Impact of systematically constructed nonorthogonal code shift keying for optical code division multiple access

Nobuyoshi Komuro; Hiromasa Habuchi

doi:10.1364/OPTCON.479147

1. Introduction

Optical wireless communications (OWCs) are expected to become a next-generation intelligent transport system and indoor communications [1–21]. The intensity modulation and direct detection (IM/DD) systems have been investigated as practical OWCs. Various methods have been investigated, such as the optical modulation and de-modulation systems [4], modeling of the optical wireless channel [5], and information modulation system in IM/DD OWC system with pseudo-noise (PN) sequences [6,9,10,14–17]. For practical use in the above application systems, it is also desirable to provide the multiple-access capability to the IM/DD systems: i.e., optical code division multiple access (OCDMA) system.

IM/DD OCDMA modulation schemes can be classified into on-off keying (OOK) type [9,10], pulse position modulation (PPM) type [5,22], sequence inversion keying (SIK) type [14–16], and code shift keying (CSK) type [18–21]. In order to solve the multiple access interference (MAI) problem, which is one of the problems of OCDMA, interference canceler techniques [3,16] and orthogonalization of spreading codes [11–13] have been considered. The spreading-code orthogonalization includes optical orthogonal codes and extended prime codes [12,13], designed to avoid chip-pulse collisions, and modified pseudo-orthogonal M-sequences [17], designed to suppress interference at the receiver. OOK and SIK types have a data transmission efficiency of 1 (bit/code). Although PPM can improve the bit error rate performance because of its orthogonal multi-level transmission characteristics, the data transmission efficiency is $\frac {\log _2 N}{N}$ (bit/code) when the number of slots per frame is $N$, which is lower than OOK.

On the other hand, the data transmission efficiency of the CSK type is $\log _2 M$ (bit/code) when the number of codes used by one user is $M$, which is higher than those of the OOK, PPM, and SIK types. Moreover, the bit error rate performance of the CSK type is similar to those of the PPM type with $N=M$. This study focuses on the CSK type because of the advantages of CSK.

Although researchers have studied the CSK type by using the group characteristics of extended prime codes [18–21], the performance of the CSK type is about $M=2$, which does not show its original performance. Komuro and Habuchi have increased $M$ by using nonorthogonal codes instead of orthogonal codes for the $M$ spreading codes given to the user, thereby increasing the data transmission efficiency [23–28].

This study examines the performance of the authors’ previously proposed system when used in a multi-user environment. This study aims to clarify the performance in a multi-user environment by focusing on nonorthogonal CSK, which can further improve the data transmission efficiency characteristics of the CSK type, which is superior in terms of data transmission efficiency and error rate characteristics. Our proposed nonorthogonal CSK system systematically constructs nonorthogonal codes. This paper analyzes the system performances of the proposed system considering the scintillation, background noise, and MAI to evaluate the fundamental performance of the proposed system. The numerical results show the effectiveness of the proposed system.

2. Proposed system structure

2.1 Construction of nonorthogonal code

The proposed system constructs a nonorthogonal code is constructed by concatenating $M_{con}$ pseudo-orthogonal M-sequences (POMs). The POM set $\boldsymbol {POM}$ is composed of M-sequence and an additional chip, which is expressed as

(1)$$\begin{aligned}\boldsymbol{POM} = \left( \begin{array}{c} POM_1\\ POM_2\\ \cdots\\ POM_{M_{pom}} \end{array} \right) = \left( \begin{array}{cccccc} a_1 & a_2 & a_3 & \cdots & a_{L_M} & -1\\ a_2 & a_3 & \cdots & a_{L_M} & a_1 & -1\\ & & \vdots & & & \\ a_{L_M} & a_1 & a_2 & \cdots & a_{L_M-1} & -1 \end{array} \right), \end{aligned}$$

where $a_i\,(i=1, 2, \ldots, L_M)$ is $+1$ or $-1$ value and $L_M$ is the length of M-sequence. The transmitter converts the -1 value into 0 when transmitting. And the receiver has $\{+1, -1\}$-valued POM set. The cross-correlation value of the $i$-th POM of transmitter, $POM_{T_i}$, and the $j$-th POM of receiver, $POM_{R_j}$, is expressed as

(2)$$\begin{aligned}\frac{1}{T} \int_{0}^{T} POM_{T_i}(t) POM_{R_j}(t) dt = \left\{ \begin{array}{cc} \frac{L_M}{2} & (T_i = R_j) \\ 0 & (T_i \neq R_j), \end{array} \right. \end{aligned}$$

where $T$ is a cycle of POM.

