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋$	1. Constellation mapping symbols divided into groups; 2. GSMP performed in time domain.
The proposed TD-QGSM	${log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋$	1. I/Q separation performed before OFDM modulation; 2. GSM performed in time domain.
The proposed TD-QGSMP	$N{log}_2\left(M\right)+⌊{log}_2\left(\mathcal{C}\left(N_t,N\right)\right)⌋$	1. Constellation mapping symbols divided into groups; 2. I/Q separation performed before OFDM modulation; 3. GSMP performed in time domain.

demonstrates the theoretical SE of the proposed schemes. In addition, the SEs of some existing OFDM-based generalized MIMO transmission schemes proposed in [15,19] are also presented here for easy comparison.

4. Detection Methods

In this section, we first review the traditional ML-MRC detection algorithm employed in most generalized MIMO schemes, and then the proposed IVC-based OMP detection algorithm for the two proposed OFDM-based QG-MIMO systems is detailed accordingly.

4.1. The Traditional ML-MRC Detection Algorithm

ML-MRC is a kind of two-step detection method. Briefly, the received TD signal vectors and the index vectors are first estimated using ML detection, and then the final estimation of the received TD samples is generated by applying MRC detection. To enhance the reception performance, it is usually necessary to perform the zero forcing (ZF) equalization on the received signal before the ML-MRC detection. More specifically, let ${\widehat{\mathbf{x}}}_l={\left[{\widehat{x}}_l^1,{\widehat{x}}_l^2,\dots ,{\widehat{x}}_l^{N_t}\right]}^T$ denote the received signal vector after ZF equalization. Firstly, sort the elements of ${\widehat{\mathbf{x}}}_l$ in the descending order to obtain the corresponding sorted index vector, and then take the first N elements of the sorted index vector to form an abstracted index vector. The final index vector ${\widehat{\mathbf{v}}}_l={\left[{\widehat{v}}_l^1,{\widehat{v}}_l^2,\dots ,{\widehat{v}}_l^N\right]}^T$ is generated by sorting the abstracted index vector in the ascending order. Employ ${\widehat{\mathbf{x}}}_l$ and ${\widehat{\mathbf{v}}}_l$ to obtain the N TD signal vectors, and then the final TD samples can be obtained through MRC operation. However, one limitation of the ML-MRC detection is that the errors occurring in the ML detection will have a direct impact on the overall MRC detection performance.

4.2. Proposed IVC-Based OMP Detection Algorithm

OMP is a greedy algorithm with much lower computational complexity than the traditional ML detection. In this study, we modified the original OMP and proposed an IVC-based detection algorithm for the OFDM-based TD-QGSM and TD-QGSMP in the following.

Let ${\mathbf{y}}_l$, ${\mathbf{s}}_l$, and ${\mathbf{n}}_l$ denote the l-th $(l=1,2,\dots L)$ column of $\mathbf{y}$, $\mathbf{s}$, and $\mathbf{n}$, respectively, and then Equation (1) can be rewritten as

$${\mathbf{y}}_l=\mathbf{H}{\mathbf{s}}_l+{\mathbf{n}}_l,$$

(10)

where ${\mathbf{s}}_l$ is an N-sparse vector, ${\mathbf{y}}_l$ is the received signal, $\mathbf{H}$ is the channel matrix and also represents the measurement matrix, and the optimization problem is to estimate the signal ${\widehat{\mathbf{s}}}_l$ from ${\mathbf{y}}_l$ and $\mathbf{H}$. Let e denote the number of iterations, and let ${\mathbf{r}}_e$ and ${\Lambda }_e$ denote the residual and the index set in the e-th iteration, respectively. Then, the initialization items can be set as $e=1,{\mathbf{r}}_0={\mathbf{y}}_1,{\Lambda }_0=\varnothing$; the corresponding atomic set, support set, and estimated signal set are $B_0=\varnothing$, ${\vartheta }_0=\varnothing$, and ${\Gamma }_0=\varnothing$, respectively. To eliminate the effect of channel gain on signal transmission and compare the signals between different channels at the same power level, it is necessary to normalize the channel matrix, shown as

$$\overline{\mathbf{H}}=\frac{\mathbf{H}}{{∥\mathbf{H}∥}_2}.$$

(11)

After computing the inner product of $\overline{\mathbf{H}}$ and ${\mathbf{r}}_0$, the support index can be obtained by

$$a_e=\underset{1⩽\mu ⩽N_t}{argmax}\left|〈{\mathbf{r}}_{e-1},{\overline{h}}_{\mu }〉\right|,$$

