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Abstract

Blade Tip Timing Technology (BTT) is a new type of blade vibration measurement method with non-contact measurement capabilities, which is associated with high efficiency and convenience. However, due to under-sampling feature of the BTT signal, the method can only measure part of the vibration information. Therefore, various BTT spectral reconstruction algorithms have been developed based on the under-sampling feature and sparsity of the blade vibration spectrum. Because of the harsh working environment of the blades, BTT signal typically contains high-level Gaussian noise in actual measurement, which significantly reduces frequency-domain sparsity and even aliases the vibration modals, resulting in a decrease in the performance of sparse reconstruction algorithms. In this paper, a multi-sequences BTT signal noise reduction method is proposed, which is based on sparse representation to suppress Gaussian noise. Through appropriately modifying the constraint in the ℓ0 optimization problem, the method filters out the noise elements in the sparse vector of the BTT signal. Then, denoising is completed by sparse inverse transformation. Finally, a global average-based singular value decomposition dictionary learning algorithm (GA-K-SVD) is proposed to generate an over-complete sparse dictionary which is adaptable to the original signal, to increase the effectiveness of GA-K-SVD. At last, simulation and experience are carried out to verify the performance optimization effect for sparse reconstruction algorithm and the effectiveness of proposed noise reduction method.

Keywords

Blade tip timing (BTT)signal denoising sparse representation dictionary learning over-complete sparse dictionary

Introduction

Rotor blades are the key components of compressors, turbines, and other aero-engine machinery. They usually work in high temperature, alternating aerodynamic pressure and high centrifugal force environments, each of which will reduce their fatigue life rapidly, and eventually lead to cracks or even blade fracture. In the case of serious engineering accidents,^1,2 blade failures are mainly caused by high cycle fatigue damage, that is, resonance-based fatigue damage. Online measurements of the vibration state of blades can effectively prevent this risk and provide early warning of any damage caused by abnormal vibration. Recently, blade tip timing (BTT) technology is widely applied in the field of blade vibration online measurement, which is a non-contact blade vibration-based measurement technology. BTT can monitor all blades at the same time without contact, which performs better than traditional measurement methods like strain gauges.³

The BTT technique records the actual time of arrival (TOA) of the blade tip by several probes mounted on a fixed casing.⁴ Assuming that the blade is rigid (no vibration), TOA can be calculated from the rotation speed, blade angle, and probe installation angle, each of which can be expressed as functions of the expected TOA. When blades vibrate, there will be a time difference between actual TOA and expected TOA; this can be easily calculated to obtain the vibration displacement. Due to the sampling characteristics of BTT, each of the measurement probe can only collect one pulse signal per revolution, so the BTT signal is seriously under-sampled.⁵

The under-sampling characteristics of BTT signals can be characterized by using compressed sensing theory during signal processing.^6–10 Lin¹¹ first proposed a multi-coset BTT sampling model applying compressed sensing to transform the infinite dimensional sparse representation model into a finite dimensional multiple measurement vector (MMV) model. The Orthogonal Matching Pursuit (OMP) method¹² was then used to reconstruct the blade multi-modal vibration spectrum. Pan¹³ proposed a sparse representation model for BTT signal based on a single measurement vector and used the Basis Pursuit Denoising (BPDN)¹⁴ algorithm. MUSIC¹⁵ is another class of blind spectral analysis methods which can be used in BTT spectrum reconstruction. Wang improves frequency identification effect of MUSIC by extending the batch length of the data set¹⁶ and reduces the computational complexity by reducing the dimensionality of the noise subspace.¹⁷

These sparse reconstruction algorithms are all implemented based on the sparsity of the blade vibration in the frequency spectrum, which have a high requirement on the quality and clarity of the original signal. Unfortunately, due to the harsh working environment of the aero-engine, the signal collected by BTT probes inevitably contains high-level Gaussian noise,¹⁸ which greatly reduces the sparsity of the blade vibration spectrum. For sparse reconstruction algorithms, low sparsity leads to the reduction of accuracy and the increase of the computational complexity. For example, when the OMP method is dealing with noisy signals, to meet the requirements of iterative error, the number of iterations increases rapidly with the raise of noise intensity, which means great increase of the calculation time, and the high-level Gaussian noise terms will alias the frequency components of the blade vibration which is seriously aliased in BTT under-sampled stage, resulting in a serious decline in accuracy. Therefore, it is very necessary to preprocess the BTT signal. However, due to the under-sampling characteristics of the BTT signal, filters, or traditional filtering techniques, such as wavelet denoising¹⁹ and empirical mode decomposition (EMD),²⁰ are very easy to mistakenly filter out the aliased blade vibration modals, which cannot be applied to BTT signal noise reduction. Therefore, new noise processing methods for BTT signals need to be developed. Cao²¹ proposed two filtering methods for the BTT signal based on sampling aliasing frequency (SAFE) map, which can extract synchronous vibration and denoise the asynchronous vibration due to the clear distribution of different components in the SAFE map. Wang¹⁸ proposed a robust sparse representation model based on Mixture of Gaussian (MoG) to reduce the impact of complex noise caused by speed fluctuation and casing vibration on the BTT sparse model. In this paper, we conduct noise reduction on BTT signals through sparse representation.

