## 3 Methodology
### Augmentation and Normalization Techniques  
**Speed Perturbation (SP)**: SP involves resampling the original raw speech signal, which results in a warped time signal. Given an audio speech signal \(s(t)\), time warping by a factor \(\beta\) gives the signal \(s(\beta t)\). The Fourier transform of \(s(\beta t)\) is \(S(\omega/\beta)/\beta\). As a result of the changes in the time domain, which affect the number of frames in the utterance, the time warping produces shifts in the frequency components (shift of the speech spectrum); thus, speed perturbation affects both tempo and pitch [21]. The adults' speech training data was perturbed at 90% and 110% of the original rate, creating a 3-fold training set.  
**Spectral Augmentation (SpecAug):** SpecAug squeezes and stretches the spectrogram locally and has been found to improve recognition performance in conversational and non-read speech type scenarios [22]. It is applied to the log mel spectrogram of the input audio rather than the raw audio itself. It consists of three augmentation policies: (1) time masking; (2) frequency masking (that masks a block of consecutive time steps or mel frequency channels); and (3) time-warping, which randomly warps the spectrogram along the time axis. SpecAug was applied with its default settings, i.e., the maximum width of each frequency mask \(F=30\), the maximum width of each time mask \(T=40\), the number of frequency and time masks \(=2\), and the masked parts are filled with the mean.  
**Vocal Tract Length Normalization (VTLN):** The vocal tract length varies from person to person and is quite different for children and adults. The differences in vocal tract length lead to differences in the spectrum, i.e., the formant frequencies shift in frequency. The process of compensating spectral variation due to the length of the vocal tract is known as VTLN. It is a normalization technique that is often used for speaker recognition and related tasks [60]. Basic normalization techniques linearly scale the center frequencies of the filter bank in the front-end feature extractor to approximate formant frequency scaling [61]. More recent approaches include calculating a linear feature transform for each VTLN warp factor, i.e., [12].  
\[x^{\alpha}=A^{\alpha}x+B^{\alpha}=W^{\alpha}X; \tag{1}\]  
where \(W^{\alpha}=[A^{\alpha};B^{\alpha}]\) is the affine transformation matrix, \(x\) is the feature vector, \(\alpha\) is the warping factor (chosen using grid search), \(x^{\alpha}\) is the transformed feature vector for warp factor \(\alpha\), \(A^{\alpha}\) is the linear transformation matrix and \(B^{\alpha}\) is the linear bias for warp factor \(\alpha\), and \(X\) is the extended feature set [62]. Once the warping factors are estimated, a piece-wise linear warping function is implemented that maps the frequency range in three segments. Let the warping function be \(W(f)\), where \(f\) is the frequency. The central segment maps \(f\) to \(f/\alpha\), where \(\alpha\) is the VTLN warp factor (typically in the range 0.8 to 1.2). The process of VTLN warps the features to that of an ideal or reference speaker (\(\alpha_{ref}=1\)). In adult male speakers, the energy in the speech spectrum tends to be concentrated towards the lower frequencies, whereas in adult female speakers, it is generally higher; hence, their estimated warping factors are around \(\alpha_{male}\geq\alpha_{r}\) and \(\alpha_{female}\leq\alpha_{r}\), respectively. For children, since their spectrum energies are typically even higher than those for female speakers, it is expected that \(\alpha_{child}<\alpha_{r}\) compresses the frequency axis closer to the reference. Feature normalization with VTLN is a two-step process, i.e.,  
1. Train a VTLN model on a given speech dataset.
2. Estimate the warping factor \(\alpha\) for an utterance and normalize its features with the factor \(\alpha\).