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The short time Fourier transform time-frequency representation (STFT-TFR) method degenerates, and the corresponding short time Fourier transform time-frequency filtering (STFT-TFF) method fails under

The time-frequency analysis method is an effective tool for nonstationary signal processing, which has been broadly used in communication, acoustics, mechanical signal processing, medical signal processing, and others [

The time-frequency filtering technology which takes advantage of time-frequency localized spectra of the data provides an adaptive filtering method for the nonstationary signals. The time-frequency filtering method applies an adaptive weighting function to separate out the signals from the noise. The higher weighting parts localize the regions which are expected to be the signal components, and the lower weighting parts attenuate the noise in the time-frequency domain. The inverse transform of time-frequency representation is used to reconstruct the original signals. Recently, an adaptive time-frequency filtering method employing the Stockwell transform time-frequency representation (ST-TFR) method and its inverse transform are used to analyze the earthquake data [

The Gaussian noise is assumed in the above time-frequency representation methods and time-frequency filtering methods, and second-order statistics is applied to analyze the signals. In general cases, Gaussian hypothesis is reasonable, and the above methods are effective. However, in some actual signals such as the mechanical bearing fault signals, electroencephalogram, and radar signal, the noise has obvious pulsing characteristics, and the noise is non-Gaussian and nonstationary

In this paper, an improved time-frequency representation method employing fractional low order moment is proposed for

The characteristic function of

PDFs of S

Fractional

The discrete form of

In this section, we introduce how to solve fractional

The squared magnitude of

Each signal has certain regions and feature in time-frequency domain. The problem is illustrated in Figure

Time-frequency representation of FM signal employing the FLOSTFT-TFR method under

Short time Fourier transform (STFT) of

The STFT-TFR method fails under

We multiply a weight function

Calculate FLOSTFT with (

Compute FLOSTFT-TFR employing (

Select appropriate filter function

Calculate inverse FLOSTFT-TFR with (

Perform inverse operation of

The real signal

In this simulation, we apply the FLOSTFT-TFR and STFT-TFR methods to demonstrate the time-frequency distribution of FM signals under Gaussian noise and S

Time-frequency distributions of

Figures

In this simulation, we set

The filtered time-frequency representation employing the STFF-TFF and FLOSTFT-TFF method under S

Figures

In Figure

The real waveforms of two FM signals in time domain. (a) The original signal. (b) The original signal contaminated by S

In this simulation,

The mixed MSEs employing the STFT-TFF and FLOSTFT-TFF methods under different

The mixed MSEs employing the STFT-TFF and FLOSTFT-TFF methods under different

Figure

The MSE analysis of the STFT-TFF and FLOSTFT-TFF methods under different

In this simulation, the experiment signal is adopted from the Case Western Reserve University (CWRU) bearing data center [

Time-frequency representations of the machine fault signals in BA under S

Time-frequency representations of the machine fault signals in DE under S

Time-frequency representations of the machine fault signals in FE under S

The waveforms of the machine fault signals in time domain. (a) The original bearing out race fault signals in BA. (b) The original bearing outer race fault signals in BA contaminated by S

The waveforms of the machine fault signals in time domain. (a) The original bearing outer race fault signals in DE. (b) The original bearing outer race fault signals in DE contaminated by S

The waveforms of the machine fault signals in time domain. (a) The original bearing outer race fault signals in FE. (b) The original bearing outer race fault signals in FE contaminated by S

Figures

Figures

In Figures

Combined with the results of upper experiments, we see that the FLOSTFT-TFF method can better demonstrate time-frequency distribution of the machine fault signals and extract their fault features than the existing STFT-TFF method under S

Fractional

The shorter the length of weight function

We can set the parameter

The FLOSTFT and IFLOSTFT methods are a linear transformation, but the FLOSTFT-TFR algorithm is a nonlinear transformation.

In this paper, we consider

The authors declare that they have no conflicts of interest.

This work is financially supported by Natural Science Foundation of China (61261046 and 61362038), the Natural Science Foundation of Jiangxi Province, China (20142BAB207006 and 20151BAB207013), the Research Foundation of Health Department of Jiangxi Province, China (20175561), and Science and Technology Project of Jiujiang University, China (2013KJ01, 2016KJ001, 2016KJ002, and 2016KJ003).