## 2 Method
### The Proposed CNN-PPG SCR System  
Figure 1A shows the proposed CNN-PPG SCR system in this study, included training and testing phases. In the training phase, the speech commands of dysarthric patients (\(x_{i}\)) and the corresponding label result (\(t_{i}\)) were used to train the CNN model [48; 49; 50], where \(i\) was the frame index. The detailed structure of the CNN model used in the CNN-PPG SCR system is shown in Appendix A (Table A1), and achieved the best performance in this study. The \(x_{i}\) was processed by the unit of feature extraction (MFCC) to obtain the \(X_{i}^{MFCC}\) (i.e., 120-dimensional), first. Next, the unit of feature extraction (PPG) was used to convert the \(X_{i}^{MFCC}\) to a 33-dimensional PPG feature (\(X_{i}^{PPG}\)); for the detailed 33-dimensional phone, refer to Appendix B (Table A2). More specifically, the PPG features were obtained from the acoustic model of a speaker-dependent ASR system in Figure 1C, in which the TDNN [51; 52] structure was used. A previous study, which used 40 coefficients, showed that the TDNN structure can effectively learn the temporal dynamics of speech signals [52]; hence, it could provide more benefits than alternative approaches to handle the issue of phonetic variation of a dysarthric patient's speech. The detailed training method of the acoustic model is described in the following section on the ASR-based SCR system. The parameters (\(\varphi_{CNN}\)) of the CNN model were trained and based on the PPG feature per frame (\(\mathbf{X}_{i}^{PPG}\)) and the command probability vector (\(\mathbf{Y}_{i}\)). More specifically, a nonlinear transfer function \(F_{CNN}(\cdot)\) was learned to predict a probability vector (\(\mathbf{P}_{i}\)) by \(\mathbf{X}_{i}^{PPG}\); note that \(\mathbf{P}_{i}\) should approach \(\mathbf{Y}_{i}\) after the CNN model is well trained. The above description can be depicted as the following Equations (1) and (2). \[F_{CNN}\Big{(}\mathbf{X}_{i}^{PPG}\Big{|}\varphi_{CNN}\Big{)}=f_{h}\Big{(}\cdots f _{3}\Big{(}f_{2}\Big{(}f_{1}\Big{(}\mathbf{X}_{i}^{PPG}\Big{|}\varphi_{1}\Big{)} \Big{|}\varphi_{2}\Big{)}\Big{|}\varphi_{3}\Big{)}\cdots\Big{|}\varphi_{h} \Big{)} \tag{1}\]  
\[F_{CNN}\Big{(}\mathbf{X}_{i}^{PPG}\Big{|}\varphi_{CNN}\Big{)}=\mathbf{P}_{i}\simeq\bm {Y}_{i} \tag{2}\]  
Each operation \(f_{h}(\cdot|\varphi_{h})\) is defined as the \(h^{th}\) layer of the network. In addition, the convolution layer can be expressed as:  
\[\mathbf{X}_{h+1}=f_{h}(\mathbf{X}_{h}|\varphi_{h})=relu(\mathbf{W}_{h}\otimes X_{h}+\mathbf{b }_{h}), \tag{3}\]  
\[\varphi_{h}=[\mathbf{W}_{h},\mathbf{b}_{h}] \tag{4}\]  
Note that \(\mathbf{W}_{h}\) and \(\mathbf{b}_{h}\) are defined as the collection of the \(h^{th}\) hidden layer's kernels (so-called filters) and the vector bias term, respectively. \(\otimes\) represents a valid convolution, \(relu(\cdot)\) is a piecewise linear activation function, and \(\mathbf{X}_{h}\) is an output vector from the \(h^{th}\) layer. In this CNN-based SCR system, we applied a fully connective layer with softmax activation function as the final output [53]. The model uses cross entropy as the objective function (\(L(\cdot)\)) to adjust the parameter's (\(\varphi_{CNN}\)), objective function, as shown in Equation (5). \[L(\varphi_{CNN})=-\frac{1}{N}\sum_{i=1}^{N}\sum_{c=1}^{19}y_{i,c}\log(p_{i,c}) \tag{5}\]In Equation (5), \(N\) represents the total number of frames used during training. \(y_{i,c}\) is an element from the \(\mathbf{Y}_{i}\) vector, which represents the ground truth probability value of the \(i^{th}\) frame being the \(c^{th}\) type, and \(p_{i,c}\) comes from the \(\mathbf{P}_{i}\) vector and represents the predicted scalar probability of the \(i^{th}\) frame being the \(c^{th}\) type. For model training, the back-propagation algorithm is used to find the suitable \(\varphi_{CNN}\) through the following equation:  
\[\varphi^{\prime}_{CNN}=arg_{\varphi}min(L(\varphi_{CNN})) \tag{6}\]  
Further details of the CNN can be found in the descriptions in [48; 49; 50].