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Detection and performance analysis for MIMO visible light communication system using joint optical spatial and pulse amplitude width modulation
EURASIP Journal on Wireless Communications and Networking volume 2024, Article number: 8 (2024)
Abstract
Conventional optical spatial modulation (SM) scheme activates one of the lightemitting diodes (LEDs) to transmit an intensitymodulated optical signal, in which the index of the activated LED is determined by spatial symbol and the emitted intensity is controlled by temporal symbol. In order to enhance the spectral efficiency (bits per channel use), we propose a joint SM and pulse amplitude width modulation (PAWM) as a novel optical spatial–temporal signaling scheme. In this paper, the proposed SMPAWM optical signaling scheme is applied in a multiinput multioutput (MIMO) visible light communication (VLC) system. Employing optimal maximum likelihood (ML) algorithm to extract the spatial and temporal symbols is computationally prohibitive; hence, we develop a novel lowcomplexity detection scheme that converts the joint optimization problem separately to decode the spatial and temporal symbols. Moreover, theoretical results in terms of the successful identification probability of activated LED as well as the overall symbol error rate are derived. Extensive computer simulations are performed to validate the analytical results. It is shown that the proposed detection scheme is a feasible alternative to the ML detector in the VLCMIMO system employing SMPAWM.
1 Introduction
Spatial modulation (SM) is a new promising MIMO technology that carries the information simultaneously over both the spatial and temporal domains [1,2,3,4,5]. Compared with conventional MIMO scheme, SM avoids the InterChannel Interference (ICI), requires no InterAntennas Synchronization (IAS), and needs one or only a few RF chains for data transmission. Hence, lowcomplexity implementation can be achieved in SM MIMO scheme. Moreover, one of the major advantages of SM is the increase in bits per channel use (bpcu) by a factor equal to the logarithm of the number of antennas at the transmitter. Recently, it has also been shown that SM technique can be applied for optical wireless communication systems [6,7,8,9,10], particularly for indoor visible light communication (VLC) system. The reason is due mainly to the static and lineofsight (LoS) characteristic of indoor VLC channel. Therefore, the locationdependent spatial constellation is plausible to be further utilized to boost the overall spectral efficiency [11,12,13,14,15].
In optical spaceshift keying (SSK) [6], the input bits are used to select only single laser source or lightemitting diode (LED) while the rest are idle to send an optical pulse over an indoor channel at each transmission instant. In the work of [16], optical generalized space shift keying (GSSK) is proposed and analyzed in which multiple LEDs are activated simultaneously. Notice that no temporal modulated symbol is transmitted by the activated LED for SSK or GSSK. To increase the bpcu, the transmitted bits stream of the SM scheme is composed of two informationcarrying blocks, in terms of spatial and temporal symbols. Like SSK scheme, spatial symbol determines one from the array of LED at the transmitter. The chosen LED is obliged for temporal data transmission. In the scheme proposed in [9], the temporal symbol is mapped to the pulse position modulation (PPM) signal constellation diagram, which is named spatial pulse position modulation (SPPM), e.g., a single PPM symbol is sent from an active transmit antenna at each signaling period, while in the work of [10], SM employing pulse amplitude modulation (PAM) is shown to outperform repetition coding (RC) when high spectral efficiencies are desired. SM (GSM) technique in which an unipolar Mary PAM (MPAM) optical signal is transmitted by the active LEDs at each time instance is applied for MIMO VLC system [12, 15, 17, 18].
To further achieve a higher spectral efficiency in optical SM systems, we joint use of multiple optical pulse amplitudes as well as widths for temporal symbol modulation. Hence, we propose a novel optical SM system, denoted as a \(\left( {N_{t} ,M,N} \right)\)ary spatial pulse amplitude width modulation (SMPAWM). We apply the SMPAWM scheme in a VLC MIMO system. Most past works for optical SM scheme employs maximum likelihood (ML) decision rule to jointly demodulate the temporal and spatial symbols. However, since ML detection requires exhaustive searches on the parameters embedded in the SMPAWM signal, the computation load is prohibitive in general scenario. For this reason, one of the contributions of this paper is to propose a novel lowcomplexity detection scheme. Rather than jointly optimization, the received vector signal at the output of photodetectors (PDs) is first linearly transformed by spatial matched filtering (SMF). Since the optical PAWM signal is intensity modulated and channel vector is location dependent, hence, energy detection is utilized at the output of SMF to identify the index of the active LED. In what follows, the (M, N)ary PAWM signal is then demodulated separately by Mary PAM and Nary pulse width modulation (PWM) multilevels circuits with modified minimum distance between the signal constellation points. Performance in terms of the average overall symbol error rate (SER) is extensively analyzed. All theoretical derivations are validated by Monte Carlo simulations which are in good agreement.
Major contributions of this work are summarized as follows:

