Fractional Superlets

The Continuous Wavelet Transform (CWT) provides a multi-resolution representation of a signal by scaling a mother wavelet and convolving it with the signal. The scalogram (squared modulus of the CWT) then represents the spread of the signal's energy as a function of time and scale. The scalogram has constant relative temporal resolution but, as the scale is compressed (frequency increased), it loses frequency resolution. To compensate for this, the recently-introduced superlets geometrically combine a set of wavelets with increasing frequency resolution to achieve time-frequency super-resolution. The number of wavelets in the set is called the order of the superlet and was initially defined as an integer number. This creates a series of issues when adaptive superlets are implemented, i.e. superlets whose order depends on frequency. In particular, adaptive superlets generate representations that suffer from "banding" because the order is adjusted in discrete steps as the frequency increases. Here, by relying on the weighted geometric mean, we introduce fractional superlets, which allow the order to be a fractional number. We show that fractional adaptive superlets provide high-resolution representations that are smooth across the entire spectrum and are clearly superior to representations based on the discrete adaptive superlets.