Analysis of multichannel sound field recording and reconstruction  359

normalized so that the integral of their square norm over direction is a unit. This normalization is for mathematical convenience, but it is different from the normalization of signals W, X, Y, and Z in Equation (6.4.1). In Equation (6.4.1), the maximal magnitudes of W, X, Y, and Z are normalized to a unit to avoid signal overload in practice. Independent signals or spherical harmonic functions can be normalized by various methods. Independent signals with different normalizations are mathematically equivalent except those for gain factors. The methods of normalization should be noticed to avoid confusion (Charpentier, 2017).

Similar to the case of horizontal recording, Equation (9.1.25) can be regarded as a recording method on the basis of directional beamforming. According to the summation formula of spherical harmonic functions given in Equation (A.17) in Appendix A, Equation (9.1.25) can be written as

where Pl[cos(ΔΩ′)] is the l-order Legendre polynomials, ΔΩ′ is the angle between Ω′ and ΩS. As the order L − 1 increases, the directivity or directional pattern of beamforming becomes sharp; accordingly, Equation (9.1.26) approaches the loudspeaker signals for ideal reproduction. However, the number of independent or encoded signals also increases with the order and system become complicated. When the order (L − 1) tends to infinity, Equation (9.1.25) or (9.1.26) reaches the case of recording with ideal beamforming given in Equation (9.1.20). Equations (9.1.25) and (9.1.26) indicate that the directional beam can be steered to arbitrary direction without altering the beam shape by changing Ω′. This feature is common in spatial Ambisonics.

Similar to the case in a horizontal sound field, an arbitrary spatial sound field in a sourcefree region can be decomposed as a linear superposition of plane waves from various direc-

tions. In this case, the directional distribution function PA in , f of incident plane wave amplitudes with respect to the origin is no longer a Dirac delta function. Spatial Ambisonic recording can also be regarded as a directional sampling on PA in , f of incident plane wave amplitudes. The outputs of various order directional microphones are the spherical harmonic components of PA in , f . When the order (L − 1) tends to infinite, loudspeaker signals in reproduction approach the limitation of the Dirac delta function.

The discrete configuration with a finite number of loudspeakers is used in practical reproduction. The loudspeaker configurations of spatial Ambisonics are more complex than those of horizontal Ambisonics. In many practical cases, the decoding equations and reproduced signals of higher-order spatial Ambisonics cannot be derived directly from Equation (9.1.25). Instead, they should be derived through sound field analysis in Sections 9.2 and 9.3.

9.2  GENERAL FORMULATION OF MULTICHANNEL SOUND FIELD RECONSTRUCTION

9.2.1  General formulation of multichannel sound field reconstruction in the spatial domain

As stated in Sections 1.9.1 and 3.1, ideal spatial sound reproduction can be achieved through the physical reconstruction of the target sound field within a region. Sound field reconstruction is an important aspect of spatial sound. It is closely related not only to conventional multichannel sound techniques but also to some methods in other fields, such as active noise

360  Spatial Sound

control and acoustical holography (Fazi and Nelson, 2010, 2013). Therefore, analysis of the reconstructed sound field is important to the evaluation of the accuracy or extent of approximation in practical multichannel sound reproduction and development of new reproduction techniques. Moreover, the mathematical expression and analysis methods of sound field reconstruction vary in different literature, although they are actually equivalent (Ahrens and Spors, 2008b; Poletti, 2005b). For unification, a general formulation of multichannel sound field reconstruction is addressed in this section and Sections 9.2.2 and 9.2.3. The formulation of sound field reconstruction in a spatial domain, or more strictly in the frequency and spatial domain, is first discussed here because a sound field can be represented by sound pressure as a function of receiver position and frequency.

Some terms are first introduced. In the analysis of sound field reconstruction, including the analyses in this chapter and those on acoustical holography and wave field synthesis in Chapter 10, the terms “secondary sources” and “driving signals” are used to represent the ideal sound sources of reproduction and the signals of these ideal sound sources, respectively. Accordingly, the term “secondary source array” is used to represent the configuration of secondary sources. In practice, the ideal sound sources are approximately realized by loudspeakers. Therefore, the term “loudspeaker” is usually referred to a practical secondary source, the corresponding configuration is called “loudspeaker configuration” and the corresponding driving signals are termed “loudspeaker signals.” In the following discussion, these terms are used flexibly and alternately.

