## 3 Processing and Analysis of Measurements  
### Signal Separation and Spectrum Analysis  
The major difference between the COP and acceleration measurements is the dimension representation. The COP signals, which are recorded through the force plate, consist of the two  
Figure 1: The facilities in the experiment design: (**1**) force plate, (**2**) referenced wall. Figure 2: The dazzling graph of the referenced wall (ref: [[http://richrock.com/illusion.html](http://richrock.com/illusion.html)]([http://richrock.com/illusion.html](http://richrock.com/illusion.html))). independent components (in X and Y directions) that are perpendicular to the gravitational direction. According to the SOT, the COP signal in X-direction is defined as the medial-lateral (ML) component and the Y-component is defined as the anterior-posterior (AP) direction. The acceleration signals that are measured by the tri-axial accelerometer consist of the three components in the typical X-Y-Z directions of the Cartesian coordinate system. Since the accelerometer is bound on the human body's waist and may tilt with human body swaying, the X-Y-Z directions of the tri-axial accelerometer are not fixed in the measurement process. Through observing the ways that the examinees adjust their posture for balance purposes, both the COP and the acceleration signals have obvious non-stationarity characteristics. The EMD approach that was proposed by [PERSON] et al. [12] is an adaptive data analysis method and can be utilized to separate the non-stationary signals. The EMD process can be simply expressed as:  
\[x(t)=\sum_{k=1}^{m}c_{k}(t)+r_{m}(t) \tag{1}\]  
where, \(x(t)\) represents any complicated non-stationary signal, \(c_{k}(t)\) is the \(k\)-th intrinsic mode function (IMF) of the signal \(x(t)\) and \(r_{m}(t)\) is the final residue which can be a constant or the signal mean trend. Based on the concept of EMD, each IMF component satisfies the following conditions [12]: (1) The number of extrema and the number of zero-crossings must be either equal or differ at most by one in the whole data set, and (2) At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zeros. The measurements of COP and acceleration were first decomposed into the IMFs by the EMD method. With the EMD method, the non-stationary COP and acceleration signals can be separated into the independent signal components of different frequency bands. In the experimental process, on the other hand, the external noises or disturbances that are mixed in the measurements may inevitably interfere with the analysis of the COP and the acceleration signals; therefore, it is crucial to remove the uncorrelated signal components from the measurements of COP and acceleration, and reserve the signal components that contain the information correlated to the balancing ability of the human body. By taking the Hilbert transform of the signal components (IMFs), the analytical signal of \(c_{k}(t)\) can be formed as:  
\[z_{k}(t)=c_{k}(t)+jH\ \left[c_{k}(t)\right]=c_{k}(t)+j\delta_{k}(t)=A_{k}(t)e^{j \phi_{k}(t)} \tag{2}\]  
where, \(H[c_{k}(t)]\) represents the Hilbert transform of \(c_{k}(t)\). The time-dependent amplitude, \(A_{k}(t)\), time-dependent phase, \(\phi_{k}(t)\), and instantaneous frequency, \(\omega_{k}(t)\), of \(c_{k}(t)\) can be formulated as:  
\[ \begin{array}{c}A_{k}(t)=\sqrt{c_{k}(t)^{2}+\delta_{k}(t)^{2}}\\ \phi_{k}(t)=\tan^{-1}\frac{\delta_{k}(t)}{\delta_{k}(t)}\\ \omega_{k}(t)==\frac{\phi_{k}(t)}{dt}\end{array}