When the number of POMs is $M_{pom}$ and the number of concatenations is $M_{con}$, the nonorthogonal codes $\boldsymbol {NS}$ are expressed as

(3)$$\begin{aligned}\boldsymbol{NS} =\left( \begin{array}{c} NS\,1\\ NS\,2\\ \vdots \\ NS\,M_{pom} \cdot 2^{M_{con}} \end{array} \right) = \left( \begin{array}{ccccc} \boldsymbol{POM}^{(1)} & \boldsymbol{POM}^{(2)} & \cdots & \boldsymbol{POM}^{(M_{con}-1)} & \boldsymbol{POM}^{(M_{con})}\\ \boldsymbol{POM}^{(1)} & \boldsymbol{POM}^{(2)} & \cdots & \boldsymbol{POM}^{(M_{con}-1)} & \overline{\boldsymbol{POM}^{(M_{con})}}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \overline{\boldsymbol{POM}^{(1)}} & \overline{\boldsymbol{POM}^{(2)}} & \cdots & \overline{\boldsymbol{POM}^{(M_{con}-1)}} & \boldsymbol{POM}^{(M_{con})}\\ \overline{\boldsymbol{POM}^{(1)}} & \overline{\boldsymbol{POM}^{(2)}} & \cdots & \overline{\boldsymbol{POM}^{(M_{con}-1)}} & \overline{\boldsymbol{POM}^{(M_{con})}} \end{array} \right),\end{aligned}$$

where $\boldsymbol {POM}^{(i)}$ is the $i$-th element and $\overline {\boldsymbol {POM}^{(i)}}$ is the negative of $\boldsymbol {POM}^{(i)}$.

In the proposed system, each transmitter has $\lfloor \frac {M_{pom}}{K} \rfloor$ POMs, where $M_{pom}$ is the number of POMs, and $K$ is the number of users. A nonorthogonal code, called a frame, is constructed by concatenating $M_{con}$ POM. POM pair sets [17] are used as primitive codes. A frame length, $L_{f}$, is $L_{om} M_{con}$, where $L_{om}$ is the length of POM. In the proposed system, a transmitter selects one of $M_{pom}$ POMs from $\lfloor \log _2 \frac {M_{pom}}{K} \rfloor$ (bit) data, DATA 1. The selected POM is concatenated by $M_{con}$ (bit) data, DATA 2, where $M_{con}$ is the concatenation pattern of the selected POM. A frame has $\left ( \lfloor \log _2 \frac {M_{pom}}{K} \rfloor + M_{con} =N_{bit} \right )$ (bit) data. In order to communicate in OWC, a transmitter converts chips with a value of $+1$ to $+1$ (mark) and converts ones with a value of $-1$ to 0 (space).

2.2 System model

Figure 1 shows the model of the proposed system. Also, Fig. 2 shows the example of a transmission signal structure. The transmitter prepares $\{+1,0\}$-valued POMs. The transmitter converts source data into $\lfloor \log _2 \frac {M_{pom}}{K} \rfloor + M_{con}$ (bit). Transmitter $k$ selects one of $M_{pom}$ POMs according to $\lfloor \log _2 \frac {M_{pom}}{K} \rfloor$ (bit) data, DATA $1^{(k)}$. Then, the transmitter $k$ concatenates the selecting POM with the concatenation pattern based on the $M_{con}$ (bit) data, DATA $2^{(k)}$. The transmitter converts the $-1$ value into 0 when transmitting. Therefore, the transmitter transmits a pulse as the $+1$-valued chip and no pulse as the $-1$-valued chip. Background light, APD, and thermal noises are assumed to cause the noise shown in Fig. 1. Also, scintillation is assumed to cause light intensity changes.

Fig. 1. System model.

