## Abstract

This study proposes the optical wireless Code Shift Keying (CSK) with nonorthogonal sequences to improve the system performances of the Intensity Modulation Direct Detection (IM/DD) Optical Wireless Communications (OWCs). The proposed scheme systematically constructs nonorthogonal sequences. The proposed scheme is expected to improve the system performances of the IM/DD OWC. Optimum parameters can be used to obtain the best system performance; thus, this paper gives analytical expressions of the data transmission efficiency and delay property of the proposed scheme as a function of the system parameters. This paper analyzes the data transmission efficiency and the delay property of the proposed scheme considering the scintillation and background noise to evaluate the fundamental performance of the proposed system. This study also assumes a single-user case (i.e., single transmitter environment). The numerical results show that the data transmission efficiency and the delay property of the proposed scheme are better than those of the conventional schemes, regardless of the effect of the scintillation and/or background noise.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical Wireless Communications (OWCs) are expected to become next-generation intelligent transport system and indoor communications [1–23]. The Intensity Modulation and Direct Detection (IM/DD) systems have been investigated as practical OWCs. Accordingly, various methods have been investigated, including optical modulation and de-modulation systems [4], optical wireless channel modeling [5], and information modulation system in the IM/DD OWC system with Pseudo-Noise (PN) sequences [6,8–12].

The On-off Keying (OOK) system [6,8,9] and the Sequence Inversion Keying (SIK) [10–12] have been investigated as information modulation schemes in IM/DD OWC systems. The OOK system is useful in optical fiber communications. However, setting the ideal threshold in OWC is difficult because of the scintillation and background noise. SIK has been investigated to solve the threshold setting problem. The SIK system uses two PN sequences. The transmission node selects one PN sequence according to the transmission data. The SIK system achieves the same symbol error rate performance as the OOK system without setting a threshold. The SIK system is suitable for IM/DD OWCs. The data transmission efficiency of the SIK system is relatively low because the number of bits per sequence of the SIK system is one. One of the issues encountered with the IM/DD OWCs is the increase of the data transmission efficiency.

In [13–16], the Code Shift Keying (CSK) system was investigated for improving the system performance in IM/DD OWC systems. In the CSK system, one of the $M$ sequences is selected according to the transmission data. Moreover, the increase in the number of sequences increases the number of bits per sequence. Most CSK systems use orthogonal sequences; hence, long sequences are necessary to increase the number of orthogonal sequences, which may cause the data transmission efficiency to decrease.

Komuro and Habuchi originally proposed the nonorthogonal CSK system for radio communications [17–19]. According to their studies reported in [17–19], compared to the conventional orthogonal CSK system, the nonorthogonal CSK system can increase the sequence number with the same or shorter sequence length, which is expected to improve the Bit Error Rate (BER) performance and the frequency utilization efficiency. Besides, the nonorthogonal CSK system can easily achieve internal synchronous timing acquisition, which is difficult to achieve with the orthogonal CSK system. The realization of those features in OWCs using only non-negative signals is expected to improve the BER performance and the utilization efficiency. Therefore, it is essential to show the superiority of the basic performance of the nonorthogonal CSK system in OWCs.

This study, which was previously presented in part at GCCE2019 [20], proposes the optical wireless CSK with nonorthogonal sequences to improve the system performances of IM/DD OWC systems. The proposed scheme systematically constructs nonorthogonal sequences. The proposed scheme is expected to improve the data transmission efficiency because the proposed system can use more sequences than the conventional orthogonal CSK under the same sequence length. Optimum parameters can be used to obtain the best system performance; thus, this paper gives analytical expressions of the system performances of the proposed scheme as a function of the system parameters. This paper analyzes the data transmission efficiency and the delay property of the proposed scheme considering the scintillation and background noise to evaluate the fundamental performance of the proposed system. This study also assumes a single-user case (i.e., single transmitter environment). The numerical results show the effectiveness of the proposed scheme.

## 2. Proposed scheme

#### 2.1 Construction of the nonorthogonal sequence

In the proposed scheme, a nonorthogonal sequence, called a frame, is constructed by concatenating $M_{con}$ orthogonal sequences. The orthogonal sequence set $\boldsymbol{OS}$ is constructed by the {$+1,-1$} Hadamard matrix $\boldsymbol{H}_{M}$ expressed as

The proposed nonorthogonal sequence set $\boldsymbol {NS}$ is expressed as follows when the number of orthogonal sequences is $M_{os}$ (i.e., $\boldsymbol {OS}=\boldsymbol {H}_{M_{os}}$) and the number of concatenations is $M_{con}$:

#### 2.2 Example of a nonorthogonal sequence

Figure 1 shows an example of the nonorthogonal sequence construction. In the example, a nonorthogonal sequences is constructed by concatenating three orthogonal sequences. In the figure, one nonorthogonal sequence is constructed by multiplying one primitive orthogonal sequence by $(+--)$-polarity signals. To communicate in OWC, the transmitter converts chips with a value of $+1$ to $+1$ (mark) and ones with a value of $-1$ to 0 (space). When the length of the orthogonal sequence and the number of the orthogonal sequences are 4, the frame length and the number of nonorthogonal sequences are 12 $(=4 \times 3)$ and 32 $(=4 \times 2^3)$, respectively.

