## 2 Introduction of Speech Feature Representation and Acoustic Model Configuration
### Hidden Markov Model with Deep Neural Networks  
In an ASR system, while observing a sequence of acoustic features \(O=[\varrho_{1},\varrho_{2},\varrho_{3},\ldots,\varrho_{T}]\) (e.g., FBANK or MFCC), the objective is to find the most likely word sequences \(W^{*}\) among all possible word sequences \(W=(w_{1},w_{2},w_{3},\ldots,w_{T})\). That is,  
\[W^{*}=\arg\max_{W}P(W|O)=\arg\max_{W}P(O|W)P(W), \tag{10}\]  
where \(P(W|O)\) is the posterior probability of \(W\), which is proportional to \(P(O|W)\) and \(P(W)\) computed by the acoustic models and language models, respectively.  
Among the various acoustic models, a hidden Markov model (HMM) is a double stochastic process consisting of states, each describing the observations (speech features). An HMM is parameterized by state-transition probability and state-dependent observation probability. In particular, the state-dependent observation probability has been usually computed via a Gaussian mixture model (GMM) in the past. However, it is more frequently evaluated with a deep neural network (DNN) in the state-of-the-art ASR. In the following, we briefly describe the DNN-HMM [30].  
1. _State-dependent observation probability [31]:_ An HMM acoustic model consists of several states, and each state corresponds to a probability function that describes the random observation vector. To train a deep neural network (DNN) that exhibits the probability of observations for each state, a common procedure is to train a conventional GMM-HMM (using a GMM to describe each state) at first and then use the GMM-HMM to provide each observation (feature) in a training set with a state label. After that, supervised learning is conducted on the observations as the input together with the state labels as the desired output to obtain the deep neural networks (DNNs) for the states of the HMM. We often use the  
Figure 1: Flow chart of MFCC calculation.  
softmax activation for the output layer of the DNN and the log-likelihood as the loss function, and thus the resulting DNN can output the state-dependent observation probability, denoted by \(b_{j}(o_{t})\), where \(j\) is the state index and \(o_{t}\) is the observation at an arbitrary time index \(t\).
2. _State-transition probability:_ Each DNN-HMM acoustic model usually corresponds to a single mono-phone or a tri-phone. To find the most likely state sequence in an HMM for an arbitrary input utterance, the Viterbi algorithm is often employed. Let \(\lambda\) refer to the model parameters of a specific HMM, and \(O=[o_{1},o_{2},o_{3},\ldots,o_{T}]\) is the observation sequence corresponding to that HMM. The Viterbi algorithm tries to find the probability: \[\delta_{t}(i)=\max_{q_{1}q_{2}\ldots q_{t-1}}[P(q_{1}q_{2}\ldots q_{t-1},q_{t} =i,o_{1}o_{2}\ldots o_{t}|\lambda)]\] (11) for \(t=1,2,\ldots,T\) in series and finally find the best state sequence that produces the highest probability \(P^{*}=\max_{1\leq i\leq N}[\delta_{T}(i)]\). It can be found that: \[\delta_{t+1}(j)=\max_{i}[\delta_{t}(i)a_{ij}]b_{j}(o_{t+1}),\] (12) where \(a_{ij}\) is the transition probability from the \(i\)th state to the \(j\)th state. According to Equation (12), \(\delta_{t+1}(j)\) considers both the observation probability \(b_{j}(o_{t+1})\) and state-transition probability \(a_{ij}\) of the HMM model.