# Two-Dimensional Permutation Vectors’ (PV) Code for Optical Code Division Multiple Access Systems

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

^{−9}.

## 1. Introduction

^{−10}at data rates 40 Giga bits per second (Gbps)) as well as long PON distance up to 41 km with increased maximum throughput equal to 285.1 Gb/s/km was achieved.

## 2. The 2D Wavelength-Hopping/Time-Spreading System

## 3. The 2-Dimensional Wavelength-Time (W-T) Permutation Vectors’ (PV) Code Construction and Properties

#### 3.1. One-Dimensional Approach

^{3}) = 3. In general, dim $\mathbb{R}$($\mathbb{R}$

^{m}) = m.

^{m}.

^{4}is represented as:

^{k}and an arbitrary permutation vector ($Perm$).

^{k}, the permuted $\mathbb{R}$

^{k}is obtained using Equation (6):

- Cross-correlation between each row is equal to 0.
- Each column is an element of the vector space $\mathbb{R}$
^{k}. - Number of Perm possibilities is equal to $P{V}_{poss}=$ $\frac{\left(\mathrm{WK}\right)!}{\left(\mathrm{w}!\left(\mathrm{L}-\mathrm{w}\right)!\right)}$.

#### 3.2. W-T Two-Dimensional PV Approach

^{k}and an arbitrary permutation vector ($Permi$), so that:

_{i}and G

_{d}), the total number of served users = K

^{2}. Therefore, the 2D W-T PV codes’ generation equation was defined as:

_{3}, λ

_{5}) but transmitted them at different time slots (t3, t7) and (t5,t6), respectively. Furthermore, when two codes used the same time slots, like ${C}_{21}{C}_{12}$ and ${C}_{22}{C}_{12}$(green colored), they sent different wavelengths, (λ

_{3}, λ

_{5}) for ${C}_{21}{C}_{12}$ and (λ

_{6}, λ

_{8}) for ${C}_{22}{C}_{12}$, thus maintaining a zero cross-correlation.

#### 3.3. Properties of 2D W-T Permutation Vectors (PV) Codes

## 4. 2-Dimensional PV System Description

#### 4.1. Design of 2D-PV System

#### 4.2. Design of 2D-PV Transmitter Part

_{b}/S), which represents code length in time spread), where t

_{b}/S stands for bit time period per second. The pulse train repetition rate is set equal to the system bit rate (1/t

_{b}). The bit stream is modulated by the input data using an amplitude modulator. In the on-off keying process, the modulator produces an optical pulse when transmitted data bit is one, or else it produces zero output.

#### 4.3. Design of 2D-PV Receiver Part

## 5. Proposed 2D-PV W-T-Based OCDMA-PON

_{b}= 1 ns).

## 6. Proof of Concept

^{−9}. In particular, multi diagonal (MD) code (M = 63, P = 3) is also plotted and results indicate an overlap with our proposed code due to their similar properties.

