Block-wise recursive APES aided with frequency-squeezing postprocessing and the application in online analysis of vibration monitoring signals
Introduction
Over the course of the past few decades, a class of nonparametric adaptive filter-bank methods has been put forward and studied extensively, including the Capon method (CM) which was originally proposed for wave-number estimation in array signal processing by J. Capon in 1969 [1], [2], and the amplitude and phase estimation of a sinusoid (APES) method derived by J. Li and P. Stoica in 1996 by means of approximate maximum likelihood (AML) approach [3]. They have become powerful tools for signal analysis that improved the resolution limit of traditional Fourier transform (FT). Afterwards, P. Stoica et al. interpreted those spectral estimators as adaptive matched-filter banks [4], and proffered a relatively simple derivation of APES filter from pure narrow-band filter considerations [5]. In essence, both CM and FT could be regarded as special cases of APES [6]. Referring to [7], by making use of a higher order expansion technique, it was proved that CM underestimates the amplitude of spectrum in samples of practical length, whereas APES is unbiased resulting in more accurate amplitude identification results especially in the case of structural modal analysis. This offers a compelling reason for preferring APES rather than CM in the following.
In fact, APES enjoys three features that highly satisfy the requirements of signal analysis in online monitoring environment:
i. APES spectrum has narrower spectral peaks and lower side-lobes compared with the traditional implementation of FT, i.e., the fast Fourier transform (FFT).
ii. The computing frequency precision (spacing between adjacent spectral lines) and the computing frequency interval (the range of all spectral lines) can be arbitrarily assigned, while in FFT, they are fixed to (T denotes the sampling duration of the data sequence) in the case of without padding zeros and ( denotes the sampling frequency), respectively.
iii. High spectral resolution and accurate amplitude estimation are available but using much shorter sample length than FFT.
The feature i is well-known and has been proven in literature such as [3], [6]. However, there is almost no work revealing and emphasizing the features ii and iii, so a numerical example is used here to illustrate them. Take the signal expressed by Eq. (1) for consideration:where, represents a Gaussian additive noise with a standard deviation of 0.2. The signal is plotted in Fig. 1(a), and three data segments with lengths of 10s, 50s, and 100s are marked. The sampling frequency is 100Hz. Spectrums obtained by APES and FFT are shown in Fig. 1(b).
Because the scales of frequency-axis are fixed at Hz in FFT, generally the analytical signal length has to be long enough to ensure the computing precision of spectral estimation. For example, as shown in Fig. 1(b), in the case of s, FFT’s frequency-axis scale misses the exact positions 3.17Hz and 4.23Hz, so the second and third peaks of the spectrum look like they are cut off at the top, while the first peak reaches the true height because the frequency 2.32Hz happens to fall on the scale. Since actual frequencies are accurate to two decimal places, the analytical signal length must be at least 100s (without padding zeros) to enable FFT’s frequency-axis scale scans the modal frequencies. In contrast, the frequency-axis scale of APES can be set freely, such as Hz used herein, which doesn’t depend on the length of analytical signal. The above shows the feature ii of APES. In addition, from the results exhibited in Fig. 1(b), it’s evident to see that the spectral resolution of APES filter using 10s data is much higher than that of FFT using the same length data, and is between the resolution of Fourier spectrum estimated by using 50s and 100s data. This indicates APES’s feature iii.
Features ii and iii, that have not been explicitly concerned in published research, emphasized here will bring great benefits in analyzing vibration monitoring signals. Specifically, in practice, individuals don’t have to pay attention to all vibration modals, feature ii provides convenience for focusing on a certain or several interested modals through local spectral analysis and is conductive to saving computing resources. According to feature iii, APES could not only greatly reduce the delay of online analysis, but also reserve accurate amplitude information available for structural vibration modals analysis.
However, the direct calculation of APES spectrum is extremely time consuming. The high computational complexity make it hard to apply to online analysis. Consequently, many efforts have been made to explore efficient implementations of APES. One strategy is to utilize the Cholesky factorization or Gohberg-Semencul factorization of the inverse covariance matrix in combination with FFT to calculate corresponding polynomial coefficients in each batch re-evaluation [8], [9], [10], [11]. But as long as FFT arises in calculations, the sampling rate and the spacing of computing frequency points have to satisfy a certain condition, that is, the same default frequency points selection as FFT, thus the feature ii of APES mentioned above is cast away in fact. Another strategy is based on recursive implementation, which can be realized in two ways: one is based on the time-variant displacement structure of the data covariance matrix [11], [12], and the other is based on the matrix inverse lemma (MIL) to update spectral estimation [13], [14]. Although the recursive structure is compatible with the analysis of time-varying signals, historical studies are all verified in offline environment without testifying the real-time requirements of online computing. We desire that the analytical approach is competent to characterize non-stationary signals, and at the same time, it should guarantee immediate calculation avoiding data accumulation so that the managers and decision-makers are able to grasp the operating status of structures in real time, achieve timely risk warning, and take prompt emergency measures. For this purpose, the paper proposes a block-wise recursive APES (BRAPES) implemented in the form of matrix operation. Different from the manner of re-estimating APES’s spectrum as a single additional sample becomes available in literature [11], [12], [13], [14] that we name it point-based updating, BRAPES conducts re-estimate once after several samples forming a sampling (or updating) block with a certain length (assumed to be S), and the time threshold of online analysis is relaxed by S times.