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The receiver prepares $\{+1,-1\}$-valued POMs, which are pairs of those of the transmitter. The receiver obtains the electric signal by chip-level Avalanche Photo Diode (APD). Figure 3 shows the receiver procedure. After APD detection, the receiver calculates the absolute value of correlations between the received $\{+1,0\}$-valued signal and each $\{+1,-1\}$-valued POM of the receiver. Then, it selects a POM with the $k$-th largest correlation value. The receiver demodulates $\lfloor \log _2 \frac {M_{pom}}{K} \rfloor$ (bit) data (DATA $1^{(k)}$) from the selected POM. The receiver also obtains each correlation-value polarity of the selected POM. Then, the receiver demodulates $M_{con}^{(i)}$ (bit) data (DATA $2^{(k)}$) from the obtained polarity.

Fig. 2. Example of a transmission signal structure.

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Fig. 3. Receiver’s procedure.

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3. Theoretical analysis

This paper analyzes the data transmission efficiency of the proposed system. Similar to the Refs. [1,22], this analysis takes into account scintillation, background, and APD noises with noise dispersion depending on signal strength as the optical channel. In order to derive the fundamental performance of the proposed system, this paper basically follows the analyses in [1,22]. This paper defines data transmission efficiency as the total number of successful bits of all transmission users per chip duration.

3.1 Frame success ratio and data transmission efficiency

The probability density function of scintillation for the desired user’s transmission $X_D$, $P(X_D)$ is expressed as

(4)$$P(X_D) =\frac{1}{\sqrt{2\pi \sigma_s^2}X_0} \exp\left[-\frac{\left(\ln X_D + \frac{\sigma_s^2}{2} \right)^2}{2\sigma_s^2} \right],$$

where the average of scintillation $X_D$ is normalized to unity and $\sigma _s^2$ is logarithm variance which is determined by the atmospheric state [22]. When he number of concurrent transmission users is $K$, the probability density function of scintillation for $K-1$ interference users’ transmissions, $P(X)$, is expressed as

(5)$$P(X) = P(X_1) \otimes P(X_2) \otimes \cdots \otimes P(X_{K-1}),$$

where $X=X_1+X_2+ \cdots +X_{K-1}$ and $\otimes$ expresses the convolution integral.

The average number of absorbed photons over $T_c$ is

(6)$$\lambda_{s} = \frac{\eta_{A} P_{w}}{hf},$$

where $\lambda _{s}$ is the photon absorption rate, $h$ is Plank’s constant, $\eta _{A}$ is the APD efficiency, and $P_w$ is 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 as

(7)$$\begin{aligned} \lambda &= \left\{ \begin{array}{ll} \lambda_{s} + \lambda_{b} + \displaystyle\frac{I_{b}}{e} & \mathrm{for\,a\,mark} \\ \displaystyle \frac{\lambda_{s}}{Me} + \lambda_{b} + \displaystyle \frac{I_{b}}{e} & \mathrm{for\,a\,space} \\ \end{array} \right. , \end{aligned}$$

where $I_b$ is APD bulk leakage current, $e$ is charge element, $I_{b}/e$ represents the contribution of the APD bulk leakage current to the APD output, $Me$ is the modulation extinction ratio of the laser diode output power in the mark and space states, and $\lambda _{b}$ is the photon absorption rate due to actual background light, which is expressed as $\lambda _{b}=\frac {\eta _{A}P_{b}}{hf}$ when the background noise per chip duration is $P_{b}$.

When the number of concurrent transmission users is $K$, the probability that a frame is demodulated correctly, $P_c(K)$, is expressed as

(8)$$\begin{aligned} P_c(K)&= \int_{0}^{\infty} P(X_D)\\ &\times \int_{0}^{\infty} P(X) \left[ 1 - \frac{1}{2} \mathrm{erfc} \left( \frac{\mu_D(X_D)}{\sqrt{2\sigma_D^2 (X_D) M_{con}}} \right) \right] \big(1-P_{pom}(X,K)\big) dX_D dX, \end{aligned}$$

where $\mathrm {erfc}(x)= \frac {2}{\sqrt {\pi }} \int _x^{\infty } \exp (-t^2) dt$ is error function complementary, $\mu _D(X)$ and $\sigma _D^2(X)$ are the average and variance of the largest correlation output of the desired user, which are expressed as

