Increasing the Speed of Multiscale Signal Analysis in the Frequency Domain

In the Mallat algorithm, calculations are performed in the time domain. To speed up the signal conversion at each level, the wavelet coefficients are sequentially halved. This paper presents an algorithm for increasing the speed of multiscale signal analysis using fast Fourier transform. In this algorithm, calculations are performed in the frequency domain, which is why the authors call this algorithm multiscale analysis in the frequency domain. For each level of decomposition, the wavelet coefficients are determined from the signal and can be calculated in parallel, which reduces the conversion time. In addition, the zoom factor can be less than two. The Mallat algorithm uses non-symmetric wavelets, and to increase the accuracy of the reconstruction, large-order wavelets are obtained, which increases the transformation time. On the contrary, in our algorithm, depending on the sample length, the wavelets are symmetric and the time of the inverse wavelet transform can be faster by 6-7 orders of magnitude compared to the direct numerical calculation of the convolution. At the same time, the quality of analysis and the accuracy of signal reconstruction increase because the wavelet transform is strictly orthogonal.