#### 2.3 System model

Figure 2 shows the model of the proposed scheme. The transmitter and the receiver are assumed to be synchronized. They prepare the same (common) $\{+1,-1\}$-valued orthogonal sequence set $\boldsymbol {OS}$ beforehand. The receiver uses $\boldsymbol {OS}$ as reference sequences. In the figure, $M_{os}$ and $M_{con}$ are the numbers of primitive orthogonal sequences and of the concatenations, respectively.

At the transmitter, $\log _2 2^{M_{con}} M_{os}$ (bit) data are split into $M_{con}$ (bit) data and $\log _2 M_{os}$ (bit) data at the data splitter. One of the $M_{os}$ orthogonal sequences is selected according to the $\log _2 M_{os}$ (bit) data. The polarity of the selected orthogonal sequence is decided according to the $M_{con}$ bits. Although the transmitter prepares an orthogonal sequence set, the transmitter transmits a nonorthogonal sequence. The transmitter converts $-1$ value into $0$ during transmission. Therefore, the transmitter transmits a pulse as the $+1$-valued chip and transmits no pulse as the $-1$-valued chip.

The receiver estimates the transmitted data without setting a threshold. In the receiver, the received optical power is correlated by each of the assigned $M_{os}$ orthogonal sequences composed of $+1$ and $-1$. The received optical power is converted into an electrical signal in every chip duration by the Avalanche Photo Diode (APD) for chip-level detection. The converted electrical signal is multiplied by each reference sequence every chip for one sequence period to obtain the correlation value between the received signal and each orthogonal sequence. The absolute values of the $M_{os}$ correlation outputs are examined. A sequence with the largest value $OS~i$ is then selected as the transmitted orthogonal sequence. Subsequently, the receiver demodulates the $\log _2 M_{os}$ (bit) data. The polarity is determined by correlating the selected orthogonal sequence with each of the positive $(OS~i)$, and negative $(\overline {OS~i})$ sequences. The receiver determines $M_{con}$ (bit).

## 3. Theoretical analysis

This study analyzes the data transmission efficiency and the delay property of the proposed scheme considering the scintillation, background noise, APD nose, and thermal noise. We basically followed the analyses in [1,22] to derive the fundamental performance of the proposed system. Furthermore, the data transmission efficiency is defined herein as the number of successful bits per chip duration, while the delay property is the number of transmissions until a successful transmission.

#### 3.1 Frame success ratio and data transmission efficiency

The probability density function of the scintillation $X$, $P(X)$ is expressed as

The average number of absorbed photons over the chip duration $T_c$ is

where $\lambda _{s}$ is the photon absorption rate; $h$ is Plank’s constant; $\eta _{A}$ is the APD efficiency; and $P_w$ is the received laser power without scintillation and background light. The total photon absorption rate due to the signal, background light, and APD bulk leakage current, $\lambda$, is expressed asThe probability that a frame is demodulated correctly, $P_c$, is expressed as

In Eq. (7), $P_{os}(X)$ is the symbol error rate of an orthogonal sequence. In other words, when the $i$th orthogonal sequence is transmitted, $P_{os}(X)$ is the probability that an absolute value for the $i$th orthogonal sequence, $|q_i|$, is larger than the other $M_{os}-1$ absolute correlation outputs, where $q_i~(i=1, 2, \ldots , M_{os})$ is a correlation output for the $i$th orthogonal sequence. Assuming that the 1st orthogonal sequence $OS~1$ is sent, $P_{os}$ is expressed as

In Eq. (9), $\otimes$ expresses the convolution integral, and $\mu _j(X)$ and $\sigma _j^2(X)$ are the average and the variance of the $j$th correlation output expressed as follows, respectively:

In Eqs. (11)–(13), $I_s$ is the APD surface leakage current; and the excess noise factor $F$ and the variance of the thermal noise $\sigma _{th}^2$ are given by

where $G$ is the average APD gain; $k_{eff}$ is the APD effective ionization ratio; $T_r$ is the receiver noise temperature; and $R_L$ is the receiver load resister.From the above, the data transmission efficiency of the proposed scheme is expressed as

