## 2 Theory  
### Advanced Transfer Path Analysis Method (ATPA)  
ATPA characterizes the topology of a mechanical system to find out the vibroacoustic paths and the contributions of the system's components, called subsystems, to the noise at any receiver. This method works based on the coefficients of the global transfer matrix \(\mathbf{T}^{G}\), which are defined as [7; 12]  
\[\mathbf{T}^{G}_{ij}=\frac{x_{j}}{x_{i}} \tag{1}\]  
where \(x_{j}\) is the signal at node \(j\), whilst an excitation is applied at node \(i\). Typically, \(x_{j}\) is an acceleration, a rotation acceleration, or a pressure. The global transfer matrix is related to the contributions of the subsystems. On the other hand, the direct transfer matrix \(\mathbf{T}^{D}\) is another coefficient that is used to characterize paths. The coefficient \(\mathbf{T}^{D}_{ij}\) has information about the path between the nodes \(i\) and \(j\) and is defined as  
\[\mathbf{T}^{D}_{ij}=\frac{x_{j}}{x_{i}} \tag{2}\]  
with all the nodes other than \(i\) and \(j\) blocked. The direct transfer matrix can also be defined from any of the nodes to an external target point \(T\) where some output of interest is defined and controlled. \[\mathbf{T}^{D}_{iT}=\frac{p_{T}}{x_{i}} \tag{3}\]  
In this case, \(p_{T}\) can be the pressure at the target point \(T\) when excitation is applied. \(\mathbf{T}^{D}_{iT}\) is calculated as the relation between the pressure at the target \(T\) and the excitation at node \(i\), with all the nodes \(j\
eq i\) blocked. The total pressure at the target point \(T\) is defined as  
\[p_{T}=\sum_{i=1}^{N}x_{i}\mathbf{T}^{D}_{iT}+p^{e}_{T} \tag{4}\]  
where \(p_{T}\) is a signal in the target, \(x_{i}\) is the measured signal in subsystem \(i\) (i.e., the acceleration of a vibrating panel), \(\mathbf{T}^{D}_{iT}\) is the direct transfer function between subsystem \(i\) and the target, and \(N\) is the number of subsystems into which the mechanical system has been divided. In Equation (4), \(p^{e}_{T}\) is the direct field of the signal that arrives at \(T\) due to an external excitation when all the \(N\) nodes are blocked. In addition, a relationship between the global transfer defined in Equation (1) and the direct transfer defined in Equation (3) can be obtained. \[\sum_{j=1}^{N}\mathbf{T}^{G}_{ij}\mathbf{T}^{D}_{jT}=\mathbf{T}^{G}_{iT}\;for \;i=1,2,\ldots,N \tag{5}\]  
The characterization of the paths is then reduced to the mathematical problem of determining the coefficients \(\mathbf{T}^{D}_{iT}\). This can be carried out, for example, using the solution of the linear system of just \(N\) equations like Equation (5) (for the case of exactly \(N\) executions of the experiment). In ATPA, three types of subsystems are defined: structural, panel, and aerial subsystems. Structural subsystems are those that link the vibroacoustic source to the target through structural linking points (like a damped or rigid joint). Panels refer to the system's components that radiate noise and contribute to the noise perceived at the target. Aerial subsystems, like leaks or holes, are elements through which only air-borne transmission paths can be defined. ATPA can synthesize the noise at a target \(T\) as a sum of noise contributions. In this regard, the total contribution from a set of panel or structural subsystems at a target position \(T\) is defined via Equation (6). \[p_{syn,T}(\omega,t)=\sum_{i=0}^{N}a_{i}(\omega,t)T_{i\to T}^{D}(\omega)= \sum_{i=0}^{N}p_{i}(\omega,t) \tag{6}\]  
where \(i\) corresponds to a panel or a structural subsystem, \(a_{i}(\omega,t)\) is the acceleration in subsystem \(i\), and \(T_{iT}^{D}(\omega)\) is the direct transfer function from subsystem \(i\) to target \(T\). In this equation, \(p_{i}(\omega,t)\) is the noise contribution of subsystem \(i\) at the target location \(T\). Furthermore, the method allows for splitting the panel noise into structure-borne and air-borne noise.