In the past decades, using of sparsity as the driving force of image and signal denoising has attracted a lot of research attention,^22–26 and sparse signal reconstruction algorithms have been introduced to solve these image denoising problems. To simulate these poor fidelity images, the signals are sparsely represented as the product of a fixed dictionary and a sparse vector by solving a ℓ0 optimization problem.²⁷ Next, a suitable constraint condition is chosen for the ℓ0 problem, and the noise modals with smaller weights in the sparse vector are discarded. The image information modals with larger weights can be retained and the noise level is reduced. The image processing methods imply that BTT signals can be denoised in a similar way.

Inspired by compressed sensing and image denoising methods, this paper proposes a multi-probe and multi-modal BTT signal denoising method. Based on the non-sparse spectral characteristics of the noise signal and the sparse characteristics of the BTT signal spectrum, the sparse representation is used to separate noise and BTT signal components on a sparse basis, which achieve noise reduction effect. To obtain a sparser solution, we propose a Global-Average-based Singular Value Decomposition dictionary learning algorithm (GA-K-SVD) to optimize the denoising process. The Global Averaging method aims to reduce complexity of the computation by dividing the multi-channel BTT signal into multiple blocks, applying sparse representation denoising to the signal, and finally obtaining the denoised signal through global averaging. The dictionary learning algorithm can update the dictionary iteratively to approach the globally optimal solution that minimizes the block’s sparse representation error using a learning dictionary. GA-K-SVD algorithm can not only improve the clarity of the BTT signal but also improve the sparseness of the signal spectrum to allow sparse reconstruction algorithms have better performance in reconstruction of blade vibration signal.

BTT sampling principle

This section will explain the BTT sampling principle and prove the under-sampling characteristics of the BTT signal in the frequency domain, which provides a theoretical basis for applying sparse representation for BTT signal denoising.

Representation of blade tip displacement

As shown in Figure 1, an optical-fiber sensor is randomly or equidistantly embedded in the casing. The sensor will continuously emit light signals and receive light signals. When each blade passes the sensor, the blade tip will reflect from the light signal, and the sensor will receive the light signal and generate a pulse. Thus, there will be a time-delay offset between the pulses ${\tilde{t}}_{i}$ measured when the blade is vibrating and the pulses $t_{i}$ measured when the blade is not vibrating, which indicates the vibrational displacement of the blade. When the blade does not vibrate, the expected time of arrival (TOA) of the nth revolution can be expressed as follows

{\tilde{t}}_{i} [n] = \frac{α_{i} - β}{2 π} T [n] + \sum_{k = 0}^{n - 1} T [k],

(1)

where T [n] is the duration of revolution n, α_i is the angle between probe i and the reference sensor, and β is the angle between the k-th blade and the reference sensor.

Figure 1.

Blade tip timing sampling principle.

The deviation of ${\tilde{t}}_{i} [n]$ from the actual TOA, $t_{i} [n]$ , indicates the blade vibration displacement. Taking ${\tilde{t}}_{i} [n]$ as a reference, the vibration displacement of the k-th blade can be obtained by the following formula

y_{i} [n] = \frac{2 π R}{T [n]} (t_{i} [n] - {\tilde{t}}_{i} [n]),

(2)

where y_i [n] is the k-th blade tip displacement vector recorded by the i-th probe and R is the distance from the blade tip to the center of the rotating shaft.

Through equations (1) and (2), it can be seen that the sampling frequency of the BTT technology is equal to the product of the number of sensors and the rotation frequency. For the high-frequency vibration that rotating blades usually occur and the limit of sensor number, the BTT signal is usually under-sampled. Accurately reconstructing the blade vibration signal from BTT signal is also a key issue, but we won’t go into the details of signal reconstruction in this paper. The multi-channel BTT signal which has collected data from N revolutions can be represented as a multidimensional vector Y

Y = [y_{1}, y_{2}, \dots y_{I}] + ε, Y \in R^{N \times I},

(3)

where ε is the additive Gaussian noise and y_i is the sampled signal recorded by i-th probe. Y can be seen as a grayscale image, and each value represents the grayscale of the corresponding pixel (Y contains negative numbers but does not affect the denoising algorithm). Y is the signal matrix that needs to be denoised.

Sparsity properties of BBT signals

The vibration of the blade is caused by broadband shock excitation, airflow shock, for example, under the influence of these exciting forces, the blade will generate multi-modal vibration, and its response can be simulated with a multi-degree-of-freedom model (MDOF)

y (t) = \sum_{i = 1}^{Q} A_{i} e^{j (2 π f_{i} t + φ_{i})},

(4)

where

{f_{i}}

is the main frequency of the blade’s modals and Q is the number of modals. A blade vibration signal’s frequency spectrum described by MDOF is shown in Figure 2. Although the actual blade response is usually not composed of multiple single frequencies, the blade vibration modals frequency band {B_i} is nevertheless extremely narrow in the frequency spectrum, where “extremely narrow” means

\sum_{i = 1}^{Q} B_{i} ≪ f_{r}

, and

f_{r}

is the rotor rotation frequency. Therefore, the fact that most elements of the blade vibration spectrum are zeros proves that the blade vibration signal is sparse in the frequency domain. Although the characteristics of under-sampling lead to the serious aliasing of BTT spectrum, the number of modals and bandwidth remain unchanged. Therefore, the BTT signal can be considered sparse in the frequency domain.

Figure 2.

Blade vibration spectrum. Various narrow bands with bandwidth { B_i } surround the dominant frequencies.

Compressed sensing

In compressed sensing theory, this property of the BTT signal means that the signal can be sparsely represented. Utilizing the sparsity of a signal x, sparse representation can represent x as a linear combination of few atoms of a sparse dictionary D, where atoms are the column vector of D. The vector α decides which atoms are used to represent the signal. Since only few atoms are selected, α is sparse. Therefore, the sparse representation process can be expressed as follows. (Figure 3).