(1)
We propose a novel \(\left( {N_{t} ,M,N} \right)\)ary SMPAWM spatial–temporal modulation and demodulation scheme that increases the bpcu of conventional SM.

(2)
Typical MLbased receiver relies on joint optimization of multiple parameters. We propose a simple detection algorithm that separates the joint optimization into individually sequential optimization problems.

(3)
The overall average SER of the proposed detection scheme is comprehensively analyzed. We first derive the spatial domain SER, i.e., the probability of misidentifying the index of the active LED. There then, we devise the \(\left( {M,N} \right)\)ary PAWM temporal domain SER given that the index of active LED has been correctly identified. Based on the two probabilities, the theoretical expression of the overall average SER is derived.

(4)
Extensive computer simulations are performed to demonstrate the analytical results.
Notation: We use upper and lower case boldface letters to denote matrices and vectors, respectively. \(\left[ {} \right]^{T} ,\left[ {} \right]^{H}\) stand for matrix or vector transpose and complex transpose, respectively. ǁaǁ denotes the l_{2}–norm of vector a. We use E{} for expectation (ensemble average). I_{K} denotes an identity matrix of size K. \({\mathbf{e}}_{k}^{L}\) denotes the kth column vector of an identity matrix of size L. A Gaussian distributed random variable with mean µ variance \(\sigma^{2}\) reads as \(N\left( {\mu ,\sigma^{2} } \right)\). \(\hat{x}\) means the estimate of x. x(i) denotes the ith element of vector x. \({\mathbf{A}}\left {_{{\left( {i,j} \right)}} } \right.\) denotes the element of the ith row and jth column of matrix A. \({\text{tr}}({\mathbf{V}})\) denotes the trace of matrix V. \(\left\ {\mathbf{C}} \right\_{F}^{2}\) means the Frobenius norm of matrix C. The tail function reads as \(Q(x) \equiv \frac{1}{{\sqrt {2\pi } }}\int\limits_{x}^{\infty } {\exp \left( {  \frac{{t^{2} }}{2}} \right){\text{d}}t}\).
2 Methods/experimental
2.1 Channel model
In this work, we consider a VLC indoor lineofsight (LoS) MIMO channel [19, 20], in which \(N_{t}\) LEDs are deployed at the transmitter and \(N_{r}\) PDs are equipped at the receiving front end. A schematic illustration of the system is depicted in Fig. 1, where the dashed lines represent the LEDs are active and the solid lines denote the LEDs are inactive to every receiving PDs at user terminal. In VLC channel, only the (dominant) component of the channel gain is considered. The MIMO channel with dimension \(N_{r} \times N_{t}\) considered in this paper is
where \({\mathbf{h}}_{i} = \left[ {\begin{array}{*{20}c} {h_{1i} } & {h_{2i} } & {...} & {h_{{N_{r} i}} } \\ \end{array} } \right]^{T}\) denotes the \(N_{r} \times 1\) channel vector seen by the array of PDs as signal is emitted from the ith LED. \(h_{ij}\) represents the channel gain of the VLC link between the jth LED and the ith PD in indoor LoS environment. We adopt the channel model as suggested in [21]
where k is the Lambertian emission order given as \(k =  \frac{\ln 2}{{\ln \left( {\cos \phi_{1/2} } \right)}}\), \(\phi_{1/2}\) is the semiangle at halfpower of the LED, \(d_{ij}^{{}}\) is the transmission distance between the jth LED and the ith PD, \(\phi_{ij}\) and \(\phi_{ij}\) are the angle of emission and incidence from the jth LED to the ith PD. A, η and \(\Psi_{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern0pt} 2}}}\) represent the PD physical area, PD responsivity and halfpower fieldofview (FOV) angle of the PD, respectively.
2.2 Signal model
