Multichannel Simplification Based on Deviation of Loudspeaker Positions

Abstract

People hope to achieve a good impression of three-dimensional (3D) spatial sound with fewer loudspeakers at home. The present method simplified the amount of loudspeakers based on the minimum area enclosed by loudspeakers while maintaining the sound pressure at the origin. However, it doesn’t consider the distortion of a sound field within some specified regions since people always use two ears to listen. In this paper, we exploit that the distortion will be affected by the deviation of positions among loudspeakers and the selection of loudspeaker positions is redefined to obtain the weighting coefficients of each multichannel simplified system. For each multichannel simplified system, simulation result indicates that the distortion within the region of head generated by the proposed method is not more than that generated by the present method. Subjective evaluation shows the proposed method is slightly better in terms of sound localization.

R. Hu—This work is supported by National Nature Science Foundation of China (No. 61231015, 61201169, 61201340), National High Technology Research and Development Program of China (863 Program) No. 2015AA016306, Science and Technology Plan Projects of Shenzhen (ZDSYS2014050916575763), the Fundamental Research Funds for the Central Universities (No. 2042015kf0206).

A Appendix

Derivation of ( 5 ) The original sound field at $\mathbf x$ generated by a loudspeaker $\mathbf y$ is represented as $S(\mathbf x ;k) = - ik4\pi \sum \limits _{n = 0}^\infty {\sum \limits _{m = - n}^n {h_n^{(2)} (k\sigma )j_n (k\gamma )\overline{Y_n^m (\hat{\mathbf{y }})} Y_n^m (\hat{\mathbf{x }})}}$. The reproduced field at $\mathbf x$ generated by three loudspeakers is $S'(\mathbf x ;k) = - ik4\pi \sum \limits _{n = 0}^\infty {\sum \limits _{m = - n}^n {h_n^{(2)} (k\sigma )j_n (k\gamma )}{\times \sum \limits _{l = 1}^3 {w_l (k)\overline{Y_n^m (\hat{\mathbf{y }}_l)} Y_n^m (\hat{\mathbf{x }})} } }$. The squared error over the unit sphere is $\int {\left| {S(\mathbf x ;k) - S'(\mathbf x ;k) } \right| ^2 d\hat{\mathbf{x }}} = (4\pi k)^2 \sum \limits _{n = 0}^\infty {\left| {j_n (k\gamma )} \right| ^2 |h_n^{(2)} (k\sigma )| ^2} {\times \sum \limits _{m = - n}^n {\left| {Y_{nm} (\hat{\mathbf{y }}) - \sum \limits _{l = 1}^3 {w_l(k)Y_{nm} (\hat{\mathbf{y }}_l)} } \right| } } ^2$, which follows from the orthogonality property in [2] of the spherical harmonics. The addition theorem of Legendre functions states $\sum \limits _{m = - n}^n {Y_{nm}^* (\hat{\mathbf{y }})Y_{nm} (\hat{\mathbf{x }})}\mathrm{{ = }}\frac{{2n + 1}}{{4\pi }}P_n (\cos \beta )$, where $\beta$ denotes the angle between $\hat{\mathbf{y }}$ and $\hat{\mathbf{x }}$ [8]. Using this addition theorem with $\hat{\mathbf{y }}=\hat{\mathbf{x }}$, $P_n (\cos 0)=1 \forall n$, gives

$$\begin{aligned} \begin{array}{l} \int {\left| {S(\mathbf x ;k) - S'(\mathbf x ;k) } \right| ^2 d\hat{\mathbf{x }}} \\ = 4\pi k^2 \sum \limits _{n = 0}^\infty {(2n + 1)(j_n (k\gamma ))^2 |h_n^{(2)} (k\sigma )| ^2} [1 + \sum \limits _{l = 1}^3 {(w_l (k))^2 - 2\sum \limits _{l = 1}^3 {w_l (k)} } P_n (\cos \beta _{l}) \\ + 2w_1 (k)w_2 (k)P_n (\cos \beta _{12}) + 2w_1 (k)w_3 (k)P_n (\cos \beta _{31}) + 2w_2 (k)w_3 (k)P_n (\cos \beta _{23})]. \\ \end{array} \end{aligned}$$

And then, $|S(\mathbf o ;k)|^2 = \frac{{e^{ - ik\left| \mathbf{y _4 } \right| } }}{{\left| \mathbf{y _4 } \right| }}\frac{{e^{ik\left| \mathbf{y _4 } \right| } }}{{\left| \mathbf{y _4 } \right| }} = \frac{1}{|\mathbf y _4 |^2} = \frac{1}{\sigma ^2}$, thus completing the derivation.