Suppose that secondary sources are arranged continuously and uniformly in space; the frequency domain deriving signals for secondary source at position r′ is E(r′, f); the frequency domain transfer function from the secondary source to arbitrary receiver position r is G(r, r′, f). The sound pressure at the receiver point is a superposition of those caused by all secondary sources:

The integral in Equation (9.2.1) is calculated over the whole region of the secondary source distribution.

Under the free-field condition, the transfer function G(r, r′, f ) in Equation (9.2.1) depends on the physical characteristic or radiation pattern of secondary sources. If secondary sources are point sources with unit strength in the free field, G(r, r′, f) is the frequency domain sound pressure at the receiver position r caused by the source at r′ and termed free-field Green’s function in a three-dimensional space (and frequency domain). Letting Qp(f) = 1 and substituting rS with r′ in Equation (1.2.3) yield

where the superscript “3D” denotes the case of a point source in a three-dimensional space, and the subscript “free” denotes the free field. In the following discussion, these superscript and subscript are preserved or omitted depending on a situation. Equation (9.2.2) indicates

that Gfree3D r, r , f only depends on the relative distance |r − r′| between the sound source

Analysis of multichannel sound field recording and reconstruction  361

and the receiver position. Gfree3D r, r , f is also invariant after an exchanging of r and r′. This invariance is the consequence of the acoustic principle of reciprocity.

In the local region of a far field where the distance between the receiver position and the source is large enough, the spherical wave caused by a point source can be approximated as a plane wave. In this case, the secondary source can be modeled by a plane wave source in the free field. If the strength of a point source is chosen to be Qp = 4π r′ or if PA(f) = 1 in Equation (1.2.3), the transfer function from a free-field plane wave source to a region close to the origin is given as

where the superscript “pl” denotes the case of a plane wave source. By choosing an appropriate initial phase of the point source in Equation (1.2.3) to cancel the factor exp(jkr′) in Equation (9.2.3), or directly letting PA(f) = 1 in Equation (1.2.6), Equation (9.2.3) becomes

where the wave vector k in Equation (9.2.4) depends on the direction of a secondary source. Similarly, if secondary sources are straight-line sources with unit strength and an infinite length arranged perpendicular to the horizontal plane (two-dimensional space) in the free field, G(r, r′, f) is the frequency domain sound pressure at the horizontal receiver position r caused by straight-line sources intersecting the horizontal plane at r′. For convenience, r′ is called the “horizontal position of the straight-line source.” G(r, r′, f) is termed the free-field Green’s function in a two-dimensional space or horizontal plane (and frequency domain).

Letting Qli(f) = 1 in Equation (1.2.7) yields

where the superscript “2D” denotes the case of a straight-line source in a two-dimensional space (horizontal plane).

Similar to the case of a point source in a three-dimensional space, in the local region of a far field where the distance between the horizontal receiver position and the straight-line source is large enough, the cylindrical wave caused by a straight-line source can be approximated by a plane wave according to the discussion in Equations (1.2.7) to (1.2.10). When an appropriate initial phase and the frequency-dependent strength of the straight-line source are chosen, the horizontal transfer function from a straight-line source to a region close to the origin is also expressed as the plane wave approximation of Equation (9.2.4).

In addition to three ideal and relatively simple secondary sources, e.g., point source, plane wave source, and straight-line source, other practical secondary sources (such as loudspeakers) exhibit more complicated radiation patterns (such as frequency-dependent directivity). In these cases, reconstructed sound pressures can be calculated using Equation (9.2.1), but G(r, r′, f) from the secondary source to the receiver position is more complicated and does not even have an analytical expression.

Equation (9.2.1) can be extended to the case of reproduction in reflective environments, such as in a listening room with reflections. In this case, the free-field transfer function in Equation (9.2.1) should be replaced with the transfer function in reflective environments, e.g., the frequency domain counterparts of room impulse responses in Section 1.2.2.

362  Spatial Sound

Practical sound field reconstruction or sound reproduction is implemented through a discrete array with a finite number of secondary sources. If M secondary sources are arranged at the position ri (i = 0, 1…M − 1), the integral over the continuous region of secondary source distribution in Equation (9.2.1) is replaced with the summation of all the discrete secondary sources:

i 0

Equation (9.2.1) or (9.2.6) is the general formulation of multichannel sound field reconstruction in the spatial domain. The driving signals Ei(ri, f) of the discrete secondary source array cannot be obtained directly by letting r′ = ri in E(r′, f) for continuous secondary source distribution. For a uniform array in a horizontal circle, the overall gain in the driving signals of continuous secondary source distribution and discrete array differs.

