Wavelet moments and time-frequency analysis

We obtain explicit expressions for the time, scale, and frequency moments of the wavelet transform in terms of the moments of the signal and wavelet. We show that generally they do not exist for common signals even when the moments of the signal and wavelet do exist. The peculiar behavior of the wavelet transform in this regard is pinpointed. The lack of existence of simple moments makes the interpretation and usefulness of the wavelet transform for time-frequency analysis problematic and it is argued that its behavior is quite poor when compared to other simple time-frequency methods, such as the short-time Fourier transform.