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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Vibration sensor data from a mechanical system are often associated with important measurement information useful for machinery fault diagnosis. However, in practice the existence of background noise makes it difficult to identify the fault signature from the sensing data. This paper introduces the time-frequency manifold (TFM) concept into sensor data denoising and proposes a novel denoising method for reliable machinery fault diagnosis. The TFM signature reflects the intrinsic time-frequency structure of a non-stationary signal. The proposed method intends to realize data denoising by synthesizing the TFM using time-frequency synthesis and phase space reconstruction (PSR) synthesis. Due to the merits of the TFM in noise suppression and resolution enhancement, the denoised signal would have satisfactory denoising effects, as well as inherent time-frequency structure keeping. Moreover, this paper presents a clustering-based statistical parameter to evaluate the proposed method, and also presents a new diagnostic approach, called frequency probability time series (FPTS) spectral analysis, to show its effectiveness in fault diagnosis. The proposed TFM-based data denoising method has been employed to deal with a set of vibration sensor data from defective bearings, and the results verify that for machinery fault diagnosis the method is superior to two traditional denoising methods.

Denoising has always been an important task in sensor data processing, and it has also become increasingly significant in the field of electronic measurement and instruments. Vibration sensor data from a mechanical system are often associated with important measurement information for machinery condition monitoring and fault diagnosis [

Generally, data denoising can be conducted in either the time domain, or the frequency domain, or the time-frequency domain. In the time domain, a typical method is the time-domain averaging method which is most suitable for analyzing a strictly periodic signal [

The WT has the merit of multi-resolution analysis, which is very suitable for detecting a transient state anomaly that is embedded in a normal signal. Hence, data denoising based on the WT is a research hotspot. Wavelet-based denoising methods are very popular at present [

Recently, we have proposed a time-frequency manifold (TFM) technique [

In the rest of the paper, the basic TFM analysis theory of vibration signal is introduced in Section 2. In Section 3, the principle and procedure of the TFM-based data denoising method is presented, and the effectiveness of this method in denoising effects and fault diagnosis is further evaluated. Then, the effectiveness of the proposed method is verified by application to a set of practical bearing vibration sensor data in Section 4. Finally, conclusions are drawn in Section 5.

The TFM is embedded on the time-frequency distribution (TFD, which can be achieved by various TFA methods such as STFT, Continuous WT and Wigner-Ville distribution) of a non-stationary signal as an intrinsic nonlinear manifold structure in the time-frequency domain. For different vibration signals, the TFM displays different time-frequency patterns that can be extracted by a technique which addresses manifold learning on a series of TFDs in the reconstructed phase space [

The TFM learning requires firstly reconstructing the manifold of signal _{i}^{m}^{×} ^{n}

The TFM is then calculated in the reconstructed phase space. In this paper, the STFT is taken to generate the TFD. Firstly, each row (with the time sense) of the data matrix _{j}

The updated TFDs will be the input into a manifold learning algorithm for TFM calculation. In this step, the Local Tangent Space Alignment (LTSA) algorithm [

The following provides a simulation example to demonstrate the TFM signature of the vibration signal. This example illustrates a signal with periodic transient impulses, which represents typical vibration pattern of rotating machinery. The simulated signal is constructed by considering a free vibration model with damping as follows:
_{0} = 1,000 Hz, _{G}_{s}

Motivated by the merits of TFM, a new data denoising method is proposed in this study. As manifold learning can keep the intrinsic nonlinear structure in dimension reduction of a high-dimensional data matrix, the TFM signature represents the time-frequency structure nature of the original signal in the sense of noise suppression. This study introduces TFM to the signal denoising field by treating the TFM signature as a processed time-frequency result. This approach is called TFM synthesis, which mainly combines the techniques of time-frequency synthesis and PSR synthesis. With TFM synthesis, the denoised signal is expected to be reconstructed from the TFM signature. The new principle of data denoising aims to reduce background noise of signals effectively, and at the same time keep the essence of transient signals to the maximum extent. Therefore, the proposed method is especially suited for denoising faulty vibration signals.