Figure 3.

Sparse Representation Schematic, where the blank parts in x represent non-zero elements and the blank columns in D represent selected atoms.

x_{N \times M} = D_{N \times K} \times α_{K \times M} .

(5)

The unknown item, sparse vector α, can be obtained by solving the following ℓ0 problem:

\hat{α} = \arg \min {‖ α ‖}_{0} s . t . x = D α

(6)

where

{‖ \cdot ‖}_{0}

represents the number of non-zero elements in the vector and α is the sparsest solution. Although this is an NP (non-deterministic polynomial)-hard problem, there are plenty ways to solve it. The OMP¹² algorithm is a simple and efficient algorithm that can approximate the sparse solution of the ℓ0 problem. Therefore, this paper first selects the OMP algorithm to solve the sparse vector.

BTT denoising principle

The sparsity of the BTT signal in the frequency spectrum means that it can be represented by a sparse vector via a Fourier-based dictionary. However, due to the harsh working environment of aero-engines, BTT signals usually contain high levels of (Gaussian) noise. The low autocorrelation of noise means that its sparsity in spectrum is very poor, as shown in Figure 4(b), to its corresponding sparse vector. The BTT signal spectrum energy is concentrated in several specific narrow frequency bands, while the energy of the additive Gaussian noise is scattered along every frequency. Therefore, the sparsity of the noisy BTT signal is greatly reduced, which seriously affects the accuracy of the sparse reconstruction algorithm. This provides us with new inspiration for BTT signal noise reduction preprocessing.

Figure 4.

(a) A sparse signal spectrum, only a few elements are not zero. (b) A Gaussian noise spectrum, which is not sparse, and the amplitudes are average.

To describe the model of equation (6) more accurately, a bounded error should be allowed to replace the original constraints, which means that equation (6) can be written as ${‖ Y - D X ‖}_{2}^{2} < σ$ , where $σ$ is a constant close to zero, that is, the smaller the value, the more similar the original signal is to the signal after anti-sparse transformed. Therefore, under the framework of compressed sensing and sparse representation, the denoising problem can be expressed in the following more general form

\hat{X} = \arg \min {‖ X ‖}_{0} s . t . {‖ Y - D X ‖}_{2}^{2} < T (σ),

(7)

where

T (σ)

is determined by

σ

and the unknown noise level. Therefore,

T (σ)

is simply an empirical value we find to perform well. The idea in equation (7) is similar to the hard threshold denoising method, which sets all the coefficients smaller than the set threshold to zero, and only keep the larger coefficients. The model described in equation (7) can filter out most of the noise components (with small coefficients) through setting them to zero. This ensures the sparsity and retains the aliasing modals in the BTT signal, which can achieve the denoising of the sampled BTT signals.

To facilitate the understanding of the follow-up content, the constraint term can be written as the penalty term

\hat{X} = \arg \min_{X} {‖ Y - D X ‖}_{2}^{2} + μ (σ) {‖ X ‖}_{0},

(8)

where

μ (σ)

is the coefficient of the penalty term and the values of

μ (σ)

and

T (σ)

have the same selection principle.

Combining this principle, we only need to solve the ℓ0 problem of equation (7) or equation (8), and then perform sparse inverse transformation on it, and then we can get the preprocessed signal with greatly reduced noise level.

GA-K-SVD algorithm

Based on the theory explained in Section 3, solving for sparse vectors is the key to BTT signal denoising. It is true that we can use traditional sparse reconstruction algorithms (OMP, BP, etc.,) and common fixed dictionaries for a simple noise reduction; however, such methods cannot meet the real-time and high-precision vibration monitoring requirements of the BTT technology. Therefore, we propose a Global Average-based K-SVD algorithm (GA-K-SVD) as a new BTT signal denoising method. Among them, the Global Average method can effectively reduce the complexity of computation without affecting the noise reduction effect, which is closer to real-time requirements; the improved K-SVD algorithm can obtain a sparse dictionary that can obtain a sparser solution and improve noise reduction effect and modal retention.

Global average method

The multiple BTT signals are integrated into an $N \times I$ matrix, where I is much smaller than N. The actual N of the BTT signal collected in one experiment is usually 5000∼10000. Then the dictionary size required for sparse representation is at least an N×N-dimensional square matrix, which contains millions of elements. The atoms use an over-complete sparse dictionary, where over-complete applies to the columns of the matrix. When applying a sparse reconstruction algorithm to solve equation (7) or equation (8), that is, OMP algorithm, each iteration needs to calculate the residual matrix and calculate its inner product with each atom of the dictionary to locate the atom most related to the residual. Thus, a large dictionary can seriously reduce the noise reduction efficiency. However, applying a small dictionary implies the locality of algorithm, which simplifies overall signal processing and reduces the noise reduction effect.

Therefore, we propose a Global Averaging method to achieve a better balance in the abovementioned problems. The method uses a sliding block to divide the original signal matrix Y into signal patches y_j, as shown in Figure 5. Figure 5 shows that an $N \times I$ BTT signal matrix can be divided into $N - I + 1$ patches by using an $I \times I$ sliding block. The coordinate matrix $R_{j}, j = 1,2, \dots, N - I + 1$ , where $y_{j} = R_{j} Y$ is recorded. The objective function after block dividing is changed from equation (8) to

{{\hat{x}}_{j}, \hat{Z}} = \arg \min_{x_{j}, Y} λ {‖ Y - Z ‖}_{2}^{2} + \sum_{j} μ_{j} {‖ x_{j} ‖}_{0} + \sum_{j} {‖ D x_{j} - R_{j} Z ‖}_{2}^{2},

(9)

Figure 5.

