## 2 Method
### Higher Frequency Region f2
#### 2.3.2 Radiation Method  
To synthesize the desired sound field \(Q_{\text{des}}(\mathbf{X},\boldsymbol{\kappa}_{a},\omega)\) with the given loudspeaker array \(\boldsymbol{\Theta}_{l}\), \(L=128\) loudspeaker signals \(R^{{}^{\prime}}_{l2}(\boldsymbol{\Theta}_{l},\omega)\) need to be calculated. This is done in two steps. First, the swept sine, Equation (1), is played trough each individual loudspeaker \(\boldsymbol{\Theta}_{l}\) and recorded in the listening region. Here,  
\[S(\omega)\times\boldsymbol{Q}(\alpha,\omega) =P_{\boldsymbol{\Theta}}(\mathbf{X}_{v,j},\omega) \tag{16}\] \[\iff\boldsymbol{Q}(\alpha,\omega) =\frac{P_{\boldsymbol{\Theta}}(\mathbf{X}_{v,j},\omega)}{S(\omega)}\]  
and  
\[Q_{\boldsymbol{\Theta}}(\mathbf{X}_{v,j},\omega)=P_{\boldsymbol{\Theta}}( \mathbf{X}_{v,j},\omega)\times U(\omega) \tag{17}\]  
describe the relationship between the source signal, the raw microphone recordings and the final impulse response. Here, the unknown propagation term \(\boldsymbol{Q}(\alpha,\omega)\) is the ratio of the source signal and the recordings. In contrast to a real harpsichord source signal \(H(\boldsymbol{\kappa}_{a},\omega)\) we know that \(S(\omega)\gg 0\) for all audible frequencies. Thus, the complex transfer function \(\boldsymbol{Q}(\alpha,\omega)\) between each loudspeaker of the array \(\boldsymbol{\Theta}_{l}\) and each listening point \(\mathbf{X}_{tar,j}\) is determined by simply recording the propagated swept sine, Equation (16), of each loudspeaker at each listening point, followed by the deconvolution, Equation (17).  
This complex transfer function is neither the idealized monopole source radiation \(G(r,\omega)\), nor the energy-optimized radiation function \(\Psi(\alpha,\omega)G(r,\omega)\). Instead, \(\boldsymbol{Q}(\alpha,\omega)\) is the actual transfer function as measured physically. It includes the frequency and phase response of the loudspeakers, the amplitude decay and the phase-shift from each loudspeaker to each receiver. It can thus be considered the true transfer function. It includes the sound radiation characteristics of the loudspeakers, which tend to deviate from \(G\) and \(\Psi\). Solving the linear equation system  
\[Q_{\text{des}}(\mathbf{X}_{v,j},\boldsymbol{\kappa}_{a},\omega)=R^{{}^{\prime} }_{l2}(\boldsymbol{\Theta}_{l},\mathbf{X}_{v,j},\boldsymbol{\kappa}_{a}, \omega)\times\boldsymbol{Q}(\alpha,\omega) \tag{18}\]  
for all \(V=11\) microphone array positions yields the impulse response for the loudspeakers \(R^{{}^{\prime}}_{l2}(\boldsymbol{\Theta}_{l},\mathbf{X}_{v,j},\boldsymbol{ \kappa}_{a},\omega)\). This procedure is referred to as _radiation method_ as it synthesizes a desired sound field by including the measured sound radiation characteristics of the loudspeakers. Accounting for the actual transfer function from each loudspeaker to each listening point has the advantage that the rows in the linear equation system described by Equation (18) tend to deviate stronger in reality compared to idealized monopole radiators. This has been demonstrated in [20; 49]. The radiation method is a robust regularization method that has been demonstrated to relax the linear equation system [9; 20; 39; 49]. It leads to (a) low amplitudes and (b) solutions that vary only slightly, when the source-receiver constellation or the source signal is varied slightly. The method synthesizes a desired sound field as long as (a) the sound field lies in the far field and (b) at least two listening points per wavelength exist. The method is only ideal for frequency region f2, as it does not account for nearfield effects and spatial aliasing [9; 20; 49].