As shown in Fig. 1, the bit stream of the proposed SM based PAWM scheme is divided into two blocks at each transmission time interval: The first block of bit stream with length k, \(\left[ {\begin{array}{*{20}c} {b_{1}^{{}} } & {b_{2}^{{}} } & {...} & {b_{k}^{{}} } \\ \end{array} } \right]\), is referred to as “spatial symbol.” The spatial symbol is used to activate a particular LED, while the other (\(N_{t}\) − 1) LEDs are kept silent. The second block of bit stream with length l + q, \(\left[ {\begin{array}{*{20}c} {b_{k + 1}^{{}} } & {b_{k + 2}^{{}} } & {...} & {b_{k + l}^{{}} } \\ \end{array} ,\begin{array}{*{20}c} {b_{k + l + 1}^{{}} } & {b_{k + l + 2}^{{}} } & {...} & {b_{k + l + q}^{{}} } \\ \end{array} } \right]\), is referred to as “temporal symbol.” According to the nature of optical signals, intensity modulation and direct detection are usually employed. A PWM waveform consists of a sequence of pulses with each pulse having a width proportional to the symbol to be transmitted. Likewise, a PAM waveform consists of a sequence of pulses with each pulse having an amplitude proportional to the symbol to be transmitted. In this paper, the optical pulses with pulse width \(n\tau ;n \in \left\{ {1, \ldots ,N} \right\}\) and amplitude \(\frac{{mP_{t} }}{M};m \in \left\{ {1,...,M} \right\}\) are used to represent the Nary PWM (NPWM) and Mary PAM (MPAM) signals, respectively. \(P_{t}\) is the optical emission intensity of the activated LED, m is determined by the first lbits subblock, \(\left[ {\begin{array}{*{20}c} {b_{k + 1}^{{}} } & {b_{k + 2}^{{}} } & {...} & {b_{k + l}^{{}} } \\ \end{array} } \right]\), while n is determined by the second qbits subblock, \(\left[ {\begin{array}{*{20}c} {b_{k + l + 1}^{{}} } & {b_{k + l + 2}^{{}} } & {...} & {b_{k + l + q}^{{}} } \\ \end{array} } \right]\). \(M = 2^{l}\) and \(N = 2^{q}\). Hence, the optical signal to be transmitted by the active LED belongs to a twodimensional (M, N)ary PAWM alphabet. An example of (2, 4)ary PAWM transmitted pulse waveform is depicted in Fig. 2.
The spectral efficiency (measured by bpcu) of SMPAWM scheme can be calculated as
Define the rectangular function with unit height and width τ as
Then the PAWM temporal symbol carried by the active LED can be expressed as
3 Detection algorithm description
As shown in Fig. 1, at the front end of the receiver, \(N_{r}\) PDs receive the optical PAWM signal and convert them into electrical signals. If the uth LED is activated and the temporal signal \(s_{m,n} (t)\) is emitted, then the received \(N_{r} \times 1\) vector can be written as
where H is the MIMO channel matrix as defined in (1). \({\mathbf{x}}_{m,n,u} (t) = \left[ {\begin{array}{*{20}c} 0 & {...} & 0 & {s_{m,n} (t)} & 0 & {...} & 0 \\ \end{array} } \right]^{T}\) is the \(N_{t}\)dimensional vector with single nonzero element at the uth element, which is \(s_{m,n} (t)\). \({\mathbf{e}}_{u}^{{N_{t} }}\) is the uth column vector of \(N_{t}\)dimensional identity matrix. Each element in \({\mathbf{v}}(t)\) represents the sum of the thermal noise and the high intensity ambient light shot noise at the PD. We model each element in \({\mathbf{v}}(t)\) as independent and identically distributed real valued additive white Gaussian noise (AWGN) having zero mean and power spectral density \(\sigma^{2}\).
Integrating the received signal within \(t \in [(j  1)\tau ,j\tau ];j = 1,...,N\) and defining \({\mathbf{r}}_{j} \equiv \int\limits_{{{(}j  1)\tau }}^{j\tau } {{\mathbf{r}}(t){\text{d}}t} ,{\mathbf{v}}_{j} \equiv \int\limits_{{{(}j  1)\tau }}^{j\tau } {{\mathbf{v}}(t){\text{d}}t}\), then we may convert (6) into matrix form
where \({\mathbf{R}},{\mathbf{V}} \in {\mathbb{R}}^{{N_{r} \times N}} ,{\mathbf{X}}_{m,n,u} \in {\mathbb{R}}^{{N_{t} \times N}} ,{\mathbf{R}} \equiv \left[ {\begin{array}{*{20}c} {{\mathbf{r}}_{1} } & {{\mathbf{r}}_{2} } & {...} & {{\mathbf{r}}_{N} } \\ \end{array} } \right],{\mathbf{V}} \equiv \left[ {\begin{array}{*{20}c} {{\mathbf{v}}_{1} } & {{\mathbf{v}}_{2} } & {...} & {{\mathbf{v}}_{N} } \\ \end{array} } \right]\). Since
There then, the observation matrix can be rewritten as