Schematic diagram of using sliding block to cut up signal patches. Each patch contains an $I^{2}$ element, and the dimension of the initialization dictionary D is $I^{2} \times 2 I^{2}$ , which effectively reduces the dimension of the oversampled matrix.

Where $x_{j}$ is the sparse vector of the j-th patch and Z (unknown) is the denoised version of Y. If we assume that the sparse dictionary D has been obtained, then ${\hat{x}}_{j}$ can be obtained by

{\hat{x}}_{j} = \arg \min_{x} μ {(σ)}_{j} {‖ x ‖}_{0} + {‖ y_{j} - D x_{j} ‖}_{2}^{2} .

(10)

Based on the principle of simplified processing, OMP algorithm is applied, which selects a most relevant atom in each iteration and stops when the residual ${‖ y_{j} - D x_{j} ‖}_{2}^{2}$ is less than $T (σ)$ . Therefore, the choice of $μ {(σ)}_{j}$ has been handled implicitly. Given information about ${\hat{x}}_{j}$ , equation (9) can be further optimized via

\hat{Z} = \arg \min_{Y} λ {‖ Z - Y ‖}_{2}^{2} + \sum_{j} {‖ D {\hat{x}}_{j} - R_{j} Z ‖}_{2}^{2} .

(11)

equation (11) has a closed-form solution of the form

\hat{Z} = {(λ I + \sum_{j} R_{j}^{T} R_{j})}^{- 1} (λ Y + \sum_{j} R_{j}^{T} D x_{j}) .

(12)

Equation (12) is meant to globally average denoised patches with some relaxation obtained by averaging the original noisy signal.

Improved K-SVD algorithm

Of course, if a sparser solution can be obtained, the screening of noise components will be easier, and the sparsity of sparse vector is closely related to the sparse dictionary. In previous work, there are many choices of sparse dictionaries, such as the DCT dictionary,²⁸ Fourier dictionary, and²⁹ Gabor dictionary.³⁰ These dictionaries are undoubtedly a convenient choice for sparse representation work, but for BTT signals, they are not optimal sparse dictionaries. When the structural matching between the dictionary and the signal is not ample enough, there will be a problem of low sparsity. Some studies have shown that the signal sparse representation using an over-complete sparse dictionary is more effective.^31–33 An over-complete sparse dictionary signifies that the number of atoms in the dictionary is much larger than the length of the signal. An over-complete sparse dictionary can obtain a sparser vector, and the restored signal has a better structural feature matching degree with the original signal. There are many methods for constructing over-complete sparse dictionaries from orthogonal basis dictionaries, such as the fractional frequency method,³⁴ but we still have a better choice. After simply initializing an over-complete sparse dictionary, we train a better sparse dictionary with the improved K-SVD dictionary learning algorithm.

We will elaborate on the improved K-SVD algorithm in combination with Global Average method. The known quantity is multi-channel BTT signal Y, as shown in equation (3). Where y_i represents the i-th BTT signal. Define x as the sparse vector corresponding to the training data Y, and the matrix D is the over-complete sparse dictionary that needs to be trained. Therefore, the process of dictionary learning can be expressed as an optimization problem:

\hat{X} = \arg \min_{X, D} {‖ X ‖}_{0} s . t . {‖ Y - D X ‖}_{2}^{2} \leq T (ε, σ)

(13)

The improved K-SVD algorithm has three stages: Initialization, Sparse Coding, and Dictionary Updating.

Initialization

The first iteration requires an initialized redundant dictionary D and sparse vectors X. According to the sparse spectrum characteristic of BTT signal, we choose to construct the following over-complete sparse dictionary D based on the fast Fourier basis dictionary.

D = {[I_{n}, Ψ_{F F T}]}_{n \times 2 n}

(14)

where

Ψ_{F F T}

is an n-dimensional fast Fourier basis dictionary and

I_{n}

is an n-dimensional identity matrix. According to Section 4.1,

N \times I

BTT signal matrix Y has been divided into

N - I + 1

sub-signal blocks

{y_{j}}

, which is the training samples.

Sparse coding

Iterations start at this stage. According to the initial dictionary, the sparse vector of each training sample is solved via

{\hat{x}}_{j} = \arg \min_{x_{j}} {‖ x_{j} ‖}_{0} s . t . \forall i, {‖ y_{j} - D x_{j} ‖}_{F}^{2} \leq T_{0} (σ) .

(15)

The optimization problem is relatively easy since it is sufficient to use the traditional OMP algorithm to solve equation (15).

Dictionary updating

In the k-th step of iterative learning, the k-th row vector that is multiplied by the sparse vector $x_{i}$ and the k-th atom $d_{k}$ of the dictionary is defined as x. Then, the objective function of equation (13) can be further written as follows

‖ Y - D X ‖_{2}^{2} = {‖ Y - \sum_{n = 1}^{K} d_{n} x_{T}^{n} ‖}_{2}^{2} = {‖ (Y - \sum_{n \neq k} d_{n} x_{T}^{n}) - d_{k} x_{T}^{k} ‖}_{2}^{2},

(16)

where

x_{T}^{n}

is the n-th row in the sparse vector matrix X.

