## 6 From Single-Point to Multi-Point Equalization
### Plane Wave Approach  
Another possibility for equalizing the sound in the low-frequency region is that offered by the plane wave approach. In rectangular rooms with a symmetric arrangement of loudspeakers in two opposite walls, it has been shown in theory [159] and experiments [160] that equalization within the entire room can be achieved at low frequencies. The approach generates a plane wave that propagatesfrom one wall to the opposite one, where it is absorbed by the loudspeakers. In the experiments of [160], the signals fed to the loudspeakers are determined with the RRE approach of [89]. The error sensors are positioned in two planes perpendicular to the direction of propagation of the simulated plane wave. The desired signal in the planes is a Dirac delta function with a delay corresponding to the time it takes the sound to travel the distance between the planes. A plane wave approach has also been studied by the authors of [161; 162; 163; 164; 165]. First, in [161], the authors developed an application based on finite-difference time domain approximation for studying low frequencies in audio reproduction. In particular, a rectangular room has been simulated by using a discrete model in time and space. Then, in [162] the application was used to study different configurations of loudspeakers in the room to reduce the effect of the acoustic modes. It is shown that by increasing the number of loudspeakers, the variation of the room response across positions is improved at the expense of an increment in the magnitude deviation at every position. The application has also been used to assess the effect of different equalization techniques, such as multi-point equalization and equalization of the acoustic radiation power of the loudspeaker. Eventually, a solution for equalizing the low-frequency sound field using multiple loudspeakers--named controlled acoustic bass system (CABS)--was proposed and studied in [163; 164; 165]. This solution creates a traveling plane wave in one side of the room and cancels it at the opposite wall using extra loudspeakers, with delayed and anti-phase response to remove back-wall reflections. Using the application of [161] and real measurements in rectangular rooms, the authors have shown that the CABS solution can produce a uniform acoustic field in the low-frequency range. In [166], the approach of [159; 163] is further extended to rooms of arbitrary shape with multiple loudspeakers _\"situated in more normal locations\"_ considering a 5.0 loudspeaker set-up. Additionally, [167] has addressed the problem of a non-rectangular room and of an asymmetric loudspeaker set-up. In [167] a multiple-input/multiple-output (MIMO) equalization technique that prescribes only the magnitude of the room response in the control points is proposed. The approach allows a smaller magnitude deviation to be obtained compared to the previous plane-wave approaches.  
To improve equalization with plane waves, a control approach called effort variation regularization was proposed in [168]. In this approach, the conventional cost function of RRE of [169]--based on the minimization of the least-squares error in multiple control points--is modified by adding a regularization term proportional to the squared deviations between source strengths. The approach can be applied both in the frequency and time domain. Simulation results show that the technique can lead to smaller global reproduction errors and better equalization performance at listening positions away from the control points, than the Tikhonov regularization or the approach based on feeding the same signal to all loudspeakers placed on the same wall.