# Whispered Speech Recognition Based on Audio Data Augmentation and Inverse Filtering
## 4 Inverse Filtering  
Inverse filtering is a powerful technique in speech processing used to isolate the excitation component from the vocal tract shape. This chapter focuses on using inverse filtering to transform normal speech into a pseudo-whisper by removing the formant structure of voiced sounds and flattening the spectrum [23; 24; 25; 26].  
Linear Predictive Coding (LPC) [27] is a method that approximates the spectral envelope of a speech signal using a finite number of coefficients. The speech signal (\(s[n]\)) can be modeled as a linear combination of its past (\(p\)) samples plus an error term (\(e[n]\)):  
\[s(n)=\sum_{i=1}^{p}a_{i}s[n-i]+e[n]\, \tag{2}\]  
where:  
* \(s[n]\) is the current value of the signal;
* \(a_{i}\) are the coefficients of the model;
* \(s[n-i]\) are the previous values of the signal up to order \(p\);
* \(e[n]\) is the error term or white noise at time \(n\).  
LPC coefficients can be calculated using different methods, including the autocorrelation method and [PERSON]'s method [28]. [PERSON]'s method estimates LPC coefficients by minimizing the forward and backward prediction errors while ensuring the model's stability.  
The vocal tract is modeled as an all-pole filter, represented as a transfer function \(H(z)\) in the \(z\)-domain, where \(a_{i}\) are the coefficients and \(p\) is the order of the system:  
\[H(z)=\frac{1}{1-\sum_{i=1}^{\mathrm{P}}a_{i}z^{-i}} \tag{3}\]  
This filter captures the resonance characteristics or formants of the vocal tract [29; 30]. The inverse filter is designed to remove the effects of the vocal tract filter:  
\[IF(z)=1-\sum_{i=1}^{\mathrm{P}}a_{i}z^{-i} \tag{4}\]  
By applying this filter to the original signal, the formant structure is removed, resulting in a flattened spectrum that emphasizes the excitation signal. From a signal processing viewpoint, the steps include frame blocking and windowing. First, the speech signal is divided into overlapping frames (25 ms with 10 ms overlap). Then, a window function (Hamming) is applied to each frame. For each frame, the LPC coefficients are computed using [PERSON]'s method. The inverse filter \(IF(z)\) is then applied to each frame:  
\[e[n]=s[n]-\sum_{i=1}^{\mathrm{P}}a_{i}s[n-i] \tag{5}\]  
Filtering can be conducted in both the time domain and the spectral domain. In this study, the signals were convolved directly with the inverse filter in the time domain. Alternatively, in the spectral domain, the signals should be transformed to the frequency domain using FFT, multiplied with the inverse filter, and then transformed back.  
The order (\(p\)) of the LPC filter affects the accuracy of the vocal tract model and the resulting spectrum after inverse filtering. Higher-order filters capture more details of the formant structure, while lower-order filters are sometimes more appropriate because they maintain some signal characteristics without fully flattening them. The LPC model of the vocal tract effectively represents the resonant frequencies. By inverse filtering, these resonances are removed, leading to a pseudo-whisper [26]. Consider a sample speech signal. The spectrums before and after inverse filtering are depicted in Figure 3.  
Inverse filtering using LPC effectively removes the vocal tract resonances, creating a flattened spectrum and transforming speech into a pseudo-whisper. This technique offers significant utility in various speech processing applications, including audio (speech) data augmentation.  
The research presented in [24] analyzes the Long-Term Average Spectrum (LTAS) of both normal speech and whisper. It demonstrates, through the calculation of cepstral distance, that the spectra of speech and whisper exhibit greater similarity after inverse filtering. Consequently, the mismatch between normal speech and whisper is reduced.