## 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.

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## Notes

- 1.
Vectors in Cartesian coordinates are denoted with a corner mark \(\llcorner \) to distinguish them from vectors in spherical coordinates, which are used throughout this book and will be introduced later in the chapter.

- 2.
This expression assumes the sign convention commonly used in electrical engineering, whereby the temporal Fourier transform is defined as \(\mathcal {F}(\omega ) = \int _{-\infty }^{\infty } f(t) e^{-i \omega t} \text {d}t\). For more information on this sign convention, the reader is referred to Sect. 2.3.

- 3.
Some texts [9] refer to the scattering effect as diffraction, although Morse and Ingard note that

*“When the scattering object is large compared with the wavelength of the scattered sound, we usually say the sound is reflected and diffracted, rather than scattered”*[28], therefore in the case of spherical microphone arrays (particularly rigid ones which tend to be relatively small for practical reasons), scattering is possibly the more appropriate term. - 4.
The sign in the powers of \(\beta \) is different from that in Allen and Berkley’s conventional image method, due to the change in the definition of that is required for the SMIR method.

- 5.
Very low frequencies are omitted due to the fact that the spherical Hankel function \(h_l(x)\) has a singularity around \(x = 0\).

- 6.
While the ray-tracing formula is frequency-independent, it has been shown [6] that ITDs actually exhibit some frequency dependence, and that because the ray-tracing concept applies to short wavelengths, this model yields only the high frequency time delay. Kuhn provides a more comprehensive discussion of this model and the frequency-dependence of ITDs [19]. It should be noted the simulation results in Fig. 4.8 are in broad agreement with Kuhn’s measured results at 3.0 kHz.

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## Appendix: Spatial Correlation in a Diffuse Sound Field

### 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]

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:

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

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

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

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

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

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]:

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Jarrett, D.P., Habets, E.A.P., Naylor, P.A. (2017). Spherical Array Acoustic Impulse Response Simulation. In: Theory and Applications of Spherical Microphone Array Processing. Springer Topics in Signal Processing, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-42211-4_4

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