Spherical Array Acoustic Impulse Response Simulation

Abstract

In order to evaluate spherical array processing algorithms comprehensively under many different acoustic conditions, it is indispensable to use simulated acoustic impulse responses (AIRs) to characterize the source–microphone acoustic channel, most typically in a room or other enclosed acoustic environment. The image method proposed by Allen and Berkley is a well-established way of doing this for point-to-point AIRs with sensors in free space. However, it does not account for the acoustic scattering introduced by a rigid sphere. In this chapter, we present a method for simulating the AIRs between a sound source and microphones positioned on a rigid spherical array. In addition, three examples are presented based on this method: an analysis of a diffuse reverberant sound field, a study of binaural cues in the presence of reverberation, and an illustration of the algorithm’s use as a mouth simulator.

Portions of this chapter were first published in the Journal of the Acoustical Society of America [17], and are reproduced in accordance with the Acoustical Society of America’s Transfer of Copyright Agreement. The content of [17] has been edited here for brevity and to standardize the notation.

Appendix: Spatial Correlation in a Diffuse Sound Field

The sound pressure at a position $\mathbf {r} = (r,\varOmega )$ due to a unit amplitude plane wave incident from direction $\varOmega _{\text {s}}$ is given by [40]

$$\begin{aligned} P(\mathbf {r},\varOmega _{\text {s}},k) = \displaystyle \sum _{l=0}^{\infty } \displaystyle \sum _{m=-l}^l \!\!\! 4 \pi \varphi (\varOmega _{\text {s}}) b_l(k) Y^*_{lm}(\varOmega _{\text {s}}) Y_{lm}(\varOmega ), \end{aligned}$$

(4.24)

where $\varphi (\varOmega _{\text {s}})$ is a random phase term and $|\varphi (\varOmega _{\text {s}})|~=~1$. Assuming a diffuse sound field, the spatial cross-correlation between the sound pressure at two positions $\mathbf {r} = (r,\varOmega )$ and $\mathbf {r}' = (r,\varOmega ')$ is given by:

$$\begin{aligned} \begin{aligned} C(\mathbf {r},\mathbf {r}',k)&= \frac{1}{4\pi } \displaystyle \int _{\varOmega _{\text {s}} \in \mathcal {S}^2} P(\mathbf {r},\varOmega _{\text {s}},k) P^*(\mathbf {r}',\varOmega _{\text {s}},k) d\varOmega _{\text {s}}\\&= \frac{1}{4\pi } \displaystyle \int _{\varOmega _{\text {s}} \in \mathcal {S}^2} \displaystyle \sum _{l=0}^{\infty } \displaystyle \sum _{m=-l}^l \!\!\! 4 \pi b_l(k) Y^*_{lm}(\varOmega _{\text {s}}) Y_{lm}(\varOmega )\\&\quad \times \displaystyle \sum _{l'=0}^{\infty } \displaystyle \sum _{m'=-l'}^{l'} \!\!\! 4 \pi b_{l'}^*(kr) Y_{l'm'}(\varOmega _{\text {s}}) Y^*_{l'm'}(\varOmega ') d\varOmega _{\text {s}}. \end{aligned} \end{aligned}$$

Using the orthonormality property of the spherical harmonics in (2.18) and the addition theorem in (2.23), we eliminate the cross terms followed by the sum over m and obtain

(4.25)

where $\varTheta _{\mathbf {r},\mathbf {r}'}$ is the angle between $\mathbf {r}$ and $\mathbf {r}'$.

In the open sphere case, we can derive a simplified expression for $C(\mathbf {r},\mathbf {r}',k)$. Firstly, we note that the expression in (4.25) is real, and therefore, for a reason which will soon become clear, $C(\mathbf {r},\mathbf {r}',k)$ can advantageously be expressed as

(4.26)

where $\mathfrak {I}$ denotes the imaginary part of a complex number. By substituting the open sphere mode strength $b_l(k) = i^l j_l(kr)$ into (4.26), we obtain

(4.27)

Using $\mathfrak {R}\{h_l^{(2)}(kr)\} = j_l(kr)$, where $\mathfrak {R}$ denotes the real part of a complex number, we can now write (4.27) as

(4.28)

As the expression marked with a $\star$ is real, its imaginary part is zero and (4.28) can be simplified to

(4.29)

Finally, using (4.7) and (4.8), we obtain the well-known spatial domain result for two omnidirectional receivers in a diffuse sound field [20, 31, 39]:

$$\begin{aligned} C(\mathbf {r},\mathbf {r}',k)= & {} - \mathfrak {I}\left\{ \frac{e^{-ik\left| \left| \mathbf {r} - \mathbf {r}'\right| \right| }}{k \left| \left| \mathbf {r} - \mathbf {r}'\right| \right| } \right\} \nonumber \\= & {} \frac{\sin (k\left| \left| \mathbf {r} - \mathbf {r}'\right| \right| )}{k \left| \left| \mathbf {r} - \mathbf {r}'\right| \right| }. \end{aligned}$$

(4.30)