(12)

where ${\overline{h}}_{\mu }$ denotes the $\mu$-th column of $\overline{\mathbf{H}}$. Put the first N inner products with the maximum absolute values from the $\mu$ inner products into the set ${\Omega }_e$, and put their indexes into the set ${\Lambda }_e$. Then, the support set can be updated as

$${\vartheta }_e={\vartheta }_{e-1}\cup {\Omega }_e,$$

(13)

the index set can be obtained as

$${\Lambda }_e=a_e\cup {\Lambda }_{e-1},$$

(14)

and the atomic set can be collected by

$$B_e=\overline{\mathbf{H}}\left(a_e\right)\cup B_{e-1}.$$

(15)

Subsequently, the estimated signal value can be obtained by

$${\widehat{s}}_e={\left({\left(B_e\right)}^T\ast B_e\right)}^{-1}\ast {\left(B_e\right)}^T\ast {\mathbf{y}}_l,$$

(16)

the residual can be calculated as

$${\mathbf{r}}_e={\mathbf{y}}_l-B_e{\widehat{s}}_e.$$

(17)

Put the estimated signal value into the set ${\Gamma }_e$, and the estimated signal set can be obtained as

$${\Gamma }_e={\widehat{s}}_e\cup {\Gamma }_{e-1}.$$

(18)

Finally, let $e=e+1$, and repeat the process (12)–(18). After N iterations, N elements from ${\vartheta }_e$ will be selected for the combination and all possible combinations will be stored in the set $\Theta$. Then, the active set is searched among $\Theta$ to find out the activated LEDs combination and thus the estimated index vector ${\widehat{\mathbf{V}}}_l$

$${\widehat{\mathbf{V}}}_l=\underset{\Delta \in \Theta }{argmax}∥{\mathbf{y}}_l-\overline{\mathbf{H}}{\mathbf{s}}_l(\Delta )∥.$$

(19)

In addition, the estimated signal vector ${\widehat{\mathbf{s}}}_l$ is obtained by arranging the elements in the set ${\Gamma }_e$ in iterative order after N iterations.

However, two issues may arise when estimating the index vector. One is that the demodulated spatial signal bits are easily turned to be all zeroes, and another is that the demodulated signal bits are not enough. Both of the above-mentioned aspects will result in illegal index vectors. In order to solve this problem, we add a correction criterion into the OMP iteration and modify the estimated index vector along with the signal detection, which can be depicted as

Criterion A

(a)

Determine the legitimacy of the obtained index vector ${\widehat{\mathbf{V}}}_l$. If the index vector ${\widehat{\mathbf{V}}}_l$ is legal, it will be directly used as the final estimated index vector ${{\widehat{\mathbf{V}}}_l}^{{}^{\prime }}$ for demapping. If the index vector ${\widehat{\mathbf{V}}}_l$ is illegal, it will be compared with each vector of the legal LED index matrix $\mathbf{I}$.

(b)

Sequentially select the vector from $\mathbf{I}$ with the highest number of the same elements in the same position as ${\widehat{\mathbf{V}}}_l$.

(c)

If there are multiple vectors in $\mathbf{I}$ that meet the condition in step (b), priority will be given to the vector that appears first in the order, and then the selected vector will be used as the final estimated index vector ${{\widehat{\mathbf{V}}}_l}^{{}^{\prime }}$ for demapping.

The detailed procedures of the proposed IVC-based OMP detection are also summarized in Algorithm 1.

Algorithm 1: The Proposed IVC-Based OMP Detection Method

Input: $\mathbf{H}$: the channel matrix, ${\mathbf{y}}_l$: the received signal vector, ${\mathbf{s}}_l$: the transmitted signal vector, N: signal sparsity or the number of the activated LEDs. Output: ${\widehat{\mathbf{s}}}_l$: the estimated signal vector, ${{\widehat{\mathbf{V}}}_l}^{{}^{\prime }}$: the activated LED index.

1. Initialization: $e=1,{\mathbf{r}}_0={\mathbf{y}}_1,{\Lambda }_0=\varnothing$, $B_0=\varnothing$, ${\vartheta }_0=\varnothing$, ${\Gamma }_0=\varnothing$.

2. Normalize the matrix $\mathbf{H}$ via  (11) and obtain $\overline{\mathbf{H}}$.

3. for $l=1toL$ do

4.     for $e=1toN$ do

5.         Compute the inner product of $\overline{\mathbf{H}}$ and ${\mathbf{r}}_0$ via  (12).

6.         Put the first N inner products with the maximum absolute values into the set ${\Omega }_e$.

7.         Update ${\vartheta }_e$, ${\Lambda }_e$, and $B_e$ via  (13),  (14), and (15), respectively.

8.         Obtain ${\widehat{s}}_e$ by using least squares described in  (16).

9.         Update ${\mathbf{r}}_e$ and ${\Gamma }_e$ via  (17) and (18), respectively.

10.          $e=e+1$.