In the principle of TFM synthesis, the TFM signature result is used to replace each of _{j}_{j}

Then time-frequency synthesis is employed on each updated STFT result to calculate a new data matrix

Assume _{c}_{w}_{w}

After getting the updated data matrix _{i}_{i}_{i}

In summary, the procedure of the proposed data denoising method can be described briefly as follows:

Given a signal

Do the STFT for each row of matrix _{j}

Select the frequency band of interest to get

Calculate the TFM of size

Update the STFT result using original phase and the element-adjusted TFM signature as new amplitude to get _{j}

A new data matrix

The denoised signal

To verify the effectiveness of the proposed data denoising method, the simulated vibration signal provided above is further analyzed for signal denoising. By using the proposed method, the reconstructed signal shown in

To further evaluate the result of the proposed method quantitatively, a new parameter is proposed as follows to assess the quality of signal denoising in the context of machinery fault diagnosis. This parameter considers the sparse property of impulses in time. As the proposed data denoising method keeps the merits of TFM signature, its result will deliver the natural transient waveform according to the well-captured time-frequency structure. The clean periodic impulsive signal considered in this paper will indicate good sparse property in the time domain. This property will be greatly beneficial to machinery fault diagnosis. Thus we propose a statistical parameter, called clustering statistical parameter (CSP), to describe the time-clustering property of periodic impulses. The CSP is calculated by the following procedure:

_{b}_{w}_{b}_{w}

To demonstrate the effectiveness of the new parameter, a simulation is conducted to show the relationship between the CSP and the SNR parameters. Specifically, the simulated signal as expressed in

Except for the relation with SNR, the CSP can also characterize well the sparse property of an ideal periodic impulsive signal, which is directly beneficial to machinery fault diagnosis. The CSP parameter is further calculated for the results of three denoising methods. As shown in

In machinery fault diagnosis, the main aim is to effectively identify the fault physics based on vibration signal analysis. The vibration signal from a rotating machine with a localized defect generally presents a periodic transient impulse property just as illustrated in the above simulation signal. As the measured vibration signal will also contain background noise coming from the working environment, signal filtering or denoising is necessary for effectively diagnosing the specific machinery fault. The previous section verifies the effectiveness and the superiority of the proposed method for data denoising. The following will validate the benefit of the TFM-based data denoising method for effective fault diagnosis.

The above CSP parameter indicates that the proposed denoising method can capture well the fault-related sparse periodic impulses. However, this parameter doesn't consider the difference of each impulse intensity. A general fault diagnosis approach, the envelope spectral analysis, will have worse performance when the impulses have a bigger difference in amplitude. A demonstration using a simulated signal with periodic impulses of different intensities is shown in _{d}

Different from the traditional approach that considers the intensity information of impulses, this FPTS-based diagnostic approach considers the distribution information of impulses in the time-frequency domain. The performance of this proposed method is demonstrated in

Using the proposed FPTS spectral analysis method, the TFM-based denoising result is further analyzed as shown in _{d}

To verify the effectiveness of the proposed TFM-based denoising method in practical engineering applications, a set of bearing data with rolling-element defect, inner-race defect and outer-race defect are analyzed. The above two traditional denoising methods are also employed for a comparison. The set of bearing data were acquired by using an experimental setup as shown in

The vibration signal with rolling-element defect is analyzed first. The waveform and the TFD of the defective signal are shown in

The proposed TFM-based data denoising method is then applied to the signal. The selected frequency band of interest is from 2,200 Hz to 4,200 Hz. To improve computational efficiency, the parts of the obtained

As a comparison, the results of band-pass filtering method and DWT-based denoising method are displayed as

To demonstrate the effectiveness of the proposed data denoising method for fault diagnosis, the denoising results are further analyzed. The TFM-based denoised signal is analyzed by the FPTS spectral analysis and the envelope spectrum analysis methods, respectively, as shown in

It can be seen that the FPTS shows similar intensities for different impulses, which contributes an absolute defective frequency component in the spectrum. However, in the envelope spectrum of the TFM-based denoising result, there exist other obvious frequency components except the defective frequency _{BSF}

