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9.4  SOME PROBLEMS RELATED TO AMBISONICS

9.4.1  Secondary source array and stability of Ambisonics

Secondary source array or loudspeaker configuration of Ambisonics is relatively flexible. Various secondary source arrays for reproduction have been developed. In Section 9.3, the minimal number of secondary sources required for different order reproductions is analyzed, and the decoding matrix or driving signals are derived. In Section 9.3.1, for horizontal Ambisonics with uniform secondary source array, each row in the matrix described in Equation (9.3.9) satisfies the discrete orthogonality of trigonometric functions. Therefore, uniform (regular) secondary source arrays are often used in horizontal Ambisonics. For nonuniform (irregular) secondary source array, the decoding matrix and driving signals can be solved with the pseudoinverse method in Equation (9.3.10).

For spatial Ambisonics, in Section 9.3.2, if the array of M secondary sources satisfies the discrete orthogonality of spherical harmonic functions given in Equation (9.3.24), an exact solution of the decoding equation and driving signals of the (L − 1)-order spatial Ambisonics can be found. However, only a few secondary source arrays satisfy the discrete orthogonality of spherical harmonic functions. Some examples are presented as follows:

1.Equiangle array. The elevation α = 90° − ϕ and the azimuth β = θ are uniformly sampled at 2L angles, respectively; M = 4L2 secondary sources are arranged in the directions of sampling.

2.Gauss–Legendre node array. Elevation is first sampled at L angles that are chosen according to the Gauss–Legendre nodes. Then, the azimuth in each elevation is sampled at 2L angles. M = 2L2 secondary sources are arranged in the directions of sampling.

3.Uniform or nearly uniform array. Directional samples are uniformly or nearly uniformly distributed on the spherical surface so that the distance between neighboring samples is constant or nearly constant. Secondary sources are arranged in the directions of sampling. The number of secondary sources is usually larger than or at least equal to the lower limit given in Shannon–Nyquist spatial sampling theorem, i.e., M ≥ L2.

The equiangle array is intuitive, but it requires a fourfold number of secondary sources than the lower limit given in the Shannon–Nyquist spatial sampling theorem. In fact, the actual angular interval between two adjacent azimuthal sampling decreases as the direction deviates from the horizontal plane to the high and low elevations. Accordingly, an elevationdependent weight λi is introduced into the driving signals in Equation (9.3.26) to reduce or avoid the overemphasis of the contribution of secondary sources in high and low elevations to the reconstructed sound field. Therefore, the efficiency of equiangle arrays is not high from the point of directional sampling and the reconstruction of sound field. As such, this array is seldom used in practical reproduction.

Gauss–Legendre node array requires twice the number of secondary sources in comparison with the lower limit given by the Shannon–Nyquist spatial sampling theorem and thus more efficient than the equiangle array. However, this array imposes more restrictions on the directions of secondary sources. In particular, no secondary sources may exist in a horizontal plane (equator), which is inconsistent with the requirement of enhancing the stability of a horizontal virtual source in reproduction. Therefore, Gauss–Legendre node array is usually inappropriate for practical uses.

A regular secondary source array usually exhibits a high efficiency and leads to a stable reconstructed sound field.A horizontal array is regular if each row of matrix [Y2D] in Equation (9.3.9) satisfies the discrete orthogonality given in Equation (4.3.18), i.e., it satisfies [Y2D]

398  Spatial Sound

[Y2D]T = const. Therefore, a horizontal uniform array with M secondary sources is regular up to the following order of Ambisonic reproduction: Q = (M − 1) / 2 if M is an odd number, or (M − 2)/2 if M is an even number. Similarly, a spatial array is regular if [Y3D] in Equation (9.3.22) satisfies the discrete orthogonality expressed in Equation (9.3.24) or (9.3.25) with a constant weight of λ0 = λ1 = … = λM−1. However, only five spatial arrays are strictly regular (Daniel, 2000). These regular arrays can be realized by arranging the secondary sources in the vertices or centers of faces of some polyhedrons, including tetrahedrons, hexahedrons (cube), octahedrons, icosahedrons, and dodecahedrons (Hollerweger, 2006). Tetrahedral, hexahedral, and octahedral arrays are regular for the first-order spatial Ambisonic reproduction. Some examples of the first-order reproduction with tetrahedral and hexahedral arrays are illustrated in Section 6.4.2. Icosahedral and dodecahedral arrays are regular up to the second-order reproduction although dodecahedral arrays provide 20 secondary sources that exceed the minimal number of (3 + 1)2 = 16 for the third-order reproduction. Moreover, secondary sources are arranged in the horizontal plane only for some polyhedral arrays, such as face-center array in a hexahedron. Various nearly uniform arrays provide more secondary sources (Appendix A) for higher-order Ambisonic reproduction. Usually, the number M of secondary sources in a nearly uniform array is 1.3–1.5 times of the lower limit given in Equation (9.3.28). If M exceeds 1.5L2, a constant weight λi = λ can be approximately chosen in the calculation of Equations (9.3.24) to (9.3.26). Therefore, the efficiency of nearly uniform array is relatively high.

