The Hilbert spectrum via wavelet projections

• Published in: Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Vol. 460; no. 2044; pp. 955 - 975
• Main Authors: Olhede, S., Walden, A. T.
• Format: Journal Article
• Language: English
• Published: The Royal Society 08-04-2004
• Subjects: Chirp; Denoising; Empirical Mode Decomposition; Energy spectrum; Fourier transformations; Hilbert Spectrum; Instantaneous Frequency; Signal bandwidth; Signal noise; Sine waves; Threshing; Transfer functions; Vibrational frequencies; Wavelet analysis; Wavelet Packets; Wavelet Transform
• ISSN: 1364-5021; 1471-2946
• DOI: 10.1098/rspa.2003.1199

Non-stationary signals are increasingly analysed in the time-frequency domain to determine the variation of frequency components with time. It was recently proposed 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-invariant) discrete wavelet transform and wavelet packet transforms are used, with the Fejér-Korovkin class of wavelet filters. These transforms produce decompositions which are conducive to statistical analysis, in particular enabling noise-reduction methodology to be developed and easily and successfully applied.