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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/).

In structural vibration tests, one of the main factors which disturb the reliability and accuracy of the results are the noise signals encountered. To overcome this deficiency, this paper presents a discrete wavelet transform (DWT) approach to denoise the measured signals. The denoising performance of DWT is discussed by several processing parameters, including the type of wavelet, decomposition level, thresholding method, and threshold selection rules. To overcome the disadvantages of the traditional hard- and soft-thresholding methods, an improved thresholding technique called the sigmoid function-based thresholding scheme is presented. The procedure is validated by using four benchmarks signals with three degrees of degradation as well as a real measured signal obtained from a three-story reinforced concrete scale model shaking table experiment. The performance of the proposed method is evaluated by computing the signal-to-noise ratio (SNR) and the root-mean-square error (RMSE) after denoising. Results reveal that the proposed method offers superior performance than the traditional methods no matter whether the signals have heavy or light noises embedded.

Vibration-based structural damage detection methods have attracted considerable attention in recent years for the assessment of health and safety of large civil structures [

Up to now, a large number of signal denoising algorithms have been proposed in the literature, such as the linear low-pass filter [

This paper is concerned with the rational selection of processing parameters in the wavelet based filtering method, and is especially relevant to a novel thresholding method, which generalizes the hard- and soft-shrinkage proposed by Donoho and Johnstone [

Considering a signal _{n}^{2}(_{n}_{j,k}_{jk}_{J,k}_{J,k}_{l,k}

The basis for the above decomposition is formed from the mother wavelet

Broadly speaking, the DWT performs a recursive decomposition of the lower frequency band obtained from a previous decomposition. As a result, a hierarchical set of approximations and details can be obtained through the various decomposition levels. This procedure, known as the multi-resolution analysis (MRA), was introduced by Mallat and it can be carried out using a computationally efficient algorithm [_{n}_{n}_{1} and a detail _{1}. Then the approximation _{1} is also split into a second level approximation _{2} and _{2}.

Suppose the measured signal _{i}_{i}_{i}_{i}_{i}_{i}_{i}

In general, the wavelet based denoising procedure can be described as follows:

Step (1) Decomposition of the input noisy signal into several levels of approximations and detailed coefficients, using the selected wavelet basis;

Step (2) Thresholding of coefficients. Extract the coefficients actually containing the true signal and discard the others;

Step (3) Reconstruction of the signal using approximations and detailed coefficients by means of the inverse wavelet transform.

The methodology can be applied to the DWT and stationary wavelet transform (SWT), as well as to the wavelet packet transform (WPT). In the present paper, we only focus on the DWT.

The choice of the mother wavelet for the aforementioned denoising procedure is theoretically arbitrary, but it is critical and important in practice, because it affects the performance of the technique. The properly selected wavelet can give rise to an orthogonal/non-orthogonal analysis; allow a fast algorithm, existence of discrete transforms, possibility of perfect reconstruction, good localization in time and/or in frequency,

Since a quantitative criterion for selecting the attenuation factor is not available, in this paper we restricted ourselves to the Daubechies wavelet by trial and error. In all our experiments listed below, the Daubechies 4 (db4) wavelet was found to perform better in preserving fine signal details. As shown in

The second factor that strongly influences the signal denoising is the level of DWT decomposition. If the level of decomposition is not enough, the improvement of the signal to noise ratio (SNR) will be limited. On the other hand, if the decomposition level is too much, the amount of computation will increase largely, and the noise reduction may be not satisfactory as well. No perfect procedure for choosing the best level of DWT decomposition has yet been proposed, here, a kind of “white noise test” is adopted (Zhang

As stated before, in the DWT a measured signal is broken down into two sub-signals, a detail and an approximate, then the detail sub-signal in turn is broken down into two other sub-signals, respectively, forming a binary tree. Since the amplitude and arrival time of every noise peak is random, this will cause a disturbance that affects the signal or masks the objective signal completely (

Step (1) Decomposition of the input noisy signal into _{1} and detailed coefficient _{1};

Step (2) Retain the approximation coefficient _{1} obtained by Step (1) and perform the white noise test on the detailed coefficient _{1}. Continue to take the decomposition of _{1} if _{1} passes the white noise test;

Step (3) Repeat the above steps,

Step (4) Abandon the last decomposition result in which the detailed coefficient hasn't passed the white noise test,

The purpose of the thresholding procedure is to eliminate or suppress small value wavelet coefficients which mainly represent the noise content. This process can be carried out following two basic rules: namely the ‘keep-or-kill’ hard thresholding and ‘shrink-or-kill’ soft thresholding introduced by Donoho and Johnstone [

The hard thresholding method is defined as follows:

Where ω_{k,j}_{k,j}

The other is called the soft thresholding:
_{k,j}

Ideal thresholding functions retain or shrink only wavelet coefficients exceeding a threshold value

The sigmoid function is so-called because it is shaped like one form of the Greek letter sigma. In general, a sigmoid function is real-valued and differentiable, having either a non-negative or non-positive first derivative which is bell shaped. The degree of approximation of the sigmoid function to the signum function can be adjusted by regulating the value of

Thereby, the modified sigmoid function is very suitable as a thresholding scheme which can be defined as follows:

It can be easily proved that the presented thresholding scheme is a kind of compromise between hard and soft thresholding, where the difference is caused by the constant

Although the presented thresholding scheme is simple and effective, selection of a rational threshold value is a crucial task as it directly affects the denoising results. For example, choosing a very large threshold will shrink almost all the coefficients to zero and may result in over smoothing of the measured signals. On the other hand, a small value of threshold will retain the sharp edges and details but may fail to suppress the noise artifacts. Among the existing methods, the most popular one is the universal threshold [

