Zoom synchrosqueezing transform and iterative demodulation: Methods with application

Synchrosqueezing transform ST, a variant of standard time-frequency reassignment method calculates directional reassignment vector in both time and frequency directions from magnitude of short-time Fourier transform STFT and Wigner-Ville distribution WVD making it possible to reconstruct the signal. Since synchrosqueezing transform is limited to process scale variable of continuous wavelength transform CWT which reveals time-frequency structure of a signal by merit of variable zooming resolution, time and frequency resolution of the synchrosqueezing time-frequency plane varies with the scale.

Synchrosqueezing transforms however has time-frequency blur problem which affects mono-component signals with time-varying instantaneous frequency as it is difficult to satisfy the desired requirement in both lower and higher frequency region of time-frequency representation.

An additional challenge is also faced with cases of multicomponent signals as continuous wavelength transform fails to separate instantaneous frequency curves which overlap in frequency domain leading to blur even before synchrosqueezing.

Researchers led by Professor Hongrui Cao from Xi’an Jiaotong University in China proposed a zoom synchrosqueezing transform ZST which generates both excellent time and frequency resolution in a specific frequency region and also analyzes the mono-component signal in the particular frequency region to obtain accurate instantaneous frequency IF estimation results. The work published in journal, Mechanical Systems and Signal Processing involved other scientists such as Songtao Xi, Xuefeng Chen and Shibin Wang.

The proposed zoom synchrosqueezing transform is composed of two parts, namely the frequency-shift operation and partial zoom synchrosqueezing operation. The zoom synchrosqueezing transform functions by using the positive frequency-shift operation to shift the signal to higher frequency in order to obtain better time resolution where after the partial zoom synchrosqueezing operation improves the frequency resolution by dividing the frequency dimension into much finer portion so as to obtain higher frequency resolution in the considered frequency region.

The proposed zoom synchrosqueezing is based on two iterative demodulation algorithm such as Viterbi algorithm and Vold-Kaman filter. The implementation steps include; step 1, where filtering of original s_o(t) to interested band signal s₁(t), step 2 involved extraction of first rough instantaneous frequency IF⁰_IK(t), step 3 involved refinement of extracted rough instantaneous frequency to IF¹_(IK)(t), in step 4, extraction of mono-component x_Ik(t) corresponding to a more accurate instantaneous frequency IF²_(IK)(t) was done, step 5 involved substitution of IF²_(IK)(t) for IF²_(IK)(t) and repetition of step 3 to step 4, step 6 which involved filtering of final accurate mono-component signals s₁(t) and the final step 7, where repetition of step 2 to step 6 took place until all components were completely extracted.

In mono-component analysis as at initial frequency f₀ of 100Hz and 400Hz, results showed that synchrosqueezed time-frequency representation had a slow oscillation portion due to worse time resolution at lower frequency region of 100Hz and it cannot accurately estimate the instantaneous frequency at 400Hz. The proposed zoom synchrosqueezing transform which only initiate partial zoom synchrosqueezing operation as central frequency of signal shifts from 100Hz to 400Hz which indicated more excellent energy-concentrated time-frequency representation with better and frequency resolution than synchrosqueezing transform.

For multicomponent analysis with same instantaneous frequency fluctuation laws and termination iterative algorithm set at 6×10^-4, it was found that the proposed zoom synchrosqueezing transform based iterative demodulation method extracts each accurate single component from multicomponent signal even at high oscillating instantaneous frequencies close to each other which provides exact instantaneous frequency estimation results.

With three different instantaneous function fluctuation laws, proposed zoom synchrosqueezing transform based iterative demodulation method provides instantaneous frequencies estimation results even with close instantaneous frequencies and different instantaneous frequency fluctuation laws which synchrosqueezing transform cannot accurately estimate. The zoom synchronizing time-frequency relationship also accurately estimated lowest frequency component which is erased due to worse time resolution in synchrosqueezing transform.

The proposed zoom synchrosqueezing transform based iterative demodulation method was used to detect a rub-impact fault occurred in a gearbox of a machine set for oxygen separation and compression in an oil refinery. With the same iterative termination set at 6×10^-4, three instantaneous frequency components located in the range of 800Hz and 2000Hz of spectrum with large amplitude showed that all three components shifted to higher frequency region by 100Hz to obtain a much better time resolution. The proposed zoom synchrosqueezing transform method efficient characterizes the instantaneous frequency of three components benefitting from its both excellent time and frequency resolution in a specific frequency region. The extracted instantaneous frequencies of the considered three components and their Fourier spectrum showed that the oscillating frequency of three instantaneous frequencies is 212.4Hz which approximately corresponds to the rotating frequency of shaft II of 213Hz

Cao et al. (2016) positive results of the proposed zoom synchrosqueezing transforms makes it available for conventional time-frequency analysis methods such as time-frequency reassignment methods and short-time Fourier transform based synchrosqueezing.