摘要
Abstract
[Objective]In practical applications,most signals are non-stationary,and time-frequency analysis serves as a vital method for analyzing these non-stationary signals.Its core principle involves filtering signals using a band-pass filter bank,where accuracy is constrained by the similarity between the filter bank and the signal.Different time-frequency analysis methods employ different strategies to construct the filter bank.However,signals may vary greatly between application domains.To accommodate diverse signal characteristics,this study proposes a time-frequency analysis method that is based on octave-symmetric wavelets(OSWs),and it uses OSWs with a half-interval overlap in the octave frequency domain as the filter bank for segmenting the signal.[Methods]The specific implementation steps are as follows:transform the minimum and maximum signal frequencies to the octave frequency domain using base-2 logarithmic operations;set the value of the octave covered by a single OSW;determine the number and center frequencies of the OSWs;according to the set octave and center frequency,select a cosine function as the amplitude spectrum shape for the OSWs in the octave frequency domain to mitigate the Gibbs effect and leverage the favorable properties of symmetric amplitude spectra;transform the OSWs defined in the octave frequency domain to the frequency domain using base-2 inverse logarithmic operations;derive the amplitude spectrum of each OSW in the frequency domain using interpolation methods;set the phase spectrum of the OSW to zero;employ the derived OSWs in the frequency domain as the filter bank;filter the signal to obtain the spectra of the decomposed signals and their time-domain representations;perform Hilbert transforms on the time-domain decomposed signals and extract their instantaneous amplitude spectra.The OSW-based time-frequency analysis spectrogram of the signal is obtained using the common time axis for all decomposed signals as the horizontal axis,the distinct dominant frequencies of each decomposed signal as the vertical axis,and the values of the instantaneous amplitude spectrum of each decomposed signal as the intensity values.For the signal used in this study,when the octave is set to 0.25 or 0.5,the resulting time-frequency spectrograms achieve a relatively balanced trade-off between time resolution and frequency resolution.[Results]The time-frequency analysis utilizing the continuous wavelet transform(CWT)and the Hilbert-Huang transform(HHT)included in MATLAB is made for identical signals.The CWT employs three distinct wavelet types(Morse,amor,and bump)as filter banks,whereas the HHT uses the intrinsic mode functions(IMFs)obtained from empirical mode decomposition(EMD)as its filtered output.Within the CWT-based time-frequency analysis,the Morse and amor wavelets produced spectra with higher time resolution but lower frequency resolution,whereas the bump wavelet produced spectra with higher frequency resolution but lower time resolution.This validates that the precision of time-frequency analysis significantly depends on the degree of similarity between the signal and the chosen filter bank.The time-frequency spectrum obtained via HHT demonstrated relatively good performance for both time and frequency resolution,but very poor horizontal continuity.Comparisons of the normalized energy and instantaneous energy plots indicated that the time-frequency spectrum based on OSWs showed no disadvantage in terms of time-frequency resolution.To analyze the noise robustness of the proposed method,weak noise and strong noise with noise-to-signal maximum amplitude ratios of 2.5 and 50,respectively,were added to the simulated signal.The results demonstrate that weak noise has a minimal impact on the time-frequency spectrum;the energy from strong noise appears in the spectrum,but the signal energy remains overwhelmingly dominant,and the time-frequency spectrum displays high resolution.[Conclusions]OSWs exhibit localization in the time domain and oscillatory behavior with alternating positive and negative amplitudes,which makes them suitable for time-frequency analysis.The OSW-based method exhibits some similarities with CWT in terms of signal decomposition.CWT utilizes a scale factor for dilation,and the OSW-based method employs the octave parameter and central frequency as dilation controls.With CWT,the scale factor is explicitly present within the time-domain expression of the wavelet function.With the OSW-based method,the octave parameter and central frequency are explicitly defined only within the analytic expression of the OSW function in the frequency domain,whereas they are implicitly contained within its numerical representation in the time domain.Dilation occurs through the variation in the central frequency for a fixed octave or the variation in the octave for a fixed central frequency.The time-shifting operation for the filter functions is implemented implicitly during the convolution between the filter and the signal.The OSW-based method bears some similarity to the HHT in terms of the processing of decomposed signals.The HHT combines EMD with the Hilbert transform.With the OSW-based method,the signals after the filtering decomposition are analogous to the IMFs obtained via EMD within the HHT framework.Consequently,the subsequent processing aligns with that of the HHT,i.e.,applying the Hilbert transform to derive the instantaneous amplitude of the signal.This study can enrich the theoretical foundation of time-frequency analysis and holds promise for broadening the applicability of time-frequency analysis methods.关键词
时频分析/非平稳信号/倍频程Key words
time-frequency analysis/non-stationary signals/octave分类
信息技术与安全科学