The universal threshold may be unwarrantedly large since its dependence on the number of samples. It will yield an overly smoothed estimate and in some cases, the pseudo Gibbs phenomena may appear [

The proposed denoising procedure is summarized in

In order to verify the performance of the proposed denoising approach, computer generated noises with variable amplitudes are added to well-known benchmark signals; moreover, the classical algorithms are performed for comparison. A number of quantitative parameters can be used to evaluate the performance of the denoising procedure in terms of the reconstructed signal quality. In this case, the following parameters are compared:

Signal-to-noise ratio (SNR):

Root-mean-square error (RMSE):

A simulation experiment is conducted to investigate the performance of the proposed filtering procedures to different signals and to various degradations. For comparison, the simulation involves four known benchmark signals (Block, Bumps, Heavy sine, and Doppler) and three kinds of degrees of degradation. The SNR of the selected signals is 2.0000 dB (heavy blurred case), 5.0000 dB (medium blurred case) and 10.0000 dB (light blurred case) for each benchmark. The computations are performed with the MATLAB for Windows version 7.4 (MathWorks, Natick, MA, USA) [

The detailed results of the three thresholding methods are illustrated in

The simulation results have shown the good performance of the proposed method for improving the signal and suppressing the noise. In order to verify its ability in an actual vibration experiment, a shaking table test was carried out for this study.

The test was conducted at the State Key Laboratory of Coastal and Offshore Engineering in Dalian University of Technology, China [

The measured and denoised acceleration response at location 2 (center of mass) of every story in the

The first three orders of mode shapes of the model before and after filtering are displayed in

How to remove the noise interference is a critical issue in vibration experiments. Especially in some tests there are not any prior knowledge of the noise, the conventional filters are not suitable. In the present work, this issue is tackled by a DWT-based denoising approach. To overcome the disadvantages of the traditional hard- and soft-thresholding method, an improved thresholding technique called the sigmoid function-based thresholding scheme is presented. Numerical and experimental results are presented to evaluate the performance of the proposed method, and the results have shown that the method is extremely efficient in eliminating noise, no matter whether the SNR is high or low. The sigmoid function-based thresholding technique can remove small coefficients and shrink large coefficients using its non-linear characteristics to reproduce peaks and discontinuities as accurately as possible, without sacrificing visual smoothness. In short, the proposed method is simple, efficient and could be easily implemented by hardware or software, which is especially suitable in vibration tests.

This research work was jointly supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 51121005), the National Natural Science Foundation of China (Grant No. 51178083), the Program for New Century Excellent Talents in University (Grant No. NCET-10-0287), and the Natural Science Foundation of Liaoning Province of China (Grant No. 201102030).

A two-level wavelet decomposition of signal.

Sketch map of the db4 wavelet.

Plot of the modified sigmoid function.

The comparison of soft, hard and sigmoid function based thresholding scheme.

Flowchart of the proposed DWT denoising method.

Benchmark signals denoising by db4 wavelet using different thresholding schemes (SNR = 2.0000), (

Benchmark signals denoising by db4 wavelet using different thresholding schemes (SNR=10.0000), (

Experimental setup for shaking table tests.

Schematic diagram of accelerator location.

Measured and denoised acceleration response.

PSD of the signals before and after filtering.

The first three orders of mode shapes of the model.

Performance evaluation of benchmark signals denoising with different thresholding schemes.

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SNR | RMSE | SNR | RMSE | SNR | RMSE | SNR | RMSE | |

Blocks | 2.0000 | 0.9826 | 12.8601 | 0.2814 | 13.0078 | 0.2767 | 13.0926 | 0.2740 |

5.0000 | 0.9826 | 14.5781 | 0.3262 | 14.7936 | 0.3182 | 14.8078 | 0.3177 | |

10.0000 | 0.9826 | 17.5151 | 0.4136 | 17.9304 | 0.3943 | 17.9304 | 0.3943 | |

Bumps | 2.0000 | 0.9826 | 13.6467 | 0.2571 | 13.6467 | 0.2571 | 13.6467 | 0.2571 |

5.0000 | 0.9826 | 15.9711 | 0.2779 | 16.0294 | 0.2760 | 16.1072 | 0.2735 | |

10.0000 | 0.9826 | 19.3004 | 0.3368 | 19.6685 | 0.3228 | 20.0682 | 0.3083 | |

Heavy sine | 2.0000 | 0.9826 | 18.9648 | 0.1393 | 18.9648 | 0.1393 | 18.9648 | 0.1393 |

5.0000 | 0.9826 | 21.1270 | 0.1535 | 21.1270 | 0.1535 | 21.1270 | 0.1535 | |

10.0000 | 0.9826 | 23.9296 | 0.1976 | 23.9296 | 0.1976 | 23.9480 | 0.1972 | |

Doppler | 2.0000 | 0.9826 | 12.4548 | 0.2949 | 13.0133 | 0.2765 | 13.4134 | 0.2641 |

5.0000 | 0.9826 | 14.0443 | 0.3469 | 14.7786 | 0.3187 | 14.8974 | 0.3144 | |

10.0000 | 0.9826 | 17.7133 | 0.4043 | 19.3587 | 0.3345 | 19.6503 | 0.3235 |

Experimental results before and after filtering.

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Before | 0.45 | −0.39 | 0.11 | 5.43 | 17.70 | 21.36 | 0.36 | 4.57 | 6.01 |

After | 0.39 | −0.37 | 0.10 | 5.43 | 17.70 | 21.36 |