Based on the assumption that channel state information (CSI) is available at the receiver, we aim to estimate the index of the active LED and demodulate the (M, N)ary PAWM temporal symbol carried by the activated LED.
3.1 Maximum likelihood (ML) detection
Assuming equal a priori probability for both spatial and temporal symbols, the maximum a posteriori (MAP) decision rule is equivalent to the maximum likelihood (ML) criterion. Under AWGN, the ML detector computes the Euclidean distance between the received vector signal, r(t), and the set of all possible received signals, and selects the minimum one. Based on (7), the active LED indices and the corresponding (M, N)ary PAWM temporal symbol carried by the activated LED can be extracted by
The ML detection procedure requires the joint exhausting searches over all the possible sets of the activated LEDs as well as all the possible (M, N)ary PAWM constellationpoint sets, e.g., to perform (10), it requires \(MNN_{t}\) trials. The load of computations required to implement (10) is to be evaluated in Sect. 5.1.
3.2 Proposed detection scheme
3.2.1 Spatial symbol detection algorithm
To make the detector feasible, we separate the joint optimization problem of (10) into sequential detection processes. The schematic diagram of the proposed detector is shown in Fig. 3. As shown in Fig. 3, the observation vector r(t) is first sent to a bank of \(N_{t}\) spatial MF (SMF) that matches to the spatial signature vector of each LED. Let the normalized \(N_{r} \times 1\) channel vector be defined as \({\tilde{\mathbf{h}}}_{i} = \frac{{{\mathbf{h}}_{i} }}{{\left\ {{\mathbf{h}}_{i} } \right\}},i = 1,...,N_{t}\), then the weight vector of each of the SMF is designed as
Upon defining \({\mathbf{W}} \equiv \left[ {\begin{array}{*{20}c} {{\mathbf{w}}_{1,MF} } & {{\mathbf{w}}_{2,MF} } & \cdots & {{\mathbf{w}}_{{N_{t} ,MF}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\tilde{\mathbf{h}}}_{1} } & {{\tilde{\mathbf{h}}}_{2} } & {...} & {{\tilde{\mathbf{h}}}_{{N_{t} }} } \\ \end{array} } \right]\), then the outputs of \(N_{t}\) spatial SMFs are \({\mathbf{y}}\left( t \right) = {\mathbf{W}}^{T} {\mathbf{r}}\left( t \right)\). Let the correlation between kth and jth normalized channel vectors be defined as \(\alpha_{kj} = \left( {{\tilde{\mathbf{h}}}_{k} } \right)^{T} {\tilde{\mathbf{h}}}_{j}\), hence we have
where \({{\varvec{\upalpha}}}_{u} \equiv \left[ {\begin{array}{*{20}c} {\alpha_{1u} } & {\alpha_{2u} } & {...} & 1 & {...} & {\alpha_{{N_{t} u}} } \\ \end{array} } \right]^{T}\) is the uth column vector of \({\mathbf{W}}^{T} {\mathbf{W}}\) in which the uth element of \({{\varvec{\upalpha}}}_{u}\) being 1. Giving that the uth LED is activated and the temporal PAWM signal \(s_{m,n} (t)\) is emitted, i.e., \({\mathbf{r}}\left( t \right){ = }{\mathbf{h}}_{u} s_{m,n} (t) + {\mathbf{v}}\left( t \right)\), then the \(N_{t} \times 1\) SMF output vector can be obtained as
where \({\tilde{\mathbf{v}}}(t) \equiv {\mathbf{W}}^{T} {\mathbf{v}}\left( t \right)\). To collect the energy in temporal domain within the observation interval, we integrate the output of each SMF over \(N\tau\), which yields
Since \(\int\limits_{0}^{N\tau } {s_{m,n} (t){\text{d}}t} { = }\int\limits_{0}^{N\tau } {\frac{{mP_{t} }}{M}p_{n\tau } (t){\text{d}}t} { = }\frac{{P_{t} mn\tau }}{M}\), and let \({\tilde{\mathbf{v}}}_{1} \equiv \int\limits_{0}^{N\tau } {{\tilde{\mathbf{v}}}(t){\text{d}}t}\), then (14) can be reformulated as
In what follows, the uth element of \({\mathbf{y}}_{1}\) is
It is evident that \({\tilde{\mathbf{v}}}_{1}\) is still Gaussian with zeromean and covariance matrix
As depicted in (16), the noise term \({\tilde{\mathbf{v}}}_{1} (u)\) is a Gaussian random variable with zeromean and variance \(\sigma^{2} N\tau\).