#### 3.2 Bit error rate

Two types of errors can occur in the proposed system: 1) the estimation error probability of the transmitted orthogonal sequence $P_{os}$; 2) the polarity-decision error probability $P_{pol}$, where $P_{os}$ is given by Eq. (8). A decision error of polarity $P_{pol}=\frac {1}{2} \mathrm {erfc}\left ( \frac {\mu _1(X)}{\sqrt {2\sigma _1^2 (X)}} \right )$ if the estimation of the transmitted orthogonal sequence is correct. The conditional bit error probability given that the estimation error of the orthogonal sequence has occurred is $\frac {1}{2}$. Therefore, the polarity-decision error probability, $P_{pol}$, is expressed as

From Eqs. (8) and (17), the BER performance of the proposed system is expressed as

#### 3.3 Delay property

The $d$th frame is successfully transmitted after the $(d-1)$ transmission failures. Thus, the delay property $D$ can be expressed as

Based on Eq. (19), the delay normalized by the number of bits is expressed as

## 4. Numerical results

Table 1 shows parameters used herein. This paper uses typical APD parameters [1,22]. Parameters in Table 1 are determined according to the values used in [1,21,22].

#### 4.1 Data transmission efficiency

Figure 3 shows the data transmission efficiency versus the transmission laser power per bit $P_{bit}$ of the SIK scheme, CSK scheme with orthogonal sequence (orthogonal CSK), and proposed scheme when $\sigma _s^2$ is 0.1, and $P_{b}$ is $-45$ (dBm), where $P_{bit} = \frac {P_{w}}{\log _2 2^{M_{con}} M_{os}}$. In the proposed scheme, the combinations of $(M_{os},M_{con})$ are $(8,3),~(8,4)$, and $(32,1)$. The frame lengths of the SIK scheme and the orthogonal CSK scheme are 32 (chip). The frame lengths of the proposed scheme are 24 (chip) for $(M_{os},M_{con})=(8,3)$ and 32 (chip) for $(M_{os},M_{con})=(8,4)$ and $(32,1)$. The lines show the analysis results, while the plots depict the simulation results. The analysis results are in a good agreement with the simulation results. Figure 3 illustrates that the data transmission efficiency of the proposed scheme is higher than those of the SIK and orthogonal CSK schemes, showing the effectiveness of the proposed scheme. The data transmission efficiencies of all schemes are low in the range where the effect of noise is large. In addition, the data transmission efficiencies of the proposed scheme with $((M_{os}=8, M_{con}=3)$ and $(M_{os}=8,M_{con}=4))$ are worse than those of the proposed scheme with $(M_{os}=32, M_{con}=1)$ and the orthogonal CSK scheme in the range where the effect of noise is large because of the inter-symbol interference. Although the data transmission efficiency of the proposed scheme with ($M_{os}=8,M_{con}=3)$ is worse than those of the other CSK schemes in the range where the effect of noise is large, it shows the best data transmission efficiency in the range where the effect of noise is small.

Table 2 shows the correlation value and the number of sequences corresponding to the correlation value. The $(M_{os},M_{con})$ combinations in the proposed scheme are $(8,3),~(8,4)$, and $(32,1)$. In Table 2, the correlation values are normalized by the frame length. The proposed scheme has more sequences than the orthogonal CSK when the frame length of the proposed scheme is equal to or shorter than that of the orthogonal CSK. Table 2 also shows that the proposed scheme uses sequences of which the correlation values are not 0, causing the inter-symbol interference.

Figure 4 depicts the BER performances versus $P_{bit}$ of the SIK, orthogonal CSK, and proposed schemes when $\sigma _s^2$ is 0.1, and $P_{b}$ is $-45$ (dBm). In the proposed scheme, the $(M_{os},M_{con})$ combinations are $(8,3),~(8,4)$, and $(32,1)$. The lines show the analysis results. The plots illustrate the simulation results. The Analysis results are in a good agreement with the simulation results. Figure 4 also exhibits that the BER performances of the orthogonal CSK and the proposed schemes are better than that of the SIK scheme, denoting the effectiveness of the CSK schemes in terms of the BER performance of IM/DD systems.

Figure 5 shows the data transmission efficiencies versus the logarithm variance of the scintillation $\sigma _s^2$ of the SIK, orthogonal CSK, and proposed schemes when $P_{b}=-45$ (dBm) and $P_{bit}=-42$(dBm). The $(M_{os},M_{con})$ combinations in the proposed scheme are $(8,3),~(8,4)$, and $(32,1)$. The lines depict the analysis results. The plots present the simulation results. Analysis results are in a good agreement with the simulation results. Figure 5 illustrates that the data transmission efficiency of the proposed, SIK, and orthogonal CSK schemes degrade as $\sigma _s^2$ increases. Moreover, the data transmission efficiency of the proposed scheme is higher than those of the SIK and orthogonal CSK schemes, regardless of $\sigma _s^2$, showing the effectiveness of the proposed scheme. The proposed scheme with ($M_{os}=8,M_{con}=3)$ shows the highest data transmission efficiency under a low-scintillation environment ($\sigma _s^2 < 0.6$).

