# A Review of Time-Scale Modification of Music Signals +
## 2 Fundamentals of Time-Scale Modification (TSM)  
As mentioned above, a key requirement for time-scale modification procedures is that they change the time-scale of a given audio signal without altering its pitch content. To achieve this goal, many TSM procedures follow a common fundamental strategy which is sketched in Figure 1. The core idea is to decompose the input signal into short _frames_. Having a fixed length, usually in the range of 50 to 100 milliseconds of audio material, each frame captures the local pitch content of the signal. The frames are then relocated on the time axis to achieve the actual time-scale modification, while, at the same time, preserving the signal's pitch.  
More precisely, this process can be described as follows. The input of a TSM procedure is a discrete-time audio signal \(x:\mathbb{Z}\rightarrow\mathbb{R}\), equidistantly sampled at a sampling rate of \(F_{\text{s}}\). Note that although audio signals typically have a finite length of \(L\in\mathbb{N}\) samples \(x(r)\) for \(r\in[0:L-1]:=\{0,1,\ldots,L-1\}\), for the sake of simplicity, we model them to have an infinite support by defining \(x(r)=0\) for \(r\in\mathbb{Z}\setminus[0:L-1]\). The first step of the TSM procedure is to split \(x\) into short _analysis  
Figure 1: Generalized processing pipeline of Time-scale modification (TSM) procedures.  
frames \(x_{m}\), \(m\in\mathbb{Z}\), each of them having a length of \(N\) samples (in the literature, the analysis frames are sometimes also referred to as _grains_, see [9]). The analysis frames are spaced by an _analysis hopsize_\(H_{\mathrm{a}}\):  
\[x_{m}(r)=\begin{cases}x(r+mH_{\mathrm{a}}),&\text{if }r\in[-N/2:N/2-1],\\ 0,&\text{otherwise}.\end{cases} \tag{1}\]  
In a second step, these frames are relocated on the time axis with regard to a specified _synthesis hopsize_\(H_{\mathrm{s}}\). This relocation accounts for the actual modification of the input signal's time-scale by a _stretching factor_\(\alpha=H_{\mathrm{s}}/H_{\mathrm{a}}\). Since it is often desirable to have a specific overlap of the relocated frames, the synthesis hopsize \(H_{\mathrm{s}}\) is often fixed (common choices are \(H_{\mathrm{s}}=N/2\) or \(H_{\mathrm{s}}=N/4\)) while the analysis hopsize is given by \(H_{\mathrm{a}}=H_{\mathrm{s}}/\alpha\). However, simply superimposing the overlapping relocated frames would lead to undesired artifacts such as phase discontinuities at the frame boundaries and amplitude fluctuations. Therefore, prior to signal reconstruction, the analysis frames are suitably adapted to form _synthesis frames_\(y_{m}\). In the final step, the synthesis frames are superimposed in order to reconstruct the actual time-scale modified output signal \(y:\mathbb{Z}\to\mathbb{R}\) of the TSM procedure:  
\[y(r)=\sum_{m\in\mathbb{Z}}y_{m}(r-mH_{\mathrm{s}}). \tag{2}\]  
Although this fundamental strategy seems straightforward at a first glance, there are many pitfalls and design choices that may strongly influence the perceptual quality of the time-scale modified output signal. The most obvious question is how to adapt the analysis frames \(x_{m}\) in order to form the synthesis frames \(y_{m}\). There are many ways to approach this task, leading to conceptually different TSM procedures. In the following, we discuss several strategies.