(9)$$\mu_D(X_D) = GT_c \left( \frac{L_f+1}{2} \lambda_s X_D + \frac{L_f-1}{2}\frac{\lambda_s X_D}{Me} + \lambda_b \right)$$

and

(10)$$\sigma_D^2(X_D) = G^2 FT_c \left[ K\frac{L_f+1}{2} \lambda_s X_D + K\frac{L_f^2-1}{2} \frac{\lambda_s X_D}{Me} + L_f\lambda_b + \frac{2I_b}{e} \right] + \frac{2I_sT_c}{e} + 2\sigma_{th}^2$$

respectively. In Eq. (8), $P_{pom}(X,K)$ is the symbol error rate of an POM. In other words, when the $i$-th POM is transmitted, $P_{pom}(X,K)$ is the probability that an absolute value for $i$-th POM, $|q_i|$, is larger than the other $\lfloor \frac {M_{pom}}{K} \rfloor -1$ absolute correlation outputs, where $q_i\,(i=1, 2, \ldots, \lfloor \frac {M_{pom}}{K} \rfloor )$ is a correlation output for $i$-th POM. Assuming that a user sends the 1st POM, $x_1$, $P_{pom}(X,K)$ is expressed as

(11)$$\begin{aligned} P_{pom}(X_D,K)&= 1-Prob[|q_1|>|q_2|, |q_1|>|q_3|, \ldots, |q_1|>|q_{\lfloor \frac{M_{pom}}{K} \rfloor}|]\\ &= 1 - \int_{-\infty}^{\infty} f(x_1) \left[ \int_{-\infty}^{x_1} f(x_{\stackrel{j}{j \neq 1}},X_D) dx_j \right]^{\lfloor \frac{M_{pom}}{K} \rfloor -1} dx_1, \end{aligned}$$

where $f(x_j,X)$ is the probability density function of correlation output for $j$-th POM $x_j$, which is expressed as

(12)$$\begin{aligned} f(x_j,X_D) = \underbrace{g(|q_j|,X) \otimes \cdots \otimes g(|q_j|,X)}_{M_{con}\,times}, \end{aligned}$$

(13)$$\begin{aligned}g(q_j,X_D) = \frac{1}{\sqrt{2 \pi \sigma_{j}^2(X)}} \exp \left[ -\frac{(q_j - \mu_j(X_D))^2}{\sqrt{2\sigma_{j}^2(X_D)}} \right]. \end{aligned}$$

In Eq. (12), $\mu _j(X)$ and $\sigma _j^2(X)$ are the average and variance of the $j$-th largest correlation output, which are expressed as

(14)$$\begin{aligned} \mu_1(X_D) = GT_c \left( \frac{L_f+1}{2} \lambda_s X_D + \frac{L_f-1}{2}\frac{\lambda_s X_D}{Me} + \lambda_b \right) \end{aligned}$$

(15)$$\begin{aligned} \mu_{j (j\neq 1)}(X_D) = GT_c \left( \frac{L_f \lambda_s X}{Me} + \lambda_b \right), \end{aligned}$$

and

(16)$$\sigma_j^2(X_D) = G^2 FT_c \left[ K\frac{L_f+1}{2} \lambda_s X_D + K\frac{L_f^2-1}{2} \frac{\lambda_s X_D}{Me} + L_f\lambda_b + \frac{2I_b}{e} \right] + \frac{2I_sT_c}{e} + 2\sigma_{th}^2$$

respectively. In Eqs. (14)–(16), $I_s$ is APD surface leakage current, $T_c$ is a chip duraion, and excess noise factor $F$ and variance of thermal noise $\sigma _{th}^2$ are given by

(17)$$\begin{aligned}F = k_{eff}G+(1-k_{eff})\frac{2G-1}{G} \end{aligned}$$

(18)$$\begin{aligned}\sigma_{th}^2 = \frac{2k_B T_r T_c}{e^2 R_L}, \end{aligned}$$

where $G$ is the average APD gain, $k_{eff}$ is APD effective ionization ratio, $T_r$ is receiver noise temperature, and $R_L$ is the receiver load resister.

