The carrier-hopping prime code a1(t) and its time-shifted versions a1’(t), a2(t), and a2’(t) are orthogonal to one another in the proposed double EWO scheme. Transmitting anyone won't cause any mutual interference. In the receiver, the MAI is produced by other (*K*-1) users, which will then bring about interference in each arm of the EWO scheme. In the presence of (*K*-1) interferers, we assume that there are *t**1* and *j**1* interferers affecting the correlators, and they are in the upper and lower arm of EWO 1, individually. In the same case, *t**2* and *j**2* interferers can be represented in EWO 2.

The received multiple access signal will be decoded by correlator and will be converted into an electrical signal by photodiode afterward. This paper assumes that the photodetection output is a random variable with Gaussian distribution as described in the references (Bazan et al 2006; Bazan et al 2007). In our proposed four arms double EWO scheme, each correlated optical signal is incident into photodiode independently; therefore, all the Gaussian random variables of photodetection outputs are mutually independent. The EWO output is the subtraction of upper arm and lower arm, and hence its mean is with subtraction while the variance is with addition because of independent Gaussian random variable (Proakis 2008).

If signature sequence a1(t) is transmitted, the means and variances in EWO 1 can be represented separately as (Bazan et al 2006; Bazan et al 2007)

\({\mu _{{x_1}}}=w{P_d}+{t_1}{P_c}\)(i. e. auto-correlation peak + MAI) | (1) |

\({\mu _{{y_1}}}={j_1}{P_c}\) | (2) |

\(\sigma _{{{x_1}}}^{2}=\sigma _{{th}}^{2}+\sigma _{{sh{t_1}}}^{2}+\sigma _{{RIN{t_1}}}^{2}+\sigma _{{sig - \operatorname{int} /{t_1}}}^{2}+\sigma _{{\operatorname{int} - \operatorname{int} /{t_1}}}^{2}\) | (3) |

\(\sigma _{{{y_1}}}^{2}=\sigma _{{th}}^{2}+\sigma _{{sh{j_{\text{1}}}0}}^{2}+\sigma _{{RIN{j_1}0}}^{2}+\sigma _{{\operatorname{int} - \operatorname{int} /{j_1}}}^{2}\) | (4) |

Within the given equation, *P**d* indicates the chip level power of the desired signal in the photodiode and *P**c* for the chip level power of the interfering signal. The code weight is represented by *w* (i.e. the number of wavelengths in the signature sequence). Furthermore, the variances of shot noise, thermal noise, beat noise (BN), and relative intensity noise (*RIN*) can be formulated as

\(\sigma _{{shls}}^{2}=2q(sw{P_d}+l{P_c}){B_e}\) | (5) |

\(\sigma _{{th}}^{2}=4{K_B}T{B_e}/{R_L}\) | (6) |

\(\sigma _{{sig - sig/l}}^{2}=2ls{P_d}{P_c}\) | (7) |

\(\sigma _{{\operatorname{int} - \operatorname{int} /l}}^{2}=\frac{2}{w}P_{c}^{2}\left( {\begin{array}{*{20}{c}} l \\ 2 \end{array}} \right)\) | (8) |

\(\sigma _{{RINls}}^{2}=RIN{(sw{P_d}+l{P_c})^2}{B_e}\) | (9) |

Here *q* represents the electron charge, *R**L* represents the receiver load resistance, *B**e* represents the electrical bandwidth of receiver, *T* represents the absolute temperature, *K**B* represents the Boltzmann constant, and *l* stands for *t**1* or *j**1*; *s* = 1 represents the correlator receiving desired signature sequence, but *s* = 0 is not.

In like manner, some other signature sequences or no signature sequence transmitted can be expressed in the above-mentioned way in EWO 1 or EWO 2.

