378  Spatial Sound

is consistent with that estimated from Equation (9.3.15). These errors are attributed to the following: spatial aliasing in the reconstructed sound field and secondary sources arranged in a circle with a finite radius of r0 = 5.0 m. The sound waves created by secondary sources can be approximated as plane waves only in a small region close to the origin.

The decoding Equation (9.3.12) is derived by matching each order of the azimuthal harmonic component of the reconstructed sound field with that of the target sound field, and the resultant decoding coefficients are independent of frequency. Although the problem of highfrequency error in the reconstructed sound field cannot be solved on the basis of a physical principle, the perceived performance of high-frequency reproduction can be improved by using some psychoacoustic methods. In particular, the perceived performance at a high frequency can be enhanced by replacing the constant coefficients κq in Equation (9.3.12) or Equation (4.3.53) with frequency-dependent coefficient κq = κq(f), which is equivalent to applying frequency windows to different azimuthal harmonic components in the Bessel– Fourier expansion (Poletti, 2000). Various optimal psychoacoustic methods for the frequency windows are available. These optimal methods preserve the decoding coefficients derived from the matching of azimuthal harmonic components at low frequencies and revising the decoding coefficients (e.g., reducing the gain of decoding coefficients for higher-order azimuthal harmonic components) at high frequencies. The shelf filters and optimization of decoding for low and high frequencies are separately illustrated as examples in Sections 4.3.3 and 4.3.4. However, according to Equation (4.3.66), frequency windows may alter the overall power spectra in reproduction and cause timbre coloration.

In summary, the decoding equation and driving signals of arbitrary Q-order horizontal Ambisonics with M uniform or non-uniform secondary source configurations are derived from the azimuthal spectrum representation of a sound field. For a uniform secondary source array, the spatial sampling and recovery theorem of a sound field is validated, and the minimal number of secondary sources of Q-order horizontal Ambisonics is proven to be (2Q + 1). The upper frequency and region of accurate sound field reconstruction increase as the order of Ambisonics increases.

9.3.2  Reconstructed sound field of spatial Ambisonics

The analysis in Section 9.3.1 can be extended to the case of spatial Ambisonics (Jot et al., 1999; Daniel, 2000; Ward and Abhayapala, 2001; Poletti, 2005b). The target or original sound field is created by a point source at ΩS and rS. It can be approximated as an incident plane wave with a unit amplitude within a region close to the origin. The sound field on a spherical surface with radius r rS around the origin is considered. According to Equation (1.2.6), the frequency domain sound pressure at the arbitrary receiver position (r, Ω) on a spherical surface is given as

where ΩS = (ΩS−Ω) is the angle between the directions of an incident plane wave and the vector of receiver position. Similar to Equation (9.2.38), Equation (9.3.16) can be decomposed by real-valued spherical harmonic functions as

l 0 m 0 1

Analysis of multichannel sound field recording and reconstruction  379

l 0 m 0 1

where jl(kr) and Pl[cos( ΩS)] are the l-order spherical Bessel function and l-order Legendre polynomials, respectively. The summation formula of a spherical harmonic function in Equation (A.17) in Appendix A is used to derive the second equality on the right side of Equation (9.3.17). Furthermore,

where Blm are a set of spherical harmonic coefficient of decomposition and proportional to various order independent signals of Ambisonics for a target plane wave source. In comparison with Equation (9.2.31), the spherical harmonic spectrum representation of the target sound pressure is evaluated using the following:

Plm r, f Blm jl kr 4 jl jl kr Ylm S l 0,1, 2 ; m 0,1, 2 l, 1, 2. (9.3.19)

In reproduction, M secondary sources are arranged on a spherical surface with radius r0. r0 is large enough so that the incident wave from each secondary source can be approximated as a plane wave at a receiver region close to the origin. The direction of the ith secondary source is Ωi. The driving signal is Ei(Ωi, ΩS, f) = Ai(ΩS) EA(f), i = 0, 1…(M − 1), and the normalized amplitude Ai(ΩS) depends on ΩS. For a target plane wave with a unit amplitude, EA(f ) = 1, substituting Equation (9.2.39) and (9.3.19) into Equation (9.2.43) yields a set of equations for the normalized amplitudes of driving signals:

