408  Spatial Sound

comparison of Equations (9.4.29) and (9.4.23) indicates that the independent signals of a target plane wave field with directional emphasis are related to those of original plane wave field by following transformation:

Therefore, directional emphasis in spatial Ambisonic reproduction can be implemented by applying a transformation of Equation (9.4.30) to the Ambisonic-independent signals, or equally, by changing the decoding matrix from [D] to [D′] with [T] = [TF] according to Equation (9.4.5) while keeping the independent signals S unchanged.

The aforementioned analysis of directional emphasis transformation is analogous to the theorem of addition of two angular moments in quantum mechanics (Zeng, 2007; Joshi, 1977). In fact, various symmetric transformations are important issues in modern physics. They are applicable to quantum mechanics, quantum field and particle theory, condensed matter physics, and even acoustics (Schroeder, 1989). Group theory is a useful mathematical tool for symmetric analysis. Gerzon (1973) applied group theory to analyze spatial Ambisonics in his early work. According to group theory (Joshi, 1977), all rotations around z-axis constitute the axial rotation group SO(2); all rotations in a three-dimensional space constitute the threedimensional rotation group SO(3); spatial reflection (inversion) and identity constitute a group S(2) of order 2; SO(3) and S(2) groups constitute the transformation group O(3) in a threedimensional space. The analysis of secondary source arrays for spatial Ambisonics in Section 9.4.1 is similar to that of crystallographic point groups. Short et al. (2007) applied the theory of the special unitary group SU(n) of the degree n to the transformation of multichannel sound signals. Moreover, the directional (beamforming) pattern of driving signals for the first-order Ambisonics with the tetrahedral array of four secondary sources in Figure 6.9 is analogous to electron distribution in a tetrahedral solid (such as silicon); furthermore, the directional pattern of driving signals for the second-order Ambisonics with the array of more secondary sources is analogous to electron distribution in transition metals (Economou, 2006). The discussion in this section shows that the methods in different branches of physics are interchangeable.

9.5  ERROR ANALYSIS OF AMBISONIC-RECONSTRUCTED SOUND FIELD

9.5.1  Integral error of Ambisonic-reconstructed wavefront

Sections 9.3.1 and 9.3.2 indicate that Ambisonics can reconstruct a target plane wave field up to a certain frequency limit and within a circular region centered at the origin. Errors in Ambisonic-reconstructed sound fields involve two parts, i.e., errors caused by approximation in truncating spatial harmonic decomposition of a sound field up to a finite order, and spa- tial-spectral aliasing errors caused by reproduction with discrete and finite secondary source array. The relationship of the errors in Ambisonic-reconstructed sound field, frequency, and size of a region is analyzed in this section.

Bamford and Vanderkooy (1995) suggested using the following (normalized) mean (integral) complex amplitude error of wavefront to evaluate the error in the reconstructed sound field:

Analysis of multichannel sound field recording and reconstruction  409

Equation (9.5.1) is the mean normalized absolute value of errors between the reconstructed pressure amplitude P′(r, θ, f) and the target pressure amplitude P(r, θ, f) over a circle of the receiver position with radius r. For a target incident plane wave with a unit amplitude, the integral in the denominator of Equation (9.5.1) is 2π.

The error in reconstructed sound field can also be evaluated with the following mean square complex amplitude error of wavefront:

For a target incident plane wave with a unit amplitude, the integral in the denominator of Equation (9.5.2) is also 2π. The error criteria in Equations (9.5.1) and (9.5.2) are appropriate for arbitrary secondary source arrays and driving signals and not limited to Ambisonics.

For Q-order horizontal far-field Ambisonics, if P(r, θ, f) is a plane wave with a unit amplitude given in Equation (9.3.1), and the reconstructed sound pressure P′(r, θ, f) is a truncation of Equation (9.3.2) up to the Q-order, and the mean square error caused by truncation is calculated from Equation (9.5.2) as

q 1

Errors are independent of the target plane wave direction. When the radius of a region and a wave number satisfies the condition of kr ≤ Q in Equation (9.3.15) for ideal reconstruction, the mean square error expressed in Equation (9.5.3) is less than 0.1 or −10 dB. In this case, a target sound field can be reconstructed accurately (Ward and Abhayapala, 2001).

