## 2 Proposed NELE Method
### Construction of a Cost Function for a Frame  
According to Figure 2 and Equation (4), the sub-frame number one of \(y_{k,f}(t)\) denoted by \(y_{k,1,f}\) has overlaps with the previous \(M\) extended frames. All of these \(M\) extended frames have sub-frames that are overlapping with \(y_{k,1,f}\). A \"summed-reverbed-short frame\" number \(k\) that includes all of sub-frames within the short window number \(k\) can be constructed using the summation of current short frame \(y_{k,1,f}(t)\) and the sub-frames of previous extended frames overlapping in frame \(k\):  
\[s_{k,f}(t)=\sum_{m=0}^{M-1}y_{k-m,m+1,f}(t). \tag{6}\]  
For the speech enhancement, a time-varying weight \(\alpha_{n,f}\) according to Figure 2 is multiplied to each short frame and subsequently its convolution with the RIR that constructs the extended frame \(y_{n,f}\). This weight will be consequently multiplied to all \(y_{n,m,f}(t)\) that construct \(y_{n,f}(t)\). By multiplication to these weights, a summed-reverbed-short frame number \(k\) is changed to a \"weighted-summed-reverbed-short frame\":  
\[ws_{k,f}(t)=\sum_{m=0}^{M-1}\alpha_{k-m,f}\ y_{k-m,m+1,f}(t). \tag{7}\]  
In Equation (7), the weights are used to reduce the amount of the overlap-masking in a reverb condition. According to the STOI [19], the temporal envelope of this weighted signal is considered for defining a cost function. A time-frequency unit norm (TF-unit) of a weighted-summed-reverbed-short frame in Equation (7) is calculated by non-coherent  
Figure 2: The schematic of three consecutive and overlapped short frames \(x_{1,f}\), \(x_{2,f}\), \(x_{3,f}\). The short frame and subsequently their extended frames \(y_{1,f}\), \(y_{2,f}\), \(y_{3,f}\) are multiplied by the weights \(\alpha_{1,f}\), \(\alpha_{2,f}\), \(\alpha_{3,f}\), respectively. The extended frame \(y_{3,f}\) and one of its sub-frames \(y_{3,m,f}\) are also shown.  
summation of the power spectral density values of its discrete Fourier transform (DFT) within a one-third octave band:  
\[WS_{k,f}=\sum_{m=0}^{M-1}\alpha_{k-m,f}\cdot\sqrt{\sum_{all\ bias}\left|DFT\left\{y _{k-m,m+1,f}(t)\right\}\right|^{2}}. \tag{8}\]  
The target signal is obtained by processing of the clean speech similar to calculation that is done for the extended frames in Equation (8). However, now only the direct part of RIR \((h_{d}(t))\) is used for the computation of the target signal:  
\[z_{n}(t)=x_{n}(t)*h_{d}(t). \tag{9}\]  
The TF-unit of this signal in a one-third octave band for frame \(k\) is calculated similar to \(WS_{k,f}\):  
\[E_{k,f}=\sqrt{\sum_{all\ bias}\left|DFT\left\{z_{k,f}(t)\right\}\right|^{2}}. \tag{10}\]  
The cost function for frame \(k\) is now defined as the squared error between the TF-unit of target \((E_{k,f})\) and the TF-unit of weighted frames \((WS_{k,f})\):  
\[CF_{k}(f)=\left(WS_{k,f}-E_{k,f}\right)^{2} \tag{11}\]  
This definition of cost function is comparable with the criteria used in the STOI that uses the correlation between temporal envelopes of the clean and degraded speech as an intelligibility score. The temporal envelope in STOI is a vector of TF-units covering a 384 ms time interval. Because increasing the correlation is equivalent to minimizing the squared error, minimizing \(CF_{k}(f)\) for consecutive frames increases the correlation and subsequently the STOI score. In addition, by defining the cost function in the form of Equation (11), the optimization is easier to handle.  
An additional factor that is being considered in the cost function is the importance of a frame denoted by \(\beta_{k}\). The importance of a frame is determined by an onset detector. If a frame is detected to be an onset, a higher \(\beta_{k}\) is multiplied by \(CF_{k}(f)\). The role of this additional weight is clarified in the following section in which the optimization procedure is explained.