# 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
According to the weighted least squares method, the objective function can be written as:  
\[y=\sum_{i=1}^{k}D_{i}\alpha_{i}+v \tag{8}\]0  
where \(Q_{j}\) is the covariance matrix in solving the \(j\)th frequency component and it can be expressed as:  
\[y=\sum_{i=1}^{k}D_{i}\alpha_{i}+v \tag{8}\]1  
\[A_{j}=\frac{a^{2}\big{(}\omega_{j}\big{)}+b^{2}\big{(}\omega_{j}\big{)}}{2} \begin{bmatrix}1&0\\ 0&1\end{bmatrix}  \tag{13}\]  
\[R=\sum_{j}^{m}D_{j}A_{j}D_{j}^{T} \tag{14}\]  
According to the weighted least square method, we have:  
\[\hat{d}_{j}=\Big{(}D_{j}^{T}Q_{j}^{-1}D_{j}\Big{)}^{-1}\Big{(}D_{j}^{T}Q_{j}^ {-1}y\Big{)} \tag{15}\]  
The inverse of \(Q_{j}\) can be obtained by:  
\[Q_{j}^{-1}=\Big{(}R-D_{j}A_{j}D_{j}^{T}\Big{)}^{-1}=R^{-1}+R^{-1}D_{j}A_{j} \Big{(}I-D_{j}^{T}R^{-1}D_{j}A_{j}\Big{)}^{-1}D_{j}^{T}R^{-1} \tag{16}\]  
By multiplying both sides of the above equation by \(D_{j}^{T}\), we have:  
\[ \begin{array}{ll}D_{j}^{T}Q_{j}^{-1}=&D_{j}^{T}\Big{(}R-D_{j}A_{j}D_{j}^{T} \Big{)}^{-1}=D_{j}^{T}R^{-1}+D_{j}^{T}R^{-1}D_{j}A_{j}\Big{(}I-D_{j}^{T}R^{-1 }D_{j}A_{j}\Big{)}^{-1}D_{j}^{T}R^{-1}\\ &=\Big{[}I+D_{j}^{T}R^{-1}D_{j}A_{j}\Big{(}I-D_{j}^{T}R^{-1}D_{j}A_{j}\Big{)}^ {-1}\Big{]}D_{j}^{T}R^{-1}\\ &=\Big{(}I-D_{j}^{T}R^{-1}D_{j}A_{j}+D_{j}^{T}R^{-1}D_{j}A_{j}\Big{)}\Big{(}I-D _{j}^{T}R^{-1}D_{j}A_{j}\Big{)}^{-1}D_{j}^{T}R^{-1}\\ &=\Big{(}I-D_{j}^{T}R^{-1}D_{j}A_{j}\Big{)}^{-1}D_{j}^{T}R^{-1}\end{array}  \tag{17}\]  
Substituting the above equation into Equation (15) yields:  
\[\hat{d}_{j}=\Big{(}D_{j}^{T}Q_{j}^{-1}D_{j}\Big{)}^{-1}\Big{(}D_{j}^{T}Q_{j}^ {-1}y\Big{)}=\bigg{[}\Big{(}D_{j}^{T}R^{-1}D_{j}\Big{)}^{-1}\Big{(}I-D_{j}^{T }R^{-1}D_{j}A_{j}\Big{)}\Big{(}I-D_{j}^{T}R^{-1}D_{j}A_{j}\Big{)}^{-1}D_{j}^{T }R^{-1}y\bigg{]}\]  
It can be seen from the above steps that the calculation of the estimated amplitude \(\hat{d}_{j}\) no longer requires solving the covariance matrix, and the complexity of the algorithm is greatly reduced. The power of the \(j\)th frequency is:  
\[P\big{(}\omega_{j}\big{)}=\frac{1}{N_{I}}\hat{a}_{j}^{T}\ \Big{(}D_{j}^{T}D_{j} \Big{)}\hat{d}_{j} \tag{18}\]  
The power of \(m\) frequencies is calculated one by one, and this step is iterated 10\(\sim\)15 times. Finally, all the peak values higher than the preset threshold \(P_{TH}\) in \(P\big{(}\omega_{j}\big{)}\), \((j=1,2,\cdots,m)\) are selected. At this time, the number of peak values is the number of harmonics, the corresponding frequency \(\omega_{j}\) is the frequency of the harmonics, and the amplitude \(a_{j}=\big{[}a\big{(}\omega_{j}\big{)},b\big{(}\omega_{j}\big{)}\big{]}^{T}\). Furthermore, the amplitude and the phase angle of the harmonics can be obtained by using the formulas of trigonometric functions. According to the IAA algorithm, the resolution of IAA is related to \(\Delta\omega\), and the smaller \(\Delta\omega\) the higher the resolution but the greater the calculation amount. Due to the wide frequency distribution of the roll eccentricity signals, if a highly accurate detection of eccentricity signals is required in the entire frequency range, the use of IAA will lead to a very large calculation amount, and it is hard to meet the requirements for real-time performance. The number of harmonics of the roll eccentricity signals is closely related to the number of rollers, i.e., the number of harmonics can only be a finite integer multiple of the number of rollers. Thus, the frequency spectra of the roll eccentricity signals only contain a finite number of non-zero coefficients, and the roll eccentricity signal is a typical sparse signal with no valid information in a large frequency band of its entire frequency range. Therefore, the distribution range of the harmonic frequency of the roll eccentricity signals must be estimated to avoid unnecessary calculations.