## 7. Conclusions

^{−9}) required for error-free transmission at minimum data rate.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Karafolas, N.; Uttamchandani, D. Optical Fiber Code Division Multiple Access Networks: A Review. Opt. Fiber Technol.
**1996**, 2, 149–168. [Google Scholar] [CrossRef] - Fathallah, H.; Rusch, L.A.; LaRochelle, S. Passive optical fast frequency-hop CDMA communications system. J. Light. Technol.
**1999**, 17, 397–405. [Google Scholar] [CrossRef] - Yang, C.-C. The application of spectral-amplitude-coding optical CDMA in passive optical networks. Opt. Fiber Technol.
**2008**, 14, 134–142. [Google Scholar] [CrossRef] - Sharma, T.; Maddila, R.K. Performance characteristics of the spectral amplitude-coding optical CDMA system based on one-dimensional optical codes and a multi-array laser. Ukr. J. Phys. Opt.
**2019**, 20, 81–90. [Google Scholar] - Abdulqader, S.G.; Aljunid, S.A.; Al-Khafaji, H.M.R.; Fadhil, H.A. Enhanced Performance of SAC-OCDMA System based on SPD Detection Utilizing EDFA for Access Networks. J. Commun.
**2014**, 9, 99–106. [Google Scholar] [CrossRef] - Lin, C.-H.; Wu, J.; Yang, C.-L. Noncoherent spatial/spectral optical CDMA system with two-dimensional perfect difference codes. J. Light. Technol.
**2005**, 23, 3966–3980. [Google Scholar] [CrossRef] [Green Version] - Sharma, T.; Maddila, R.K.; Aljunid, S.A. Simulative Investigation of Spectral Amplitude Coding Based OCDMA System Using Quantum Logic Gate Code with NAND and Direct Detection Techniques. Curr. Opt. Photonics
**2019**, 3, 531–540. [Google Scholar] - Ahmed, H.Y.; Zeghid, M.; Imtiaz, W.A.; Sghaier, A. Two dimensional Fixed Right Shift (FRS) code for SAC-OCDMA systems. Opt. Fiber Technol.
**2019**, 47, 73–87. [Google Scholar] [CrossRef] - Yin, H.; Ma, L.; Li, H.; Zhu, L. A new family of 2D wavelength/time codes with large cardinality for incoherent spectral amplitude coding OCDMA networks and analysis of its performance. Photonic Netw. Commun.
**2010**, 19, 204–211. [Google Scholar] [CrossRef] - Yang, C.-C.; Huang, J.-F. Two-dimensional M-matrices coding in spatial/frequency optical CDMA networks. IEEE Photon-Technol. Lett.
**2003**, 15, 168–170. [Google Scholar] [CrossRef] - Yegnanarayanan, S.; Bhushan, A.; Jalali, B. Fast wavelength-hopping time-spreading encoding/decoding for optical CDMA. IEEE Photon-Technol. Lett.
**2000**, 12, 573–575. [Google Scholar] [CrossRef] - Chang, T.-W.; Sargent, E. Optical CDMA using 2-D codes: The optimal single-user detector. IEEE Commun. Lett.
**2001**, 5, 169–171. [Google Scholar] [CrossRef] - Yang, G.C.; Kwong, W.C. Prime Codes with Applications to CDMA Optical and Wireless Networks; Artech House: Norwood, MA, USA, 2002. [Google Scholar]
- Kandouci, C.; Djebbari, A. Design of new hybrid wavelength hopping / time spreading codes for optical CDMA by combining OCC and BIBD ZCC codes. Optik
**2017**, 133, 73–79. [Google Scholar] [CrossRef] - Bazan, T.M.; Harle, D.; Andonovic, I. Performance analysis of 2-D time-wavelength OCDMA systems with coherent light sources: Code design considerations. J. Light. Technol.
**2006**, 24, 3583–3589. [Google Scholar] [CrossRef] - Chang, Y.-T.; Huang, J.-F.; Yen, C.-T.; Wang, C.-C.; Cheng, H.-C.; Hsu, K.-C. A new shared AWG-based OCDMA scheme implemented with time-spreading and wavelength-group-hopping embedded M-sequence code. Opt. Fiber Technol.
**2010**, 16, 114–123. [Google Scholar] [CrossRef] - Kandouci, C.; Djebbari, A.; Taleb-Ahmed, A. A new family of 2D-wavelength-time codes for OCDMA system with direct detection. Optik
**2017**, 135, 8–15. [Google Scholar] [CrossRef] - Panda, S.; Palai, G. Design and Performance analysis of data encrypted two dimensional coding technique for wavelength hopping time spreading Optical CDMA. Optik
**2020**, 207, 163864. [Google Scholar] [CrossRef] - Robert, M.G. Entropy and Information Theory; Springer: New York, NY, USA, 2011. [Google Scholar]
- Mrabet, H.; Dayoub, I.; Attia, R. Comparative study of 2D-OCDMA-WDM system performance in 40-Gb/s PON context. IET Optoelectron.
**2017**, 11, 141–147. [Google Scholar] [CrossRef] - Jellali, N.; Najjar, M.; Ferchichi, M.; Rezig, H. Development of new two-dimensional spectral/spatial code based on dynamic cyclic shift code for OCDMA system. Opt. Fiber Technol.
**2017**, 36, 26–32. [Google Scholar] [CrossRef] - Mrabet, H.; Mhatli, S.; Dayoub, I.; Giacoumidis, E. Performance analysis of AO-OFDM-CDMA with advanced 2D-hybrid coding for amplifier-free LR-PONs. IET Optoelectron.
**2018**, 12, 293–298. [Google Scholar] [CrossRef] - Mrabet, H.; Dayoub, I.; Haxha, S.; Attia, R. Performance analysis of 2D-OCDMA system in long-reach passive optical network. Opt. Laser Technol.
**2019**, 117, 64–72. [Google Scholar] [CrossRef] - Tancevski, L.; Andonovic, I. Hybrid wavelength hopping/time spreading schemes for use in massive optical networks with increased security. J. Light. Technol.
**1996**, 14, 2636–2647. [Google Scholar] [CrossRef] - Baby, V.; Glesk, I.; Runser, R.; Fischer, R.; Huang, Y.-K.; Bres, C.-S.; Kwong, W.; Curtis, T.; Prucnal, P. Experimental demonstration and scalability analysis of a four-node 102-Gchip/s fast frequency-hopping time-spreading optical CDMA network. IEEE Photon-Technol. Lett.
**2004**, 17, 253–255. [Google Scholar] [CrossRef]

**Figure 1.**Classification of 2D W-T PV codes (

**a**) (λ

_{1}λ

_{3}0 λ

_{2}0 λ

_{4}) (

**b**) ({λ

_{1}, λ

_{4}} 0 0 λ

_{2}0 λ

_{4}) (

**c**) (λ

_{1}0 λ

_{3}0 λ

_{3}0 λ

_{2}). * λ = Wavelength.

**Figure 2.**Code’s illustration ${\mathrm{C}}_{21}{\mathrm{C}}_{11}$, ${\mathrm{C}}_{21}{\mathrm{C}}_{12},$ and ${\mathrm{C}}_{22}{\mathrm{C}}_{12}$.

