# A Feasibility Study for a Hand-Held Acoustic Imaging Camera
## 5 Array Optimization Methodology
### Robustness Constraints  
We require quantitative metrics to evaluate the algorithm's beamforming performance. The key metrics utilized are the frequency-dependent directivity \(D(f)\) and white noise gain \(WNG(f)\), computed for steering angles \(\theta_{0}\) and \(\phi_{0}\). For a planar array, the directivity (in dB) is defined as:  
\[D(f)=\frac{|B(\theta_{0},\phi_{0},f)|^{2}}{\frac{1}{4\pi}\int_{0}^{2\pi}\int_{0} ^{\pi}|B(\theta,\phi,f)|^{2}\sin(\theta)d\theta d\phi} \tag{16}\]  
The white noise gain (in dB) quantifies robustness towards array imperfections:  
\[WNG(f)=\frac{|B(\theta_{0},\phi_{0},f)|^{2}}{\sum_{n=1}^{N}|w_{n}(f)|^{2}} \tag{17}\]  
We propose the expected beam pattern power (EBPP) metric to statistically evaluate the impact of variance in array gain and phase on the beam pattern \(B(f)\):  
\[\begin{array}{l}B_{e}^{2}(\theta,\phi,f)=E|B(\theta,\phi,f)|^{2}=\\ \int_{A_{0}}\cdots\int_{A_{N-1}}|B(\theta,\phi,f)|^{2}\cdot f_{A_{0}}(A_{0}) \ldots f_{A_{N-1}}(A_{N-1})dA_{0}\ldots dA_{N-1}\end{array} \tag{18}\]  
Microphone imperfections can distort the array away from the ideal modelled response. As in [38], robustness is incorporated by averaging the cost function over possible gain and phase errors through the expectation operator \(E\{.\}\).  
\[E[B(\theta,\phi,f)]\approx|B(\theta,\phi,f)|^{2}+\frac{1}{WNG(f)}(\sigma_{g}^{ 2}+\sigma_{\psi}^{2}) \tag{19}\]  
where \(\sigma_{g}^{2}\) and \(\sigma_{\psi}^{2}\) are variances of normally distributed microphone magnitude and phase mismatches, and \(WNG(f)\) is a white noise gain term. Minimizing cost tolerance to modelled errors helps ensure reliable performance. This full optimization framework allows the designing of microphone array geometries and beamformers customized for compact acoustic imaging over desired signal bands. The unconventional configurations maximize power focused toward look directions while suppressing artefacts and minimizing off-axis contributions to enable resolving spatial sound fields from small apertures. We next utilize this approach in a simulation case study of miniature array optimization. With the  
Figure 11: Flow chart of Simulated Annealing algorithm for function cost minimization.  
proposed metrics in place, we evaluate the directivity \(D(f)\), white noise gain \(WNG(f)\), and expected beam pattern power (EBPP) for our initial simulated array design. We model the microphone mismatches as Gaussian distributions with \(\sigma_{g}=0.03=3\%\) for gain error and \(\sigma_{\psi}=0.035\) rad \(\cong 2^{\circ}\) for phase error. This preliminary simulation provides favourable results across the three figures of merit, indicating promising performance in reshaping the sensor array for the _Dual Cam_ 2.0 system. The directivity quantifies the main lobe sharpness, white noise gain captures robustness, and the expected beam pattern incorporates statistical variations--together assessing the shaped array's directional sensitivity, imperfections tolerance, and expected real-world behaviour. Further refinements to the array geometry and element tuning will build upon these initial positive findings, working toward an optimal miniature microphone configuration.