Two De-Noising Methods Based on Gabor Transform

Two de-noising methods, named as the averaging method in Gabor transform domain (AMGTD) and the adaptive filtering method in Gabor transform domain (AFMGTD), are presented in this paper. These two methods are established based on the correlativity of the source signals and the background noise in time domain and Gabor transform domain, that is to say, the uncorrelated source signals and background noise in time domain would still be uncorrelated in Gabor transform domain. The construction and computation scheme of these two methods are investigated. The de-noising performances are illustrated by some simulation signals, and the wavelet transform is used to compare with these two new de-noising methods. The results show that these two methods have better de-noising performance than the wavelet transform, and could reduce the background noise in the vibration signal more effectively.


Introduction
Generally the measured vibration signals contain important information for the prognostic and fault diagnosis purposes.However, the unavoidable background noise may influence, or even distort the diagnosis results.Thus the research on de-noising methods for vibration signals has attracted much attention and many valuable works were presented over the past years.The Fast Fourier Transform (FFT) and other FFT-based spectrum analysis methods, of course, are the most commonly used signal analysis methods for fault diagnosis, which could tell us some useful information about the frequency contents and their magnitude relations, so as to help us understand the system condition [1][2].Unfortunately, the FFT and other FFT-based methods are only suitable for stationary signals whose frequency contents do not change over time, and could not present satisfactory results if the vibration signals are non-stationary.For example, when the running speed of a machine is varied, such as speedup or speed-down, its bandwidth in frequency domain would become much wider which may present less helpful information.
Accordingly, a lot of time-frequency methods suitable for non-stationary signal, such as wavelet transform, Wigner distribution, Gabor transform, etc., have been applied in fault diagnosis [3][4][5][6][7][8][9][10][11].For example, Peng and Chu [3] presented a summary about the application of the wavelet transform in machine fault diagnosis, and gave some useful prospects of the wavelet transform combined with other tools in condition monitoring and fault diagnosis.Wang and Mcfadden [5] applied the wavelet transform to represent all possible types of transients in vibration signals generated by faults in a helicopter gearbox.Gelle and Colas [9] gave preliminary research on the application of Blind Source Separation (BSS) in fault diagnosis.Roan, etc. [10] presented a non-linear BSS approach and applied it to failure detection of gear tooth.Shen and Yang [11] presented a novel BSS method based on Fractional Fourier Transform, which could be used to separate the mixed non-stationary signals successfully and was applied for fault diagnosis of rolling bearing in freight train successfully.
In these de-noising methods based on time-frequency, an unavoidable problem is to select the appropriate threshold values in time-frequency plane.As we all know, the undesirable threshold values would deteriorate the de-noising effect.In this paper, two de-noising methods without consideration on the selection of threshold values are presented, which are based on the Gabor transform.The first method is the averaging method in Gabor transform domain (AMGTD), where the de-noised signal could be obtained by applying inverse Gabor transform to the averaged signal in Gabor transform domain through many groups of measured signals.The second one is named as the adaptive filtering method in Gabor transform domain (AFMGTD), which means that one could adaptively obtain the threshold values in Gabor transform domain by averaging many groups of measured signals.

Gabor Transform Principle
The Gabor transform was proposed in 1946 by Gabor to perform simultaneous time-frequency analysis of signals.After that many researchers improved the computation method of Gabor transform and its inverse transform, and some de-noising methods based on Gabor transform were presented [12][13].Based on the Gabor transform, the union time-frequency function of a non-stationary signal could be established by the time and frequency shift.Moreover, the union time-frequency function could be expressed by other two separated sampling mesh parameters (m and n) in time-frequency plane, where these two sampling mesh parameters are related with time t and frequency f respectively.
The continuous Gabor expansion of the signal ( ) where mn a is the continuous Gabor coefficients and ( ) Here the time shift T and the frequency shift Ω satisfy the relationship , and m and n may take all integer values.The continuous Gabor coefficients is , the discrete Gabor expansion is given in the following way [12-13]   ( ) and the discrete Gabor coefficients is where T and Ω is the time and frequency shift, M and N is the frequency and the time sampling total number.Obviously, Gabor transform is a kind of linear transform.More detailed materials about the computation schemes of discrete Gabor transform could be found in the famous references [12][13].

De-noising Methods based on Gabor Transform
Averaging Method in Gabor Transform Domain (AMGTD)Numbers.
Obviously, the Gabor coefficients of signal ) (t s are two belt-shaped impulse functions along the two frequency lines of signal ) (t s .The result shows that the periodic signal generally distributes in some narrow bands in Gabor time-frequency plane, so that the signal energy would be very concentrated in the bands.And the similar research shows that other narrow-band signals, such as modulated signals, have the same property.However, the white noise signal shows disorderly distribution in Gabor time-frequency plane, which means the energy of the noise would bestrew the whole Gabor time-frequency plane.
According to the above situations, one could establish the second step of AMGTD.That means, the average for the groups of Gabor coefficients ) , ( n m Z j could be computed by ) Then the second step is to construct the adaptive filter shown as Expanding the numerator in Eq. ( 11) one could obtain . According to the aforementioned property of Gabor transform, it could be concluded that the last three parts in Eq. (12a) and Eq.(12b) would be very small when l becomes very large, and only the first part determined by the source signal ) (t y would be kept.frequency lines obtained by AMGTD and AFMGTD are more clear-cut than that by wavelet transform.If the noise level becomes very large, the de-noising performance by wavelet transform is too bad to observe useful information either in time or frequency domain, while the frequency lines, even the time histories of the de-noised signals by AMGTD and AFMGTD could still tell us more information about the source signal.

Conclusion
According to the basic theory of Gabor transform, two de-noising methods named as AMGTD and AFMGTD based on Gabor transform are put forward.The de-noising performance of the two new methods are validated by simulation signal, and the results show that the new methods have better de-noising effect than wavelet transform.The two new methods could be applied in fault diagnosis, and may provide a new direction for fault diagnosis.

Fig. 1 .Fig. 3 .
Fig. 1.The SNRs of four signals Fig. 2. The source signal and its frequency spectrum ) From the description of these two new methods, one could find there is a key problem to overcome.The problem is how to obtain the group of signals ) (t z j .Here we give two possible ways.One way is to measure a longtime signal and then divide it into a group of signals with equal length based on the known period, if the source signal Of course, in this measurement process one must keep the measurement condition unchanged and adopt the same sampling parameters.