The Hilbert spectrum via wavelet projections

Non-stationary signals are increasingly analysed in the time-frequency domain to determine the variation of frequency components with time. It was recently pro- posed in this journal that such signals could be analysed by projections onto the time-frequency plane giving a set of monocomponent signals. These could then be converted to 'analytic' signals using the Hilbert transform and their instantaneous frequency calculated, which when weighted by the energy yields the 'Hilbert energy spectrum' for that projection. Agglomeration over projections yields the complete Hilbert spectrum. We show that superior results can be obtained using wavelet- based projections. The maximal-overlap (undecimated/stationary/translation in- variant) discrete wavelet transform and wavelet packet transforms are used, with the Fejer-Korovkin class of wavelet filters. These transforms produce decomposi- tions which are conducive to statistical analysis, in particular enabling noise reduc- tion methodology to be developed and easily and successfully applied.