10.4.3  Further analysis of the relationship between acoustical holography and Ambisonics

490  Spatial Sound

Equation (10.4.20) can be derived through the following steps: (1) substituting Equations (10.4.18) and (10.4.19) into Equation (10.4.17); (2) using the integral orthogonalities of trigonometric functions in Equations (4.3.19) and (4.3.20); (3) applying the relationship Hq(ξ) = Jq(ξ)−jYq(ξ) among the Hankel function of the secondary kind, the Bessel function, and the Neumann function; and (4) using the Wronskian formula in Equation (10.4.13)

where

Equation (10.4.20) indicates that the interior pressure can be equivalently created by an array of secondary monopole straight-line sources only, and driving signals are expressed in Equation (10.4.21). For a target straight-line source with unit strength, driving signals of secondary sources are equal to their normalized amplitudes, e.g., E(θS, rS, r0, θ′, f) = A(θS, rS, r0, θ′, f). Equation (10.4.21) is the driving signal of the horizontal near-field-compensated Ambisonics with an infinite order. Equation (10.4.21) is consistent with Equation (9.3.53) except for a normalized gain. The difference in normalized gain is due to the variation in continuous and discrete secondary source arrays.

In conclusion, if only the target interior sound field is controlled, secondary monopole and dipole sources in acoustical holography or Kirchhoff–Helmholtz boundary integral equation have closely related radiation. Therefore, the target interior sound field can be equivalently reconstructed by an array of secondary monopole sources only. In other words, transition from acoustical holography to Ambisonics can occur without forcing the driving signals of secondary dipole sources to vanish. This analysis can be extended to the case of spatial Ambisonics (Daniel et al., 2003; Poletti, 2005b).

10.4.4  Comparison between WFS and Ambisonics

WFS and higher-order Ambisonics, which use an array of a single type of secondary sources, can be derived from the simplification of acoustical holography or Kirchhoff–Helmholtz boundary integral equation. However, the conditions and methods for simplification differ in two cases.

In WFS, Kirchhoff–Helmholtz boundary integral equation is approximately calculated by Rayleigh integrals or appropriate (Neumann) Green’s function to simplify the types of

Spatial sound reproduction by wave field synthesis  491

secondary sources. WFS can be theoretically achieved by an arbitrary array of secondary sources. When a curved array is used, a target source direction-dependent spatial window should be applied to driving signals to reconstruct the target sound field correctly. For example, a spatial window enables secondary sources in half of the horizontal-circular array to participate in the reconstruction of a target plane wave. Therefore, this process can be regarded as a local signal mixing method and is similar to local Ambisonic signal mixing in Section 5.2.4. Moreover, driving signals in WFS are not spatially bandlimited. In case of horizontal WFS, a stationary phase method enables the substitution of secondary straightline sources with point sources. However, mismatched secondary sources lead to errors in the reconstructed spectrum and the overall magnitude of pressure. The former can be pre-equal- ized by applying a special filter to the driving signals, but the latter can only be equalized at a special reference position or line.

Ideally, a horizontal Ambisonics requires an array of secondary straight-line sources arranged in a circle. For far-field approximation, secondary monopole straight-line sources can be substituted by point sources. Spatial Ambisonics requires an array of secondary monopole point sources. For secondary source arrays in a horizontal circle and a spherical surface, the driving signals of secondary monopole and dipole sources for controlling an interior sound field are dependent. Therefore, an array of secondary monopole sources is enough to control the interior sound field. Corresponding driving signals can be obtained from a combination of the pressure and normal velocity of a medium on the boundary of a circle or spherical surface (Section 9.8.3). When the target sound field is decomposed by spatial harmonics, driving signals are represented by a weighted combination of these spatial harmonics. In actual Ambisonics, spatial harmonic decomposition is truncated up to a certain order; thus, driving signals are spatially bandlimited. Ambisonic driving signals pertain to global signal mixing. All secondary sources in the circular or spherical array take part in the reconstruction of a target sound field, and the spatial window for these driving signals is usually not required. Given an upper frequency limit, Ambisonics can reconstruct the target sound field within a local region centered around the origin rather than an extended region within an array.

