# Roll Eccentricity Signal Detection and Its Engineering Application Based on SFFT-IAA
## 3 Roll Eccentricity Signal Detection Based on SFFT-IAA  
### Detection of Eccentricity Signal Parameters Based on IAA  
The characteristics of eccentricity signals determine the difficulties in eccentricity signal detection, i.e., effectively suppressing the effects of various random interferences and achieving a high enough resolution. However, these difficulties are hard to overcome by conventional signal detection methods. For IAA, since it has a high frequency  
Figure 2: Influences of roll eccentricity on the thickness. resolution and each frequency component is solved one by one, all other components are regarded as noise in solving a frequency component, thus avoiding the influences of random interferences and other components. Meanwhile, the resolution is a parameter in the calculation process, and a high enough resolution is needed. Additionally, by using IAA, the frequency, amplitude, and phase angle of a signal can be obtained at the same time, which simplifies the calculation process compared with the conventional methods. Hence, the use of IAA can meet the requirements of good interference resistance and high resolution for eccentricity signal detection. According to Equation (7), an eccentricity signal is generated by the superposition of a series of cosine signals with different frequencies; each frequency component in cosine form can be decomposed into the sum of a sine signal and a cosine signal. Thus, Equation (7) can be rewritten as:  
\[y=\sum_{i=1}^{k}D_{i}\alpha_{i}+v \tag{8}\]  
where  
\[D_{i}=[c_{i},s_{i}],\alpha_{i}=[a(\omega_{i}),b(\omega_{i})]^{T}\]  
\[c_{i}^{T}=[cos(\omega_{i}\Delta t),\cdots,\ cos(\omega_{i}n\Delta t)]\]  
\[s_{i}^{T}=[sin(\omega_{i}\Delta t),\cdots,\ sin(\omega_{i}n\Delta t)]\]  
\[v^{T}=[v(1),\cdots,\ v(N_{l})]\]  
\[(i=1,2,\cdots,k)\]  
Based on the measurable data containing the eccentricity information (e.g., rolling force, roller gap, and thickness deviation), a numerical sequence of length \(N_{l}\) can be measured; it is denoted as \(y\):  
\[y^{T}=[y(1),\cdots,\ y(N_{l})] \tag{9}\]  
Let the frequency range of eccentricity signals be \([\omega_{min},\omega_{max}]\). Within the frequency range, a set of virtual frequency grids is generated with a frequency spacing of \(\Delta\omega\). Obviously, the number of generated grids is:  
\[m=1+\frac{\omega_{max}-\omega_{min}}{\Delta\omega}\]  
Therefore, the frequency of each grid is \(\omega_{j}=\omega_{min}+(j-1)\Delta\omega\), \(j=1,2,\cdots,m\). The overall framework of the IAA algorithm is shown in Algorithm 1. ```
Input: The numerical sequence of length \(N_{l}\). Output: A set of the frequencies of the harmonics and the amplitudes. 1 Initialize IAA parameters;
2 Initialize the amplitudes corresponding to the frequency of each virtual grid by the basic least squares;
3 Set the number of iterations to 0;
4 While maximum number of iterations is not reacheddo
5 forj = 1: mdo
6 Calculate the amplitude of \(\omega_{j}\) by the weighted least squares;
7
8 end for
9 number of iterations = number of iterations + 1;
10
11 end for
12 forj = 1 : mdo
13 Calculate the power spectrum of \(\omega_{j}\);
14
15 end for
16
17 Search for the top \(k\) peak values in the power spectrum. ```  
**Algorithm 1**The overall framework of IAAAccording to the principle of the least squares method, the objective function at the \(j\)th frequency grid is \(min\Big{(}\|y-D_{j}a_{j}\|_{j}^{2}\Big{)}\). Then, we have:  
\[\hat{a}_{j}=\Big{(}D_{j}^{T}D_{j}\Big{)}^{-1}D_{j}^{T}y \tag{10}\]  
To avoid the interferences between different frequency components, in solving a grid frequency, the other grid frequencies are considered as noise. Therefore, in the iteration process, the weighted least squares method is employed to solve the objective function.