**Performance Evaluation of Adaptive Algorithms for Wave Field Analysis/Synthesis Using Sound Field Simulations**

Paolo Peretti, Stefania Cecchi, Laura Romoli and Francesco Piazza *Università Politecnica delle Marche Italy*

#### **1. Introduction**

542 Computational Simulations and Applications

Majchrzak, E., Mochnacki, B. & Suchy, J.S. (2008b). Estimation of boundary heat flux

Majchrzak, E., Mochnacki, B., Dziewoński, M. & Jasiński, M. (2008c). Identification of

*Archives of Foundry Engineering*, Vol. 8, No. 1, pp. 193-198.

*Engineering*, Vol. 8, No. 4, pp. 105-110.

between casting and mould using the global function specification method,

boundary heat flux on the continuous casting surface, *Archives of Foundry* 

The main objective of a multi-channel sound reproduction system is to give an optimal acoustical sensation to the listener. These systems are designed to produce sounds that are as natural as possible, so that the listener does not realize that they are generated by a loudspeakers system. For this purpose, the knowledge of only audio temporal information is not sufficient: also spatial information is needed. Furthermore, since the invention of stereophony, it is well known that at least two loudspeakers are needed in order to generate a virtual source that is not spatially localized at the speaker position. However, the stereo reproduction is very limited because the optimal source localization is focused on one point, called sweet spot. Starting from this assumption, in the recent literature, research efforts have focused on reproduction techniques that use an extremely high number of loudspeakers in order to reproduce not only a simple audio source but a complete sound field. Various techniques, able to record and to reproduce the entire sound field, have been proposed in the literature (Berkhout et al., 1993; Daniel et al., 2003; Fazi et al., 2008).

One of the most studied techniques is Wave Field Synthesis (WFS) that is directly based on the Huygens' principle. It has been introduced in the late '80s by Berkhout, who showed that audio reproduction can be linked to audio holography concepts (Berkhout, 1988). WFS is based on Kirchhoff-Helmholtz integral which permits the calculation of the pressure field inside a volume by knowing pressure and normal particle velocity on the enclosing surface. The underlying idea of WFS is to generate a sound field inside a volume bounded by an array of loudspeakers. Actually, the surface is reduced to a 2D curve positioned on the ear plane. The number of loudspeakers on this curve depends on the desired localization quality. Similarly, Wave Field Analysis (WFA) implements a sound field recording technique based on microphone arrays (Hulsebos et al., 2002). Therefore, this approach allows to record the entire sound field in the recording room (WFA) and subsequently to reproduce it in the listening room (WFS) more or less accurately depending on loudspeakers/microphones number.

As it happens in traditional reproduction systems, digital algorithms based on adaptation processing have to be applied inside the WFS/WFA in order for these techniques to be used in real applications (music, cinema, theatre, etc.). Examples of adaptive algorithms that are very useful in real applications are Acoustic Echo Cancellation (AEC), Active Noise Control (ANC), room compensation, etc. (Haykin, 1996). A straightforward implementation of these algorithms in WFA/WFS systems is not feasible due to the dramatically high

(a) (b)

On the other hand, the second and the third approaches need a processing of the recorded signals. In these cases the loudspeakers can be positioned in an arbitrary way and the loudspeakers number is not compulsorily equal to the microphones number. Even if their processing needs a higher computational load, these techniques present more flexibility and can be easily used in a real application. Sound field extrapolation can be obtained by the combination of WFA and WFS techniques. These approaches allow to record the entire sound field in the recording room (WFA) and subsequently to reproduce it in the listening room (WFS) more or less accurately depending on loudspeakers/microphones number (Fig. 1(b)). It was found that circular microphone arrays permit a very good sound field extrapolation and the problem can be treated more easily in circular coordinates (Hulsebos et al., 2002),

<sup>545</sup> Performance Evaluation of Adaptive Algorithms

for Wave Field Analysis/Synthesis Using Sound Field Simulations

The basic idea of WFS derives from Huygens' Principle (1678). It states that, at any time *t*, all the points on the wave front due to a point source can be taken as point sources for the production of secondary wavelets. Following this principle, the sound field of an arbitrary primary source *p* can be reproduced by a series of secondary sources *s* positioned on the primary wavelet. These sources can be obtained by considering the medium local vibrations

*Considering a source-free volume V, the sound pressure inside the volume generated by external sources can be calculated if the pressure and the normal particle velocity on the enclosing surface*

The Kirchhoff-Helmholtz integral is the mathematical formulation of the Kirchoff's theorem. With reference to Fig. 2, considering a sound propagation in a volume *V* enclosed by a surface

*∂P<sup>r</sup>*<sup>0</sup> (*r*, *ω*)

where *r*<sup>0</sup> and*r* denote the generic position inside *V* and on *S* respectively, *ω* is the angular frequency, *G<sup>r</sup>*<sup>0</sup> (*r*, *ω*) is the Green function which synthesizes the secondary sources located in

*<sup>∂</sup><sup>n</sup>* <sup>−</sup> *<sup>P</sup>*(*<sup>r</sup>*, *<sup>ω</sup>*)

*∂G<sup>r</sup>*<sup>0</sup> (*r*, *ω*) *∂n*

*dS*, (1)

on *s* due to *p*. Kirchhoff's theorem is the generalization of the Huygens' Principle:

*G<sup>r</sup>*<sup>0</sup> (*r*, *ω*)

*S*, the Kirchhoff-Helmholtz integral is given by (Williams, 1999)

4*π S* 

*<sup>P</sup>*(*<sup>r</sup>*0, *<sup>ω</sup>*) = <sup>1</sup>

*r*0, while *P* defines the sound pressure level.

Fig. 1. Schematic representation of audio recording and reproduction techniques. (a)

Holophony. (b) WFA/WFS.

**2.1 Sound field reproduction**

*S are known.*

hence in the following this geometry is considered.

number of inputs/outputs causing an unreasonable computational complexity. This led to the introduction of Wave Domain Adaptive Filtering − WDAF (Buchner, Spors & Rabenstein, 2004). WDAF is an extension of Frequency Domain Adaptive Filtering − FDAF (Shynk, 1992): the filtering is not only performed in temporal frequency domain (as in FDAF), but also in angular frequency domain that takes into account the spatial components of the acoustic field shape. Several WDAF applications can be found in the recent literature (Peretti et al., 2007; 2008; Spors et al., 2005).

In order to evaluate the performance of WDAF-based algorithms, all the points of the sound field have to be analyzed in order to give a complete view of the acoustic scene at several time instants. The main topic of this chapter concerns with a detailed explanation of the steps regarding numerical sound fields simulations during the application of WDAF-based algorithms in order to analyze their performance in terms of sound field reconstruction.

In section 2 the theory of WFS and WFA is summarized and their discrete versions are derived. Then, the basic concepts of WDAF are reviewed and the involved transformations are described in section 3. Therefore, in section 4, the overall simulation processing is described. The discussion starts from the derivation of the sound field of a simple monopole source. Then, the sound field of a static source is virtually reproduced by an array of loudspeakers through the WFS algorithm. In the next step, starting from the simulation of sound field recording with an array of microphones, the PC simulation of the WDAF algorithm is performed. Variations of the whole set of involved parameters are taking into account. Finally, a WDAF application to the attenuation of undesired sources is derived from the well known mono channel approach. The results of the cancellation of the entire sound field of a virtual source through an adaptive algorithm based on WDAF approach is shown. Conclusions are reported in section 5.