In the considered SMPAWM scheme, only single LED is activated during the observation interval and the optical signal is intensity modulated. Hence, as depicted in Fig. 3, the index of active LED is determined by choosing the largest among the outputs of \(N_{t}\) SMFs. The spatial information bits, \(\left[ {\begin{array}{*{20}c} {b_{1}^{{}} } & {b_{2}^{{}} } & {...} & {b_{k}^{{}} } \\ \end{array} } \right]\), can then be decoded by the following SM inverse mapper.
3.2.2 (M, N)ary PAWM temporal symbol detection algorithm
As shown in Fig. 3, to demodulate the Mary PAM (MPAM) symbol, we integrate the output of each SMF over \(t \in [0,\tau ]\), which yields
Since \(\int\limits_{0}^{\tau } {s_{m,n} (t){\text{d}}t} { = }\int\limits_{0}^{\tau } {\frac{{mP_{t} }}{M}p_{n\tau } (t){\text{d}}t} { = }\frac{{P_{t} m\tau }}{M}\), and let \({\tilde{\mathbf{v}}}_{2} \equiv \int\limits_{0}^{\tau } {{\tilde{\mathbf{v}}}(t){\text{d}}t}\), then we arrive at
Similar to the derivation of (16), we have that \({\tilde{\mathbf{v}}}_{2}\) is still Gaussian with zeromean and covariance matrix
As depicted in (18), the uth element of \({\mathbf{y}}_{2}\) is
It is evident from (20) that given the uth LED is active, the distance between Mary PAM adjacent signal constellation points is
Under AWGN and equal a priori probability, the optimum Mary PAM demodulator is a multilevel decision circuit according to the following decision rule.
As derived in (15), given that the uth LED is active, the uth element of \({\mathbf{y}}_{1}\) is
As shown in Fig. 3, to demodulate the Nary PWM (NPWM) symbol, the information of \(\hat{m}\) obtained from (22) should be exploited. Therefore, the spacing between any two adjacent NPWM constellation points at the uth element of \({\mathbf{y}}_{1}\) is
Substituting (24) into (23), we have
There then, a multilevel decision circuit with the following decision rule can be implemented to demodulate the Nary PWM symbol.
4 Average symbol error rate (SER) analysis
4.1 SER of the ML detector
In general, deriving the exact SER of the optimum ML detector is intractable. Nevertheless, the union bound technique can be applied to express the upper bound of the average SER of a point to point optical SM MIMO VLS system as
where \(P_{e} ({\mathbf{x}}_{m,n,u} (t) \to {\mathbf{x}}_{{\hat{m},\hat{n},\hat{u}}} (t))\) denotes the pairwise error probability (PEP) of transmitting \({\mathbf{x}}_{m,n,u} (t)\) and detecting erroneously as \({\mathbf{x}}_{{\hat{m},\hat{n},\hat{u}}} (t)\). The event of a symbol error occurs in which the ML estimate \({\mathbf{x}}_{{\hat{m},\hat{n},\hat{u}}} (t)\) is different from the actual data vector \({\mathbf{x}}_{m,n,u} (t)\). Based on the ML algorithm as depicted in (10), we have
Substituting (7) into (28), we have that a symbol error occurs when
Let \({\mathbf{C}} \equiv {\mathbf{X}}_{m,n,u}  {\mathbf{X}}_{{\hat{m},\hat{n},\hat{u}}}\), and exploit the fact that \(\left\ {\mathbf{A}} \right\_{F}^{2} = tr({\mathbf{A}}^{T} {\mathbf{A}})\), (29) can be reformulated as
Let random variable Z be defined as \(Z \equiv  {\text{tr}}({\mathbf{C}}^{T} {\mathbf{H}}^{T} {\mathbf{V}})\), we can derive that \(Z \sim N\left( {0,\sigma^{2} \left\ {{\mathbf{HC}}} \right\_{F}^{2} } \right)\). Therefore, the PEP of transmitting \({\mathbf{x}}_{m,n,u} (t)\) and erroneously detecting as \({\mathbf{x}}_{{\hat{m},\hat{n},\hat{u}}} (t)\) can be calculated as
Substituting (31) into (27), the upper bound of \(P_{e,ML}\) can be obtained.
4.2 SER of the proposed algorithm
Let \(P_{e,u}\) be the SER when the uth LED is selected to transmit information, then under equal a priori probability assumption, the average overall SER can be expressed as
Let \(P_{e,s\left u \right.} ,P_{e,MPAM\left u \right.} ,P_{e,NPWM\left u \right.}\) denote the probability of making error decision of spatial symbol, MPAM and NPWM temporal symbols, respectively. We may separate the error types of \(P_{e,u}\) into the following parts:

(a)
Active LED is erroneously detected (spatial symbol error); both MPAM and NPWM temporal symbols are error.

(b)
Both spatial and MPAM symbols are error, while NPWM symbols are correct.

(c)
Both spatial and NPWM symbols are error, while MPAM symbols are correct.

(d)
Spatial symbol is error, while both temporal symbols are correct.

(e)
Spatial symbol is correct, while both temporal symbols are error.

(f)
Spatial and NPWM symbols are correct, while MPAM symbol is error.

(g)
Spatial and MPAM symbols are correct, while NPWM symbol is error.
In what follows, we have
At high SNR, the term \(P_{e,MPAM\left u \right.} P_{e,NPWM\left u \right.}\) is usually small compared with the other terms and thus can be neglected. Hence, (33) can be approximately as
To derive \(P_{e,s\left u \right.}\), we should base on the spatial symbol detection algorithm as described in Sect. 3.2.1. Given that uth LED is activated and the symbol being \(s_{m,n} (t)\), the correct detection probability of activated LED can be derived as
Under equal a priori probability assumption, the average correct detection probability of active LED can then be obtained as
Therefore
Based on the decision rule of (22), the average SER of Mary PAM can be derived as
Similarly, according to the decision rule of (26), the average SER of Nary PWM can be derived as
Toward this end, \(P_{e,u}\) can then be obtained by substituting (37)–(39) into (34).
Finally substituting (40) into (32), the theoretical expression of the average SER of the proposed scheme can be derived.
5 Results and discussion
5.1 Complexity analysis
In this subsection, we evaluate and compare the complexity of the proposed MFbased scheme with the ML detector. The computation load is measured using the total number of floatingpoint operations (flops) [22] required at the SM receiver, e.g., one flop is carried out for real addition and multiplication. Therefore, the multiplication of \(m \times n\) and \(n \times p\) real matrices requires \(m \times n \times p\) flops.
Based on the ML algorithm as depicted in (10), we can obtain that the number of flops required to implement the ML detector is \(MNN_{t} (N_{r}^{{}} N_{t} N + N_{{}}^{2} N_{r} )\) flops (in which totally \(MNN_{t}\) trials required with \(N_{r}^{{}} N_{t} N + N_{{}}^{2} N_{r}\) flops for each trial). Alternatively, as depicted in Eqs. (13–14, 17), only \(N_{t} N_{r} { + }N_{t} N\) flops are required to realize the proposed algorithm. Taking a general parameters’ setting as a working example: \(N_{r}^{{}} = 10,N_{t} = 8,\) M = 16, N = 8, the number of flops required for the proposed, and ML algorithms are 144 and 1,310,720, respectively. It is shown that the computation load of the proposed algorithm is extensively reduced compared to the ML algorithm.
5.2 Performance comparison of the ML and proposed detection algorithms
