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Part of the book series: Springer Topics in Signal Processing ((STSP,volume 9))


In this chapter, we review some of the fundamentals of acoustics and introduce the spherical harmonic expansion of a sound field, which is the basis for the spherical harmonic processing framework used with spherical microphone arrays. This material provides an introduction to the key theory and equations that are required in the rest of the book.

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  1. 1.

    In this section, vectors in Cartesian coordinates are denoted with a corner mark \(\llcorner \) to distinguish them from vectors in spherical coordinates, which will be introduced in Sect. 2.2.

  2. 2.

    In this book, for consistency with spherical array processing literature, we refer to l as the order and m as the degree of the spherical harmonics and associated Legendre functions (or polynomials). However, it should be noted that in other fields, l is referred to as the degree, and m as the order. This reflects the fact that the words degree and order are used interchangeably when referring to polynomials.

  3. 3.

    The factor \(\sin \theta \) compensates for the denser sampling near the poles (\(\theta = 0\) and \(\theta = \pi \)).

  4. 4.

    As noted earlier in the chapter, the scalar product of vectors in spherical coordinates is applied after these vectors have been converted to Cartesian coordinates using (2.12).


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Correspondence to Daniel P. Jarrett .

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Jarrett, D.P., Habets, E.A.P., Naylor, P.A. (2017). Theoretical Preliminaries of Acoustics. In: Theory and Applications of Spherical Microphone Array Processing. Springer Topics in Signal Processing, vol 9. Springer, Cham.

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