# Even Dispersion Design for a Compact Linear Loudspeaker Array with Adaptive Genetic Algorithm
## 3 Constant SPL Distribution Design for Broadband Signals
### Frequency Division  
Because of the different radiation characteristics, the frequency band should be divided properly to get better performance in each sub-band. With an equispaced array, the radiation characteristic varies with the interspace of the adjacent loudspeakers, the number of loudspeakers, and the excitation frequency. We take the arrays with 4, 6, and 8 loudspeakers, respectively, as examples to describe the division strategy with a fixed interspace of the two adjacent loudspeakers. In the specific sub-band, the same delay should be applied since the phase is a frequency dependent parameter. With the phase \(\alpha(\mathbf{x}_{0},k_{ct})\) computed by adaptive GA at the central frequency of the same sub-band, the phase at the noncentral frequency is  
\[\alpha(\mathbf{x}_{0},k^{\prime})=\alpha(\mathbf{x}_{0},k_{ct})\frac{k^{\prime}}{k_{ ct}}, \tag{14}\]  
where \(k^{\prime}\) and \(k_{ct}\) refer to the wave number at the noncentral and that at the central frequency, respectively. One can see that the phase at the band edge may be significantly influenced when the bandwidth is too large. The SPL in the target line can be calculated by  
\[\widetilde{R}(\mathbf{x}_{0},k^{\prime})=R(\langle\mathbf{x}_{C},\mathbf{x}_{B}\rangle,k^{ \prime})+20\text{log}_{10}(\cos\langle\mathbf{x}_{C},\mathbf{x}_{B}\rangle), \tag{15}\]  
where \(R(\langle\mathbf{x}_{C},\mathbf{x}_{B}\rangle,k^{\prime})\) is the directivity computed by Equation (5) with the optimized phase \(\alpha(\mathbf{x}_{0},k^{\prime})\). \(\mathbf{x}_{C}\) is the position at the point \(C\) as shown in Figure 1. \(\mathbf{x}_{B}\) denotes the position of point B. To optimize the proper bandwidth, the rule for the bandwidth decision can be given by  
\[B_{res}=\max_{|\widetilde{R}(\mathbf{x}_{0},k^{\prime})-\widetilde{\widetilde{R}}( \mathbf{x}_{0},k^{\prime})|<1.5 dB}(k^{\prime}-k_{ct})c/\pi, \tag{16}\]  
where \(c\) denotes the speed of sound. \(\widetilde{\widetilde{R}}(\mathbf{x}_{0},k^{\prime})\) is the average value of \(\widetilde{R}(\mathbf{x}_{0},k^{\prime})\) over \(\mathbf{x}_{0}\). The SPL distrubution \(\widetilde{R}(\mathbf{x}_{0},k^{\prime})\) should meet the predefined performance requirement with maximum bandwidth, where the SPL difference over the target line should be less than \(\pm 1.5\) dB [(42)] in each sub-band. Besides, the band division strategy should obey the feasible condition in the NUFBs design [(34)]: For the \(l\)th band, \(\sum_{i=0}^{l-1}B_{res,i}\) equals to \(B_{res,l}v_{l}\) with some integer \(v_{l}\). Hence, there are only a finite number of bandwidth choices in each sub-band design which can be solved with the enumeration method. As a result, for the line loudspeaker array with four loudspeakers, the relative SPL distributions of the lower and the upper frequencies in the first band are shown in Figure 2a,b, respectively. With Equation (16), we can obtain the bandwidth in each band. For the central frequency from 200 Hz to 600 Hz, the bandwidth is 100 Hz. While for 600 Hz-1200 Hz, the bandwidth is 200 Hz. When the central frequency ranges from 1200 Hz to 2000 Hz, the bandwidth is 400 Hz. The remainder frequency band is divided into three parts, which are 2000 Hz-2500 Hz, 2500 Hz-3000 Hz, and 3000 Hz-4000 Hz. The lower and the upper frequencies of each band for 4 loudspeakers are summarized in Table 1.  
For the line loudspeaker array with six and eight loudspeakers, the same frequency band division strategy can be applied with Equation (16). Specifically, for six loudspeakers, the relative SPL distributions of the lower and the upper frequencies in the first band are plotted in Figure 2c,d. For eight loudspeakers, the bandwidth of first sub-band is 30 Hz, where its relative SPL distributions of the lower and the upper frequencies are shown in Figure 2e,f. For the all the frequencies ranging from 200 Hz to 4000 Hz, the lower and the upper frequencies of each band are presented in Table 1 for six and eight loudspeakers.