## 3 Identification of Automotive Seat Rattle Noise
### ICA-Based Coherence Analysis Technique
#### 3.2.2 Estimation of Number of Signals  
Typically, the relationship between the input (excitation) and output (response) in a linear system is expressed by the following equation:  
\[Y_{i}=\sum_{j}H_{ij}X_{j} \tag{18}\]  
where \(Y_{i}\) refers to the output spectrum at point \(i\), \(X_{j}\) refers to the input spectrum at point \(j\), and \(H_{ij}\) refers to the frequency response function between the degrees of freedom \(i\) and \(j\).  
In order to estimate the input by measuring the output in such a linear system, an inverse transform technique is typically used. Inverse transform technology has been applied to many fields. An error occurs in a complex system when the inverse transform technique is applied, and such errors are called ill-condition or ill-posed problems.  
The process of finding the solutions to these problems can be represented by the following linear equation and linear least squares equation:  
\[Ax=b \tag{19}\]  
\[\min_{x}\lVert Ax-b\rVert_{2} \tag{20}\]  
Here, if the ratio of the maximum and minimum values is large among the singular values of A, the ill condition appears, which suggests that the solution responds highly sensitively to small changes. Thus, in this input restoration problem, unnecessary matrix terms may result in very large errors. In this respect, the estimation of the number of signals is crucial in restoring signals. In this paper, the number of signals is estimated using Bayesian information criterion, which is a method widely used to estimate the number of input signals. [PERSON] proposed a method to estimate the dimensionality of data using Bayesian model selection, which is called Minka Bayesian Model Selection (MIBS), expressed in the following equation:  
\[\text{MIBS(n)}\approx p_{n}\left(\prod_{j=1}^{n}\lambda_{j}\right)^{-\frac{N}{ 2}}\widetilde{\sigma_{n}}^{-N(m-n)}|A_{n}|^{-\frac{1}{2}}(2\pi)^{\frac{dn+n}{2 }}N^{-n/2} \tag{21}\]  
Here,  
\[p_{n}=2^{-n}\prod_{i=1}^{n}\Gamma\left(\frac{m-i+1}{2}\right)\pi^{-(m-i+1)/2} \tag{22}\]\[|A_{n}|=\prod_{i=1}^{n}\prod_{j=i+1}^{m}\Big{(}\hat{\lambda}_{j}^{-1}-\hat{\lambda }_{i}^{-1}\Big{)}\Big{(}\lambda_{i}-\lambda_{j}\Big{)}N \tag{23}\]  
\[\widetilde{\sigma}_{n}^{2}=\left(\sum_{j=n+1}^{m}\lambda_{i}\right)m-n=1 \tag{24}\]  
\[d_{n}=mn-n(n+1)/2 \tag{25}\]  
where \(\lambda\) is the inherent value of the covariance matrix of signal \(x\), m refers to the number of measured signals, n refers to the number of original signals, and N refers to the length of the data.  
To estimate the number of rattle noises, a 48-channel microphone array (Array-Ring 48-75, Gfal, AcSoft, Bedford, UK) was used (Figure 10a). The results showed that noises occurred mainly from the headrest stays (Figure 10b). MIBS was applied to estimate how many noise sources were present. Figure 11 shows the application results of MIBS. The relative value was confirmed to converge when the estimated number of sources was two or more, which suggests that there were two major sources of rattle noise. Because the results of the noise source visualization confirmed that the headrest stays were the major noise source, two microphones (B&K TYPE 4192) were installed at the joints of the headrest stays (Figure 10c).