Figure 6 exhibits the data transmission efficiency of the proposed scheme as a function of the frame length, $L_{f}$, when $\sigma _s^2$ is 0.1; $P_{b}$ is $-45$ (dBm); $M_{con}=1, 2, 3, 4$, and $5$; $P_{bit}$ is $-45$ (dBm). A peak transmission efficiency exists for each $M_{con}$. The data transmission efficiency is low at a low $L_{f}$ because of the low noise tolerance. On the contrary, the data transmission efficiency decreases as $L_{f}$ increases because of the waste of increasing the number of chips.

#### 4.2 Delay property

Figure 7 shows the normalized delay versus $P_{bit}$ of the SIK, orthogonal CSK, and proposed schemes when $\sigma _s^2$ is 0.1, and $P_{b}$ is $-45$ (dBm). The $(M_{os},M_{con})$ combinations in the proposed scheme are $(8,3),~(8,4)$, and $(32,1)$. The lines show the analysis results. The plots depict the simulation results. Analysis results are in a good agreement with the simulation results. Figure 7 illustrates that the delays of the proposed and orthogonal CSK schemes are lower than that of the SIK scheme, showing the effectiveness of the CSK schemes in terms of the delay property.

Figure 8 exhibits the normalized delay versus the logarithm variance of the scintillation $\sigma _s^2$ of the SIK, orthogonal CSK, and proposed schemes when $P_{b}=-45$ (dBm) and $P_{bit}=-42$(dBm). The $(M_{os},M_{con})$ combinations in the proposed scheme are $(8,3),~(8,4)$, and $(32,1)$. The lines show the analysis results. The plots depict the simulation results. Analysis results are in a good agreement with simulation results. Figure 8 also shows that the delays of the proposed, SIK, and orthogonal CSK schemes degrade as $\sigma _s^2$ increases. In addition, the delay of the proposed scheme with ($M_{os}=8,M_{con}=4)$ shows the lowest delay property regardless of $\sigma _s^2$.

Figure 9 shows the normalized delay of the proposed scheme as a function of $L_{f}$ when $\sigma _s^2$ is 0.1; $P_{b}$ is $-45$ (dBm); $M_{con}=1, 2, 3, 4$, and $5$; and $P_{bit}$ is $-45$ (dBm). The delay decreases as $L_{f}$ increases because of the improvement of the noise tolerance.

#### 4.3 Parameter optimization

Next, we investigate the optimum parameter of the proposed scheme. We use an objective function that considers the data transmission efficiency and the delay property to obtain the optimal parameters of the proposed scheme. The objective function $F^{*}$ is defined as follows:

The optimal system parameters of the proposed scheme can be predicted by obtaining the combination of $M_{os}$ and $M_{con}$, such that the objective function is minimized. Fig. 10 shows the objective function $F^{*}$ of the proposed scheme versus $L_{f}$ when $\sigma _s^2$ is 0.1; $P_{b}$ is $-45$ (dBm); $M_{con}=1, 2, 3, 4$, and $5$; and $P_{bit}$ is $-45$ (dBm). The combination ($M_{os}=4$, $M_{con}=4$, $L_{f}=16$) depicts the optimum system parameters in the proposed scheme.

## 5. Conclusion

This study proposed the optical wireless CSK with nonorthogonal sequences to improve the data transmission efficiency and the delay property of IM/DD OWCs. The proposed scheme systematically constructed nonorthogonal sequences, and the system characteristics can be comprehended analytically with some assumptions. The optimal parameter set, including the numbers of orthogonal sequences and concatenations can be predicted, by using the analytical expressions. The following results were drawn from the numerical results:

- • The proposed scheme shows a better data transmission efficiency than the conventional IM/DD OWC system, regardless of the effect of the scintillation and/or background noise.
- • The proposed scheme shows a better delay property than the conventional IM/DD OWC system, regardless of the effect of the scintillation and/or background noise.
- • Considering both the data transmission efficiency and the delay property, the combination ($M_{os}=4$, $M_{con}=4$, $L_{f}=16$) is the optimum system parameters in the proposed scheme.

These results confirmed that the proposed nonorthogonal sequence is effective in terms of improving the data transmission efficiency and the delay property of the IM/DD OWC system.

Future work includes the experimental investigation.

## Funding

Telecommunications Advancement Foundation.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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