Define $E_{k}$ as the error generated by all atoms except the k-th atom and then equation (16) is expressed as follows

‖ Y - D X ‖_{2}^{2} = {‖ E_{k} - d_{k} x_{T}^{k} ‖}_{2}^{2},

(17)

where

E_{k} = Y - \sum_{n \neq k} d_{n} x_{T}^{n} .

(18)

In the iterative process, our goal is to find a {d_k, x_k} that will best approximate the residual, that is, minimize the error ${‖ E_{k} - d_{k} x_{T}^{k} ‖}_{2}^{2}$ . And through singular value decomposition, this approximation that minimizes the error can be found. This is also the reason why the K-SVD algorithm is used, which can guarantee the decrease of the residual error in every iteration of updating the dictionary.

Let $ω_{k} = {i ∣ 1 \leq i \leq K, x_{T}^{k} (i) \neq 0}$ represent the index of the sample signal Y using atom d_k. Let $Ω_{k}$ be an $N \times ω_{k}$ matrix, where $(ω_{k} (i), i)$ of $Ω_{k}$ is the non-zero value, and the rest of the elements are all zero values. Then, equation (17) is equivalent to

{‖ E_{k} Ω_{k} - d_{k} x_{T}^{k} Ω_{k} ‖}_{2}^{2} = {‖ E_{k}^{R} - d_{k} x_{R}^{k} ‖}_{2}^{2},

(19)

where

x_{R}^{k} = x_{T}^{k} Ω_{k}

represents the row vector of

x_{T}^{k}

with the zero-valued items removed and

E_{k}^{R} = E_{k} Ω_{k}

represents the error column of the atom d_k used in the sparse coding stage.

Applying singular value decomposition (SVD) to $E_{k}^{R}$ , we get the following decomposition expression

E_{k}^{R} = U Δ V^{T} .

(20)

The term d_k is updated in the initial dictionary by decomposing to get the first column of U. The sparse representation coefficient $x_{R}^{k}$ is replaced by the product of the first column of matrix V and $Δ (1,1)$ . Then, back to the second stage with the updated dictionary. The iterations will continue until E_K is less than the iteration threshold or the number of iterations reaches the upper limit, and the denoised sparse vector blocks {x_j} is the output.

Through the improved K-SVD algorithm, we are able to obtain a sparse vector set with less noise and more accurate preservation of blade vibration modal information. Finally, according to equation (12), the Global Average method is performed to obtain a complete denoising signal Z with the acknowledge of {x_j}. So far, equations (12) and (14)–(20) include several requirements of BTT signal denoising method based on the Global Average-based K-SVD algorithm (GA-K-SVD), which is summarized by Table 1.

Table 1.

GA-K-SVD algorithm.

Simulation

In Section 5, simulations are conducted to validate the effectiveness of the GA-K-SVD algorithm. The parameters of simulation blades are shown in Table 2. Assuming the blade vibration

y (t)

is set as a multi-degree of freedom model mentioned in Section 2,

y (t)

can be expressed as follows:

y (t) = \sum_{i = 1}^{Q} \frac{1}{i} \cos (2 π f_{i} t + φ_{i}),

(21)

where

φ_{i}

are set to zero for simplicity. The original blade vibration signal obtained by equation (21) and the BTT signal of probes No.1 are shown in Figure 6.

Table 2.

Parameters of simulation model.

Parameter	Value
The number of blades	5
The number of vibration modals Q	3
The vibration frequencies {f_i}	500 Hz, 940 Hz, and 1673 Hz
The rotating speed	6000 rpm
The number of probes I	6
The probe angles	9.7°, 29.2°, 58.4°, 107°, 175.1°, and 282.1°

Figure 6.

(a) Blade vibration spectrum; (b) under-sampled BTT spectrum.

In the condition of non-noise, the spectral sparsity of blade vibration is excellent, and the sparse reconstruction algorithm can also reach the threshold after few iterations and completely reconstruct the signal spectrum, as shown in Figure 7. However, in the condition of noise, the reconstruction accuracy of the sparse reconstruction algorithm will decrease, and the number of iterations (calculation amount) will also increase greatly, as shown in Figure 8. It can be seen that the reconstruction result of the OMP algorithm has error modals, which leads to the failure of the reconstruction; and the lower the signal-to-noise ratio, the longer the calculation time of the OMP algorithm. Therefore, it is necessary to deal with the high level noise in the BTT signal. Then we will simulate the GA-K-SVD algorithm to verify its denoising performance and the improvement of the effect of the sparse reconstruction algorithm.

Figure 7.

(a) Blade vibration spectrum without noise; (b) under-sampled BTT signal without noise sampled by one of the probes; (c) reconstructed blade vibration spectrum by the OMP algorithm.

Figure 8.

(a) Reconstruction blade vibration spectrum by the OMP algorithm containing noise (SNR = 5); (b) time of computation in different noise levels.

