# Even Dispersion Design for a Compact Linear Loudspeaker Array with Adaptive Genetic Algorithm
## 1 Introduction  
Live sound reinforcement aims to maintain constant listening experience for a large amount of audiences in real listening environments such as theater, cinema, auditoria, and stadium. The idea of the \"democracy of sound\" is a recent theme within the live sound reinforcement community [1]. Audiences should receive constant content regardless of the seat location. That is to say, for all audiences, the level and the frequency balance should stay nearly constant in the area that the loudspeakers are designed to cover. This paper studies the use of a compact linear loudspeaker array to achieve the even dispersion goal for practical applications. Sound pressure level (SPL) distribution is one of the most popular objective measurements in studying the even/uneven dispersion of sound fields. To obtain a constant SPL in the audience area, line loudspeaker arrays have been well designed for sound reinforcement by many researchers. There are mainly two alternative groups of approaches in this field [2]. One is to compute the loudspeaker driving signals with a reproduction equation such as Ambisonics [3, 4], the least squares (LS)-based solution [5, 6], delay-and-sum beamformer [7, 8], and so on. The excitation parameters of loudspeakers are calculated for matching the desired and the reproduced SPL distribution with the control points and the loudspeaker locations. However, these approaches are quite unstable because of the ill-conditioned problem [9] and sufficient only for the specific sound field with high accuracy [10]. The other is the well-known wave field synthesis (WFS) [11], which is derived from the Kirchhoff-Helmholtz integral [12]. For the planar and the linear arrays, the formulation is equivalent to [PERSON]'s first integral formula [13]. However, it is a quite complicated system with a large number of loudspeakers which must be set up carefully and driven individually [14, 15]. In this paper, an adaptive geneticalgorithm (GA)-based approach is proposed to compute the excitation parameters of loudspeakers. GA, firstly proposed by [PERSON] [16], is an adaptive and competent approach for evolutionary search. It has been successfully applied in many fields; array optimization is an example [17; 18; 19; 20; 21], and can be treated as a kind of symbolic regression [22; 23]. However, for real audio applications, it is necessary to consider the performance in a wide frequency range. Broadband designs are addressed in sound reproduction [24; 25] and beamforming techniques [26]. There are some inevitable limitations for broadband sound field control with a uniformly spaced array [27]. With the increase of frequency, difficulties increase dramatically for constant sound radiation characteristics design. Firstly, the number of loudspeakers should be increased with the range of the frequency band. An array, which consists of 300 loudspeakers, is proposed by [PERSON] [25] for the signals up to 4000 Hz. However, only a limited number of loudspeakers is available for many practical applications. Lasso-based methods [6; 27; 28; 29] are approved for loudspeaker location optimization with a reduced set of loudspeakers. Due to the complicated computation mentioned above, it is valuable to exploit a feasible system in real applications. In this paper, we focus on providing the broadband constant SPL distribution in the audience area using a linear loudspeaker array ranging from 200 Hz to 4000 Hz. Firstly, an adaptive GA, which can overcome both the low speed and the premature convergence problems [30], is introduced for the even dispersion goal in narrowband. The proposed method shows good performance compared with conventional approaches such as delay and sum beamformer [7; 8], contrast maximization [31], and LS-based pressure matching [32; 33].