The windowed Fourier transform (short time Fourier transform) and the S-transform are widely used signal processing tools for extracting frequency information from non-stationary signals. Previously, the windowed Fourier transform had been adopted for signals on graphs and has been shown to be very useful for extracting vertex-frequency information from graphs. However, high computational complexity makes these algorithms impractical. We sought to develop a fast windowed graph Fourier transform and a fast graph S-transform requiring significantly shorter computation time. The proposed schemes have been tested with synthetic test graph signals and real graph signals derived from electroencephalography recordings made during swallowing. The results showed that the proposed schemes provide significantly lower computation time in comparison with the standard windowed graph Fourier transform and the fast graph S-transform. Also, the results showed that noise has no effect on the results of the algorithm for the fast windowed graph Fourier transform or on the graph S-transform. Finally, we showed that graphs can be reconstructed from the vertex-frequency representations obtained with the proposed algorithms.

中文翻译：

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图上信号的顶点频率表示的一种快速算法

加窗傅里叶变换（短时傅里叶变换）和 S 变换是广泛使用的信号处理工具，用于从非平稳信号中提取频率信息。以前，加窗傅里叶变换已被用于图上的信号，并且已被证明对于从图中提取顶点频率信息非常有用。然而，高计算复杂度使得这些算法不切实际。我们试图开发一种快速加窗图傅立叶变换和一种需要显着缩短计算时间的快速图 S 变换。所提出的方案已经用合成测试图信号和从吞咽期间进行的脑电图记录得出的真实图信号进行了测试。结果表明，与标准窗口图傅立叶变换和快速图 S 变换相比，所提出的方案提供了显着更低的计算时间。此外，结果表明噪声对快速加窗图傅立叶变换或图 S 变换的算法结果没有影响。最后，我们展示了可以从使用所提出的算法获得的顶点频率表示重建图。