### Vibration Signal De-Noising and Noise Modeling

Efficient data preprocessing is essential for early diagnosis of fault conditions and better prediction of Remaining Useful Lifetime. The goal here is to improve the signal-to-noise ratio through careful noise estimation and removal.

A comprehensive scheme for dealing with the noise in a process consists of three main blocks; noise estimation, noise modeling, and noise characterization. These processes are explained as follows.

• Noise Estimation: Noise estimation entails careful analysis of the system to identify all possible noise sources. Noise effecting the signals might originate in the environment, the sensors or through cross-talk besides other possibilities. The nature of each noise source needs to be analyzed in detail before devising a strategy for noise removal. A good understanding of the clean signal characteristics is also essential since it helps identify the clean signal model. If such a model can be generated, it becomes easier to estimate the cumulative effect of all noise sources mentioned above.

• Noise Modeling: Having estimated the noise signal from the raw sensor data, we proceed by building a comprehensive model for the noise signal. A library of noise models is evaluated which quantifies the nature of noise under all possible operating conditions presented for the process. Such a modeling exercise has multiple benefits. This information can be used for artificial noise generation in simulation systems as well as a means to estimate noise in online real-time diagnostics and prognostics systems.

• Noise characterization: Finally, the estimated noise signals need to be analyzed thoroughly for extracting useful information about the system condition. Besides condition monitoring, an increase or decrease in system noise can also be used to automatically detect a change in the operating condition of the system.

##### De-Noising and its impact on feature

The overall de-noising scheme is as follows:

The accelerometer collects the vibration signal and Time Synchronous Averaging signal *s(t)* is calculated. The blind deconvolution de-noising algorithm is carried out in the frequency domain and hence *s(t)* is Fourier transformed to get *S(f)*. Then, the de-noising algorithm is applied to *S(f)* and outputs the de-noised vibration signal in the frequency domain *B(f)*. If the time domain signal is required, *B(f)* can be inverse Fourier transformed to get *b(t)*. From *B(f)* and *b(t)*, features can be extracted and fused and will be used for fault diagnosis and failure prognosis. The failure prognostic algorithm generates an estimate of crack length on the planet gear carrier. This estimated crack length and load profile of the helicopter serve as inputs of vibration model to generate the noise-free modelled vibration signal *m(t)*. This modelled vibration signal is Fourier transformed into the frequency domain and its frequency spectra is normalized to get the weighting factor vector *W(f)*, which will be used in the nonlinear projection of blind deconvolution de-noising algorithm.

The architecture of the blind deconvolution de-noising algorithm is decomposed and shown as follows.

In this scheme, nonlinear projection, which is based on vibration analysis in the frequency domain, and cost function minimization are critical components, which will be described in later sections. At first, an initial inverse filter needs to be defined. This inverse filter is an initial estimate of the modulating signal in the frequency domain and will converge to a filter through optimization routine to recover the vibration signal from the measured vibration signal *S(f)*. The initial inverse filter is convoluted with *S(f)* to obtain a rough estimate of the noise-free vibration signal . The signal passes through the nonlinear projection, which maps to a subspace which contains only known characteristics of the vibration signal, to yield *Bnl(f)*. The difference between and *Bnl(f)* is denoted as *E(f)*. By adjusting iteratively to minimize the *E(f)* and when *E(f)* reaches a minimal value, the signal –>*B(f)* can be regarded as the de-noised vibration signal. At the same time, converges to *Z(f)*. Through an inverse Fourier transform, the de-noised vibration signal in the time domain can be obtained as well.

#### Some Experimental Results

1. When this scheme is applied to the vibration signals from a gearbox of a UH-60A BlackHawk helicopters with a seeded fault which evolves with the operation of the system, the measured signal, the de-noised vibration signal, and the noise signal show as follows:

(a) measured signal (b) de-noised vibration signal (c) noise signal

2. The signal-to-noise ratios, before and after the de-noising, are shown as follows:

3. Feature Sideband Ratio is investigated before and after the de-noising. The performance indexes used to evaluate the quality of features include:

• correlation coefficient between the raw feature and crack growth curve (CCR, a rough overall evaluation);

• correlation coefficient between the smoothed feature vector and crack growth curve (CCS, overall accuracy measurement);

• percent mean deviation (PMD, overall precision measurement);

• running correlation coefficient in a window (“local” accuracy measurement);

• running percent mean deviation (“local” precision measurement).

The feature and its performance are shown as follows:

Sideband Ratio and its running performance indexes, before and after the de-noising;

CCR, CCS, and PMD of TSA feature are 0.953, 0.971, and 5.57%, respectively;

CCR, CCS, and PMD of de-noised feature are 0.983, 0.991, and 3.57%, respectively.