# Instantaneous Properties Of Nonlinear And Non Stationary Signals Biology Essay

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This paper describes a comparison of several time-frequency analysis methods for estimating and characterising the instantaneous properties of nonlinear and non-stationary signals. The aim of this research is to establish an efficient method for signal demodulation which will be used to extract instantaneous characteristic from nonlinear signals. The Hilbert Transform, Hilbert Huang Transform and Wavelet Analysis are employed and compared to find the most effective techniques specifically developed for analysing instantaneous frequency and amplitude. Hence, local energy density of the signals in time-frequency domain can be interpreted. Several numerical simulated signals were used to validate and compare the capabilities and performances of the methods. These signals which have different composition of basic sinusoidal waveform were tested on each of the aforementioned method and the results in terms of its instantaneous properties have been evaluated.

## Keywords: Hilbert transform, Hilbert Huang transform, Wavelet transform, Instantaneous frequency

## 1. Introduction

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Traditional method of characterizing signal properties is basically based on Fourier transform. By analysing its spectrum, energy distribution of the signal can be determined. However these methods rely on the assumption of linearity and assume that the data are strictly periodic or stationary in time which limits their applicability to real problems. In addition, Fourier spectrum defines uniform harmonic components globally and therefore needs many additional harmonic components to simulate nonstationary data. The idea of using Time-Frequency Signal Analysis (TFSA) for nonlinear and non-stationary signals has widely been practiced by many researchers [1, 2, 3]. The TFSA of a signal has been used to show the energy distribution and frequency variation of the signal over time. It has been proven to localize individual components of a multicomponent signal.

Interpreting signals by its Instantaneous Frequency (IF) is very popular approach recently and has been discussed in many research areas [4]. More reviews can be found in Boashash [5, 6]. IF can be considered as the most intuitive concept in TFSA. It can provide new or additional information about the local content of the signal in time-frequency domain. The interpretation of instantaneous frequency of a real signal, defined as the derivative of the phase of a complex signal representation was first given by Gabor [19] and Ville [20], has been a subject of investigation and debate for years. This interpretation has been argued extensively in which it may be physically appropriate only for monocomponent signals, where there is only one spectral component or a narrow range of frequencies varying as a function of time. Another interpretation of the IF comes from the TFSA point of view, where the IF of a signal is the weighted average frequency at each time in the signal. The main paradox is that the IF often ranges beyond the spectral support of many signals and it is generally difficult to use in practice because there is no systematic and general method for determining the individual properties of a signal, which is itself a challenging problem. The principle of computing instantaneous frequency of multicomponent sinusoidal model for an arbitrary signal by TFSA techniques has been widely discussed in [7, 8, 9, 10]. Some deep and detailed discussion about IF can be found in literatures [13, 14].

Hilbert Huang Transform (HHT) method, proposed by Huang [11] used Empirical Mode Decomposition (EMD) to decompose signals adaptively and is applicable to nonlinear and non-stationary data. Fundamental theory on nonlinear time series and the detailed discussion and justification can be found in [12]. The recent improved HHT method, Ensemble EMD (EEMD) has also been highlighted [14, 15]. Finally, the Wavelet Transform (WT) has been proposed to interpret signals. A complete description of the time-frequency and time-scale analyses can be found in [16]. A theoretical treatment of wavelet analysis is given in [17]. Previous work in comparing the performance of WT and HHT was performed by Kijewski and Kareem [18].

## 2. Methodology for Signal Analysis - Theoretical Background and Algorithms

Most of the real world signals such as human speech, communication signals and vibration in machines are nonlinear, non stationary and stochastic. Such signals may have a distinct average spectral structure that reveals important information such as for speech recognition or early detection of damage in machinery. The traditional signal processing technique, Fourier Transform is dealing with deterministic process where the signals are linear or stationary. This classical statistical spectrum analysis approach (periodogram and auto-correlation) is used to estimate the power spectral density (PSD) across frequency of the stochastic signal. Spectrum analysis of any single block of data using window-based deterministic spectrum analysis, however, produces a random spectrum that may be difficult to interpret. It has been prove in many researches that for any applied method of nonlinear signal have to be adaptive and involve finite length of data.

