# A Measure Based on Beamforming Power for Evaluation of Sound Field Reproduction Performance
## 3 Proposed Method  
### Beam-Power and Directional Filter  
Beamforming methods can distribute the acoustic energy in a sound field along the propagating directions. When the beamforming methods are applied to a successfully-reproduced field in a free-field condition, the beam-power should have the maximum value at the intended direction of the virtual monopole source. Even in non-anechoic conditions, the direction of the maximum value does not vary considerably because the beam-power of the direct sound is still dominant, while room reflections are distributed in many directions. The effect of the reflections can be reduced by using a directional filter that gives a greater weighting towards the monopole direction.  
To obtain a consistent measure across the frequency, the main lobe needs to have a constant width. This width depends on the beamforming method and the aperture size of the sensor array. If HOA is used, the reproducible region is determined as \(kr<N\)[5], where \(k\) is the wave number, \(N\) is the order of the HOA, and \(r\) is the radius of the region. This means that the aperture size decreases with the frequency. Delay-and-sum (DAS) beamforming is chosen [43], because DAS beamforming also has the narrower main lobe width at the higher frequency for a constant aperture size. Thus, HOA is used together with DAS beamforming. If the reproducible region is frequency-invariant, constant directivity beamforming methods need to be used.  
The desired beamformer output with respect to the assumed azimuth angle \(\phi_{c}\) for the DAS beamforming can be calculated in frequency domain as:  
\[\widetilde{B}(\phi_{c})=\int_{0}^{2\pi}\int_{0}^{a}\widetilde{P}_{f}(r,\phi) \exp(-ikR_{c})2\pi rdrd\phi, \tag{7}\]  
where \(a\) is the radius of the controllable region of the \(N\)-th order HOA (\(a=N/k\)), and \(R_{c}\) is the distance between the measurement point \((r,\phi)\) and the assumed position of the source \((r_{c},\phi_{c})\), as shown in Figure 3. The intended position of the virtual monopole is known (\(\overrightarrow{r}_{v}=(r_{c},\phi_{v})\)) and the radius \(r_{c}\) is assumed to be identical to the distance of the monopole \(r_{v}\). If \(r_{v}\) is assumed to be much greater than the radius of the reproducible region \(a\) (\(r_{v}>>a\)) for simplicity, inserting Equation (2) into Equation (7) leads to:  
\[\widetilde{B}(\phi_{c})=\left|\frac{8\pi a^{2}A}{{r_{v}}^{2}}\frac{J_{1}\left( 2 ka\sin\left(\frac{\phi_{0}-\phi_{v}}{2}\right)\right)}{2 ka\sin\left(\frac{ \phi_{0}-\phi_{v}}{2}\right)}\right|, \tag{8}\]  
where \(J_{1}\) is the first-order Bessel function of the first kind. See Appendix A for more details. This beam-former output is frequency-invariant as \(ka\) is constant as \(N\). To make it independent of the distance, the normalized beam-power \(\widetilde{\beta}_{N}\) can be defined as:  
\[\widetilde{\beta}_{N}(\phi_{c})=\frac{\widetilde{B}(\phi_{c})}{\max\left\{ \widetilde{B}(\phi_{c})\right\}}=\left|\frac{J_{1}\left(2 ka\sin\left(\frac{ \phi_{0}-\phi_{v}}{2}\right)\right)}{ka\sin\left(\frac{\phi_{0  
reducing the weighting at remote directions from \(\phi_{v}\). In this study, a directional filter that has the maximum at \(\phi_{c}=\phi_{v}\) is defined as:  
\[W(\phi_{c})=0.5[1+\cos(\phi_{c}-\phi_{v})] \tag{11}\]  
The beam-power error is then defined as:  
\[\vec{e}_{BF}^{2}=\frac{\int_{0}^{2\pi}\left|W(\phi_{c})\beta_{N}(\phi_{c})-W( \phi_{c})\widetilde{\beta}_{N}(\phi_{c})\right|^{2}d\phi_{c}}{\int_{0}^{2\pi} \left|W(\phi_{c})\widetilde{\beta}_{N}(\phi_{c})\right|^{2}d\phi_{c}} \tag{12}\]  
where \(\beta_{N}(\phi_{c})\) is the normalized beam-power of the reproduced field.