Discrete arrays with a finite number of secondary sources are used in practical horizontal Ambisonics, which cause mirror spatial spectra and aliasing in driving signals. The overall error in the reconstructed sound field is a mix of truncation errors and spatial-spectral aliasing errors. In Sections 9.3.1 and 9.3.2, driving signals are derived by matching each azimuthal harmonic component of the reconstructed sound field with that of the target sound field up to the Q-order. This operation is equivalent to minimizing the mean square error of the reconstructed sound field in Equation (9.5.2). For example, M secondary sources are arranged uniformly in a horizontal circle and satisfy the condition of M ≥ (2Q + 1), the target sound field is a plane wave with a unit amplitude expressed in Equation (9.3.1), and driving signals are given in Equation (9.3.12). Ambisonic-reconstructed sound pressure is calculated from Equations (9.3.12) and (9.2.14). Then, the mean square error of the reconstructed sound field can be calculated from Equation (9.5.2). However, analytic results are relatively complicated (Poletti, 2000). Instead, some numerical results have been described (Ward and Abhayapala, 2001; Poletti, 2005b).

Figure 9.7 illustrates the mean square errors of the reconstructed sound field for Q = 1-, 2-, and 3-order horizontal Ambisonics with M = 8 secondary sources and the target plane wave from θS = 22.5°. Errors are expressed in decibels. They increase as kr increases at least for kr ≤ 4. That is, the higher the frequency is and the larger the distance from the origin is, the larger the error will be. However, for a given kr, errors decrease as the Ambisonic order increases, or given the errors, the maximal allowable kr increases with the order of Ambisonics. For example, given the error of Err2(kr) ≤ −14 dB, Q = 1-, 2-, and 3-order reproduction have the

Analysis of multichannel sound field recording and reconstruction  411

calculated on the basis of the mean error in a fixed receiver position over a variation in the incident plane wave direction −180° < θS ≤ 180°. The results are illustrated in Figure 9.8. For kr ≤ 2 that does not exceed the limit given by Shannon–Nyquist spatial sampling theorem in Equation (9.3.15), errors in different numbers of secondary sources are almost identical. They increase as kr increases, but they are still less than −14 dB. For kr > 2, they are obvious; for kr = 4, they reach 0 dB. When kr > 3, errors depend on the number M of secondary sources. Moreover, they reduce slightly or at least do not increase when M increases from 6 to 8. They further reduce when M increases from 8 to 12.

Below the limit of kr < Q of Shannon–Nyquist spatial sampling theorem, Ambisonics can theoretically reconstruct a target sound field with slight errors. If the number of secondary sources is larger than the lower limit of M = (2Q + 1), errors are basically independent from the number of secondary sources. Above the limit of Shannon–Nyquist spatial sampling theorem, i.e., for kr > Q, the error of the reconstructed sound field increases obviously. Errors generally depend on the order and number of secondary sources and target source (or plane wave) directions because the reconstructed sound field is the coherent superposition of those caused by multiple secondary sources. In the above case, increasing the number of secondary sources slightly reduces the error in the reconstructed sound field for Q > kr. In other cases, increasing the number of secondary sources may increase the error. The analysis of binaural pressures in Section 12.1.3 yields similar results.

Solvang (2008) analyzed the relationship between spectral distortion (SD) in a reconstructed plane wave and the number of secondary sources in horizontal Ambisonics, where M ≥ (2Q + 1) secondary sources are arranged uniformly in a circle for Q-order reproduction. The conclusion for kr < Q is similar to that observed in Figure 9.8, i.e., the SD in the reconstructed sound pressure is small and basically independent of the number of secondary sources. SD is obvious for kr > Q. In this case, further increasing the number of secondary sources to M > (2Q + 1) increases SD. However, using M > (2Q + 1) secondary sources reduces the SD around the region of kr = Q. Therefore, Solvang concluded that the number of secondary sources should be chosen as a compromise of the SD at kr > Q and around kr = Q. Solvang’s conclusion is different from the observation in Figure 9.8 at kr > Q and around the region of kr = Q.

However, Solvang used a mean SD over receiver positions as the error criterion. SD is defined as the ratio between the power spectra of the reconstructed sound pressure and the target sound pressure. A deviation of SD from a unit (0 dB) indicates an error in reconstructed sound pressures. When the mean SD over receiver positions is calculated, errors in different receiver positions may be wiped out. For example, if the SDs at two receiver positions are 1.5 and 0.5, respectively, the mean SD is a unit (0 dB), which is obviously inappropriate. Therefore, the mean SD in Solvang’s analysis is low, and the conclusion should be revised. By contrast, the mean square complex amplitude error of wavefront in Equation (9.5.2) is used for the calculation in Figure 9.8, and the problem is avoided.

The aforementioned error analysis can be extended to spatial reproduction. For P(r, Ω, f) and P′(r, Ω, f), Equation (9.5.2) can be extended to the integral overall a spherical surface with radius r:

For a target incident plane wave with unit amplitude, the integral in the denominator of Equation (9.5.4) is 4π.