**Figure 11.**Bit error rate vs. number of active users for 2D PV (K1 = 63, K2 = 3), 2D diagonal eigen value unity code (M = 63, P = 3), 2D diluted perfect difference code (M = 63, P = 3), and 2D Multi Diagonal code (K1 = 62, K2 = 3) with 0 dBm effective transmitted power at data rate 622 Mbit/sec.

**Table 1.**1-dimensional-PV codes generated with code weight (w) = 2 and number of generated codes (K) = 4.

${\mathit{R}}_{\mathit{g},\mathit{h}}^{\mathbf{0}}\mathbf{\left(}\mathit{d},\mathit{i}\mathbf{\right)}$ | ${\mathit{R}}_{\mathit{g},\mathit{h}}^{\mathbf{1}}\mathbf{\left(}\mathit{d},\mathit{i}\mathbf{\right)}$ | ${\mathit{R}}_{\mathit{g},\mathit{h}}^{\mathbf{2}}\mathbf{\left(}\mathit{d},\mathit{i}\mathbf{\right)}$ | ${\mathit{R}}_{\mathit{g},\mathit{h}}^{\mathbf{3}}\mathbf{\left(}\mathit{d},\mathit{i}\mathbf{\right)}$ | |
---|---|---|---|---|

$d=g,i=h$ | w | $0$ | $0$ | $0$ |

$d=g,i\ne h$ | $0$ | w | $0$ | $0$ |

$d\ne g,i=h$ | 0 | $0$ | w | $0$ |

$d\ne g,i\ne h$ | $0$ | $0$ | $0$ | w |

**Table 3.**Allocation of respective wavelengths $\left({\lambda}_{s}\right)$ and time delays $\left({\tau}_{s}\right)$.

${\mathit{A}}_{\mathit{g},\mathit{h}}$ | ${\mathbf{\lambda}}_{\mathbf{1}}$ | ${\mathbf{\lambda}}_{\mathbf{2}}$ | ${\mathbf{\lambda}}_{\mathbf{3}}$ | ${\mathbf{\lambda}}_{\mathbf{4}}$ | ${\mathbf{\lambda}}_{\mathbf{5}}$ | ${\mathbf{\lambda}}_{\mathbf{6}}$ | ${\mathbf{\lambda}}_{\mathbf{7}}$ | ${\mathbf{\lambda}}_{\mathbf{8}}$ | ${\mathbf{\lambda}}_{\mathbf{9}}$ | ${\mathbf{\lambda}}_{\mathbf{10}}$ | ${\mathbf{\lambda}}_{\mathbf{11}}$ | ${\mathbf{\lambda}}_{\mathbf{12}}$ | ${\mathbf{\lambda}}_{\mathbf{13}}$ | ${\mathbf{\lambda}}_{\mathbf{14}}$ | ${\mathbf{\lambda}}_{\mathbf{15}}$ | ${\mathbf{\lambda}}_{\mathbf{16}}$ | |

0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | ||

${\mathit{\tau}}_{\mathbf{1}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{2}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{3}}$ | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{4}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{5}}$ | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{6}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{7}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{8}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{1}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{2}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{3}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{4}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{5}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{6}}$ | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{7}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mathit{\tau}}_{\mathbf{8}}$ | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |

System Component | Value |
---|---|

LED spectrum | 30 nm |

LED frequency | 1490 nm |

EDFA Gain | 9 dB |

Encoder bandwidth | 0.4 nm |

MZM extinction ratio | 30 dB |

SMF attenuation | 0.25 dB/km |

SMF dispersion | 18 ps/nm/km |

FF ring length | 20 Km |

DF length | 5 Km |

Decoder bandwidth | 0.4 nm |

PIN responsitivity | 0.75 A/W |

PIN dark current | 10 nA |

PIN thermal noise | 1e^{−22} W/Hz |

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**MDPI and ACS Style**

Yousif Ahmed, H.; Zeghid, M.; A.Imtiaz, W.; Sharma, T.; Chehri, A.; Fortier, P.
Two-Dimensional Permutation Vectors’ (PV) Code for Optical Code Division Multiple Access Systems. *Entropy* **2020**, *22*, 576.
https://doi.org/10.3390/e22050576

**AMA Style**

Yousif Ahmed H, Zeghid M, A.Imtiaz W, Sharma T, Chehri A, Fortier P.
Two-Dimensional Permutation Vectors’ (PV) Code for Optical Code Division Multiple Access Systems. *Entropy*. 2020; 22(5):576.
https://doi.org/10.3390/e22050576

**Chicago/Turabian Style**

Yousif Ahmed, Hassan, Medien Zeghid, Waqas A.Imtiaz, Teena Sharma, Abdellah Chehri, and Paul Fortier.
2020. "Two-Dimensional Permutation Vectors’ (PV) Code for Optical Code Division Multiple Access Systems" *Entropy* 22, no. 5: 576.
https://doi.org/10.3390/e22050576