In the simulation, original BTT signal matrix Y is made up by I-columns of the BTT signal with length N = 1000. Gaussian noise is then added to Y, and the GA-K-SVD algorithm is applied for noise reduction. Firstly, signal needs to be split (see in Section 4.1). If the number of probe is 6, a 6×6 sliding window is set. As shown in Figure 5, the process of signal division starts from the first column of Y, and Y is divided into N+1-I blocks expressed as a set ${(y_{j}, R_{j})}$ , $j = 1,2, \dots (N + 1 - I)$ , that is, j = 1∼995, where $y_{j}$ is the sub-block and $R_{j}$ is the coordinate matrix, satisfying $y_{j} = Y \cdot R_{j}$ . Then, we are ready to train the denoising sparse representation matrix D using ${(y_{j}, R_{j})}$ as the training sample. According to the data amount of $y_{j}$ (6*6), a 36*72 initialization Fourier dictionary is constructed according to equation (14) shown in Figure 9(a). Then GA-K-SVD algorithm enters iteration, including the Sparse Coding stage and Dictionary Updating stage (see in Section 4.2). According to equation (15), in each iteration, the sparse vector of each $y_{j}$ in the current sparse representation dictionary must be calculated through the OMP algorithm firstly, then the residual is calculated according to equations (16)–(20), and the dictionary is updated. In this simulation, we choose to iterate 10 times and set the threshold function to $T (σ) = 1.15 \cdot σ$ . The trained dictionary is shown in Figure 9(b). Finally, calculate the sparse vector of each $y_{j}$ according to equation (10) and perform sparse inverse transformation to realize noise reduction reconstruction of ${y_{j}}$ and perform global average processing according to equation (12) to eliminate the overlap between blocks and obtain the noise-reduced blade vibration signal.

Figure 9.

(a)The initialization Fourier dictionary; (b) trained sparse dictionary.

To evaluate the performance of different reconstruction algorithms, two metrics are defined. The first is the Relative Root Mean Square (RRMS), defined as follows.

RRMS = \sqrt{\sum_{i = 1}^{n} {(\hat{y} [i] - y [i])}^{2} / \sum_{i = 1}^{n} y {[i]}^{2}} .

(22)

The second is the Signal Optimization Rate (SOR), defined as follows.

SOR = (1 - \frac{RRSM (denoised signal)}{RRSM (signal with noise)}) \times 100 %,

(23)

where

y [i]

and

\hat{y} [i]

are noise-free simulated signals and GA-K-SVD denoised signals (or non-denoised signals), respectively. Note that the closer the RRMS is to 0, the better the noise reducing effect (or the smaller the noise content). The closer the SOR is to 100%, the more significant the optimization effect on the noisy signal. In this section, for each noise level, we randomly generated 100 groups of Gaussian noise and added them to the simulation signal. Then, we performed the GA-K-SVD algorithm on noise-added signal to calculate the average RRSM and SOR.

To visually demonstrate the noise reduction results of GA-K-SVD, Figure 10(c)–(e) shows the signal before and after denoising, and Figure 11 shows the average RRSM and SOR before and after denoising in different noise levels. The improvement effect of the GA-K-SVD algorithm on high-noise level signals is significant, but for low-noise level signals, the algorithm has little effect (of course, the significance of preprocessing low-noise signals is low).

Figure 10.

(a) Simulated under-sampled BTT signal without noise; (b) simulated under-sampled BTT frequency spectrum; (c) simulated noisy under-sampled BTT signal (SNR = 5); (d) simulated noisy under-sampled BTT frequency spectrum (SNR = 5); (e) simulated under-sampled BTT signal after denoising through the GA-K-SVD algorithm; (f) simulated under-sampled BTT frequency spectrum after denoising through the GA-K-SVD algorithm.

Figure 11.

RRMS of different noise levels.

We perform a Fourier transform on the original BTT signal, noisy BTT signal, and denoised BTT signal to observe the spectrums, shown in Figure 10(b), (d), and (f). The vibration modals of the BTT signal are completely preserved after applying the sparse representation transformation, and only the amplitude of each main frequency changes slightly. This shows that the GA-K-SVD algorithm can retain most of the vibration modal information of the blade while reducing the noise level, which is important for the reconstruction of BTT signals or the identification of vibration parameters. Then, we performed sparse reconstruction using the denoised BTT signal, as shown in Figure 12. Compared to Figure 7(c) and Figure 8(a), the optimized signal can correctly reconstruct all modals of the blade vibration in 0.116 s, and the efficiency and the reconstruction accuracy have been greatly improved. Therefore, the GA-K-SVD algorithm is also of great significance to the sparse reconstruction of BTT signal.

Figure 12.

Spectrum reconstruction result through OMP after applying the GA-K-SVD algorithm.

The selection of the $σ$ is also important. Figure 13 shows how different $σ$ affect the noise reduction effect. We notice that the noise reduction effect is low if the $σ$ is very small, while an excessively high $σ$ will even lead to negative optimization of the signal, making the signal more difficult to identify. According to equation (15), a small $σ$ causes more noise modals to be retained in the sparse vector, which leads to unhealthy samples. Thus, the trained dictionary can only filter out noise limitedly, resulting in the low effect of noise reduction; and a large $σ$ would probably cause that aliased blade vibration modals are filtered out along with the noise, resulting in a negative optimization. In this example, $σ$ takes a value near 100 which has the best noise reduction performance. However, the principle to choose $σ$ still needs to be studied urgently. In the future, we will focus on the setting of the threshold parameter $σ$ and determine the relationship among the data length, noise level, and the $σ$ through formula derivation.

Figure 13.

Effect of σ parameter on noise reduction performance of the GA-K-SVD algorithm.

Experience

We use BTT data obtained in a test rig of a certain type of aero-engine to further verify GA-K-SVD. In this experiment, the rotational speed was 3600 rpm, and five blades are installed evenly on the rotor. Five BTT sensors are installed on the casing unevenly, and their installation angles are {0°, 42.2°, 109°, 130.2°, 322.8°}. The experiment carried out a long-term test. For the No. 2 blade, 5-channel time pulsed signals were collected by each BTT probe, and the vibration displacement of the No. 2 blade was calculated according to equation (2). The test data is shown in Figure 14(a) with a length of 1000.

