# A Multiple-Input Multiple-Output Reservoir Computing System Subject to Optoelectronic Feedbacks and Mutual Coupling

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## Abstract

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## 1. Introduction

## 2. Numerical Simulation Model

_{r}(t) (r = 1, 2,..., M). Finally, the reservoir states x

_{r}(t) are acquired at the sampling rate of τ/M at the output layer. After a linear sum of the reservoir states is taken, the output value is obtained [20,21,22,23]. These coefficients in this linear sum are called the output weights, which are the only ones that need be determined by training in the system. A simple linear regression algorithm can be employed to minimize the difference between the actual computing output value and the desired output value [24]. Their difference is represented by the normalized root mean square error (NRMSE) and the word error rate (WER), as described in Equations (1) and (2). The smaller the values of NRMSE and WER, the closer the actual output value is to the desired output value.

_{all}is the number of all the input signals, and N

_{correct}is the number of signals correctly recognized. In our numerical simulations, 10 training repetitions will be carried out with different arrays, and the average values of the results of the 10 repetitions will be taken as their final values.

_{N + 1}). Each input signal to be identified, input

_{i}(i = 1, 2, 3, ……, N), is added to every route loop, and together they are fed back to the radio frequency (RF) terminal of each MZM. Consequently, N-route input signals can be recognized simultaneously. The optoelectronic delays are composed of the optical delay lines (ODL

_{i}, i = 1, 2, 3, ……, N), photodetectors (PD

_{i}, i = 1, 2, 3, ……, N), band-pass filters (BPF

_{i}, i = 1, 2, 3, ……, N), electric power amplifiers (AMP

_{i}, i = 1, 2, 3, ……, N), and electric power distributers (D

_{i}, i = 1, 2, 3, ……, N). ODL

_{i}is used for the time delay. PD

_{i}converts an optical signal into an electrical signal. BPF

_{i}is employed for band limitation. AMP

_{i}amplifies an electrical signal. D

_{i}is used for collecting the reservoir states.

_{i}(t) = πv

_{i}(t)/2v

_{πRF}is the normalized bias voltage of MZM, v

_{i}(t) is the voltage of each loop, and v

_{πRF}is the input half-wave voltage of MZM. In order to facilitate the numerical solution, the integral of x

_{i}(t) is set as y

_{i}(t), as shown in Equation (4). ϕ

_{i}is the phase offset, which is determined by the direct current (DC) offset voltage of the MZM. f

_{Hi}and f

_{Li}are the cutoff frequencies for the high frequency and the low frequency, respectively, τ

_{Hi}= 1/2πf

_{Hi}, τ

_{Li}= 1/2πf

_{Li}. β

_{i}is the feedback strength. τ

_{j}(t) (j = 1, 2, 3, ⋅⋅⋅, N) is the feedback delay time determined by the length of ODL

_{i}. γ

_{i}(i = 1, 2, 3, ⋅⋅⋅, N) is the gain coefficient of the input signal. J

_{i}(t) (i = 1, 2, 3, ⋅⋅⋅, N) is the serial input signal of the reservoir. n

_{i}(t) is the white noise added to the signal. The values of the system parameters are shown in Table 1. The main noise of the whole setup includes the noise of the input signals, the insertion loss of the MZMs, and the thermal noise of photodetectors. For the performance analysis of the RC system, the influence of the noise of the input signals is mainly analyzed. The noise of the MZMs and photodetectors is ignored for the sake of simplification here.

## 3. Simulation Results of Signal Recognitions

#### 3.1. Four-Route Optical Packet Header Recognition

#### 3.2. Digital Speech Recognition

#### 3.3. Eight-Input Eight-Output Optoelectronic RC

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic of the RC (reservoir computing) based on a single nonlinear node subject to delay feedback; (

**b**) Multiple-input multiple-output RC based on optoelectronic feedbacks and mutual coupling. LD: laser diode, EDFA: erbium-doped fiber amplifier, OC: optical coupler, MZM: Mach–Zehnder modulator, ODL: fiber-optic delay line, PD: photodetector, BPF: band-pass filter, AMP: electric amplifier, D: electric power divider.

**Figure 2.**Relationship between the optical packet header recognition errors and the feedback strength β.

**Figure 3.**4-input 4-output RC recognition results of the 3-bit optical packet headers for the signal-to-noise ratio (SNR) of 20 dB: (

**a**) Desired output; (

**b**) Actual output.

**Figure 4.**Relationships between the optical packet header recognition errors and the different SNRs of the inputs.

**Figure 5.**Digital speech recognition results for SNR of 20 dB: (

**a**) Desired output; (

**b**) Actual output.

**Figure 6.**8-input 8-output RC recognition results of 3-bit optical packet headers when the SNR is 20 dB. (

**a**) Desired output; (

**b**) Actual output.

Symbol | Parameter | Value |
---|---|---|

ϕ_{i} | offset phase of the MZM | –π/4 |

τ_{Hi} | high-frequency cutoff characteristic time | 19.89 ps |

τ_{Li} | low-frequency cutoff characteristic time | 51.34 ps |

β_{i} | feedback strength | 0.5~5 GHz |

τ_{i} | feedback delay time | 2.5 ns |

γ_{i} | input gain | 1 |

**Table 2.**Recognition results for the 8-bit, 16-bit, 32-bit optical packet headers when the SNR (signal-to-noise ratio) is 20 dB or 35 dB. NRMSE: normalized root mean square error.

35 dB of SNR | 8-bit Optical Packet Header Recognition | 16-bit Optical Packet Header Recognition | 32-bit Optical Packet Header Recognition | ||||
---|---|---|---|---|---|---|---|

20 dB of SNR | |||||||

Training NRMSE | 0.04880 | 0.0879 | 0.0953 | ||||

0.0568 | 0.2003 | 0.2553 | |||||

Testing NRMSE | 0.0870 | 0.1977 | 0.1650 | ||||

0.0954 | 0.3605 | 0.3725 | |||||

WER | 0% | 0% | 0% | ||||

0% | 0% | 0% |

SNR of Input | Without Noise | 30 dB | 20 dB | 10 dB |
---|---|---|---|---|

Training NRMSE | 0.0509 | 0.0717 | 0.0729 | 0.1128 |

Testing NRMSE | 0.1051 | 0.1136 | 0.1195 | 0.2485 |

WER | 1.4% | 1.6% | 1.6% | 14.6% |

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**MDPI and ACS Style**

Bao, X.; Zhao, Q.; Yin, H.
A Multiple-Input Multiple-Output Reservoir Computing System Subject to Optoelectronic Feedbacks and Mutual Coupling. *Entropy* **2020**, *22*, 231.
https://doi.org/10.3390/e22020231

**AMA Style**

Bao X, Zhao Q, Yin H.
A Multiple-Input Multiple-Output Reservoir Computing System Subject to Optoelectronic Feedbacks and Mutual Coupling. *Entropy*. 2020; 22(2):231.
https://doi.org/10.3390/e22020231

**Chicago/Turabian Style**

Bao, Xiurong, Qingchun Zhao, and Hongxi Yin.
2020. "A Multiple-Input Multiple-Output Reservoir Computing System Subject to Optoelectronic Feedbacks and Mutual Coupling" *Entropy* 22, no. 2: 231.
https://doi.org/10.3390/e22020231