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2.1 Hilbert Transform

The Hilbert transform (HT) analysis provides a method for determining an explicit expression of the IF of the signal and to obtain the analytic signal. For a given real valued signal x(t), the Hilbert transform of a function x(t) is defined as

(1)

where H{x(t)} is the Hilbert transform of x(t) and τ=variable of integration. The Cauchy principle value of the integral is used in equation (1). Given x(t) and H{x(t)}, a complex analytic signal z(t) can be defined as,

z(t) = x(t) + j H{x(t)} (2)

which can be expressed as,

z(t) = x(t) + j H{x(t)}= A(t)ejφ(t) (3)

where A(t) is the time-varying amplitude or the envelope of z(t) given as

A(t) = â”‚x(t) + j H{x(t)}â”‚ (4)

and φ(t) is the phase of z(t) given as

φ(t) = tan-1 H{x(t)}/x(t) (5)

By verifying the orthogonality of signal H{x(t) of x(t), the instantaneous frequency, ω(t) of z(t) can be obtained by

ω(t) =d φ(t)/dt (6)

where φ(t) is the continuous, unwrapped phase of the signal.

2.2 Hilbert Huang Transform (EMD)

The HHT basically involve two basic process, Empirical Mode Decomposition (EMD) and Hilbert spectral analysis. EMD or simply known as the sifting process, was proposed by Huang[12, 13], is a process of decomposing any nonstationary and nonlinear physical signals adaptively into a set of different simple intrinsic modes of oscillations called an Intrinsic Mode Function (IMF). The decomposition is based on the direct extraction of the energy associated with various intrinsic time scales in the physical signal. Each intrinsic mode, which represents a simple oscillation, can be used to obtain the instantaneous frequency of the signal by applying the Hilbert transform to the IMF component. The following conditions must be fulfilled in order to successfully compute the IF without losing any physical meaning.

In the whole dataset, the number of extrema and the number of zero-crossings must either equal or differ at most by one, and

At any point, the mean value of the envelope deï¬ned by the local maxima and the envelope deï¬ned by the local minima is zero.

The two conditions are necessary to ensure that an IMF is a nearly periodic function and the mean is set to zero. The instantaneous frequency derived from the IMF's through the Hilbert transform give a full energy distribution of the data in time-frequency domain which is designated as the Hilbert Spectrum. An analytic signal of each simulation signals are derived through HT and the local energy concentration/distribution were represented on the time-frequency domain. In the sifting process, the first component of IMF, contains the finest scale (or the shortest period component) of the time series. The residue after extracting IMF contains longer period variations in the data. Therefore, the modes are extracted from high frequency to low frequency. Thus, EMD can be used as a filter to separate high frequency (fluctuating process) and low frequency (slowing varying component) modes.

The decomposition process of EMD is described as follows:

i) Extract all the local maxima and minima of the time series signal x(t);

ii) Generate the upper and lower envelopes, emin(t) and emax(t), by cubic spline lines

interpolation.

iii) Calculate the mean m(t) of the upper and lower envelopes:

m(t) = (emin(t) + emax(t))/2

iv) Obtain the first component c1(t) by subtracting the mean m(t) from the original signal:

c1(t) = x(t) − m(t)

v) Check the status of c1(t),denote ci(t) as the ith IMF and replace x(t) with the residual if it

is an IMF.

r(t) = x(t) − ci(t)

vi) Repeat all the above steps until a predefined condition of standard deviant between consecutive components is met if it is not an IMF. The residual satisfies some stopping criterion where r(t) becomes monotonic or ci(t) or r(t) has too small effect.

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Examples of our work2.3 Hilbert Huang Transform (EEMD)

EMD has proved to be quite versatile in a broad range of applications for extracting nonlinear and non-stationary signals. However, the EMD sometimes cannot reveal signal characteristics accurately because of the mode mixing problem. Mode mixing is defined as a single IMF either consisting of signals of widely disparate scales, or a signal of a similar scale residing in different IMF components. When the mode mixing problem occurs, an IMF may wrongly be interpreted. To overcome the problem, Wu and Huang [13] have developed a new noise-assisted data analysis method called Ensemble Empirical Mode Decomposition (EEMD). The basic idea of EEMD is to define the true IMF components as the mean of an ensemble of trials. Each trial consists of the decomposition results of the signal with added white noise of finite amplitude. The effect of the added white noise is to provide a uniform reference frame in the time-frequency space so that signals of different scale are projected to the proper IMF.