In this subsection, we will validate the theoretical derivation in previous section by computer simulation. Moreover, we will compare the performance of the optimum ML and the proposed detection scheme. The parameters setting of the indoor MIMO VLC channel follow those parameters given in [22]: \(\eta\) = 0.53, \(\phi_{1/2} = 37.5^\circ\), \(\Psi_{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern0pt} 2}}} = 90^\circ , \, A = 50{\text{ mm}}^{2}\), and \(d_{ij}^{{}}\) is set to be uniformly distributed within 2 ~ 3 m. Note that the signaltonoise power ratio (SNR) exploited for simulation is defined as \({\text{SNR = }}10\log_{10} \frac{{P_{t} }}{{\sigma^{2} }}\left( {{\text{dB}}} \right)\) and the simulation result is obtained from the average of 100,000 independent trials.
Since identification of the index of active LED at the transmitter is essential for the proposed algorithm, hence, we aim to evaluate the performance of the correct identification probability of the proposed algorithm. Figure 4 presents the simulation and theoretical [we use (35) and (36) to evaluate \(P_{c,s\left u \right.}\)] results of the probability of successful identification with respect to SNR, where the spatial–temporal modulation order is set as \((N_{t} ,M,N) = (8,4,8)\), respectively. The scenarios for different receiver array size, \(N_{r} = 8,16,24\), are provided for comparison. Alternatively, we fix the receiver array size as \(N_{r} = 12\), and the results for different spatial modulation order \(N_{t} = 8,16,32\) are presented in Fig. 5. From the results depicted in Figs. 4 and 5, we have the following observations:

(1)
As SNR increases, the correct identification probability of the index of activated LED increases as well.

(2)
Theoretical results are slightly better than the simulation results, whereas ML detection significantly outperforms the proposed detection algorithm. Nevertheless, as described in the previous subsection, the price for ML algorithm is higher and thus impractical computation load.

(3)
Performance degrades for larger \(N_{t}\). It is as expected since it is more probable to misidentify the index of the activated LED for larger \(N_{t}\). On the other hand, larger \(N_{r}\) leads to better performance. This is due mainly to the fact that as receive array size increases, larger degrees of freedom enhance the separability of the active LED from the inactive ones.
In the second part of simulations, we aim to evaluate the overall average SER (\(P({(}u,m,n) \ne (\hat{u},\hat{m},\hat{n}));u,\hat{u} = 1,...,N_{t} ,m,\hat{m} = 1,...,M,n,\hat{n} = 1,...,N\)) of the optimal ML and the proposed detection algorithms. Moreover, the performances of simulation and theoretical [we use (40), (35) and (32) to evaluate \(P_{e}\)] results of the proposed algorithm are also provided for comparison. As we set \((N_{t} ,M,N) = (8,4,8)\), Fig. 6 presents the overall SER with respect to SNR for the cases of \(N_{r}\) = 8, 16, and 24, respectively, while in Fig. 7, we set \((N_{r} ,M,N) = (12,4,8)\) and evaluate the overall SER for \(N_{t} = 8,16,32\). Based on the results depicted in Figs. 6 and 7, we have

