## 2 Methods and Procedure Analysis
### Enhance Ibrahim Time Method (EITM)  
To identify the dynamic properties, the Ibrahim time-domain method (ITM) is one of the first techniques developed, while the systems/machines are under running conditions. There are several concerns with the OMA method, based on ITM, especially the difficulties regarding the detection of the resonances (closely to harmonics), due to rotating components. The standard ITD was initially developed for a simple input system, and only a simple decomposition analysis was permitted [1]. EITM was used to reduce the effects of closely spaced modes and apply for MIMO systems [1]. This limitation of the method can be evaded if we realize that the matrices can worked as Toeplitz block matrices, which essentially contain correlation data. This means that the MIMO version of the ITD can be obtained by EITD by calculating the Toeplitz arrays of all outputs [1].  
The following equations describes the ITM's analysis procedure [1].  
\[y(t)=c_{1}a_{1}e^{\lambda_{1}k\Delta t}+c_{2}a_{2}e^{\lambda_{2}k\Delta t}=c_{ 1}a_{1}\mu_{1}^{\ k}+c_{1}a_{1}\mu_{2}^{\ k} \tag{1}\]  
where \(\Delta t\) is the time delay, \(\lambda_{n}\) are the continuous time resonances, \(a_{n}\) are the mode shapes, and \(\mu_{n}\) are the discrete resonances. The following equation describes the Hankel matrix [1]:  
\[\mathrm{H}=\left[\begin{array}{cccc}y(1)&y(2)&\ldots&y(np-3)\\ y(2)&y(3)&\ldots&y(np-2)\\ y(3)&y(4)&\ldots&y(np-1)\\ y(4)&y(5)&\ldots&y(np)\end{array}\right]=\left[\begin{array}{c}\mathrm{H}_{ 1}\\ \mathrm{H}_{2}\end{array}\right] \tag{2}\]  
If the free decomposition has \(np\) numbers of data in the signal (taking into account \(\mathrm{H}_{1}\)), it can be expressed as [1]:  
\[\mathrm{H}_{1}=\Psi\Lambda \tag{3}\]  
where \(\Psi\) is a matrix which contains the mode shapes.  
\[\Psi=\left[\begin{array}{cccc}a_{1}&a_{2}&\ldots\\ \mu_{1}a_{1}&\mu_{2}a_{2}&\ldots\end{array}\right] \tag{4}\]  
\(\Lambda\) is the matrix containing the discrete time poles raised to different powers and multiplied by the corresponding modal amplitudes [1].  
\[\Lambda=\left[\begin{array}{cccc}c_{1}\mu_{1}^{0}&c_{1}\mu_{1}^{1}&\ldots&c _{1}\mu_{1}^{np-3}\\ c_{2}\mu_{2}^{0}&c_{2}\mu_{2}^{1}&\ldots&c_{2}\mu_{2}^{np-3}\\ \vdots&\vdots&&\vdots\end{array}\right] \tag{5}\]  
The Hankel matrix \(\mathrm{H}_{2}\) can be expressed with a delay of two time steps:  
\[\mathrm{H}_{2}=\Psi[\mu_{n}]^{2}\Lambda \tag{6}\]  
In order to eliminate expression \(\Lambda\), Equations (1) and (3) can be used, and this gives:  
\[\Psi^{-1}\mathrm{H}_{1}=\mathrm{H}_{2}\Psi^{-1}[\mu_{n}]^{-2} \tag{7}\]  
Multiplying both sides of Equation (7) with \(\Psi[\mu_{n}]^{2}\) gives:  
\[\Psi[\mu_{n}]^{2}\Psi^{-1}\mathrm{H}_{1}=\mathrm{H}_{2} \tag{8}\]  
It is defined the system matrix:  
\[\mathrm{A}=\Psi[\mu_{n}]^{2}\Psi^{-1} \tag{9}\]Finally, the ITD estimation values can be seen in Equations (10)-(13) [1].  
\[\mathrm{AH_{1}}=\mathrm{H_{2}} \tag{10}\]  
\[\widehat{A_{1}}=\mathrm{H_{2}}\mathrm{H_{1}^{T}}\big{(}\mathrm{H_{1}}H_{1}^{T} \big{)}^{-1} \tag{11}\]  
\[\widehat{A_{2}}=\mathrm{H_{2}}\mathrm{H_{2}^{T}}\big{(}\mathrm{H_{1}}H_{2}^{T} \big{)}^{-1} \tag{12}\]  
\[\widehat{A}=\big{(}\widehat{A_{1}}+\widehat{A_{2}}\big{)}/2 \tag{13}\]  
The standard ITD was initially developed for a simple input system, and only a simple decomposition analysis was permitted. This method's limitation can be evaded if we realize that the matrices can worked as Toeplitz block matrices, which essentially contain correlation data. This means that the MIMO version of the ITD can be obtained by calculating the Toeplitz arrays of all output. For obtaining the EITD method, the readers may refer to more details in [1].