This new approach utilizes the full advantage of the statistical characteristics of white noise to perturb the signal in its true solution neighbourhood [13], and to cancel itself out after serving its purpose via ensemble averaging. Hence, it significantly reduces the effect of mode mixing and represents a substantial improvement over the original EMD.

The EEMD algorithm is described as follows:

i) Initialize the number of ensembles, M, the amplitude of the added white noise, and k = 1.

ii) Perform the kth trial on the white noise-added signal.

Add a white noise series with the given amplitude to the original signal

Xk(t) = x(t) + nk(t)

where nk(t) indicates the kth added white noise series, and xk(t) represents the

noise added signal of the kth trial.

Decompose the noise-added signal, xk(t), into I IMFs, ci,k (i = 1, 2, . . . , I ), using the EMD method, where ci,k denotes the ith IMF of the kth trial, and I is the number of IMFs.

If k < M, then go to step (i) with k = k + 1.

Repeat steps (i) and (ii) iteratively, but with different white noise series each time.

iii) Calculate the ensemble mean, mi, of the M trials for each IMF

mi =1/M ΣM, k=1 ci,k, i = 1, 2,...., I , k = 1, 2,...,M

iv) Let the mean, mi (i = 1, 2,..., I ), of each of the I IMFs as the final IMF.

2.4 Wavelet Analysis

The Wavelet transform (WT) is a time-frequency method that can provide time varying characteristics of the processed signal. It is a short wavy function that is stretched or compressed and placed at many positions on the signal to be analyzed. It was introduced by Jean Morlet (January 13, 1931 - April 27, 2007) at the beginning of the 1980s that used it to evaluate seismic data. Since then, various types of wavelet transforms have been developed, and most of its applications found in data analysis. Mathematically, the continuous-time wavelet transform (CWT) can accomplishes the multi-resolution tasks by time-shifting and time-scaling a window function ψa,τ(t) or simply called mother wavelet. The shifting of ψa,τ(t) is denoted by the scale a, the dilation parameter that the wavelet is stretched or compressed, for changing the oscillating frequency, and τ is the translation of the coefficient when the wavelet is moved from position to position.. The a-scaled and τ -shifted basis element is given by

(7)

As such, it is usually said that wavelets perform linear transform that decomposes an arbitrary signal via basis functions in which the dilations and translations of the mother wavelet (16) are involved. The Morlet wavelet is a classic example of the CWT and the most frequently used wavelet function which has been applied in this research. It employs a windowed complex exponential as the mother wavelet:

(8)

The squared magnitude of the coefficients can be plotted via the scalogram as energy content in time and frequency domain through a two-dimensional and three-dimensional perspective view. In the physical interpretation, the modulus of the wavelet transform shows how the energy of the signal varies with time and frequency.

## (9)

SGx(a, τ ; ψ) = â”‚Wx (a, b; ψ)â”‚2

The aim of the wavelet transform is to estimate spectral content of a signal and describe its change over time. As with the windowed Fourier transform, local maxima, called ridges, give the frequency content as a function of time. The time-frequency resolution of the wavelet transform depends on the frequency of the signal. At high frequencies, the wavelet reaches at a high time resolution but a low frequency resolution, whereas, at low frequencies, high-frequency resolution and low time resolution can be obtained. Such adaptive ability of time-frequency analysis reinforces the important status of the wavelet transform in many application areas.

## 3. Simulated Signal and Analysis

In this section, the performances of the HT, HHT (EMD), HHT (EEMD) and the WT have been compared with each other through several numerical cases. The simulation studies have been conducted to assess the applicability of the proposed technique in analyzing composite oscillations resulting from the selected simulated signals. In these studies, the multicomponent signals that contain two and three frequency components were chosen for analysis. All signals have been tested on each of the above mentioned methods to evaluate the ability of the method to extract frequency components and verify the accuracy of assessing the properties of nonlinear signals. By analyzing the influence on the main-frequency distribution caused by nonlinearity of the signal, an approximate IF of an approximately periodic signal which contains multifrequency components is established. Previous studies have shown that conventional analysis fails to separate the frequency components, thus making physical interpretation difficult. The decomposing capabilities of each method and the resulting IF have been focused to test the ability of the technique to deal with nonlinear and nonstationary signals. MATLAB program is used thoroughly in this research. The results for applying HHT method on simulated signals are plotted in time-frequency diagram while for WT are presented in a scalogram which display 2D and 3D view of IF versus time of investigated signal.

