# An Improved Hilbert-Huang Transform for Vibration-Based Damage Detection of Utility Timber Poles
## 2 Limitation of Original Hilbert Huang Transform:  
### Hilbert-Huang Transform  
[PERSON] et al. [38] first introduced Hilbert-Huang transform(HHT) as an adaptive method for analysing nonstationary and nonlinear signals in 1998. Hilbert transform of an arbitrary signal \(x(t)\) has the local properties \(x(t)\) of is well-defined as the convolution of the signal \(x(t)\) with \(1/t\) as follows:  
\[y(t)=\frac{P}{\pi}\int_{-\infty}^{+\infty}\frac{x(\tau)}{t-\tau}\;d\tau \tag{1}\]  
where \(P\) is the Cauchy principal value. Analytical signal \(z(t)\) of \(x(t)\) can be found through coupling \(x(t)\) and \(y(t)\), as  
\[z(t)=x(t)+iy(t)=a(t)e^{i\varphi(t)} \tag{2}\]  
\[a(t)=\sqrt{x^{2}(t)+y^{2}(t)} \tag{3}\]  
\[\varphi(t)=\arctan\left(\frac{y(t)}{x(t)}\right) \tag{4}\]  
where \(a(t)\) is the instantaneous amplitude of \(x(t)\), which can reflect how the energy of the signal \(x(t)\) changes with time, and \(\varphi(t)\) is the instantaneous phase of \(x(t)\). Main property of Hilbert Transform is the instantaneous frequency which has a meaning only for mono component signals. It can be defined as the presence of only one frequency value at a given time frame or signal. The physical meaning of instantaneous frequency \(\omega(t)\) of a signal \(x(t)\) can be found from the time derivative of instantaneous phase \(\varphi(t)\) as the following:  
\[\omega(t)=\frac{d\varphi(t)}{dt} \tag{5}\]  
However, practical signals obtained from the field are complex and contains more than one frequency component at a time. In order to obtain physically meaningful instantaneous frequencies, original signal captured from field should have one frequency or a narrow band of frequencies that vary as a function of time. Therefore, captured complex multicomponent signal must be divided into subcomponents. [PERSON] et al. [38] proposed a new decomposition technique called Empirical mode decomposition (EMD) for this purpose. EMD is capable of decomposing a complex multicomponent signal into nearly mono component signals which are referred as intrinsic mode functions (IMFs). These IMFs usually satisfy two conditions:  
* The number of extrema and the number of zero crossings should be equal or differ by at most one when considering whole data set. * The mean value of the envelope defined by the local maxima and local minima is zero at any point. A given signal \(x(t)\) can be decomposed into different IMFs through EMD process. At first, all the local maxima and local minima points from a given signal \(x(t)\) are identified. Using cubic spline method, these points are interpolated into upper and lower envelope of the signal. Secondly, the mean value of upper and lower envelope is calculated which is denoted as \(m_{1}(t)\), and the difference between the signal \(x(t)\) and \(m_{1}(t)\) is the first component of shifting process denoted as \(h_{1}(t)\)[38] as follows,  
\[x(t)-m_{1}(t)=h_{1}(t) \tag{6}\]  
where \(h_{1}(t)\) should satisfy the condition of IMF but ideally that is not the case. \(h_{1}(t)\) is considered as new signal and the whole shifting process is repeated until \(h_{1}(t)\) satisfy the condition of a true IMF, denoted as \(c_{1}(t)\). Hence, 1 st IMF is obtained which is the highest frequency component in the original signal. Later, \(c_{1}(t)\) is removed from rest of the signal and the residue of the signal \(r_{1}(t)\) is obtained which is expressed as follows:  
\[x(t)-c_{1}(t)=r_{1}(t) \tag{7}\]  
Then this \(r_{1}(t)\) is considered as a new signal and the whole procedure is repeated to find \(c_{2}(t)\), \(c_{3}(t)\), \(c_{4}(t)\),..., \(c_{n}(t)\) and \(r_{2}(t)\), \(r_{3}(t)\), \(r_{4}(t)\),..., \(r_{n}(t)\). Stopping criteria for the shifting process is also defined by [PERSON] et al. [38] for the standard deviation value which is calculated from two consecutive shifting results and the value is limited between 0.2 and 0.3.