Given the characteristics, the array and driving signals of secondary sources, Equation (9.2.1) or (9.2.6) can be used to analyze the reconstructed sound field. Conversely, given the target sound field, Equations (9.2.1) and (9.2.6) can be used to search for the type, array, and driving signals of secondary sources and design the reproduction system. This discussed content is the main point of reconstructed sound field analysis in the following sections.

9.2.2  Formulation of spatial-spectral domain analysis of circular secondary source array

Equations (9.2.1) and (9.2.6) are formulated in a spatial domain or more strictly in frequency and spatial domains, where the reconstructed sound pressure is a function of frequency f and receiver position r. The reconstructed sound field can also be analyzed in a spatial-spectral domain or more strictly a frequency and spatial-spectral domain. For some regular secondary source array and appropriate coordinate systems, transforming Equations (9.2.1) and (9.2.6) to the spatial-spectral domain for analysis is convenient and may lead to significant results. This phenomenon is similar to the transformation of the signals in the time domain to the frequency domain for analysis in signal processing. However, the appropriate spatial spectrum representation of the reconstructed sound field depends on the secondary source array and chosen coordinate system.

A usual array involves arranging secondary sources uniformly on a horizontal circle, which is often used for horizontal Ambisonic reproduction. Spatial-spectral domain analysis of circular secondary source array is discussed in this section (Bamford and Vanderkooy, 1995; Daniel, 2000; Ward and Abhayapala, 2001; Poletti, 1996, 2000). In this case, a polar coordinate system is convenient for analysis.

The secondary sources are arranged uniformly and continuously on a horizontal circle at r′ = r0. The position of secondary sources is denoted by the polar coordinate (r0, θ′). An arbitrary receiver position inside the circle is denoted by (r, θ). The line element on the circle is r0dθ′. The line integral in Equation (9.2.1) is calculated over the circle r′ = r0. If r0 is merged into the driving signals E(r′, f) as an overall gain, Equation (9.2.1) can be expressed as a onedimensional convolution over the azimuth θ:

Analysis of multichannel sound field recording and reconstruction  367

azimuthal Fourier or harmonic components or vice versa. The method of solving driving signals via Equation (9.2.23) or (9.2.24) is a mode-matching method. Equation (9.2.23) or (9.2.24) is valid when the two sides of the equality in these equations do not vanish. In addition, Ei(θi, f) of discrete secondary sources cannot be directly obtained by letting θ′ = θi in E(θ′, f) for continuous secondary sources in Equation (9.2.7). An overall gain or normalized factor should be supplemented.

Through azimuthal spectrum representation, Equation (9.2.7) can be converted to a form different from Equation (9.2.23). The driving signal E(θ′, f) is a periodic function of azimuth θ′ with a period 2π, so it can be expanded as a realor complex-valued Fourier series:

The realor complex-valued azimuthal Fourier coefficients can be calculated similarly to Equation (9.2.9) or (9.2.10). In the complex-valued Fourier expansion, when Equations (9.2.8), (9.2.12), and (9.2.25) are substituted into Equation (9.2.7), a convolution between two functions in the spatial domain becomes a multiplication between two corresponding

This equation is the formulation of multichannel sound field reconstruction in the azi- muthal-spectral domain.

Equation (9.2.26) can be expressed in the form of real-valued azimuthal Fourier coefficients, but it is relatively complicated. For a secondary source with its main axis pointing to

the origin and symmetric against the main axis, it has Gq2 r, r0, f 0 . In this case, Equation (9.2.26) is expressed by real-valued azimuthal Fourier coefficients as

According to Equation (9.2.26), given the driving signals and transfer functions from secondary sources to receiver positions in the azimuthal-spectral domain, the reconstructed sound pressure in the azimuthal–spectral domain can be evaluated. Or, given the azimuthal

spectrum representation Pq r, f or Pq(r, f) of the target sound field, the driving signals of secondary sources can be found by substituting the azimuthal spectrum representation of reconstructed sound pressures in Equation (9.2.26) with Pq r, f or Pq(r,  f):