## 2 Principles
### Principle of NNWSC-Based Receiver  
A trained NN model can directly convert the received signal waveform, which is heavily affected by ISI and IBI, into complex symbols. This converter is referred to as a waveform-to-symbol converter, and it can replace conventional stages of matched filtering, down-sampling, and symbol-level post-equalization. Compared to the conventional multi-stage processing at the receiver, the NNWSC optimizes the entire receiver chain and leverages the NN's powerful capability to uncover implicit relationships between the received waveform and the symbols, thereby achieving enhanced performance.  
At the receiver side, the received signal is fed into the NNWSC, whose structure is shown in Figure 2. The NNWSC consists of an input layer, two hidden layers, and an output layer with 2N nodes, each corresponding to the real and imaginary components of each sub-band. To account for ISI and IBI, a sliding window is applied to the received signal r(t). The slice for the k-th symbol can be expressed as follows:  
\[\mathrm{X(k)}=\left[\mathrm{r}\left(\mathrm{sk}-\frac{1-1}{2}\right),\ldots, \mathrm{r(sk-1)},\mathrm{r(sk)},\mathrm{r(sk+1)},\ldots,\mathrm{r}\left( \mathrm{sk}+\frac{1+1}{2}\right)\right], \tag{9}\]  
where 1 is an odd number representing the length of the sliding window, and \(\mathrm{s}\) is the step size of the sliding window, which is set equal to the upsampling rate. For instance, in this paper, with 20\(\times\) upsampling, the step size is set to 20. Each slice is then processed sequentially through the two hidden layers and finally connected to the 2N output nodes, producing the real and imaginary parts.  
The target symbol vector for the N sub-bands is represented as follows:  
\[\mathrm{D(k)}=\left[\mathrm{d}_{\mathrm{I}}^{1}(\mathrm{k}),\mathrm{d}_{ \mathrm{Q}}^{1}(\mathrm{k}),\ldots,\mathrm{d}_{\mathrm{I}}^{\mathrm{N-1}}( \mathrm{k}),\mathrm{d}_{\mathrm{Q}}^{\mathrm{N-1}}(\mathrm{k}),\mathrm{d}_{ \mathrm{I}}^{\mathrm{N}}(\mathrm{k}),\mathrm{d}_{\mathrm{Q}}^{\mathrm{N}}( \mathrm{k})\right], \tag{10}\]  
where \(\mathrm{d}_{\mathrm{I}}^{\mathrm{n}}(\mathrm{k})\) and \(\mathrm{d}_{\mathrm{Q}}^{\mathrm{n}}(\mathrm{k})\) denote the I and Q components of the symbol in the n-th sub-band. The relationship between the inputs and outputs of the NNWSC can be described as follows:  
\[\mathrm{y(k)}=\mathrm{W}^{3}\left(\mathrm{W}^{2}\left(\mathrm{W}^{1}(\mathrm{ X(k)}+\mathrm{b}^{1}\right)+\mathrm{b}^{2}\right)+\mathrm{b}^{3}, \tag{11}\]  
Figure 2: The schematic diagram of the coordinated MISO photonic mmWave communication system using an NNWSC-based receiver.  
where W\({}^{1}\), W\({}^{2}\), and W\({}^{3}\) are the weight matrices for each layer, and W\({}^{3}\) denotes the biases for each layer. In this model, the Adam optimizer is used, which is a common technique for adaptive learning rate adjustment [29]. After forward propagation, the MSE between the predicted symbols and the transmitted symbols is calculated as the loss function, which can be expressed as follows:  
\[\mathrm{L}=\frac{1}{\mathrm{m}}\mathrm{\sum}\|\mathrm{X(k)}-\mathrm{D(k)}\|^{ 2}, \tag{12}\]  
where m is the batch size. Thus, the NNWSC learns a set of filters that can down-convert the NM-CAP sub-bands while mitigating ISI and IBI. In contrast, cascaded filter parameters are updated independently, which does not yield a globally optimal result.