(1)
Figures 6 and 7 demonstrate that the ML detector outperforms the proposed detection scheme. However, the benefits of the proposed algorithm are that the complexity reduction overwhelms slightly degradation in SER performance.

(2)
Figures 6 and 7 verify that the SER performance of the theoretical and computer simulation results is in good agreement.

(3)
Figures 6 and 7, respectively, demonstrate that larger \(N_{r}\) and/or smaller \(N_{t}\) yields better SER performance. The reasons are as we have claimed that as \(N_{r}\) is large, the spatial signatures of \(\left\{ {{\mathbf{h}}_{i} } \right\}_{{i = 1,...,N_{t} }}\) are more separable and larger \(N_{t}\) corresponds to larger spatial SER.
Finally, we compare the overall SER performance between the proposed SMPAWM and conventional optical spatial modulations, such as SMPAM or SMPWM. To make the comparison fair, the SER of the three schemes should be evaluated under the same bpcu. For example, a \(\left( {N_{t} ,M,N} \right){ = }\left( {8,4,8} \right)\) SMPAWM system should be compared with \(\left( {N_{t} ,M} \right) = \left( {8,32} \right)\) SMPAM or \(\left( {N_{t} ,N} \right) = \left( {8,32} \right)\) SMPWM system since their bpcu are all equal to 8 (\(\eta { = }\log_{2} 8{ + }\log_{2} 4 + \log_{2} 8 = \log_{2} 8{ + }\log_{2} 32 = 8\)). Figure 8 presents the overall SER with respect to SNR for the \(\left( {N_{t} ,M,N} \right){ = }\left( {8,4,8} \right)\) SMPAWM,\(\left( {N_{t} ,M} \right) = \left( {8,32} \right)\) SMPAM, and \(\left( {N_{t} ,N} \right) = \left( {8,32} \right)\) SMPWM systems in which the number of PDs at the receiving end is set as \(N_{r}\) = 16. As verified by the results presented in the figure, the proposed SMPAWM scheme outperforms the conventional SMPAM and SMPWM schemes.
6 Conclusions
In this paper, we have proposed a novel optical modulation and signaling scheme, named SMPAWM, operating in VLC MIMO communication system. The proposed SMPAWM scheme has higher spectral efficiency or bpcu over conventional SM based optical communication systems. Moreover, we have developed a novel detection scheme that are computationally efficient compared to the existing optimum ML detector. The theoretical analysis of the overall average SER has been comprehensively derived. Through extensive computer simulations, we have verified that the analytical and numerical results of the proposed algorithm are closely matched. It has been verified that the proposed detector can work reliably in VLC MIMO channel as the number of receiver array size is large and/or the number of LEDs is small. Therefore, with a much lower computational complexity, the proposed detection scheme is a feasible alternative to the existing ML detector for SM MIMO VLC system.
Data availability
The data that support the findings of this study are available on request from the corresponding author.
Abbreviations
 VLC:

Visible light communication
 SM:

Spatial modulation
 MIMO:

Multiinput multioutput
 SMF:

Spatial matched filtering
 PAWM:

Pulse amplitude Width Modulation
 LED:

Lightemitting diodes
 bpcu:

Bits per channel use
 ML:

Maximum likelihood
 SER:

Symbol error rate
 PD:

Photodetector
 PAM:

Pulse amplitude modulation
 AWGN:

Additive white Gaussian noise
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Wu, WC. Detection and performance analysis for MIMO visible light communication system using joint optical spatial and pulse amplitude width modulation. J Wireless Com Network 2024, 8 (2024). https://doi.org/10.1186/s1363802402335x
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DOI: https://doi.org/10.1186/s1363802402335x