11.   end

12.          Obtain ${\widehat{\mathbf{V}}}_l$ via  (19).

13.          if ${\widehat{\mathbf{V}}}_l\in$ legal index matrix $\mathbf{I}$ then

14.          Obtain ${{\widehat{\mathbf{V}}}_l}^{{}^{\prime }}$ via the step (a) of Criterion A.

15.        else

16.          Obtain ${{\widehat{\mathbf{V}}}_l}^{{}^{\prime }}$ via the step (b) and (c) of Criterion A.

17.       end

18.          Obtain ${\widehat{\mathbf{s}}}_l$ by arranging the elements in the set ${\Gamma }_e$ in iterative order.

19. end

4.3. Complexity Analysis

To demonstrate the advantages of the proposed algorithm, we will analyze the computational complexity of ML-MRC detection algorithm and the proposed IVC-OMP detection algorithm separately from the perspective of time complexity.

As for the ML-MRC detection, which is a two-step detection algorithm, the computational complexity consists of both ML and MRC components. In ML detection, in order to obtain the index vector ${\widehat{\mathbf{v}}}_l$, we successively used descending and ascending sorting methods, and the computational complexity required for this process can be approximately estimated as $O\left(N{N}_r\right)$. In the ML stage of estimating the signal vector ${\widehat{\mathbf{s}}}_l$, considering the modulation order and generalized operation of QAM, its computational complexity can be approximated as $O\left(2^{⌊{log}_2\left(C\left(N_t,N\right)\right)⌋}{N}_r{M}^{N}N\right)$. In MRC detection, the computational complexity mainly focuses on the calculation of weight factors, which can be approximated as $O\left(N_{r}M\right)$. Hence, the total computational complexity of the ML-MRC method can be expressed as $O\left(2^{⌊{log}_2\left(C\left(N_t,N\right)\right)⌋}{N}_r{M}^{N}N\right)$.

As for the proposed IVC-based OMP detection, a total of N iterations are required in Algorithm 1, and each iteration’s operations are mainly concentrated in two parts, including the inner product operation of (12) and solving least squares problem of (16). The computational complexity of the inner product operation part is related to the number of LEDs and PDs and can be approximately evaluated as $O\left(N_r\right)$, while the computational complexity of solving least squares problem is $O\left(N_r^2\right)$. However, the illegal vector correction operation occurs outside the iteration and is only required once, leading to the negligible computation complexity. Therefore, the total computational complexity of the proposed detection method can be expressed as $O\left(N_r^{2}N\right)$.

Based on the above analysis of the complexity of the two algorithms, we find that the complexity of the ML-MRC algorithm increases exponentially with the increase in the number of activated LEDs. However, the proposed IVC-based OMP algorithm takes advantage of the sparse characteristics of the received signal and will first detect the indexes of the activated LEDs and then estimate the received symbols according to these indexes by solving the least squares problem, which makes the impact on its complexity when the number of active LEDs increases much smaller than that of ML-MRC.

Table 4. Complexity comparison.

Detection Algorithm	Activated LED Number	Floating Point Operation Count
ML-MRC	1	512
	2	12,288
	3	262,144
	4	1,310,720
The proposed IVC-based OMP	1	64
	2	160
	3	288
	4	448

provides a brief comparison of the average complexity in the cases of $N_t=4$, $N_r=4$, and $M=16$.

5. Simulations

In this section, the SE and the BER performance of the two proposed OFDM-based QG-MIMO schemes are investigated in the case of a 4 × 4 VLC-MIMO system. In addition, a typical 4 m × 4 m × 3 m room is considered where the 2 × 2 LED array is mounted at the center of the ceiling and the receiving plane is set 0.85 m above the floor. The relevant simulation parameters are listed in

Table 5. Simulation parameters.

Parameter	Value
Number of LEDs	4
Number of PDs	4
Room dimension	4 m × 4 m × 3 m
Height of ceiling	3 m
LED spacing	2 m
Height of receiving plane	0.85 m
PD spacing	10 cm
Semi-angle at half power of LED	60${}^{\circ }$
Responsivity of the PD	0.53 A/W
Active area of PD	${10}^{-4}$ cm2
Refractive index	1.5
Half-angle FOV of optical lens	60${}^{\circ }$
Gain of the optical filter	0.9
Number of activated LEDs	2

. The SEs of the systems listed in Table 3 are first compared, and then Monte Carlo simulations are used to compare the BER performance of these systems under different conditions. Also, an OFDM modulation involving 255 sub-carriers and 512 IFFT/FFT points is used. In order to make the performance evaluation more fair, we assume in the simulation that the data transmission of each channel is synchronous.

