## 4 Methods
### Dilated Causal Convolution  
The PPG signal, continuously collected over time, is a physiological indicator that reflects the filling and emptying processes of blood vessels. This process is intricately linked to heartbeats and fluctuations in blood pressure. Both PPG and ambulatory blood pressure signals possess periodic, autocorrelated, nonlinear, and non-stationary traits [19]. Consequently, cuff-less continuous blood pressure estimation solely based on PPG signals qualifies as a time series analysis. In previous investigations, RNNs and their variations, including long short-term memory (LSTM) and gate recurrent unit (GRU), have proven effective in capturing the temporal dependencies inherent in PPG signals, surpassing the accuracy achieved by conventional machine learning techniques and CNNs in blood pressure estimation. Nevertheless, conventional RNNs encounter challenges such as gradient vanishing or exploding when dealing with extended sequence data. Additionally, their serial computation nature hinders efficiency, restricting their applicability to extensive datasets. The emergence of TCNs [20] adeptly addresses the limitations of traditional RNNs in handling long sequences. By leveraging the architecture of convolutional neural networks, TCNs offer enhanced performance and broader applicability in sequence modelling endeavours.  
Dilated causal convolution [21] stands as a pivotal component for harnessing the strengths of TCN. In the context of time series data, causal convolution signifies that the convolution kernel solely observes data preceding the present time step, preventing access to current or subsequent data. This precautionary measure ensures that the model refrains from utilizing future information when anticipating the present time step. Conversely, dilated convolution broadens [22] the receptive field of the kernel by introducing gaps in sampling, termed dilation factors, thereby facilitating the capture of more extended temporal dependencies. Dilated causal convolution [23] seamlessly integrates the attributes of both causal and dilated convolutions, effectively grasping long-term dependencies within time series data. Typically, the architecture of a dilated causal convolution block takes the following form.  
A dilated causal convolution block showed in Figure 4 is a module composed of multiple dilated causal convolution layers, commonly used in the construction of network structures such as TCN. Each dilated causal convolution layer possesses distinct dilation factors but shares the same convolution kernel size [24]. The operation of one-dimensional dilated causal convolution can be represented as:  
\[y[t]=\sum_{k=0}^{K}x[t-r\cdot k]\cdot h[k] \tag{2}\]  
In the formula, \(y_{[t]}\) represents the \(t\)-th element of the output sequence, \(x_{[t]}\) represents the \(t\)-th element of the input sequence, \(h_{[k]}\) represents the weight parameters of the convolution kernel, \(r\) represents the dilation coefficient, and \(K\) represents the size of the convolution  
Figure 4: Dilated causal convolution.  
kernel. For each element, the \(y_{|t|l}\) of the output sequence is obtained by weighted summation of specific positions in the input sequence. The weights \(h_{[k]}\) of the convolution kernel are multiplied element-wise with the corresponding positions in the input sequence and then summed to obtain the elements of the output sequence. In dilated causal convolution, the dilation factor d determines the skipping interval of the convolution kernel, controlling the position of the convolution kernel by multiplying i with d. This allows the convolution kernel to expand its receptive field while maintaining causality, i.e., relying only on current and past input data. Relevant studies have shown that dilated causal convolution significantly outperforms RNN ([PERSON] et al., 2020) in sequence modelling tasks such as text generation and speech synthesis.