## 3 Time-Scale Modification  
In order to address the second challenge of ASR for children's speech described in Section 1--the mismatch caused by the lack of adequate training data from child speakers--the present study investigates modifying the time-scale structure of children's speech in the system testing phase. The goal of the time-scale modification is to make the testing data of child speakers more similar to the training data of adult speakers by modifying two prosodic features of speech, \(F_{0}\) and speaking rate. Both the \(F_{0}\) modification and the speaking rate modification were conducted using the same method, the real-time iterative spectrogram inversion with look-ahead (RTISI-LA) algorithm [35; 46; 47]. RTISI-LA was originally developed as a method for estimating time-domain signals from overlapping magnitude spectra that have been computed frame by frame. RTISI-LA is a similar kind of phase recovery technique as the Griffin-Lim algorithm [46; 48]. RTISI-LA is, however, much faster than the Griffin-Lim algorithm and therefore justified to be used in applications like the current study, where large numbers of speech data need to be processed. The RTISI-LA method consists of the following steps. * The speech signal is processed in frames of \(L\) samples by computing the short-time Fourier transform magnitude (STFTM) spectrum using the FFT with the Hamming window. The frame shift (\(S\)) is selected as \(S=L/4\) so that each frame overlaps with three previous and three following frames. In the following, the frame index is denoted by \(m\) and the window function by \(w(n)\). * To reconstruct the speech signal from its STFTM, an iterative frame-by-frame signal estimation process is applied. Let us suppose that the first \(m-1\) frames of the speech  
Figure 2: A block diagram of the data augmentation method. signal have already been reconstructed from STFTM, and let us denote this signal by \(x_{m-1}(n)\). The task is to synthesize \(x_{m}(n)\). * In order to estimate the \(m\)th frame, a partial analysis frame is created using overlapping (OLA) for the \((m-1)\)th, \((m-2)\)th, and \((m-3)\)th frame of \(x(n)\) considering an overlap of 75%. The fourth quarter of this partially filled frame is filled with zeros. Let the partial frame be denoted by \(x_{m-1}(n)w(n-mS)\). In RTISI-LA, the future \(k\) frames influence the reconstruction of the \(m\)th frame. After the \(m\)th frame is generated, it is kept uncommitted until the \((m+k)\)th frame is generated. * Next, the Fourier transform of the partial frame is computed using a scaled Hamming window. * The phase information computed from the Fourier transform of the partial frame is then combined with the STFTM for the \(m\)th frame. * The inverse Fourier transform of the derived frequency-domain signal produces a new estimate for the \(m\)th frame. In each iteration, the estimation of \(x(n)\) is updated. As described in [46], the RTISI-LA algorithm can be used to modify both the \(F_{0}\) and speaking rate of speech signals. By using the notations of [46], these two modifications were conducted in the current study as follows by using child speech as input. \(F_{0}\) ModificationThe \(F_{0}\) modification was computed by first re-sampling the input speech signal in the time domain. To modify the \(F_{0}\) of the input speech signal downwards by factor \(q\), where \(0<q<1\), the input frame of \(L^{\prime}\) samples was re-sampled to obtain a longer frame of \(L\) samples (i.e., \(L^{\prime}=qL\)), which was then used in the STFTM computation. We used simple linear interpolation in re-sampling because it is computationally inexpensive and was reported in [46] to provide reasonable sound quality. The value of \(L\) was fixed to 160 samples. The STFMs of the overlapping frames (i.e., the magnitude spectrogram) were then processed using the RTISI-LA algorithm (steps 1-6 above) to obtain the \(F_{0}\)-modified time domain output signal.