## 2 Proposed NELE Method
### Preprocessing  
As a first step, the time-domain signal is transformed to the frequency domain on a frame-by-frame basis. The main goal of this preprocessing unit is the separation of speech signal into one-third octave bands. This frame wise frequency analysis is needed for optimization and also for onset detection. First, the clean speech \(x(t)\) is framed using \(\tau=30\) ms Hann windows with 50% overlapping to construct N overlapping frames \(x_{n}(t)\):  
\[x(t)=\sum_{n=1}^{N}w(t-n\frac{\tau}{2})x(t)=\sum_{n=1}^{N}x_{n}(t), \tag{1}\]  
in which \(N\) is determined according to the length of signal. Then, each frame is convolved with the RIR:  
\[x(t)*h(t)=\sum_{n=1}^{N}x_{n}(t)*h(t)=\sum_{n=1}^{N}y_{n}(t). \tag{2}\]  
Here, \(y_{n}(t)\) is called \"extended frame\" number \(n\). An extended frame can mask the upcoming frames dependent on the reverberation time \(T_{60}\). Each extended frame is analyzed using a Fast Fourier Transform (FFT) and subsequently the frequency bins are separated according to the one-third octave bands and finally are synthesized using the inverse FFT (IFFT):  
\[\begin{split}& y_{n}(t)\smash{\mathop{\longrightarrow}\limits^{ \text{FFT}}}Y_{n}(F)\\ &\smash{\mathop{\longrightarrow}\limits^{\text{one-third octave band bin separation}}_{\text{non-third octave band bin separation}}}Y_{n,f}(F)\smash{\mathop{\longrightarrow}\limits^{\text{IFFT}}}y_{n,f}(t), \end{split} \tag{3}\]  
where \(y_{n,f}(t)\) is the synthesized signal in one-third octave band number \(f\). The \(y_{n,f}(t)\) can be considered as the convolution of a short frame \(x_{n,f}(t)\) with the RIR that is filtered by the one-third octave band number \(f\). The \(x_{n,f}(t)\) is named a \"short frame\". The \(y_{n,f}(t)\) can also be described by smaller frames called \"sub-frames\", such that the length of each sub-frame  
Figure 1: Block diagram of the proposed NELE approach. The preprocessing includes signal windowing, convolving short frames with the RIR to construct extended frames, FFT processing and bin separation to divide a speech into one-third octave bands and onset detection. In the cost function block, the extended frames and data of the onset detection unit are used to construct a cost function for each short frame. The constructed cost functions are iteratively optimized using the data of onset detection unit to compute the gains of short frames. Finally, the enhanced signal is reconstructed using the obtained weights by OLA and summation over one-third octave bands.  
is equal and temporally aligned to that of a short frame. The length of an extended frame is \(M\) times the length of a short frame:  
\[y_{n,f}(t)=\sum_{m=1}^{M}y_{n,m,f}(t),\quad M=\left\lceil\frac{l_{w}+l_{h}}{l_{w} }\right\rceil, \tag{4}\]  
where \(l_{w}\) and \(l_{h}\) are the length of a short frame and the RIR respectively. A sub-frame \(y_{n,m,f}(t)\) is defined as the frame number \(m\) of an extended frame \(y_{n,f}(t)\):  
\[y_{n,m,f}(t)=\left\{\begin{array}{ll}y_{n,f}(t)&,\ (m-1)\ \tau\leq t\leq m \tau\\ 0&,\ \text{otherwise}\end{array}\right.. \tag{5}\]  
The extended frame that is transformed into its corresponding sub-frames allows one to calculate the total signal power that can be observed within the reverberant environment within a particular time frame and frequency band. Since in this approach, a time variant filtering is applied on a frame-by-frame basis to the input signal \(x\), the effect of the filtering can be evaluated in weighted summations of \(y_{n,m,f}(t)\) that are affected by all extended frames in a short frame interval. This resulting summation will be considered in a cost function. The weight of short frame \(x_{n,f}(t)\) and subsequently an extended frame \(y_{n,f}(t)\) within a one-third octave band is denoted by \(\alpha_{n,f}\). The short frames and their respected weights, an extended frame, and a sub-frame are schematically illustrated in Figure 2.