A signal decomposition theorem with Hilbert transform and its application to narrowband time series with closely spaced frequency components

Empirical mode decomposition and Hilbert spectral analysis have been extensively studied in recent years for the system identification of structures. It often encounters three challenges: (1) unable to decompose a signal with closely spaced frequency components such as wave groups in ocean engineering and beating responses in structural and mechanical systems, (2) difficult to distinguish the frequency components in a narrowband signal that is commonly seen in the free vibration of structures, and (3) unable to separate small intermittent fluctuations from a large wave. In this paper, a new analytical mode decomposition theorem based on the Hilbert Transform of a harmonics multiplicative time series is developed to address the challenges. The theorem can be applied with two procedures based on the decomposition of the original signal only or the previously decomposed (modified) signals in sequence. Numerical examples for four representative engineering applications indicate that the new theorem is superior to existing methods in decomposing a time series into many signals whose Fourier spectra are non-vanishing over mutually exclusive frequency ranges separated by bisecting frequencies. It is simple in concept, efficient in computation, consistent in performance, and reliable in signal processing. The accuracy of the new theorem is insensitive to noise. The discernable frequency spacing between the dominant frequencies of decomposed signals is theoretically near zero but practically equal to twice the frequency resolution of a finite length time series. Each bisecting frequency can be selected as an average of its two nearby frequencies of interest and is insensitive to other choices between 80% and 120% of the average value. The modified signal decomposition procedure can be less accurate than the original signal decomposition procedure due to potentially accumulated numerical errors in Hilbert transforms.