In addition to the aforementioned arrays, some other arrays satisfy the discrete orthogonality expressed in Equation (9.3.24; Lecomte et al., 2015). These arrays may be mathematically perfect, but they are restricted in practical uses. In particular, arranging secondary sources in the bottom is usually inconvenient. Irregular or non-uniform arrays are often used in accordance with the practical reproduction space, and the pseudoinverse method in Equation (9.3.23) is used to solve the driving signals. However, non-uniform arrays may cause instability in the reconstructed sound field. Therefore, feasibility and stability should be considered comprehensively in the design of practical spatial arrays. Instability may also occur in horizontal Ambisonic reproduction with irregular arrays.

Some perturbations on reproduction systems, such as slight errors in secondary source positions and slight differences in the characteristics of secondary sources, are inevitable. Instability means that the reconstructed sound field is sensitive to these small perturbations. Stability depends on the number and configuration of secondary sources in reproduction and is closely related to the pseudoinverse calculation of [Y2D] in Equation (9.3.10) or [Y3D] in Equation (9.3.22) for deriving the decoding matrix. According to numerical analysis theory, stability can be evaluated on the basis of the condition number of [Y2D] or [Y3D] (Sontacchi,

where γmax[Y3D] and γmin[Y3D] are the largest and smallest singular values of [Y3D] in Equation (9.3.22). Because [Y3D] is an L2 × M matrix, [Y3D][Y3D]T is an L2 × L2 real symmetric matrix

to unit) the condition number is, the more stable the system will be.

As simple cases, the hexahedral array of eight loudspeakers and the tetrahedral array of four loudspeakers in Section 6.4.2 are first analyzed. For the (L − 1) = 1 order spatial

Analysis of multichannel sound field recording and reconstruction  399

Ambisonics, the condition numbers of [Y3D] for two arrays are 1.00 and 1.01; therefore, reproduction is stable. However, for the (L − 1) = 2 order spatial Ambisonics, the condition number for both arrays are infinite. In fact, the number of secondary sources (four and eight) in both arrays does not reach the lower limit of the second-order reproduction in Equation (9.3.28).

Three kinds of layer-wise arrays are further analyzed (Liu and Xie, 2013a):

1.Three-layer array of 28 secondary sources (three-layer 28)

Upper, middle (horizontal), and bottom layers are located at ϕ = 45°, 0°, and −45°, respectively. Eight secondary sources with a uniform azimuth of θ = 0°, 45°, …, 315° are arranged in each of the upper and bottom layers. Twelve secondary sources with a uniform azimuth of θ = 0°, 30°, …, 330° are arranged in the middle layer.

2.Three-layer array of 32 secondary sources (three-layer 32)

Upper, middle (horizontal), and bottom layers are located at ϕ = 45°, 0°, and −45°, respectively. Eight secondary sources with a uniform azimuth of θ = 0°, 45°, …, 315° are arranged in each of the upper and bottom layers. Sixteen secondary sources with a uniform azimuth of θ = 0°, 22.5°, …, 337.5° are arranged in the middle layer.

3.Five-layer array of 36 secondary sources (five-layer 36)

Twelve secondary sources with a uniform azimuth of θ = 0°, 30°, …, 330° are arranged in the horizontal plane at ϕ = 0°; eight secondary sources with a uniform azimuth of θ = 0°, 45°, …, 315° are arranged in each elevation plane at ϕ = ±30°; four secondary sources with a uniform azimuth of θ = 0°, 90°, 180°, and 270° are arranged in each of the elevation plane at ϕ = ±60°.

The three aforementioned arrays are easy to implement and often used for research on Ambisonics. For comparison, a nearly uniform array of 36 secondary sources (nearly uniform 36), equiangle array of 36 secondary sources (equiangle 36), and Gauss–Legendre node array of 32 secondary sources (Gauss–Legendre 32) are also analyzed. The condition numbers of [Y3D] of various arrays for the preceding four-order Ambisonics are listed in Table 9.1 (Liu, 2014).

Nearly uniform 36 is the best among various arrays in Table 9.1. It is available up to the fourth-order reproduction with appropriate condition numbers of [Y3D]. Five-layer 36 exhibits similar results, but its condition numbers are higher than those of the nearly uniform 36. Gauss–Legendre 32 is available up to the third-order reproduction. Equiangle 36, threelayer 32, or three-layer 28 are available up to the second-order reproduction. Noticeably, after secondary sources in the horizontal plane at ϕ = 0° increase, the condition numbers of

Table 9.1  Condition numbers of various secondary source arrays