3.1 Simulation 01: The Frequency Modulation Signal (FM)

To illustrate the performance of the proposed method, the FM signal was first be considered and observed. It has two frequency components in which the high frequency of the signal is modulated by the low frequency signal.

Figure 1: The FM signal

Figure 1 shows the original FM signal depicts as x(t) =cos(2πfct+ sin(2πfmt)) where fc = 600 Hz is the carrier frequency and fm = 50Hz is the modulation signal. The sampling frequency fs is 8000 Hz. The analytic signal produced by HT with its IF and spectrum are shown in Figure 2. The IF plotted in time frequency plane shows that the signal energy is mainly concentrated on the main frequency. Their waveforms which are distorted from the sine waveform can be treated as the results caused by nonlinearity.

Figure 3a shows the IMFs and the residue of FM signal produced by the EMD and each component reflects a different oscillation mode with different amplitude and frequency content. The first IMF has the highest frequency content; frequency content decreases with the increase in IMF component until the 4th IMF component, which is almost a linear function of time. The IF corresponding to each individual IMF is shown in Figure 3b. Clearly, it can be shown from the IF diagram that the main frequency oscillate at 600 Hz is generated from IMF1 while the low frequency component oscillate at 50Hz which is generated from IMF4.

## Figure 2. The IF and spectrum of FM signal by HT

## Mode mixing on HHT(EMD

## Figure 3. Analysis of FM signal by HHT(EMD), The IMFs (a) and The IFs (b)

The same signal has been applied to HHT (EEMD) and the result of their IMF, IF and spectrum of those method are plotted in Figure 4a,b,c,d. Finally the WT plot the scalograms of its IF in 2D and 3D view as in figure 5a and 5b respectively.

## b)

## a)

## c)

## d)

## Figure 4. Analysis of FM signal by HHT(EEMD), The IMFs (a), The IF of each IMF (b), The power

## spectrum (c) and The overall IF (d)

## a)

## b)

## Figure 5. The Wavelet Scalogram of FM signal, The 2D Scalogram (a) and The 3D Scalogram (b)

3.2 Simulation 02: The Complex Piecewise Signal

In the second example, the performance of the proposed methods has been tested on the Complex Piecewise (CPW) signal. Consider the combination of a simple sinusoidal and FM signal as the CPW with the following expression

Where f1 = 50 Hz, f2 = 500 Hz, Sampling freq fs = 5000 Hz.

Figure 6 shows that the CPW sinusoidal signal has a combination of a single sinusoidal waveform with FM signal but separated into two different frequency components in two time frames.

Figure 6. The Complex Piecewise Signal(CPW)

Figure 7. The IF and spectrum of CPW signal by HT

a)

The results shows that it is acceptable to use HT for this type of signal as f1 and f2 are clearly visible in time-frequency plane as in figure 7. By decomposing of the CPW signal using the EMD, the IMFs of some monocomponent signal are shown in Figure 8a.

The IMFs are then analysed individually using HHT to produce analytic signal. After decomposing the signal and obtaining IMFs, the IF algorithm is applied and each frequency component of IMF is evaluated as shown in Figure 8b.

## a)

## b)

c)

Figure 8. Analysis of CPW signal by HHT (EMD), The IMFs (a), The IFs (b) and The Power spectrum (c)

The results based on HHT (EEMD) method on CPW signal are shown in Figure 9a,b,c,d while the scalograms of WT analysis are plotted in Figure 10a,b.

## a)

## b)

## c)

## d)

## Figure 9. Analysis of CPW signal by HHT(EEMD), The IMFs (a), The IF of each IMF (b),

## The power spectrum (c) and The overall IF (d)

## b)

## a)

## Figure 10. The Wavelet Scalogram of CPW signal, The 2D Scalogram (a) and The 3D Scalogram (b)

3.3 Simulation 03: The Complex Composite Signal

For more complex example of simulated signal, the Complex Composite Signal (CCS) is evaluated which can be expressed as x(t)= sin(2πf2t)+cos(2πf1t+sin(2πf2t)), where f1=600Hz, f2=50Hz, fs=5000Hz.

