Applications of an Improved Time-Frequency Filtering Algorithm to Signal Reconstruction

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 α stable distribution noise environment. A fractional low order short time Fourier transform (FLOSTFT) which takes advantage of fractional p order moment is proposed for α stable distribution noise environment, and the corresponding FLOSTFT time-frequency representation (FLOSTFT-TFR) algorithm is presented in this paper. We study vector formulation of the FLOSTFT and inverse FLOSTFT (IFLOSTFT) methods and propose a FLOSTFT time-frequency filtering (FLOSTFT-TFF) method which takes advantage of time-frequency localized spectra of the signal in time-frequency domain. The simulation results show that, employing the FLOSTFT-TFR method and the FLOSTFT-TFF method with an adaptive weight function, time-frequency distribution of the signals can be better gotten and timefrequency localized region of the signal can be effectively extracted from α stable distribution noise, and also the original signal can be restored employing the IFLOSTFTmethod.Their performances are better than the STFT-TFR and STFT-TFFmethods, and MSEs are smaller in different α and GSNR cases. Finally, we apply the FLOSTFT-TFR and FLOSTFT-TFF methods to extract fault features of the bearing outer race fault signal and restore the original fault signal from α stable distribution noise; the experimental results illustrate their performances.


Introduction
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 [1][2][3][4][5].Short time Fourier transform time-frequency representation (STFT-TFR) has a rich structure with dependencies along both the time and frequency axes, which is widely applied in speech [6,7], communication [8,9], and others [10,11].A speech enhancement algorithm employing the STFT-TFR method is presented in [6], which takes advantage of the time and frequency dependencies of speech signals.An improved method which can reconstruct the spectral phase of the speech signal is proposed in [7]; the method only depends on the fundamental frequency and the noisy observation from the STFT domain.An interference excision algorithm is proposed for direct-sequence spread spectrum communication systems [8,9], which analyze the interference in STFT timefrequency domain and detect and estimate the high-power jamming signal employing inverse STFT (ISTFT).Abdullah et al. apply the STFT-TFR method to analyze the switches faults of three-phase voltage source inverter [10].Zhang et al. propose a time-frequency MUSIC algorithm employing the STFT-TFR method in [11], which is applied to estimate the DOAs of the closely spaced sources.
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 [12,13]; WANG et al. propose a new time-frequency filtering method employing normalized window S transform and TT transform in [14], which is applied to attenuate high frequency noise and random noise for the radar echo signal.An adaptive filtering method employing generalized S transform is presented in [15], which is constructed by a new adaptive time-frequency filtering factor, and the method is applied to restore the LFM signals from the noise.A generalized S transform is presented in [16], which is used to extract the special signal components and filter out the noise.CHI et al. propose an adaptive time-frequency filtering method based on the STFT-TFR method and apply it to filter out the noise in FM signal; the results demonstrate that the time-frequency filtering method is effective for FM signal filtering [17].A FM interference suppression method employing the STFT-TFR and time-frequency filtering is presented in [18], which is applied to suppress the interference in the FM signals and improve the correlation output.Yang proposed an inverse STFT method and studied its vector formulation in [19].
The Gaussian noise is assumed in the above timefrequency 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  stable distribution [20][21][22][23][24][25][26].When  = 2, the noise belongs to Gaussian distribution, and when 0 <  < 2, it is low order  stable distribution.In low-order  stable distribution environment, the mentioned methods in [12][13][14][15][16][17][18][19] are based on second-order statistics, and their performance will degenerate when Gaussian model is employed to analyze the non-Gaussian signals.Recently,  stable distribution model is used for the bearing fault signals diagnosis in [27][28][29][30].Fractional lower order statistic is applied to extract the evoked potential in [31,32].In  stable distribution environment, the STFT-TFR method in the literature [8][9][10][11] degrades and even fails.Therefore, the corresponding time-frequency filtering method employing the STFT-TFF method in [17][18][19] degenerates.Hence, we propose fractional lower order short time Fourier transform and its inverse transform and propose a fractional lower order short time Fourier transform time-frequency representation (FLOSTFT-TFR) method for  stable distribution noise environment.Based on the proposed FLOSTFT-TFR and IFLOST methods, we present a novel fractional lower order short time Fourier transform time-frequency filtering (FLOSTFT-TFF) algorithm which takes advantage of time-frequency localized spectra of the signal in time-frequency domain in this paper.The simulations show that the FLOSTFT-TFR method can better demonstrate time-frequency distribution of the signals, the FLOSTFT-TFF method can effectively restore the original signal under  stable distribution noise environment, their performances are better than the STFT-TFR and STFT-TFF methods, and their MSEs are smaller under different  and GSNR.We apply the FLOSTFT-TFR and FLOSTFT-TFF methods to extract fault features of the bearing outer race fault signal and restore the original fault signal from  stable distribution noise, which illustrate their performances.Hence, the FLOSTFT-TFR method provides an approach for the special time-frequency analysis cases.IFLOSTFT is inverse transform of FLOSTFT, which provides a computational efficient way to restore the original signal from its time-frequency distribution when the undesired parts are removed.
In this paper, an improved time-frequency representation method employing fractional low order moment is proposed for  stable distribution noise environment, and a novel time-frequency filtering method is presented.The paper is structured in the following manner. stable distribution is introduced in Section 2. In Section 3 we demonstrate a fractional low order short time Fourier transform and its inverse transform, the vector formulation of FLOSTFT and IFLOSTFT are derived, and the improved time-frequency representation is studied.The improved fractional lower order time-frequency filtering method is demonstrated in Section 4. The simulations are performed to demonstrate justifiability of the proposed time-frequency representation and time-frequency filtering methods based on FLOSTFT and IFLOSTFT, and the simulations of the outer race fault signals diagnosis are demonstrated in Section 5. Finally, the discussions and conclusions are given in Sections 6 and 7, respectively.