From the above, the data transmission efficiency of the proposed system is expressed as

(19)$$S_{sys} = \frac{K N_{bit} P_{c}}{L_f}.$$

3.2 Bit error rate

The BER performance is derived based on [28]. Two types of errors can occur in the proposed system: 1) the estimation error probability of the transmitted POM $P_{e1}(K)$; 2) the polarity-decision error probability $P_{e2}(K)$, where $K$ is the number of concurrent transmission users. $P_{e1}(K)$ is derived based on Eqs. (4), (5), and (11);

(20)$$\begin{aligned} P_{e1}(K)&= \int_0^{\infty} P(X_D) \int_0^{\infty} P(X) P_{pom}(X_D,K) dX_D dX\\ &= \int_0^{\infty} P(X_D)\\ &\times\int_0^{\infty} P(X)\left\{ 1 - \int_{-\infty}^{\infty} f(x_1) \left[ \int_{-\infty}^{x_1} f(x_{\stackrel{j}{j \neq 1}},X_D) dx_j \right]^{\lfloor \frac{M_{pom}}{K} \rfloor -1} dx_1 \right\} dX_D dX. \end{aligned}$$

If the estimation of the transmitted orthogonal sequence is correct, a decision error of polarity is expressed as

(21)$$P_{e2}(K)=\int_0^{\infty} P(X_D) \int_0^{\infty} P(X) \left[ \frac{1}{2} \mathrm{erfc}\left( \frac{\mu_D(X)}{\sqrt{2\sigma_D^2 (X)}} \right) \right] dX_D dX.$$

The conditional bit error probability given that the estimation error of the POM has occurred is $\frac {1}{2}$. Therefore, the polarity-decision error probability, $P_{e2}(K)$, is expressed as

(22)$$\begin{aligned} P_{e2}(K)&= \int_0^{\infty} P(X_D)\\ &\times \int_0^{\infty} P(X) \left[ 1 - \frac{1}{2} \mathrm{erfc} \left( \frac{\mu_D(X_D)}{\sqrt{2\sigma_D^2 (X_D)}} \right) \right] \bigg(1-P_{pom}(X_D,K) \bigg) dX_D dX\\ & + \frac{1}{2}P_{e1}(K). \end{aligned}$$

From Eqs. (20) and (22), the BER performance of the proposed system is expressed as

(23)$$BER(K) =\displaystyle \frac{\log_2 M_{pom}}{\log_2 M_{pom}+M_{con}} \left( \frac{2^{\log_2 M_{pom} -1}}{M_{pom}-1 } \right) P_{e1}(K) + \displaystyle \frac{M_{con}}{\log_2 M_{pom}+M_{con}} P_{e2}(K).$$

4. Numerical results

This study evaluates the proposed system’s performance through analysis and simulation. Figure 4 shows the simulation diagram. This study did the simulation by Monte Carlo simulation. In the simulation, first, each transmitter generates data bits and transmits them. Then, the simulation multiplies each light intensity of transmission signals by a random value based on a log-normal distribution (i.e., scintillation). Also, the simulation adds a random value based on a normal distribution to those signals (i.e., noise). In other words, the simulation artificially gives scintillation and noise, similar to the Refs. [1,22]. The simulation calculates the data transmission efficiency from the demodulated signal at the receiver. Also, the simulation calculates the BER by comparing the transmitted data with the demodulated data. The simulations were done 10000000 times in order to calculate BER and data transmission efficiency.

Fig. 4. Simulation diagram.

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Table 1 shows the parameters which are used in this paper. This paper uses typical APD parameters [1,22]. Parameters in Table 1 are determined according to the values used in [1,22,29].

Table 1. Notations

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Figure 5 shows the BER performances versus transmission laser power per bit $P_{bit}$ of the proposed, orthogonal CSK, and SIK systems when $\sigma _s^2$ is 0.01, $P_{b}$ is -45 (dBm), and $K$ is 2. In the proposed system, the combinations of $(M_{pom},M_{con})$ are $(7,3),\,(7,4)$, and $(31,1)$. The frame length of the conventional CSK system is 32 (chip). The frame lengths of the proposed system are 24 (chip) for $(M_{pom},M_{con})=(7,3)$, 32 (chip) for $(M_{pom},M_{con})=(7,4)$ and $(31,1)$. Lines show the analysis results, and plots show the simulation results obtained. Analysis results are qualitatively agreements with simulation results. It is seen from Fig. 5 that the BER performances of the proposed system are better than those of the orthogonal CSK and SIK systems, denoting the effectiveness of the nonorthogonal CSK system in terms of the BER performance of OCDMA system.

