## 3 The Presented New Method for Noise-Robust Feature Extraction
### The Procedures of the Presented Method  
In Section 1, we discuss the issue of using the mean squared error (MSE) as the loss function to train the speech enhancement model. The resulting model does not necessarily benefit speech quality and intelligibility in terms of the PESQ and STOI evaluation scores. The underlying reason is the objective mismatch, i.e., minimizing the MSE loss function versus maximizing the evaluation scores, PESQ and STOI. Inspired by this observation, in this study, we present a training algorithm for noise-robust speech feature extraction in which the loss function is directly related to the recognition accuracy. Briefly speaking, the main idea of our proposed method is to find the acoustic speech features \(o_{t}^{\prime}\) in noisy environments that maximize the accuracy of state sequences \(q\) in HMMs with respect to the clean noise-free condition. The flowchart of the presented method is shown in Figure 2, and it consists of the following steps:  
1. We compute the FBANK and MFCC features for every utterance in the training set. Then, the features are further processed by mean and variance normalization (MVN), resulting in the final features \(o_{t}\), where \(t\) is the frame index. 2. With the MFCC features in the training set, the GMM-HMM acoustic models (mono-phones and tri-phones) are trained following the standard recipes of [PERSON] [32]. Moreover, linear discriminant analysis (LDA) [33], maximum likelihood linear transform (MLLT) [34], and speaker-adaptive training (SAT) [35] are applied to the speech features during the model training. After training, we obtain the GMM-HMM aligned state sequences \(\tilde{q}\) for each feature sequence \(o_{t}\) in the training set, and the state sequence includes mono-phone labels and tri-phone labels. 3. With the FBANK features and their state labels for the training set in hand, we train the corresponding DNN-HMM acoustic models [31]. It is a multi-classification task since we aim to find the transformation between the speech features \(o_{t}\) and their corresponding state labels \(q_{t}\). The final layer of each DNN can produce the state-dependent observation probability of each \(o_{t}\). It is worth noting that here we adopt multi-condition speech training set in the DNN learning stage. As such, the resulting DNN-HMMs are expected to be more noise-robust than the original GMM-HMMs. With the more accurate state sequences for each frame, combined with the trained HMM model in Step 2, we could obtain the word error rates (WER, %) which are the evaluation scores in our experiment. * This step is the core of our presented method. Here, we train a denoising deep neural network to transform the original speech feature \(o_{t}\) to another representation \(o^{\prime}_{t}\) as follows: \[o^{\prime}_{t}=f_{DN}(o_{t}),\] (13) where \(f_{DN}(. )\) denotes the mapping function of the trained network. The objective of the trained network is to make the resulting new features \(o^{\prime}_{t}\) predict a state sequence from the DNN-HMM created in Step 3 that is closest to the ground-truth state sequence. We express the underlying idea in the following: \[f_{DN}=\arg\max_{f}(Acc(q,q^{\prime}=g(f(o_{t})|\lambda)))\] (14) where \(\lambda\) is the DNN-HMM acoustic model, \(g\) is the function to produce the highest-likelihood state sequence, \(q^{\prime}\), of new features \(f(o_{t})\) given the model \(\lambda\), \(q\) is the ground-truth state sequence, and \(Acc\) is the log-likelihood function to measure the accuracy of \(q^{\prime}\) relative to \(q\). Once the denoising deep neural network \(f_{DN}\) is trained, in the recognition process we use it to map the speech features \(o_{t}\) in a noise-corrupted utterance to new features \(o^{\prime}_{t}\). Then, we feed \(o^{\prime}_{t}\) to the _original_ (not re-trained) DNN-HMM acoustic models together with language models in order to produce the maximum-likelihood state sequence and the word sequence.