𝛼 Stable Distribution
The characteristic function of  stable distribution can be defined as [20,21] where  ∈ (0, 2] is the characteristic index; it determines the pulse degree of  stable distribution.If  = 2,  is Gaussian distribution; if  ∈ (0, 2),  is non-Gaussian fractional lower order distribution.Parameter  is the dispersion coefficient, parameter  is the location parameter, and parameter  is the symmetry coefficient.If the parameter  = 0,  is called symmetric  stable distribution (SS), its characteristic function can be expressed as (2), and when  = 2, which is the same as the characteristic function of Gaussian distribution, Probability density functions (PDFs) of SS are given under  = 0.5,  = 1.0,  = 1.5, and  = 2.0 in Figure 1.

Computation of Inverse FLOSTFT.
In this section, we introduce how to solve fractional  order moments x() of () from FLOSTFT.Both sides of ( 18) are multiplied × matrix Γ  [I  ⊗ (Ψ  Φ −1 )]; we can get Then where x is the estimation of ] is × matrix.We call (20) inverse fractional low order short time Fourier transform (IFLOSTFT).

FLOSTFT Time-Frequency Filtering Algorithm
Each signal has certain regions and feature in time-frequency domain.The problem is illustrated in Figure 2, which shows that the FM signals gather in the time regions from  1 to  2 and the frequency regions from  1 to  2 .Then, we can separate the signal from the noise or eliminate undesired parts of the signal in time-frequency domain.In this section, we will study the time-frequency filtering algorithm based on the ISTFT and IFLOSTFT method under  stable distribution noise.
(, ) is designed according to time and frequency characteristics of the signal in time domain.From (25), we calculate inverse transform, yielding where STFT   (  ,   ) is the estimation of STFT  (  ,   ), which is filtered by the filtering function (, ) in timefrequency domain.When all STFT   (  ,   ) is solved, we obtain STFT   (, ).Following, we can calculate its inverse STFT employing the method in [19], and the estimated   () is gotten.The time-frequency filtering method is called short time Fourier transform time-frequency filtering (STFT-TFF) algorithm.

FLOSTFT-TFF Algorithm.
The STFT-TFR method fails under  stable distribution noise environment, and the STFT-TFF time-frequency filtering method based on the STFT-TFR method degenerates; hence, the FLOSTFT-TFR method has been demonstrated in Section 3. In this section, we will study how to restore the signal from its time-frequency representation employing the FLOSTFT-TFR method.
We multiply a weight function (, ) on both sides of ( 21 (, ) is a filter function, which is selected according to time-frequency characteristics of the signal.We define the estimation of FLOSTFT  (, ) as where (  ,   ) is th time point and th frequency point.When all points FLOSTFT   (  ,   ) are calculated, we can obtain FLOSTFT   (, ).Then, we can solve x() employing the IFLOSTFT method in Section 3.3.For the real signal (), we have From (29), we obtain x () in ( 30) is the estimation of ().We call the filtering method fractional low order short time Fourier transform time-frequency filtering (FLOSTFT-TFF) algorithm.
Step 3. Select appropriate filter function (, ) and extract the time-frequency regions of the signal employing (27).