5.1. Spectral Efficiency

Figure 4 and Figure 5 show the SE versus ${log}_{2}M$ for different OFDM-based generalized MIMO schemes when $N=2$ and $N=3$ are considered, respectively. When $N=2$ and ${log}_{2}M=8$, the SE of the proposed TD-QGSM is, respectively, improved by 56.5%, 56.5%, and 99.8% compared to TD-GSM, FD-GQSM, and FD-GSM, while the SE of the proposed TD-QGSMP is, respectively, improved by 72.3% and 99.9% compared to TD-GSMP and FD-GSMP. The TD-QGSMP shows the highest SE among all the OFDM-based generalized MIMO schemes. When $N=3$ and ${log}_{2}M=8$, the improvement in SE of the proposed TD-QGSM is almost the same as when $N=2$, while that of the TD-QGSMP is more significant, at least 81.6% higher than the other two multiplexing schemes. According to the analysis in Section 3.3, since the the proposed TD-QGSMP uses N LEDs to transmit different signals, resulting in an $N-1$ fold increase in the SE of the constellation part, the SE’s improvement will be more significant as the number of activated LEDs increases.

5.2. BER Performance

In this study, we compare the BER performance of the OFDM-based generalized MIMO schemes under different conditions, including the comparisons among the schemes with different target SEs and different modulation orders. In addition, the BER performance is evaluated when employing different detection algorithms. It is worth noting that the channel gain in a typical indoor VLC system always varies owing to the distinct distance and transmit/incidence angle for each LED/PD pair, leading to a significant difference in the received SNR.

Figure 6 and Figure 7 compare the BER performance of various TD-generalized MIMO schemes with the same SE. It should be noted that different QAM should be deployed in these schemes so as to achieve the same SE for fair comparison. As clearly shown in these two figures, the BER performances of TD-QGSM and TD-QGSMP are basically the same under the conditions of SE of 4 bits/s/Hz and 6 bits/s/Hz, and both of them are significantly better than TD-GSM and TD-GSMP. This fact indicates that the TD-QGSM and TD-QGSMP require less modulation order to transmit the same number of bits per unit time than TD-GSM and TD-GSMP, which not only reduces the wastage of spectral resources but also makes the system more resistant to interference, resulting in better BER performance.

Figure 8 and Figure 9 compare the BER performance of different OFDM-based generalized MIMO schemes under the same modulation order when $N=2$ is employed. When $M=64$, the BER performance of TD-QGSM is almost the same as that of FD-GSM and outperforms that of all other OFDM-based generalized MIMO schemes, while the BER performance of TD-QGSMP is similar to that of FD-GQSM. In addition, the BER advantage over TD-GSM and TD-GSMP becomes especially significant. In the case of $M=256$, the TD-QGSM still shows the best BER performance. However, the advantage of TD-QGSMP is not significant due to the fact that the signal stream has been divided into many groups prior to OFDM modulation, which results in reduced spacing between neighboring symbols and increases the probability of inter-symbol interference. In addition, as the modulation order increases, the constellation points will become denser and the Euclidean distance will decrease, resulting in poorer BER performance. Combined with the previous analysis of SE, compared with the other generalized MIMO schemes, TD-QGSM not only has higher SE but also has excellent BER performance. For TD-QGSMP, its BER performance is slightly inferior to that of FD-GSM and FD-GSMP but still demonstrates a great improvement over TD-GSM and TD-GSMP. Moreover, it also has a huge SE advantage.

Figure 10 shows the BER performance comparison of the TD-generalized MIMO systems when the ML-MRC detection and the proposed IVC-based OMP detection are deployed, respectively, under the same modulation order ($M=16$). It can be seen that the BER performance of TD-QGSM and TD-QGSMP using the proposed IVC-based OMP detection method is significantly improved by about 62.5% as compared to the one using the ML-MRC method. As for TD-GSM, a much lower BER can also be achieved when using the IVC-based OMP, and the improvement is even greater than that of TD-QGSM and TD-QGSMP. In summary, the proposed IVC-based OMP algorithm can achieve computational complexity reduction as well as BER performance improvement, which is beneficial for the proposed QG-MIMO schemes.

6. Conclusions

In this paper, two OFDM-based QG-MIMO schemes, TD-QGSM and TD-QGSMP, are proposed in a VLC system for the improvement in the SE and BER performance. The simulation results show that applying IQ decomposition of the constellation signals prior to OFDM modulation substantially enhances the SE of the system. In addition, since the required modulation orders are less than the other schemes for the same transmission rate, the system has a stronger anti-interference capability. Moreover, an IVC-based OMP detection algorithm is proposed to further improve the reception performance of the systems. With the aid of the illegal vector correction criterion, the receiver’s capacity to estimate the index vector is augmented and the BER performance is also improved. The results demonstrate that the BER performance of the proposed detection algorithm has a dramatic advantage as compared to the traditional ML-MRC. Introducing differential modulation into the OFDM-based generalized MIMO systems to reduce the symbol interference and further improve the system performance is left for future work.