Shock and Vibration

Volume 2017 (2017), Article ID 7286946, 12 pages

https://doi.org/10.1155/2017/7286946

## A Generalized Demodulation and Hilbert Transform Based Signal Decomposition Method

Department of Civil Engineering, Hefei University of Technology, Hefei, Anhui Province 23009, China

Correspondence should be addressed to Wei-Xin Ren

Received 9 January 2017; Revised 16 March 2017; Accepted 10 April 2017; Published 31 May 2017

Academic Editor: Pedro Galvín

Copyright © 2017 Zhi-Xiang Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes a new signal decomposition method that aims to decompose a multicomponent signal into monocomponent signal. The main procedure is to extract the components with frequencies higher than a given bisecting frequency by three steps: (1) the generalized demodulation is used to project the components with lower frequencies onto negative frequency domain, (2) the Hilbert transform is performed to eliminate the negative frequency components, and (3) the inverse generalized demodulation is used to obtain the signal which contains components with higher frequencies only. By running the procedure recursively, all monocomponent signals can be extracted efficiently. A comprehensive derivation of the decomposition method is provided. The validity of the proposed method has been demonstrated by extensive numerical analysis. The proposed method is also applied to decompose the dynamic strain signal of a cable-stayed bridge and the echolocation signal of a bat.

#### 1. Introduction

Vibration and sound signals contain intrinsic information of dynamic systems. The famous Fourier analysis can be used to project the signal into the frequency domain and identify natural frequencies of linear time-invariant systems. However, the Fourier analysis fails to study time-varying or nonlinear systems due to the nonstationarity of the signals. Therefore, numerous time-frequency analysis methods have been proposed seeking to solve this problem. The time-frequency analysis methods can be roughly classified into two categories: energy distribution and signal decomposition.

As one of the most representative methods of the energy distribution category, the wavelet transform (WT) is essentially an adjustable window Fourier spectral analysis method. With the aid of the WT, Ruzzene et al. identified natural frequencies and damping with real world data from a bridge, and Wang et al. identified instantaneous frequency (IF) of time-varying structures [1, 2]. Although the WT method has many successful engineering applications, it is difficult to achieve high resolutions in time and frequency domains simultaneously due to the Heisenberg-Gabor uncertainty principle [3]. Notwithstanding, the WT is a powerful tool for nonstationary signals in the time-frequency domain and has motivated many analogous time-frequency energy distributions such as the transform, the chirplet transform, and the synchrosqueezed wavelet transforms [4–6]. The synchrosqueezed wavelet transforms developed by Daubechies et al. are a new time-frequency analysis tool with a special reassignment method [6]. It can offer better time-frequency resolution than many other methods, and its successful applications in dynamic signal reconstruction and gearbox fault diagnosis and so forth can be found in [7–9]. However, versatile as these energy distribution category methods are, the main problem is their nonadaptive nature, since these methods utilize a family of preselected oscillatory bases to represent signals. In spite of this, the WT and other methods of energy distribution category are still important for nonstationary signal processing. Therefore, we will use the WT method in this paper to preprocess the signal for subsequent decomposition.

Empirical mode decomposition (EMD) proposed by Huang et al. in 1998 has become a representative signal decomposition method [10]. The EMD can decompose a multicomponent signal into intrinsic mode functions whose amplitude and IF can be demodulated by Hilbert transform. Due to its adaptivity, the EMD has received increasing attentions in the field of signal processing and been applied in a broad domain such as vibration signal analysis, acoustic signal analysis, and geophysical studies [11–13]. Similar to EMD, the local mean decomposition (LMD) proposed by Smith decomposes signals into a set of functions, each of which is the product of an amplitude and a pure frequency modulation signal. The LMD method has been used for electroencephalogram (EEG) analyzing [14]. However, as semiempirical methods, EMD and LMD are heuristic in nature and lack solid mathematical foundation [8]. Huang and Wu also pointed out that the Hilbert transform of the intrinsic mode functions may contain error if Bedrosian’s theorem on Hilbert transform of product functions is not established [13, 15].