Simulations
The real signal () contaminated by the noise may be written as And two FM signals are expressed as where  = 0.004,  = −0.025, = 1.72,   performance of the proposed methods, we define mixed mean square error (MSE) as where  is the number of Monte-Carlo experiments and x1 () and x2 () denote the signals estimated by the STFT-TFF or FLOSTFT-TFF methods.The figures show that the methods can also obtain timefrequency distribution of ().Time-frequency distributions of () employing the STFT-TFR and FLOSTFT-TFR methods under  stable distribution noise environment are shown in Figures 3(c) and 3(d), respectively.The results show that the STFT-TFR method fails, and the FLOSTFT-TFR method has good performance.Hence, the FLOSTFT-TFR method has better performance in demonstrating the time-frequency distribution of the signals than the STFT-TFR method under  stable distribution noise environment and is robust.

Study of the FLOSTFT-TFF and STFT-TFF Algorithm.
In this simulation, we set V() as SS distribution noise,  = 0.8,  = 1.2, GSNR = 18dB, and  = 32.We apply the FLOSTFT-TFF and STFT-TFF methods to reconstruct the original FM signals from SS distribution noise; the simulations are shown in Figure 4.
Figures 4(a) and 4(b) are the time-frequency representations of () filtered by the STFF-TFF and FLOSTFF-TFF methods under SS distribution noise environment, respectively.The results show that the time-frequency representation filtered by the STFF-TFF method is incorrect, but the FLOSTFF-TFF method can better extract the time-frequency distribution of ().Hence, the FLOSTTF-TFF method has better performance in extracting time-frequency distribution of the signal under SS distribution noise environment.
In Figure 5 we show from (a) to (d) the original FM signal, the original FM signal contaminated by SS distribution noise, the estimated FM signal employing the STFT-TFF method, and the FM signal estimated by the FLOSTFT-TFF method.The experimental results show that the contaminated original signal is overwhelmed by SS distribution noise.The restored signal employing the STFT-TFF method has larger deviation, but the signal reconstructed by the FLOSTFT-TFF method is similar to the original signal.Hence, the FLOSTFT-TFF method has a better quality of signal reconstruction under  stable distribution noise environment.

MSE Analysis under Different 𝛼 and GSNR.
In this simulation, V() is set as SS distribution noise,  = 0.8,  = 1.2, GSNR = 18 dB, and  = 32.We apply the FLOSTFT-TFF and STFT-TFF methods to restore the original FM signals and compare MSEs of the methods under different GSNR and ; the simulations are shown in Figures 6 and 7.
Figure 6 is MSE analysis under different GSNR; the experimental results show that MSEs employing the STFT-TFF method vary from 8 dB to 80 dB when GSNR change from 26 dB to 14 dB, but MSEs employing the FLOSTFT-TFF method are in the range of −20 dB to −40 dB.Hence, the FLOSTFT-TFF method has obvious advantage to restore the original signals under different GSNR.In particular, the advantage of the FLOSTFT-TFF method is more obvious when GSNR < 18 dB.
The MSE analysis of the STFT-TFF and FLOSTFT-TFF methods under different  is shown in Figure 7.We see that MSEs of the STFT-TFF method have a large variation, which vary from −20 dB to 220 dB, but MSE of the FLOSTFT-TFF method is stable about −30 dB.Hence, the FLOSTFT-TFF method has better stability in reconstructing the original signals.

Applications of FLOSTFT-TFF Method to Machine Fault
Signals.In this simulation, the experiment signal is adopted from the Case Western Reserve University (CWRU) bearing data center [33].The faulted bearings with fault diameters of 0.007 inches are reinstalled into the test motor, and the motor speed is 1797 RPM (revolutions per minute).The digital data is collected at 12,000 samples per second, and 0.2-second data is selected as test signal from the outer race bearing fault in BA, DE, and FE; then  = 2400.SS distribution noise ( = 0.8, GSNR = 15dB) is added in the fault signals, which is assumed as the actual working environment 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 SS distribution noise environment.