Fig. 5. Bit error rate obtained from analyses (lines) and simulation results (plots) of the proposed, orhogonal CSK, SIK systems as a function of $P_{bit}$ ($\sigma _s^2=0.01$, $P_{b} =-45$ (dBm), and $K=2$).

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Figure 6 shows the data transmission efficiency versus $P_{bit}$ of the proposed, orthogonal CSK, and SIK systems when $\sigma _s^2$ is 0.01, $P_{b}$ is -45 (dBm), and $K$ is 2. In the proposed system, the combinations of $(M_{pom},M_{con})$ are $(7,3),\,(7,4)$, and $(31,1)$. Lines show the analysis results, and plots show the simulation results. Analysis results qualitatively agree with simulation results. It is seen from Fig. 6 that the data transmission efficiency of the proposed system is higher than those of the orthogonal CSK and SIK systems, denoting the effectiveness of the proposed system. In addition, the data transmission efficiency of the proposed system with $((M_{pom}=7, M_{con}=3)$ is worse than that of the proposed systems with $((M_{pom}=31, M_{con}=1)$ and $((M_{pom}=7, M_{con}=4)$ in the range where the effect of noise is large because of the inter-symbol interference. Although the data transmission efficiency of the proposed system with ($M_{pom}=7,M_{con}=3)$ is worse than those of the other nonorthogonal CSK systems 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.

Fig. 6. Data transmission efficiencies obtained from analyses (lines) and simulation results (plots) of the proposed, orthogonal CSK, and SIK systems as a function of $P_{bit}$ ($\sigma _s^2=0.01$, $P_{b} =-45$ (dBm), and $K=2$).

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Figure 7 shows the data transmission efficiency versus $P_{bit}$ of the proposed, orthogonal CSK, and SIK systems when $\sigma _s^2$ is 0.01, $P_{b}$ is -45 (dBm), and $K$ is 3. In the proposed system, the combinations of $(M_{pom},M_{con})$ are $(7,3),\,(7,4)$, and $(31,1)$. Lines show the analysis results, and plots show the simulation results. Analysis results are qualitatively agreements with simulation results. As well as $K=2$, although the data transmission efficiency of the proposed system with ($M_{pom}=7,M_{con}=3)$ is worse than those of the other nonorthogonal CSK systems 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. It is seen from Figs. 6 and 7 that the data transmission efficiency for $K=3$ is worse than that for $K=2$ in the range where the effect of noise is large because of the inter-user interference. On the other hand, the proposed system with $K=3$ shows higher data transmission efficiency than that with $K=2$ in the range where noise is small.

Fig. 7. Data transmission efficiencies obtained from analyses (lines) and simulation results (plots) of the proposed, orthogonal CSK, and SIK systems as a function of $P_{bit}$ ($\sigma _s^2=0.01$, $P_{b} =-45$ (dBm), and $K=3$).

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Figure 8 shows the data transmission efficiency versus $\sigma _{s}^2$ of the proposed, orthogonal CSK, and SIK systems when $P_{bit}$ is -44 (dBm), $P_{b}$ is -45 (dBm), and $K$ is 3. In the proposed system, the combinations of $(M_{pom},M_{con})$ are $(7,3),\,(7,4)$, and $(31,1)$. Lines show the analysis results, and plots show the simulation results. Analysis results are qualitatively agreements with simulation results. It is seen from Fig. 8 that the data transmission efficiency of the proposed and the orthogonal CSK system degrade as $\sigma _s^2$ increases. Figure 8 also illustrates that the proposed system with ($M_{pom}=7,M_{con}=3)$ shows the highest data transmission efficiency when the effect of scintillation is small ($\sigma _s^2 < 0.1$).

Fig. 8. Data transmission efficiencies obtained from analyses (lines) and simulation results (plots) of the SIK, orthogonal CSK, and proposed systems as a function of $\sigma _s^2$ ($P_{bit}=-44$ (dBm) and $P_{b} =-45$ (dBm)).