Figure 11. The Complex Composite Signal (CCS)

The CCS signal is a mixed mode signal of a sinusoidal signal plus the FM signal as shown in Figure 11. Figure 12 shows that the high frequency component f1 and low frequency f2 signal was entirely buried under the single complex IF produced by HT. As the HT can only produce a single analytic signal in time-frequency domain, it seems that this method fails to extract the local IF information from CCS signal which have more than one frequency component. This example illustrates that the HT analytic signal is only applicable for monocomponent signal analysis. Logically it is impossible to analyze multicomponent signal with single instantaneous frequency plane.

## Figure 12. The IF and spectrum of CCS signal by HT

## b)

## a)

## Figure 13. Analysis of CCS signal by HHT(EMD), The IMFs (a) and The IFs (b)

Figure 13a is the IMF components of the CCS decomposed by HHT method. Obviously, there are three frequency components exist in CCS signal, thus it is easy to note that the centre frequency oscillate at around 600 Hz and 50 Hz respectively. Figure 13b shows the IF associated with each IMF for CCS signal and it is clearly shown that the frequency components (f1 and f2) appear in time-frequency plane.

## a)

## b)

## d)

## c)

## Figure 14. Analysis of CCS signal by HHT (EEMD), the IMFs (a), the IF of each IMF (b),

## the power spectrum (c) and the overall IF (d)

The IMFs and the residue derived by applying EEMD to the CCS are shown in Figure 14a-d. 2D and 3D scalogram of the WT analysis are shown in Figure 15a,b.

## b)

## a)

## Figure 15. The Wavelet Scalogram of CCS signal, the 2D Scalogram (a) and the 3D Scalogram (b)

## 4. Results and Discussions

The analysed results have shown that HT alone was not suited for analysing multi-component signals in such a way that it was only produced single instantaneous frequency of multi-frequency component signal. Hence, it application was only limited to linear and stationary data but it still be used thoroughly in the research as a preliminary tools for advanced signal processing purposes.

Empirically, all tests indicate that HHT is a superior tool for time-frequency analysis of nonlinear and nonstationary data. All numerical simulations and experimental results confirm these capabilities of the proposed method with the use of HHT. Clearly, the EMD returns component IMFs of a very close visual match to the original signal components, with nearly indistinguishable amplitudes and frequencies. Being different from wavelet decomposition, HHT analysis is based on an adaptive basis, and the frequency is deï¬ned through the HT. Consequently, there is no need for the spurious harmonics to represent nonlinear waveform deformations as in any of the a priori basis methods. Its basis is adaptively produced depending on the signal itself, which brings not only high decomposition efficiency but also sharp frequency and time localization. Furthermore, there is no uncertainty principle limitation on time or frequency resolution from the convolution pairs based also on a priori basis. Besides creating an analytic signal by phase-shifting mode, the HHT method has an advantage of decomposing signal for getting more accurate result. However extensive testing is required to determine whether or not the technique proposed are sufficiently flexible to process real data from experimental work.

Compared with wavelet components, the IMF components have lower-frequency contents, which are useful to analyze low-frequency oscillation. Obviously, the first IMF component extracted from HHT contains the discontinuities and noise in the original signal, its time-varying frequency and amplitude which become excellent signals for indicating time instants of signal abnormality. However, the discontinuity induced Gibbs' phenomenon that makes HHT analysis inaccurate at two data ends. The Gibbs' phenomenon caused ripple which always occurs when the obtained IMF cannot satisfy the monocomponent conditions strictly. Further study in HHT analysis is needed in order to enhance its accuracy and robustness. This phenomenon is clearly noticeable in Figure 8 for analysing CPW signal using HHT (EMD) method. The effects have tremendously reduced when HHT (EEMD) method was applied as shown in Figure 9.

Mode mixing problem which has been discussed earlier are also highlighted in Figure 3a. This phenomenon occurs during EMD operation where some low energy components will be masked by the high energy components. The effect is only occur if HHT (EMD) is used but no longer exists in HHT (EEMD) as an improvement has been made on this method.

