# Adaptive Linearization for the Sub-Nyquist Photonic Receiver Based on Deep Learning
## 3 Results and Discussion  
To ensure reasonable learning capacity during the training process, the loss functions of the training dataset and validation dataset were monitored. Figure 3a depicts the training and validation loss curves of the Resnet model. The training loss and validation loss were calculated by the absolute error between the network output and the reference data on the training set and validation set, respectively. Essentially, due to the early stopping scheme, which remedied the overfitting and saved the training time, the training process experienced 5535 epochs. The training loss was reduced from 14.450 to 0.0560, while the validation loss was stable at 0.0604. The training loss showed the same convergence trend as the validation loss. Furthermore, Figure 3a also elucidates the curve of the learning rate, in which the optimal model was obtained when the learning rate was reduced to \(10^{-6}\). From the convergence of the two loss functions of the Resnet model, the learning mode was explicitly correct where there was no abnormal state of overfitting or underfitting. Figure 3b,c depicts the distribution of the samples on the training set and testing set in different loss values, respectively. This ensured that the sample distribution of the training set and testing set was basically the same, and most samples were concentrated between 0.005-0.020. The training process took about 71,244.4 s. \begin{table}
\begin{tabular}{c c} \hline \hline
**Hyperparameter** & **Initial Value** \\ \hline Learning Rate & \(10^{-3}\) \\ Epoch & 10,000 \\ Batchsize & 32 \\ \(\alpha\) for _Leaky Relu_ & 0.03 \\ Patience for ReduceLROnPlateau & 50 \\ Factor for ReduceLROnPlateau & 0.1 \\ Patience for Early Stopping & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Hyperparameters in the Training Process. In the sampled analog photonic link, all the high-frequency components were supposed to be down-converted within the first Nyquist zone (0-515 MHz). Hence, it was sufficient to set the frequency observation window, which just accommodated the first Nyquist zone. According to the down-conversion principle of the sampled analog photonic link, the mapping relationship of the input frequency to the output frequency is described by Equation (7), where \(n\) is the comb of the optical frequency comb, which is the nearest comb to the signal, and \(f_{s}\) is the repetition frequency of the OFC. \[f_{out}=|f_{in}-nf_{s}| \tag{7}\]  
Figure 4 displays the effects of the deep learning-assisted spurious elimination spectrum of the single-tone sinusoidal signals formed by splicing the spectrums of several representative frequency bands from 0-40 GHz. All the input frequency components were down-converted to the first Nyquist zone (0-515 MHz) after being received by the sub-Nyquist photonic link. It should be noted that when the power of the input signal is high, the harmonic and intermodulation distortions will also fall into the first Nyquist zone. As shown in Figure 4, with the aid of deep learning linearization, intricate distortions like the down-converted second-order harmonics and third-order harmonics were automatically suppressed under the condition of unknown signal frequency information. Significantly, the deep learning method also reduced the noise floor (NF) to improve the \(SNR\) in all covered frequency bands. Figure 4: Effects of deep learning-based spurious elimination in the frequency domain with the input signal of various input power, ranging from 0 to 40 GHz. The selected displayed input frequencies are 0.2 GHz, 2.7 GHz, 5.8 GHz, 10.0 GHz, 14.0 GHz, 20.0 GHz, 24.5 GHz, 27.5 GHz, 32.5 GHz, and 40.0 GHz, which were all down-converted to the first Nyquist zone (0–515 MHz) with the frequencies of 200 MHz, 390 MHz, 380 MHz, 300 MHz, 420 MHz, 430 MHz, 220 MHz, 310 MHz, 460 MHz, and 170 MHz, respectively. \(N\) represents the Nyquist zone of each input signal.