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Table 2 shows the achievable data transmission efficiency and the maximum capacity of users when the frame length is $L_{f} \,(=(M_{pom}+1)M_{con})$. Table 2 shows that while those of the orthogonal CSK and SIK systems approach 0.5 as $L_f$ increases, the data transmission efficiency of the proposed system approaches 1.0. Table 2 also shows that the maximum capacity of the proposed system with $M_{con}=1$ is the best among SIK and CSK systems.

Table 2. Achievable data transmission efficiency and maximum capacity of users when the frame length is $L_f \,(=(M_{pom}+1)M_{con})$

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Our obtained results show that the proposed system with $M_{con}=1$ is the best from the user-capacity perspective, while the data transmission efficiency of the proposed system with $M_{con}=3$ is the best at a small number of users.

5. Conclusion

This study proposed the nonorthogonal CSK system in order to clarify the performance improvement of the proposed system on the OCDMA system. Our proposed nonorthogonal CSK system systematically constructs nonorthogonal codes. This paper analyzed the system performances of the proposed system considering the scintillation, background noise, and multiple access interferences to evaluate the fundamental performance of the proposed system. The numerical results show the effectiveness of the proposed system. The numerical results showed that the proposed system outperformed the conventional OCDMA systems. Our obtained results that the systematically constructed nonorthogonal code is effective for enhancing OCDMA system performance is a useful finding for future research approaches.

Funding

Japan Society for the Promotion of Science (22K19770).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Parameter	Value
Chip duration, $T_{c}$	$4.0 \times 10^{- 4}$
APD effective ionization ratio, $k_{e f f}$	0.02
Average APD gain, $G$	100
Modulation extinction ratio of the laser diode output power	100
in the mark and space states, $M e$
Boltzmann’s constant, $k_{B}$	${1.38}^{- 23}$
Receiver noise temperature, $T_{r}$	300 (K)
Receive load resistor, $R_{L}$	1030 ( $Ω$ )
Elementary charge, $e$	$1.6 \times 10^{- 16}$ (C)
Planck’s constant, $h$	$6.626 \times 10^{- 34}$
APD Bulk leakage current, $I_{b}$	0.1 (nA)
APD surface leakage current, $I_{s}$	10 (nA)
Quantum efficiency, $η$	0.6
Laser wavelength, $λ_{l}$	830 (nm)

System	Data transmission efficiency	Maximum capacity of users
Proposed	Approximately 1.0	$\frac{L_{f} - 1}{M_{c o n}} (= M_{p o m})$
Orghogonal CSK	Approximately 0.5	$\frac{L_{f} - 1}{2} (= \frac{M_{p o m}}{2})$
SIK	Approximately 0.5	$\frac{L_{f} - 1}{2} (= \frac{M_{p o m}}{2})$

Parameter	Value
Chip duration, $T_{c}$	$4.0 \times 10^{- 4}$
APD effective ionization ratio, $k_{e f f}$	0.02
Average APD gain, $G$	100
Modulation extinction ratio of the laser diode output power	100
in the mark and space states, $M e$
Boltzmann’s constant, $k_{B}$	${1.38}^{- 23}$
Receiver noise temperature, $T_{r}$	300 (K)
Receive load resistor, $R_{L}$	1030 ( $Ω$ )
Elementary charge, $e$	$1.6 \times 10^{- 16}$ (C)
Planck’s constant, $h$	$6.626 \times 10^{- 34}$
APD Bulk leakage current, $I_{b}$	0.1 (nA)
APD surface leakage current, $I_{s}$	10 (nA)
Quantum efficiency, $η$	0.6
Laser wavelength, $λ_{l}$	830 (nm)

System	Data transmission efficiency	Maximum capacity of users
Proposed	Approximately 1.0	$\frac{L_{f} - 1}{M_{c o n}} (= M_{p o m})$
Orghogonal CSK	Approximately 0.5	$\frac{L_{f} - 1}{2} (= \frac{M_{p o m}}{2})$
SIK	Approximately 0.5	$\frac{L_{f} - 1}{2} (= \frac{M_{p o m}}{2})$

Impact of systematically constructed nonorthogonal code shift keying for optical code division multiple access

Abstract

1. Introduction

2. Proposed system structure

2.1 Construction of nonorthogonal code

2.2 System model

3. Theoretical analysis

3.1 Frame success ratio and data transmission efficiency

3.2 Bit error rate

4. Numerical results

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (2)

Equations (23)

Optics Continuum