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TECHNICAL PAPERS

An Adaptive Time–Frequency Representation and its Fast Implementation

[+] Author and Article Information
Bao Liu

Department of Mathematics, West Virginia University, Morgantown, WV 26506bliu@math.wvu.edu

Sherman Riemenschneider

Department of Mathematics, West Virginia University, Morgantown, WV 26506

Zuowei Shen

Department of Mathematics, National University of Singapore, Singapore 117543, Singapore

J. Vib. Acoust 129(2), 169-178 (Sep 19, 2006) (10 pages) doi:10.1115/1.2424969 History: Received December 03, 2004; Revised September 19, 2006

This paper presents a fast adaptive time–frequency analysis method for dealing with the signals consisting of stationary components and transients, which are encountered very often in practice. It is developed based on the short-time Fourier transform but the window bandwidth varies along frequency adaptively. The method therefore behaves more like an adaptive continuous wavelet transform. We use B-splines as the window functions, which have near optimal time–frequency localization, and derive a fast algorithm for adaptive time–frequency representation. The method is applied to the analysis of vibration signals collected from rotating machines with incipient localized defects. The results show that it performs obviously better than the short-time Fourier transform, continuous wavelet transform, and several other most studied time–frequency analysis techniques for the given task.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 9

The top plot shows the bearing vibration signal and the bottom plot, its Fourier transform

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Figure 8

Time–frequency planes of the signal in Fig. 7 obtained using different methods: (a) short-time Fourier transform with a window of length 0.01s; (b) short-time Fourier transform with a window of length 0.12s; (c) continuous wavelet transform; (d) Wigner–Ville distribution; (e) Choi–Williams distribution; (f) cone-shaped kernel distribution; (g) Jones and Baraniuk’s adaptive STFT; and (h) the proposed adaptive time–frequency representation

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Figure 10

Time–frequency planes of the signal in Fig. 9 obtained using different methods: (a) short-time Fourier transform with a window of length 0.1ms; (b) continuous wavelet transform; (c) Choi–Williams distribution; and (d) the proposed method

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Figure 1

Comparison between cubic B-spline (dashed line) and Gaussian (solid line)

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Figure 2

The top plot shows the simulated signal and the bottom plot, its Fourier transform

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Figure 3

Time–frequency planes of the signal in Fig. 2 obtained using different methods: (a) short-time Fourier transform with a window of length 0.018s; (b) short-time Fourier transform with a window of length 0.5s; (c) continuous wavelet transform; (d) Wigner–Ville distribution; (e) Choi–Williams distribution; (f) cone-shaped kernel distribution; (g) Jones and Baraniuk’s adaptive STFT; and (h) the proposed adaptive time-frequency representation

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Figure 4

(a) Zoomed-in representation of Fig. 3; and (b) zoomed-in representation of Fig. 3

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Figure 5

(a) Window length as a function of time selected with Jones and Baraniuk’s adaptive STFT; and (b) window length as a function of frequency selected with the proposed method

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Figure 6

(a) Zoomed-in time–frequency representation obtained using the proposed method with the window length sequence smoothed; and (b) smoothed window length as a function of frequency

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Figure 7

The top plot shows the gearbox vibration signal and the bottom plot, its Fourier transform

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