Deep Learning-Aided Faster-Than-Nyquist Rate Optical Spatial Pulse Position Modulation

Fig. 5. Performance comparison of OSPPM and OSPPM-FTN systems. (a) BER performance of OSPPM and OSPPM-FTN systems with different $τ$ ; (b) transmission rate and spectral efficiency of OSPPM and OSPPM-FTN systems with different $τ$

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Fig. 6. Relationship between loss and training rounds of MNN decoder

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Fig. 7. Comparison of computational complexity of different decoders

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Fig. 8. BER performance of ML and MNN decoders under different turbulence

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Table 1. Mapping table of OSPPM-FTN system

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Table 1. Mapping table of OSPPM-FTN system

Input bits	LD index	4-PPM signal	4PPM-FTN signal
0000	$x_{s} = {[1 0 0 0]}^{T}$	$x_{p p m} = [P_{1} 0 0 0]$	$x_{m} = [ο_{1} \dots P_{1} + ο_{9} \dots 0 + ο_{q} \dots 0 + ο_{27} \dots 0 + ο_{B}]$
0001	$x_{s} = {[1 0 0 0]}^{T}$	$x_{p p m} = [0 P_{1} 0 0]$	$x_{m} = [ο_{1} \dots 0 + ο_{9} \dots P_{1} + ο_{q} \dots 0 + ο_{27} \dots 0 + ο_{B}]$
0010	$x_{s} = {[1 0 0 0]}^{T}$	$x_{p p m} = [0 0 P_{1} 0]$	$x_{m} = [ο_{1} \dots 0 + ο_{9} \dots 0 + ο_{q} \dots P_{1} + ο_{27} \dots 0 + ο_{B}]$
0011	$x_{s} = {[1 0 0 0]}^{T}$	$x_{p p m} = [0 0 0 P_{1}]$	$x_{m} = [ο_{1} \dots 0 + ο_{9} \dots 0 + ο_{q} \dots 0 + ο_{27} \dots P_{1} + ο_{B}]$
$⋮$	$⋮$	$⋮$	$⋮$
1110	$x_{s} = {[0 0 0 1]}^{T}$	$x_{p p m} = [0 0 P_{1} 0]$	$x_{m} = [ο_{1} \dots 0 + ο_{9} \dots 0 + ο_{q} \dots P_{1} + ο_{27} \dots 0 + ο_{B}]$
1111	$x_{s} = {[0 0 0 1]}^{T}$	$x_{p p m} = [0 0 0 P_{1}]$	$x_{m} = [ο_{1} \dots 0 + ο_{9} \dots 0 + ο_{q} \dots 0 + ο_{27} \dots P_{1} + ο_{B}]$

Table 2. Turbulence model parameters
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Table 2. Turbulence model parameters
Parameter $C_{n}^{2}$ / $m^{- 2 / 3}$
Strong turbulence $2.5 \times 10^{- 11}$
Middle turbulence $6.0 \times 10^{- 13}$
Weak turbulence $9.0 \times 10^{- 16}$

Table 3. Spectral efficiency and transmission rate of different schemes
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Table 3. Spectral efficiency and transmission rate of different schemes
Modulation Spectral efficiency /（bit · s^-1· Hz^-1） Transmission rate /（bit/channel）
OSPPM $(l o g_{2} N_{t} D) / D$ $l o g_{2} N_{t} + l o g_{2} D$
OSPPM-FTN $[l o g_{2} N_{t} + (l o g_{2} D) / τ] / (τ D)$ $l o g_{2} N_{t} + (l o g_{2} D) / τ$

Table 4. Parameters of MNN decoder

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Table 4. Parameters of MNN decoder

Parameter	Content
Hidden layer activation	ReLU
Output layer activation	Sigmoid
Loss function	Cross entropy loss
Optimizer	SGD
Epoch	30
Learning rate	0.001
Number of training set	$6 \times 10^{6}$
Number of validation set	$1 \times 10^{6}$
Hidden nodes	16, 24, 16

Table 5. Relationship between hidden layer number and decoding accuracy
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Table 5. Relationship between hidden layer number and decoding accuracy
Hidden layer number Accuracy /%
2 99.930015
3 99.978038
4 99.953223
5 99.821631

Table 6. Relationship between learning rate and decoding accuracy
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Table 6. Relationship between learning rate and decoding accuracy
Learning rate Accuracy /%
0.01000 99.640017
0.00100 99.996005
0.00010 99.987218
0.00001 99.705215

Table 7. Computational complexity of different decoders
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Table 7. Computational complexity of different decoders
Decoder Complexity /FLOPs
ML $N_{r} D (2 N_{t} N_{r} D + 2 N_{r} D - 1)$
MNN $2 N_{r} D G_{1} + 2 G_{1} G_{2} + 2 G_{2} G_{3} + 2 G_{3} N_{t} D + N_{t} D$

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Yue Zhang, Xiangwen Ye, Minghua Cao, Huiqin Wang. Deep Learning-Aided Faster-Than-Nyquist Rate Optical Spatial Pulse Position Modulation[J]. Acta Optica Sinica, 2024, 44(5): 0506003

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