Figure 14.

(a) Experiment signal with Gaussian noise; (b) denoised experiment signal by GA-K-SVD.

The blade vibration frequency was estimated via the dynamic stress measurement and analysis of the blade, as shown in Table 3. Table 3 demonstrates that the vibration of the blade contains six modals with center frequencies of 157.3 Hz, 378.8 Hz, 577.3 Hz, 630.1 Hz, 743.3 Hz, and 997.4 Hz, and this is important information to verify the GA-K-SVD method.

Table 3.

Reconstruction results of blade modals.

	Number of modals	Dynamic stress analysis modals (Hz)	Noisy signal modals (Hz)	Denoised signal modals (Hz)
Modals	f ₁	157.3	204.5	157.3
	f ₂	378.8	281.3	378.8
	f ₃	577.3	563.5	577.3
	f ₄	630.1	630.1	630.1
	f ₅	743.3	743.3	743.3
	f ₆	997.4	997.4	997.4

The data preprocessed by GA-K-SVD is shown in Figure 14(b); the block size is 5×5, the dictionary size is 25×50, and σ = 100. Figure 15 shows the results of the OMP reconstruction of the noise-containing signal and the noise-reduced signal, where the modals under 100 Hz are probably caused by vibration of rotor, casing, and probes. It can be seen that the aliased modals of the BTT signal are preserved intact after denoising process. As shown in Figure 15(a), in the case of no denoising, the modal identifications at 157.3 Hz, 378.8 Hz, and 577 Hz have frequency deviation errors. As shown in Figure 15(b), the spectral reconstruction of the preprocessed data using GA-K-SVD can retain all aliased modals, while the preprocessing algorithm greatly reduces the number of iterations for the OMP algorithm to find the most relevant column. Meanwhile, the processing time is shortened from 17.81 s to 7.00 s, while GA-K-SVD only takes 2.07 s to complete the denoising. This verifies the preprocessing method proposed in this paper.

Figure 15.

(a) Reconstructed experiment blade vibration spectrum from noisy BTT signal; (b) reconstructed experiment blade vibration spectrum from denoised BTT signal.

Conclusion

This work proposes a novel method for denoising BTT signals with high noise level based on sparse representation. The method reduces Gaussian noise by creating sparse representations that can filter out noise components. The proposed algorithm GA-K-SVD can reduce noise in under-sampled BTT signals while preserving blade vibration modals. Simulations and experiments were performed to validate GA-K-SVD. The key procedures and results are summarized as follows:

(1) A novel noise reduction method based on sparse representation for BTT signals is proposed. Leveraging the sparsity of the BTT signal in the frequency spectrum, the main modals of the vibration are found via a sparse representation model. By removing small modals in the sparse vector that are close to zero and performing a sparse inverse transformation on the optimized sparse vector, we can obtain a denoised BTT signal.

(2) The Global Average method is proposed to improve denoising efficiency. An oversized sparse dictionary created by the large length of the BTT signal will decrease the denoising speed, while a small-sized sparse dictionary leads to the loss of the modals. To solve this problem, the method represents the large multi-channel BTT signal with a small sparse dictionary by splitting BTT signal into blocks with a sliding window. Finally, Global Average method splices each noise-reduced signal blocks together and eliminates overlaps to complete signal reconstruction and noise reduction. As an important part of GA-K-SVD, it is of great significance to improve the efficiency of the proposed algorithm.

(3) An over-complete sparse dictionary is built based on the improved K-SVD algorithm to increase the denoising effect. When applying a fixed Fourier-based dictionary, the sparsity of sparse vectors is not ideal and the aliased modals can be lost for aero-engine blades with diverse vibration modalities. Instead, the improved K-SVD algorithm can obtain an over-complete sparse dictionary with a better structural feature matching degree of the original signal. Main modals can be easily distinguished from small modals when sparse vectors of the signal blocks are found. Combining the Global Average method and the improved K-SVD algorithm, the efficient noise reduction of BTT signal can be realized.

(4) Simulation and experiment show that the GA-K-SVD algorithm has a great denoising effect on the high noise level of the BTT signal; it can preserve the aliased modals as well. Besides, by applying the GA-K-SVD algorithm, the efficiency and the reconstruction accuracy of a reconstruction algorithm, such as OMP, can be greatly improved for high-level noise scenarios.

Supplemental Material

Supplemental Material - A novel method for noise reduction of blade tip timing signals based on sparse representation and dictionary learning

Supplemental Material for A novel method for noise reduction of blade tip timing signals based on sparse representation and dictionary learning by Hua Zheng, Guanyu Fang, Shiqiang Duan, and Hang Li in Journal of Low Frequency Noise, Vibration, and Active Control.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hua Zheng

Supplemental Material

Supplemental material for this article is available online.

References

Zhang

Liu

Niu

. Nonlinear dynamics of rotating blades with variable cross-section. IOP Conference Series: Materials Science and Engineering 2019; 531(1): 012051.

Heath

Slater

Mansfield

, et al. Turbomachinery blade tip measurement technioues. AGARD Conference Proceedings 1998; 32: 1–32.

Ping

. Application review of blade tip timing method for foreign aeroengine blade vibration measurement. Avi-ation Science and Technology 2013; 6: 5–9.

Figaschewsky

Hanschke

Kühhorn

. Efficient generation of engine representative tip timing data based on a reduced order model for bladed rotors. American Society of Mechanical Engineers 2018; 51159: V07CT35A023.