WT analysis is becoming the most promising tools for analyzing nonlinear data within a time series. By decomposing a time series into time-frequency space, the IF of the signal can be determined. Unfortunately, many studies using WT analysis have suffered from an apparent lack of quantitative results as it involves a transform from a one-dimensional time series (or frequency spectrum) to a diffuse two-dimensional time-frequency domain. This diffuseness has been worsened by the use of arbitrary normalizations and the lack of statistical significance tests. Another problem of wavelet transforms is that it is not adaptive. The new parameters are needed when the properties of an analysed signal varied caused by the use of function orthogonality and predetermined basis functions to extract components of different time scales. However the results from WT analysis in this research for all simulated signal are very encouraging which indicate very accurate and sharp energy concentration. The scalograms in figure 5a, 10a and 15a have illustrated an exact separation of high and low frequency components in terms of their resolution. Decomposing component with high frequency is shown as a wider frequency band than the one with low frequency; on the other hand, the Hilbert-Huang spectrum has uniform resolution for all frequency part and all sinusoidal components have the same wide frequency bands as it does not involve the concept of the time and frequency resolution but represents the instantaneous frequency.

## 5. Conclusions

Three different known signals have been analysed and the simulations computed for the proposed signal processing techniques have indicated positive results. Compared the HT, HHT with the wavelet decomposition of the evaluated signal, the following conclusions have been made:

The overall results show that both the HHT and WT analysis methods lead to the same conclusions concerning the comparisons of the three simulated signal related to their respective IF. It has been proven from the results that both the HHT and the WT have represented the true frequency patterns of this signal and both methods have clearly shown the main characteristics of signal. However, the HHT shows more details that may be obscured by the scalogram.

The Hilbert spectrum can clearly present the energy distribution with time and frequency. Most energy of Hilbert spectrum is much more concentrated in the definite range of time and frequency than that of Morlet spectrum.

The HHT is indeed a powerful method suitable for analyzing non-linear and non-stationary data and make no assumptions of linearity and/or stationarity as it is necessary with other traditional signal analysis method like WT analysis. It is based on self-adaptive basis; the frequency is derived by differentiation rather than convolution; therefore, it is not limited by the uncertainty principle.

Another advantage of the HHT is that it can reduce any complex non-stationary and nonlinear signal into simple independent IMFs. The decomposition of signal is based on the local characteristic time scales revealed by the signal's local maxima and minima and cubic splines of the extrema to sequentially sift components of different time scales, starting from high frequency to low frequency component. It does not use predetermined basis functions and function orthogonality for component extraction and it allows the use of adaptive distorted harmonics. Hence, components are extracted without distortion and their time-varying amplitudes and frequencies can be accurately computed using the HT to reveal signal characteristics and nonlinearities.

The IMF of HHT is directly decomposed from original data, while wavelet components are decomposed according to mother wavelet, which are influenced strongly by the selected mother wavelet. The IMFs have shown a clear IF as the derivative of the phase function and hence it successfully reflects the intrinsic physical property of original data.

The HHT method has better computing efficiency than the scalogram, which means that it is more suitable for analysing a large size of data.

## 6. Further work

A number of advanced algorithmic variations for computing instantaneous amplitude and instantaneous frequency have been discussed. It has been proven that the appropriate methods of analyzing nonlinear and non-stationary data are the Hilbert Huang Transform and Wavelet analysis and these techniques performed well on simulating artificial signal. However extensive testing is required to determine whether or not the technique proposed are sufficiently flexible to process real data from experimental work. Further work is required for testing the right method on real experimental data for detecting structural damage using the information contained in vibration signatures. Theoretical and experimental works of nonlinear acoustics phenomena in metallic structure will be studied in the final stage of this research. Several complex issues related to signal processing will be investigated. The main objective of this research is to develop and validate new efficient methods for the simulation and experimental works of various signal processing techniques used for crack detection. The basic idea is to extract and analyse modulations sidebands around the acoustical spectral component and used to detect the crack. This is made possible by modelling and simulating the Vibro-Acoustics response signals obtained analytically, from experimental measurements. Different analytical models of the test specimens will be examined in order to study the complexity and error of both methods. The performance of the proposed HHT method will be compared to the performance of WT developed on the same benchmark problem using the same algorithm that have been used in simulation. The measurements from the damaged and undamaged structure will also be used to validate the responses. Using these approaches, a comparative study of various signal processing techniques can be implemented to find the most accurate and efficient methods for crack detection. The results will also indicate that both methods are useful in practice and can provide a basis for future research and development of methods capable of handling more complex nonlinear systems.