The recursive structure enables BRAPES time-dependent, and due to its relatively concentrated spectral density, it can be regarded as a promising time–frequency (TF) analysis method analogized to the short-time Fourier transform (STFT) known as sectioned windowing FT. In terms of non-stationary signal processing, besides STFT, there are classical approaches such as wavelet transform (WT) and Winger-Ville distribution (WVD) [15], [16]. Due to the Heisenberg uncertainty principle, they suffer from poor resolution in TF representation. To date, massive efforts have been spent on improving the readability of spectrogram, and advanced post processing methods such as the reassignment method (RM) [17], [18] and the synchrosqueezing transform (SST) [19], [20] have been propelled to the forefront investigations of this respect recently. RM redistributes the coefficients of STFT on both the time-axis and frequency-axis according to the center of the local energy distribution of spectrogram. SST, which can be implemented based on the STFT framework or the WT framework, is a special case of RM by squeezing spectrogram only in the frequency direction [21], [22]. The strength of SST lies in two aspects, first, it allows reconstruction of the original signal, and second, its calculation efficiency is superior to RM because SST involves one-dimensional (1D) integration while RM involves two-dimensional (2D) integration [23]. As a powerful post-processing means that tremendously enhances the resolution of traditional TF analysis (TFA) methods, SST has been frequently accepted in many fields, such as machine fault diagnosis [24], [25], [26], [27], medicine [28], [29], [30], vibration signals [31], [32] and so on. Inspired by the concept of RM and SST, this paper develops a frequency squeezing postprocessing (FSP) technique for APES spectrum. The essence of FSP is to shift spectral lines toward the natural frequencies according to the spectral shape. Then, evenly spaced spectral lines are going to be redistributed non-uniformly, resulting in high density at the natural frequencies and fairly sparse elsewhere. During the post-processing, the amplitude value keeps unchanged to ensure the availability in structural modal analysis. Although FSP doesn’t enjoy solid mathematical foundation like RM, especially SST, it exhibits favorable performance of concentrating TF representation in practice.
In summary, this paper derives a block-wise recursive APES (BRAPES) method for high-resolution non-stationary signal analysis, and develops a frequency-squeezing postprocessing (FSP) technique to concentrate the TF representation obtained by BRAPES, and forms the complete algorithm named BRAPES-FSP. The block-like recursive structure loosens the time threshold of online analysis and is helpful to continuous computation. The introduction of FSP could further shorten the sampling length needed for spectral estimation, reducing the computing burden directly and enhancing the time sensitivity of the algorithm. Therefore, we realize the extension of APES filter for online spectral estimate of time-variant signals. Furthermore, the application potential of APES in mechanical and civil engineering communities has yet to be exploited, in spite of its increasing maturity in the domains of medical imaging, synthetic aperture radar imaging, etc. To the best of the authors’ knowledge, to date, the researches of APES in civil engineering realm are only seen in [33], [34], [35]. This paper effectively broadens the application scenarios of APES, indicating its strength in processing structural vibration monitoring signals.
The overall structure of the study takes the form of six sections, including this introductory section. Section 2 begins by a brief review of APES and looks at its matrix realization. Section 3 is concerned with the derivation of BRAPES algorithm and the designation of FSP technique in details. Then, a numerical validation on the proposed BRAPES-FSP is examined in Section 4. Further, experimental investigations on the cable acceleration data from a cable-stayed bridge and from a cable vibration test are carried out in Section 5. Finally, Section 6 provides a summary of the full text.
Section snippets
A retrospect of APES and its matrix form
APES can be derived from different perspectives [3], [4], [5]. First, this section takes a short revisit to APES on the account of the matched filter-bank interpretation to make the paper as self-contained as possible, though it has been a familiar topic to relative scholars. Then, we introduce the matrix implementation of APES.
Block-wise recursive APES aided with frequency-squeezing postprocessing (BRAPES-FSP)
Last section gives the matrix expression of the amplitude spectrum at time t, which is also the initialization step of recursive APES. On this ground, this section is going to construct the block-wise recursive structure of APES and detailedly introduce the postprocessing technology characterized by frequency squeezing. At the end of this section, the computing framework of the whole BRAPES-FSP is addressed.
Performance analysis
A synthetic time-varying signal is employed to compare the performance of proposed BRAPES-FSP with STFT and SST. The signal with a sampling frequency of 100Hz and a duration of 50s is defined as follows:which consists of two amplitude-modulated (AM) and frequency-modulated (FM) components , two pure FM components and a Gaussian noise. The signal-to-noise ratio (SNR) equals to:
Cable monitoring signals of Yongjiang Bridge
Yongjiang Bridge is a prestressed concrete single-tower cable-stayed bridge with a span of m, which adopts the fixed pier-pylon-beam system. Two cable planes are arranged in a harp style with a cable spacing distance of 8m and a total of 44 cables along the bridge. Cables on the outside of each plane are equipped with full-length LED lamps, and cables on the inside are naked. A total of 24 cables are monitored and the positions of them are indicated in Fig. 16. Among them, 20 acceleration
Conclusion
In this paper, we firstly construct the block-wise recursive APES (BRAPES) implemented in matrix, which endows conventional APES the ability to deal with non-stationary signals. And then we develop the frequency-squeezing postprocessing (FSP) technique that immensely concentrates the energy of spectrogram obtained by BRAPES, making the TF representation more readable. Finally, BRAPES for spectral analysis and FSP for post-treatment are integrated forming a complete online solution naturally,
CRediT authorship contribution statement
Xuewen Yu: Methodology, Methodology, Investigation, Writing - original draft, Writing - review & editing, Visualization. Danhui Dan: Conceptualization, Validation, Resources, Writing - review & editing, Supervision, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work is supported by the Opening Foundation of National Key Laboratory of Bridge Structural Health and Safety; the National Nature Science Foundation of China (Grant No. 5187849); the National Key R&D Program of China (2017YFF0205605); Shanghai Urban Construction Design Research Institute Project ”Bridge Safe Operation Big Data Acquisition Technology and Structure Monitoring System Research”.
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