**5. Conclusions**

16 Will-be-set-by-IN-TECH

(a)

(b)

(a) (b)

Fig. 15. Adaptive Noise Cancellation with WDAF approach. (a) Corrupted input sound field.

(b) Noise-free output sound field.

Fig. 14. Adaptive noise cancellation schemes. (a) Traditional mono-channel approach. (b) Wave domain extension for WFA/WFS systems; in case of loudspeakers positions different

from microphones positions, a further extrapolation is needed for signal **u**.

Wave Field Analysis and Synthesis are two techniques that permit direct sound field recording and reproduction using microphones and loudspeakers arrays. In order to use these techniques in real world applications (e.g., cinema, home theatre, teleconferencing), it is necessary to apply multi-channel Digital Signal Processing algorithms, already developed for traditional systems. A straightforward implementation needs an extremely high computational complexity, hence Wave Domain Adaptive Filtering (WDAF) has been introduced extending the well-known Fast LMS algorithm. Since WDAF is derived from transforms related to WFA/WFS, it inherits their strengths: the extension of optimal listening area and a better source localization in the environment. It is based on cylindrical harmonics decomposition. In this chapter, the numerical implementation of WDAF transformations have been described and results of simulations have been presented. In particular, considering circular arrays, the area, the sound field is correctly extrapolated on, becomes larger increasing the maximum order of cylindrical harmonic and decreasing the reproduced frequency. Through the only cylindrical harmonics decomposition, some numerical problems could arise nearby the origin of the circular array. Therefore, plane wave decomposition, based on far field approximations, can be used in order to overcome this problem. Numerical simulations of the application of WDAF to adaptive noise cancellation show the effectiveness of the algorithm.