The vibration signal with inner-race defect is shown in

To remove noise, the proposed TFM-based data denoising method is then applied to the signal. In the TFM signature calculation, the 2,000–4,300 Hz frequency band of interest, is selected to improve computational efficiency. The obtained TFM signature (

For comparison, the results of the other two traditional methods are displayed as

The above denoising results are further analyzed to verify the effectiveness of the proposed data denoising method for fault diagnosis. The FPTS spectrum and envelope spectrum of the TFM-based denoised signal are shown in _{BPFI}

The outer-race defective vibration signal is analyzed to verify the proposed method.

Two traditional denoising methods, band-pass filtering method and DWT-based denoising method, are also used to process this signal. The results are displayed in

The above denoising results are again analyzed to verify the fault diagnosis effectiveness of the proposed method. As seen in _{BPFO}

This paper presents a novel vibration sensor data denoising method which employs TFM to reconstruct a clean signal from the noisy raw signal by combining techniques of time-frequency synthesis and PSR synthesis. Based on the discussions above, following conclusions can be drawn:

The proposed denoising method inherits the merits of TFM in noise suppression and resolution enhancement to represent an intrinsic time-frequency structure, so it does not only reduce background noise effectively, but also keeps the intrinsic time-frequency structure of the periodic transient impulses in rotating machinery fault signal, which is significant for intrinsic vibration data characteristics and reliable fault diagnosis.

The proposed FPTS spectral analysis method is beneficial to utilizing the good sparse property of the TFM in impulse characteristics, and hence can be combined well with the proposed data denoising method for an improved fault diagnosis.

The performance of the proposed data denoising method and its FPTS spectral analysis has been verified by means of defective bearing data in comparison with the band-pass filtering and the DWT-based denoising methods. The proposed method shows great benefits and value in vibration sensor data denoising for effective machinery fault diagnosis.

This work was supported by the National Natural Science Foundation of China (Grant No. 51005221) and Program for New Century Excellent Talents in University (NCET-13-0539). The authors would like to thank Case Western Reserve University for offering free download of the bearing data, and the anonymous reviewers for their valuable comments.

The authors declare no conflict of interest.

Representation of TFD and TFM for the simulated vibration signal: (

Result of the proposed TFM-based data denoising method for the simulated signal: (

Denoising results of two traditional methods: (

Demonstration of the FPTS spectral analysis for a simulated signal with periodic impulses of different intensities: (

Fault diagnosis results of the simulation case for different methods: (

The bearing test stand.

Denoising results of the rolling-element defective vibration signal: (

Fault diagnosis results of rolling-element defective vibration signal: (

Denoising results of the inner-race defective vibration signal: (

Fault diagnosis results of the inner-race defective vibration signal: (

Denoising results of the outer-race defective vibration signal: (

Fault diagnosis results of the outer-race defective vibration signal: (

The relationship between the CSP and the SNR.

+∞ (Clean signal) | 30 | 20 | 10 | 0 | −5 | −∞ (White noise) | |

0.3021 | 0.0774 | 0.0399 | 0.022 | 0.018 | 0.0172 | 0.0168 |

The CSP values for the denoising results of different methods.

CSP | 0.2268 | 0.0193 | 0.0237 |

Parameters of tested bearings.

Rolling-Element Defect | 0.011 × 0.021 | 1,796 | _{BSF} |

Inner-Race Defect | 0.011 × 0.007 | 1,797 | _{BPFI} |

Outer-Race Defect | 0.011 × 0.014 | 1,749 | _{BPFO} |

The CSP values for the denoising results of different methods for rolling-element defective signal.

CSP | 0.0441 | 0.1893 | 0.0458 | 0.049 |

The CSP values for the denoising results of different methods for inner-race defective signal.

CSP | 0.0617 | 0.1587 | 0.0619 | 0.1137 |

The CSP values for the denoising results of different methods for outer-race defective signal.

CSP | 0.0381 | 0.2693 | 0.0413 | 0.1303 |