# A Review of Time-Scale Modification of Music Signals +
## 4 TSM Based on Waveform Similarity Overlap-Add (WSOLA)  
### The Procedure  
One of OLA's problems is that it lacks signal sensitivity: the windowed analysis frames are copied from fixed positions in the input signal to fixed positions in the output signal. In other words, the input signal has no influence on the procedure. One time-domain strategy to reduce phase jump artifacts as caused by OLA is to introduce some flexibility in the TSM process, achieved by allowing some tolerance in the analysis or synthesis frames' positions. The main idea is to adjust successive synthesis frames in such a way that periodic structures in the frames' waveforms are aligned in the overlapping regions. Periodic patterns in the input signal are therefore maintained in the output. In the literature, several variants of this idea have been described, such as _synchronized OLA_ (SOLA) [10], _time-domain pitch-synchronized OLA_ (TD-PSOLA) [11], or autocorrelation-based approaches [12]. Figure 4: Illustration of a typical artifact for an audio signal modified with OLA. In this example, the input signal \(x\) is stretched by a factor of \(\alpha=1.8\). The OLA procedure is visualized for two frames indicated by window functions. The periodic structure of \(x\) is not preserved in the output signal \(y\). Another well-known procedure is _waveform similarity-based OLA_ (WSOLA) [4]. This method's core idea is to allow for slight shifts (of up to \(\pm\Delta_{\max}\in\mathbb{Z}\) samples) of the analysis frames' positions. Figure 5 visualizes the principle of WSOLA. Similar to OLA, WSOLA proceeds in an iterative fashion. Assume that in the _mth_ iteration the position of the analysis frame \(x_{m}\) was shifted by \(\Delta_{m}\in[-\Delta_{\max}:\Delta_{\max}]\) samples. We call this frame the _adjusted analysis frame_\(x^{\prime}_{m}\) (Figure 5a):  
\[x^{\prime}_{m}(r)= \begin{cases}x(r+mH_{\mathrm{a}}+\Delta_{m}),&\text{if }r\in[-N/2:N/2-1],\\ 0,&\text{otherwise}.\end{cases}  \tag{6}\]  
The adjusted analysis frame \(x^{\prime}_{m}\) is windowed and copied to the output signal \(y\) as in OLA (Figure 5a). Now, we need to adjust the position of the next analysis frame \(x_{m+1}\). This task can be interpreted as a constrained optimization problem. Our goal is to find the optimal shift index \(\Delta_{m+1}\in[-\Delta_{\max}:\Delta_{\max}]\) such that periodic structures of the adjusted analysis frame \(x^{\prime}_{m+1}\) are optimally aligned with structures of the previously copied synthesis frame \(y_{m}\) in the overlapping region when superimposing both frames at the synthesis hopsize \(H_{\mathrm{s}}\). The _natural progression_\(\tilde{x}_{m}\) of the adjusted  
Figure 5: The principle of Waveform Similarity Overlap-Add (WSOLA). (**a**) Input audio signal \(x\) with the adjusted analysis frame \(x^{\prime}_{m}\). The frame was already windowed and copied to the output signal \(y\); (**b,c**) Retrieval of a frame from the extended frame region \(x^{+}_{m+1}\) (solid blue box) that is as similar as possible to the natural progression \(\tilde{x}_{m}\) (dashed blue box) of the adjusted analysis frame \(x^{\prime}_{m}\); (**d**) The adjusted analysis frame \(x^{\prime}_{m+1}\) is windowed and copied to the output signal \(y\). analysis frame \(x^{\prime}_{m}\) (dashed blue box in Figure 5b) would be an optimal choice for the adjusted analysis frame \(x^{\prime}_{m+1}\) in an unconstrained scenario:  
\[\tilde{x}_{m}(r)= \begin{cases}x(r+mH_{\text{a}}+\Delta_{m}+H_{\text{s}}),&\text {if }r\in[-N/2:N/2-1],\\ 0,&\text{otherwise}.\end{cases}  \tag{7}\]  
This is the case, since the adjusted analysis frame \(x^{\prime}_{m}\) and the synthesis frame \(y_{m}\) are essentially the same (up to windowing). As a result, the structures of the natural progression \(\tilde{x}_{m}\) are perfectly aligned with the structures of the synthesis frame \(y_{m}\) when superimposing the two frames at the synthesis hopsize \(H_{\text{s}}\) (Figure 5b).