# A Measure Based on Beamforming Power for Evaluation of Sound Field Reproduction Performance
## 1 Introduction  
Sound field reproduction methods, such as higher-order ambisonics (HOA) [1; 2; 3; 4; 5; 6; 7; 8] and wave field synthesis (WFS) [9; 10; 11], attempt to reproduce a sound field that is similar both physically and perceptually to what is intended. Multi-zone sound field reproduction methods have the same purpose for each sound zone [12; 13; 14; 15]. Sound field reproduction systems using those reproduction methods have been implemented in several places [16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27], and there have been a number of studies on physical validations [17; 18; 25; 26; 28; 29; 30] and perceptual evaluations [17; 18; 20; 22; 28; 31; 32; 33] with the reproduction systems. There are spatial [17; 18; 25; 26; 28; 30], temporal, and spectral features [29] in the physical validations, which are related to perceptual attributes, such as localization, source width, and sound quality. This paper is concerned with a physical evaluation for the spatial feature, which is related to human localization. For the physical evaluation of the spatial feature, the spatially-averaged squared pressure error between the desired and the reproduced fields, called the spatial error in what follows, has been widely used [5; 6; 7; 8; 12; 14; 15; 25; 34], whilst listening tests are conducted for perceptual evaluation of human localization. However, the spatial error does not always correspond to human localization [29]. For example, if a sound source is reproduced in non-anechoic rooms, room reflections significantly increase the spatial error, although human localization is affected to a lesser degree. In addition, differences in the amplitude decay with the distance increase the spatial error consistently, but play a less important role from a perceptual view point [35; 36; 37; 38]. The aim of this paper is to propose aphysical measure to evaluate the spatial feature of the reproduced sound fields that potentially has better agreement with human localization than the spatial error does. Since it is nearly impossible that a physical measure perfectly reflects human localization as long as human perception of source localization is not fully understood, this paper attempts to reduce the effects of the room reflections and amplitude decay with distance while evaluating the directions of sound waves as effectively as the spatial error. Room reflections and spatial amplitude decay are relevant issues to practical reproduction systems. Most reproduction systems are installed in highly-damped rooms, not in anechoic conditions, although most reproduction methods strictly assume a free-field condition. There are a few studies that attempt to make use of room reflections [39; 40], but these methods are feasible only if the room geometry and absorption properties are known, which is not always possible in practice. Moreover, linear arrays and circular arrays are implemented in many cases, which is called a 2.5-dimensional (2.5D) reproduction. When a 2.5D reproduction is used to reproduce a monopole source, the reproduced monopole does not have the amplitude decay of \(1/r\), where \(r\) is the distance to the monopole. Unwanted reflections and unsuccessful reproduction of the amplitude decay would consistently magnify the spatial error, of which the perceptual effect is found to be rather limited [35; 36; 37; 38]. In order to overcome the limitations of the spatial error, the present study proposes a measure based on the beamforming power, or the beam-power. The beam-powers are calculated based on the sound field in a control region, whereas the other binaural measures, such as inter-aural time difference (ITD) or inter-aural level difference (ILD), are specific to the position and direction of the listener at each time. The main reason why beamforming is used for the measure is that it can concentrate the direct sound energy in a control region along its propagating direction, whereas the reflections are, more or less, uniformly distributed.