Feldman introduced a very simple signal decomposition method called Hilbert vibration decomposition (HVD), which decomposes an initial signal into a sum of components with slow varying instantaneous amplitudes and frequencies [16, 17]. Gianfelici et al. introduced an iterated Hilbert transform (IHT) method to obtain slow varying amplitude and its corresponding oscillatory signal by filtering and implement the method iteratively to the residue [18]. Qin et al. have successfully utilized the IHT method for mechanical fault diagnosis [19]. The idea to decompose a multicomponent signal into monotones is very useful and deserves further study.

More recently, Chen and Wang developed a new signal decomposition method named analytical mode decomposition (AMD) [20]. The AMD method is an efficient and accurate method that separates a signal into two parts below and above the bisecting frequency [21]. Wang et al. successfully applied the AMD method to many structural vibration signal decomposition cases for modal parameters identification [7, 22, 23]. However, an error that cannot be neglected arises when the AMD method is applied for processing discrete signals [24]. The reason of the error is that the AMD method involves multiplication of signal and makes the frequencies of some components of the signal exceed the Nyquist frequency [25]. An improved multistep AMD, or an interpolation of the discrete signal, can be adopted to reduce the error [24], but the computation cost is increased significantly.

In this study, we introduce a generalized demodulation and Hilbert transform (GDHT) based signal decomposition method, which possesses the capacity of AMD but avoids computational error. The generalized demodulation is first developed by Olhede and Walden aiming to track the time-dependent frequency content of each component in a multicomponent signal [26, 27]. Using generalized demodulation, monocomponent signals with curved IF profile can be converted to another analytical signal with a constant frequency, which is very useful for enhancing time-frequency representation [8, 9]. With this in mind, components with lower frequencies are projected onto negative frequency domain so that they can be eliminated by Hilbert transform. And an inverse generalized demodulation is conducted to restore the components with higher frequencies. This procedure operates like a high-pass signal filter and can be used to extract recursively all monocomponent signals in a multicomponent signal. In the next section, the generalized demodulation theory is introduced. In Section 3, a comprehensive derivation of the decomposition method is provided. Finally, the proposed method is validated by numerical analysis and applied to practical cases such as vibration signal filtering and echolocation signal decomposition.

#### 2. Generalized Demodulation

Consider a monocomponent signal expressed aswhere and are the amplitude and the IF of , respectively. Define the quadrature signal of asWith this definition, a complex signal can be formed asThe generalized demodulation of the signal is achieved by multiplying it with a mapping function , which givesIf a proper phase makes the signal become a component with constant frequency , that is, , the IF of the original signal can be obtained byConversely, the inverse generalized demodulation recovers the original signal by multiplying the signal with the conjugate of the mapping function; that is, , which restores the original signalThe above six equations are exactly rigorous formulas so far. In practice, however, since the phase of the signal is unknown, the Hilbert transform is always used to obtain a substitution for the complex signal . The complex signal defined by Hilbert transform is given bywhere represents the Hilbert transform of the signal .

It should be noted that substituting by implies that the Bedrosian identity is established and is an analytic signal [15], so that the signal satisfiesThis condition can be well satisfied in signals where the amplitudes and the instantaneous frequencies (IFs) are slow varying functions. Otherwise, only approximated results will be obtained if the signals contain abruptly changes caused by sudden events (such as a brittle fracture of a structural component).

#### 3. Signal Decomposition Method

In the following content, the multicomponent signal is investigated, which is defined bywhere and are the amplitude and the IF of the th component , respectively. In many practical applications, the amplitude and the IF of the signal components are always slow varying functions. The multicomponent signal is said to be well separated if the Fourier transform of each amplitude can be neglected for and the IFs satisfy This relationship of the th IF and the th IF is illustrated in Figure 1. Thus, the phase and bisecting frequency of the mapping function can be chosen asGiven the bisecting frequency the signal can be decomposed into two parts by 3 steps.