In practical WFS, M secondary sources are utilized to control the pressure or normal velocity of a medium on the boundary and reconstruct the target sound field in the entire region inside the boundary. As indicated in Section 10.1.5, Shannon–Nyquist spatial sampling theorem requires that the arc length between adjacent secondary sources on the boundary should not exceed half of the wavelength in the worst case. Therefore, more secondary sources are needed for reproduction in a larger region. By contrast, as indicated in Section 9.6.3, horizontal Ambisonics can be equivalent to a scheme of controlling the pressures at O uniform receiver positions in a circle with radius r through a uniform array of M secondary sources arranged in a circle with radius r0 > r. In Equation (9.3.15), Shannon–Nyquist spatial sampling theorem requires that the number of secondary sources should satisfy M ≥ O, and the arc length between adjacent receiver positions should not exceed half of the wavelength. The required spatial samples on a circle with radius r is smaller than that on a circle with radius r0 > r to satisfy the spatial sampling theorem. In other words, Ambisonics reconstructs the target sound field in a smaller region (rather than the entire region inside the array) through fewer secondary sources than WFS. A similar method is used in local WFS in Section 10.2.4. That is, when a smaller number of secondary sources are used, a local WFS improves the accuracy in sound field reconstruction at the cost of reducing the reproduction region.

The reconstructed sound field of WFS and Ambisonics exhibits different physical and perceptual characteristics because of the aforementioned differences between them (Spors and Wierstorf, 2008). However, WFS and Ambisonics can be analyzed using similar methods because their reconstructed sound field and driving signals are closely related to each other.

492  Spatial Sound

A horizontal-circular array of secondary monopole straight-line sources is considered, and its radius is r0. Spatial spectral analysis of a circular array is discussed in Section 9.2.2, and the problems of spatial aliasing and mirror spatial spectra are addressed in Section 9.5.2. The discussions in Chapter 9 focus on Ambisonics, but some general methods and results are applicable to WFS.

The reconstructed sound field of WFS and Ambisonics can be evaluated by substituting the driving signals into Equation (9.2.1) or (9.2.6). In the case of a target plane wave, the driving signals of WFS and Ambisonics are expressed in Equations (10.2.32) and (9.3.53), respectively. Analyses on the horizontal-circular array of secondary monopole straight-line sources, including the calculation of a relative energy error in Equation (9.5.15), lead to the following results.

For WFS, the following conditions are observed:

1. The active secondary sources do not constitute a close curve or curved surface because a spatial window is applied to the driving signals of secondary sources. Therefore, the problem of interior eigen modes in an enclosed space does not occur. A unique solution of driving signals is available to all frequencies or more strictly for all (kr0). The problem of instability in the solution of driving signals does not take place. However, the spatial window causes an edge effect.

2. Even if the spatial aliasing error is ignored, an error occurs in the reconstructed sound field inside an array. This result is also observed in other arrays of secondary sources.

3. For a target plane wave, driving signals are not spatially bandlimited. The analysis in Section 9.5.2 indicates that the discrete array of secondary sources leads to a spatial aliasing error in the reconstructed sound field. Above a certain frequency limit, spatial aliasing leads to the obvious interference of a sound field.

4. The spatial distribution of errors caused by a discrete array is irregular. Above a certain frequency limit, spatial aliasing occurs in the entire receiver region. However, when the receiver position is far from the secondary source array, the spatial aliasing error is reduced.

5. Spatial aliasing in the reconstructed sound field may lead to perceivable timbre coloration.

For higher-order Ambisonics, the following conditions are presented:

1. Active secondary sources constitute a close curve or curved surface. Interior eigen modes occur at some frequencies or more strictly at some (kr0). At these frequencies, the solutions of driving signals are not unique [see the discussion after Equation (9.3.5)]. In other words, the interior sound field cannot be controlled at some frequencies, and the problem of instability in the solution of driving signals occurs.

2. The driving signals of a Q-order Ambisonics are spatially bandlimited. If the number of secondary sources satisfies M ≥ (2Q + 1), driving signals do not cause spatial aliasing. However, the mirror spatial spectra of the driving signals caused by a discrete array may cause an error in the reconstructed sound field.

3. The spatial distribution of an error caused by mirror spatial spectra is regular. When the number of secondary sources satisfies the condition of M ≥ (2Q + 1), all v ≠ 0 terms in the summarization of Equation (9.5.12) can be omitted if kr is smaller than a certain value [Equation (9.3.14)] because the Bessel function Jq(kr) oscillates and decays when its order q is not less than [exp(1)kr/2]. Therefore, Ambisonics can reconstruct the target sound field in a circular region centered at the origin and up to a certain frequency. The radius of this region and the upper frequency limit, or more strictly the