Chen

Sheng

Xia

, et al. A comprehensive review on blade tip timing-based health monitoring: status and future. Mech Syst Signal Process 2021; 149: 107330.

Spada

Nicoletti

. Applying compressed sensing to blade tip timing data: A parametric analysis. Proceedings of the 10th international conference on rotor dynamics–IFToMM. Berlin, Germany: Springer International Publishing, 2019, pp. 121–134.

Chen

Sheng

Xia

. Multi-coset angular sampling-based compressed sensing of blade tip-timing vibration signals under variable speeds. Chin J Aeronaut 2021; 34(9): 83–93.

Sheng

Chen

Xia

, et al. Compressed sensing-based blade tip-timing vibration reconstruction under variable speeds. 2020 international conference on sensing, measurement & data analytics in the era of artificial intelligence (ICSMD), Xi'an, China, 15–17 October 2020.

Liu

Duan

Niu

, et al. Reconstruction of blade tip-timing signals based on the MUSIC algorithm. Mech Syst Signal Process 2022; 163: 108137.

10.

Chen

Sheng

Liu

, et al. Deep compressed sensing-based accurate blade vibration reconstruction using sub-sampled tip-timing measurements. Changzhou, China: Changzhou Institute of Technology, 2022.

11.

Lin

Chen

, et al. Sparse reconstruction of blade tip-timing signals for multi-mode blade vibration monitoring. Mech Syst Signal Process 2016; 81: 250–258.

12.

Chen

Huo

. Theoretical results on sparse representations of multiple-measurement vectors. IEEE Trans Signal Process 2006; 54(12): 4634–4643.

13.

Pan

Yang

Guan

, et al. Sparse representation based frequency detection and uncertainty reduction in blade tip timing measurement for multi-mode blade vibration monitoring. Sensors 2017; 17(8): 1745.

14.

Chen

Donoho

Saunders

. Atomic decomposition by basis pursuit. SIAM Rev 2001; 43(1): 129–159.

15.

Tenneti

Vaidyanathan

. iMUSIC: a family of MUSIC-like algorithms for integer period estimation. IEEE Trans Signal Process 2019; 67(2): 367–382.

16.

Wang

Yang

, et al. An improved multiple signal classification for nonuniform sampling in blade tip timing. IEEE Trans Instrum Meas 2020; 69(10): 7941–7952.

17.

Wang

Yang

, et al. Subspace Dimension Reduction for Faster Multiple Signal Classification in Blade Tip Timing. IEEE Trans Instrum Meas 2021; 70: 1–10.

18.

Wang

Yang

, et al. Robust sparse representation model for blade tip timing. J Sound Vib 2021; 500: 116028.

19.

Borsdorf

Raupach

Flohr

, et al. Wavelet based noise reduction in CT-images using correlation analysis. IEEE Trans Med Imag 2008; 27(12): 1685–1703.

20.

Maheshwari

Kumar

. Empirical mode decomposition: theory & applications. Int J Electron Eng 2014; 7: 873–878.

21.

Cao

Tian

Yang

, et al. Blade tip timing signal filtering method based on sampling aliasing frequency map. IEEE Trans Instrum Meas 2022; 71: 1–12.

22.

Mairal

Bach

Ponce

, et al. Non-local sparse models for image restoration. 2009 IEEE 12th international conference on computer vision, Kyoto, Japan, 29 September 2009.

23.

Zhang

Dong

Zhang

, et al. Two-stage image denoising by principal component analysis with local pixel grouping. Pattern Recogn 2010; 43(4): 1531–1549.

24.

Dong

Zhang

Shi

, et al. Nonlocally centralized sparse representation for image restoration. IEEE Trans Image Process 2013; 22(4): 1620–1630.

25.

Mahdaoui

Ouahabi

Moulay

. Image denoising using a compressive sensing approach based on regularization constraints. Sensors 2022; 22(6): 2199.

26.

Zhao

Yang

. Hyperspectral image denoising via sparse representation and low-rank constraint. IEEE Trans Geosci Rem Sens 2015; 53(1): 296–308.

27.

Zhang

Gao

Tao

, et al.

Multi-scale dictionary for single image super-resolution

. 2012 IEEE conference on computer vision and pattern recognition, Providence, RI, USA, 16–21 June 2012.

28.

Thapa

Raahemifar

Lakshminarayanan

. A new efficient dictionary and its implementation on retinal images. 2014 19th International Conference on Digital Signal Processing, Hong Kong, China, 20–23 August 2014.

29.

Saito

. Local Fourier dictionary: a natural tool for data analysis. SPIE 1999; 3813: 610–624.

30.

Dao

Griffin

. Compressed sensing of EEG with Gabor dictionary: effect of time and frequency resolution. 2018 40th annual international conference of the ieee engineering in medicine and biology society (EMBC), Honolulu, HI, USA, 18–21 July 2018.

31.

Bruckstein

Donoho

Elad

. From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev 2009; 51(1): 34–81.

32.

Duarte-Carvajalino

Sapiro

. Learning to sense sparse signals: simultaneous sensing matrix and sparsifying dictionary optimization. IEEE Trans Image Process 2009; 18(7): 1395–1408.

33.

Rubinstein

Zibulevsky

Elad

. Double sparsity: learning sparse dictionaries for sparse signal approximation. IEEE Trans Signal Process 2010; 58(3): 1553–1564.

34.

Yang

Wang

, et al. Effect of intranasal arginine vasopressin on human headache. Peptides 2012; 38(5): 100–104.

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