After subtraction, means and variances of two EWO outputs can be expressed respectively as

$${\mu _{{z_1}}}={\mu _{{x_1}}} - {\mu _{{y_1}}}$$

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$$\sigma _{{{z_1}}}^{2}=\sigma _{{{x_1}}}^{2}+\sigma _{{{y_1}}}^{2}$$

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and

$${\mu _{{z_2}}}={\mu _{{x_2}}} - {\mu _{{y_2}}}$$

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$$\sigma _{{{z_2}}}^{2}=\sigma _{{{x_2}}}^{2}+\sigma _{{{y_2}}}^{2}$$

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Finally, the resultant output *Z* (i.e. two EWO outputs with addition) is sent to the coded modulation decoder for decoding, and its mean and variance can be represented as

$${\mu _z}={\mu _{{z_1}}}+{\mu _{{z_2}}}$$

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$$\sigma _{z}^{2}=\sigma _{{{z_1}}}^{2}+\sigma _{{{z_2}}}^{2}$$

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To produce + 2 or -2 data symbol, two signature sequences (i.e. a1(t) and a’1(t); a2(t) and a’2(t)) will be sent simultaneously. While no signature sequence is transmitted represents data symbol 0. In Fig. 3, we are able to observe that the probabilities of transmitting data symbol {+2, -2} and data symbol 0 are equally probable. Therefore, under (*K*-1) interferers the means and variances of *Z* with different data symbols transmitted can be expressed as

\({\mu _{{z_{\{ +2, - 2\} }}}}= \pm 2w{P_d}\) | (16) |

\(\begin{gathered} \sigma _{{z{\{ _{+2, - 2}}\} }}^{2}=\sum\limits_{{i=1}}^{{K - 1}} {\left( {\begin{array}{*{20}{c}} {K - 1} \\ i \end{array}} \right)} {2^{ - \left( {K - 1} \right)}} \times \sum\limits_{{j=1}}^{i} {\left( {\begin{array}{*{20}{c}} i \\ j \end{array}} \right)} {\left( {{h_{av}}} \right)^j}{\left( {1 - {h_{av}}} \right)^{i - j}} \hfill \\ {\text{ }} \times \left( {\sigma _{{{z_1}}}^{2}+\sigma _{{{z_2}}}^{2}} \right) \hfill \\ \end{gathered}\) | (17) |

\(\begin{gathered} \sigma _{{{z_0}}}^{2}=\sum\limits_{{i=1}}^{{K - 1}} {\left( {\begin{array}{*{20}{c}} {K - 1} \\ i \end{array}} \right)} {2^{ - \left( {K - 1} \right)}} \times \sum\limits_{{j=1}}^{i} {\left( {\begin{array}{*{20}{c}} i \\ j \end{array}} \right)} {\left( {{h_{av}}} \right)^j}{\left( {1 - {h_{av}}} \right)^{i - j}} \hfill \\ {\text{ }} \times \left( {4\sigma _{{{y_1}}}^{2}} \right) \hfill \\ \end{gathered}\) | (18) |

Transmitting one signature sequence will incur interference on other correlators with average hit probability\(\frac{w}{N}\)(Bazan et al 2006; Bazan et al 2007), where *N* stands for code length and *w* stands for code weight. While in our double EWO scheme, two signature sequences are simultaneously transmitted, and therefore the average hit probability would be \({h_{av}}=2 \times \frac{w}{N}\).

The average signal to noise ratio is a measure of the quality of a communication channel, and it is a figure of merit in the performance calculation. Since data symbol {+2, -2} and data symbol 0 are with equal probability, the average mean and variance can be represented as

$${\mu _{{z_{average}}}}=\frac{1}{2} \times 2w{P_d}$$

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$$\sigma _{{{z_{average}}}}^{2}=\frac{1}{2}(\sigma _{{{z_{\{ +2, - 2\} }}}}^{2}+\sigma _{{{z_0}}}^{2})$$

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The proposed coded modulation scheme utilizes a trellis structure, and the system performance will be determined by free Euclidean distance. Provided that there are (*K*-1) interferers present in the optical network, the average bit error rate can be demonstrated as (Proakis 2008; Wicker 1995; Moon 2005)

$${P_b} \approx \frac{{{B_{d{}_{{free}}}}}}{m}Q\left( {\frac{{{d_{free}}}}{2} \times \frac{{{\mu _{{z_{average}}}}}}{{\sigma _{{{z_{average}}}}^{{}}}}} \right)$$

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In which, \({B_{{d_{free}}}}\) is the average number of erroneous information bits among all the minimum distance error events, and *m* is the information bits in each trellis node. In Fig. 3, we are able to observe that each minimum distance error event leads to one information bit error and each node contains one information bit.