The error occurs because a high QualityFactor results in wavelets that are too narrowly focused in frequency, potentially violating the necessary conditions for stable wavelet scattering. Reducing the QualityFactor increases the wavelets' bandwidth, ensuring they are appropriately scaled relative to the highest wavelet center frequency.
The necessary condition is that the wavelet's frequency standard deviation must not be too narrow compared to the scaling function's frequency support to ensure numerical stability and effective signal representation.
Consider the following example:
- Assume the center frequency of the mother wavelet is around 1 kHz.
- With a QualityFactor (Q) of 15, the 3-dB bandwidth results in a very small frequency standard deviation (1000/15 = 66.7 Hz) relative to the highest wavelet center frequency.
- The scaling function's frequency support is determined by the invariance scale. For an invariance scale of 0.005 seconds, the scaling function captures frequencies up to approximately 1/(2*0.005) = 100 Hz.
This is problematic because the frequency standard deviation is smaller than the scaling function's frequency support (66.7 Hz < 100 Hz).
By using a lower QualityFactor (Q = 9, Standard Deviation = 1000/9 = 111.11 Hz), the wavelet's frequency standard deviation becomes sufficiently large, satisfying the condition relative to the scaling function's frequency support. This adjustment ensures that the wavelets can effectively cover the necessary frequency range without becoming too narrow, thus resolving the error and allowing the wavelet scattering transform to operate correctly.