Discussion
(i) Fractional  order moment  ⟨⟩ () of the nonstationary signal () is stationary and integrable within the time window ℎ(); however, the traditional STFT method is still no-stationary and integrable because when  < 1, [||] = ∞.
(ii) The shorter the length of weight function , the higher time resolution, but the lower the frequency resolution.In a practical application, we should set a reasonable balance value between time resolution and frequency resolution and select the proper length of window function.
(iii) We can set the parameter  according to characteristic index of  stable distribution noise ; the smaller , the smaller .When  = 2, the FLOSTFT method degenerates into the STFT method, and the corresponding IFLOSTFT evolves into the ISTFT method,;hence, the improved FLOSTFT is a generalized algorithm.
(iv) The FLOSTFT and IFLOSTFT methods are a linear transformation, but the FLOSTFT-TFR algorithm is a nonlinear transformation.

Conclusions
In this paper, we consider  stable distribution noise as the background noise.Hence, the improved FLOSTFT and IFLOSTFT methods employing fractional p order moment are proposed, and FLOSTFT-TFR is presented for timefrequency analysis.We take advantage of time-frequency localized spectra of the signal and propose a novel FLOSTFT-TFF method, which can restore the original signal from  stable distribution noise.We apply the FLOSTFT-TFR method to get time-frequency distribution of two FM signals and use the FLOSTFT-TFF method to extract the time-frequency distribution of two FM signals from SS distribution noise; finally, the IFLOSTFT algorithm is used to restore the original FM signals; the results demonstrate their effectiveness.We compare the performance of the STFT-TFF and FLOSTFT-TFF methods; the results show that the performance of the FLOSTFT-TFF method is better than the STFT-TFF method, which can effectively suppress SS distribution noise and work in low GSNR.In practical applications from  stable distribution noise employing FLOSTFT-TFF method, and reconstruct the original fault signal employing IFLOSTFT.

Figure 2 :
Figure 2: Time-frequency representation of FM signal employing the FLOSTFT-TFR method under  stable distribution noise.

Figure 3 :
Figure 3: Time-frequency distributions of () employing the STFT-TFR and FLOSTFT-TFR methods.(a) Time-frequency distribution employing the STFT-TFR method under Gaussian noise environment.(b) Time-frequency distribution employing the FLOSTFT-TFR method under Gaussian noise environment.(c) Time-frequency distribution employing the STFT-TFR method under SS distribution noise environment.(d) Time-frequency distribution employing the FLOSTFT-TFR method under SS distribution noise environment.
Methods.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 SS stable distribution noise environment, respectively.If V() is Gaussian noise ( = 2), we set SNR = 12dB,  = 1.8.And if V() is SS distribution noise, we set the characteristic index of SS distribution noise  = 0.8, fractional low order moment parameter  = 1.1, GSNR = 16dB, and length of the weighted function  = 32.The experimental results are shown in Figures 3(a)-3(d).Figures 3(a) and 3(b) are the time-frequency distributions of () employing the STFT-TFR and FLOSTFT-TFR methods under Gaussian noise environment, respectively.

Figure 8 :
Figure 8: Time-frequency representations of the machine fault signals in BA under SS distribution noise environment.(a) Time-frequency representation of the machine fault signals in BA employing the STFT-TFR.(b) Time-frequency representation of the machine fault signals in BA employing the FLOSTFT-TFR.(c) The machine fault signals in BA restored by the STFT-TFF method.(d) The fault signals in BA reconstructed by the FLOSTFT-TFF method.

Figure 9 :Figure 10 :
Figure 9: Time-frequency representations of the machine fault signals in DE under SS distribution noise environment.(a) Time-frequency representation of the machine fault signals in DE employing the STFT-TFR method.(b) Time-frequency representation of the machine fault signals in DE employing the FLOSTFT-TFR method.(c) The machine fault signals in DE restored by the STFT-TFF method.(d) The fault signals in DE reconstructed by the FLOSTFT-TFF method.

Figure 11 :
Figure 11: 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 SS distribution noise.(c) The restored bearing outer race fault signals in BA employing the STFT-TFF method.(d) The bearing outer race fault signals in BA resconstructed by the FLOSTFT-TFF method.

Figure 12 :
Figure 12: 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 SS distribution noise.(c) The restored bearing outer race fault signals in DE employing the STFT-TFF method.(d) The bearing outer race fault signals in DE reconstructed by the FLOSTFT-TFF method.

MathematicalFigure 13 :
Figure 13: 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 SS distribution noise.(c) The restored bearing outer race fault signals in FE employing the STFT-TFF method.(d) The bearing outer race fault signals in FE reconstructed by the FLOSTFT-TFF method.