## 4 Proposed Hearing Aid System
### Wiener Filter (WF)  
To eliminate noise and improve the enhancement of the input signal, we applied the Wiener filter to noisy speech \(Y_{i}\) as the first step in the HC-DDAEs as shown in Figure 6.  
Figure 5: The proposed HC-DDAEs model-based speech enhancement with the Wiener filter and a composite of three DDAEs. The model includes an (**a**) offline phase (i.e., training stage), and an (**b**) online phase (i.e. testing stage). IFFT, inverse fast Fourier transform; FFT, fast Fourier transform; LPS, log power spectrum; MFCC, Mel frequency cepstral coefficient. Figure 6: Wiener filter at a noisy speech signal. A random clean signal \(x_{i}\) is subject to distortion by the gain \(G\) and additive noise \(n_{i}\) (Equation (2)). The frequency-domain Wiener filter can be defined as:  
\[G(\succ,i)=\frac{\xi(\succ,i)}{1+\xi(\succ,i)} \tag{5}\]  
where \(G(\succ,i)\) is the gain function for the \(i\)-\(th\) sub-band in the frame \(\lambda\) and \(\xi(\lambda,i)\) is the estimate of a priori SNR, which is the main parameter to reduce speech distortion in a noisy environment. In this work, we used the posteriori and a priori SNR estimates for the noise power spectral density \(\hat{P}^{t}(. )\) at each frequency component \(f_{k}\):  
\[\hat{P}^{t}_{B}(f_{k})=\succ\hat{P}^{t-1}_{B}(f_{k})+(1-\succ)\big{|}B^{t}(f_{ k})\big{|}^{2} \tag{6}\]  
\[S\hat{N}R^{t}_{Post}(f_{k})=\frac{|X_{k}|^{2}}{\hat{P}^{t}_{B}(f_{k})} \tag{7}\]  
where \(P(. )\) represents half-wave rectification at the actual time interval \(t\), and \(B\) is the spectrum of the noise. The presented priori SNR has a definition below:  
\[S\hat{N}R^{t}_{prio}(f_{k})=(1-\beta)P\big{[}S\hat{N}R^{t}_{prio}(f_{k})-1 \big{]}+\beta\frac{\big{|}\hat{X}^{t-1}(f_{k})\big{|}^{2}}{\hat{P}_{B}(f_{k})} \tag{8}\]  
We set the values of the main parameters to \(\lambda=\beta=0.98\). Higher values may make the noise even quieter, and may result in damaging the audio that remains. Further details are provided in [19]. Fast Fourier transform (FFT) is applied to transform signals from the time domain to the frequency domain. Firstly, input elements are rearranged in bit-reversed order, then the transform output (time extraction) is generated. The basic idea is to break up a transform of length points \(M\) into two transforms of length \(M/2\):  
\[ \begin{array}{l}X[i]=\sum\limits_{m=0}^{M-1}x_{m}e^{-2\pi imk/M},\ k=0,1, \ldots,M-1\\ X[i]=\sum\limits_{m=0}^{M/2-1}x_{m}e^{-2\pi i(2m)k/M}+\sum\limits_{m=0}^{M/2 -1}x_{m}e^{-2\pi i(2m+1)k/M}\\ X[i]=\sum\limits_{m=0}^{M/2-1}x_{m}^{odd}e^{-2\pi imk/(M/2)}+\sum\limits_{m=0} ^{M/2-1}x_{m}^{even}e^{-2\pi imk/(M/2)}.\end{array}  \tag{9}\]  
where \(i\) denotes the frame in the FFT transform, more details can be found in [25]. The frequency range is 0.1 to 1 Hz. Then, the two popular sound features, namely, the Mel frequency cepstral coefficients (MFCC) [30, 31] and the log power spectrum (LPS) [28], are used as the acoustic features in the proposed approach for better results. The feature extraction task to extract characteristic features out of speech utterances takes the signal's frame every 16-32 ms and updates every 8-16 ms. When MFCC constraints are applied to the output layer, clean LPS characteristics can better predict:  
\[\theta^{*}=\text{argmin}(f(\theta)+\eta^{1}\parallel W^{1}\parallel^{2}_{2}+ \ldots+\eta^{1}\parallel W^{1}\parallel^{2}_{2} \tag{10}\]  
where \(\theta=\{W_{i},b_{i}\}\) is the network parameter \(\left\{\eta^{1},\ldots,\ \eta^{l}\right\}\), which controls the trade-off between the reconstruction accuracy and regularization of the weighting coefficients. In this study we set \((\eta^{1}=\ldots=\ \eta^{l}=0002)\). Using the estimated parameters of the DDAE model, noisy speech is reconstructed to enhanced speech in the online phase.