Equation (9.3.20) is valid for jl(kr) ≠ 0 similar to the case of horizontal reproduction. Ai(ΩS) is also decomposed by real-valued spherical harmonic functions to solve the driving

signals:

l 0 m 0 1

In extreme cases of uniform and continuous secondary source array on a spherical surface, Dlm is proportional to Ylm , where Ω′ is the continuous direction of secondary source distribution. After multiplication with the signal waveform EA(f ) in the frequency

domain, EA f Ylm is proportional to Elm , f in Equation (9.2.44).

Substituting Equation (9.3.21) into Equation (9.3.20) yields a matrix equation for unknown decoding coefficients:

380  Spatial Sound

where S3D Y001 S , Y111 S , Y112 S , Y101 S . T is a ∞ × 1 column matrix or vector composed of the normalized amplitude of independent signals of spatial Ambisonics, and [D3D] is an M × ∞ decoding matrix whose entries are Dlm i . The rows of this matrix are arranged in the order of Ω0, Ω1… ΩM−1, and its columns are arranged in the order of (l, m, σ) = (0,0,1), (1,1,1), (1,1,2), (1,0,1)… etc. [Y3D] is an ∞ × M matrix whose entries are Ylm i .

The rows of this matrix are arranged in the order of Y001 , Y111 , Y112 ,Y101 . etc., and its columns are arranged in the order of Ω0, Ω1… ΩM−1. Here, each entry in the signal S3D is normalized so that the integral of their square amplitude in all directions is a unit. This normalization is different from that expressed in Equation (6.4.1), where the maximal magnitudes of W, X, Y, and Z are normalized to a unit. Accordingly, [Y3D] and [D3D] here are different from those in Equations (6.4.5) to (6.4.8). However, the results of different normalizations are equivalent.

For arbitrary (L − 1) ≥ 1 order reproduction, the summation in Equation (9.3.21) is truncated up to the order l = (L − 1), and S3D, [D3D], and [Y3D] become an L2 × 1 column matrix, an M × L2 decoding matrix, and an L2 × M matrix, respectively. When M > L2 and when [Y3D][Y3D]T is well-conditions, the decoding coefficients or matrix in Equation (9.3.22) can be solved using the following pseudoinverse method:

Equation (9.3.23) is an extension of Equation (6.4.8) for the first-order spatial Ambisonics to an arbitrary higher-order spatial Ambisonics, and the results satisfy the condition of con- stant-amplitude normalization.

Driving signals are obtained by substituting decoding coefficients into Equation (9.3.21). The decoding coefficients and normalized amplitude of driving signals depend on secondary source configuration. The problem of secondary source configuration on a spherical surface is generally complicated. However, for a special secondary source array or discrete directional sampling on a spherical surface, the spherical harmonic functions of secondary source directions satisfy the discrete orthogonality given in Equation (A.20) in Appendix A up to the order (L − 1). For real-valued spherical harmonic functions, discrete orthogonality yields

M 1

iYl m i Ylm i ll mm l,l L 1 , 0 m,m l 1,2, (9.3.24) i 0

where δll′ is the Kronecker delta function, the weight λi depends on the configuration or directional sampling scheme of secondary sources. Equation (9.3.24) can be written in a matrix form:

where [Λ] = diag [λ0, λ1,…λM−1] is an M × M diagonal matrix, and [I] is an L2 × L2 identify matrix. When the secondary source configuration satisfies the aforementioned discrete

orthogonality of spherical harmonic functions, the exact solution of the decoding equation

Analysis of multichannel sound field recording and reconstruction  381

and the driving signals in Equation (9.3.20) for arbitrary (L − 1) order reproduction are obtained as

where A(Ωi, ΩS) is directly obtained by letting Ω′ = Ωi in the driving signals of Equation (9.1.25) for uniform and continuous secondary array; Ω′i = ΩS − Ωi is the angle between the directions of the ith secondary source and the target source; and Pl[cos( Ω′i)] is the l-order Legendre polynomials. The summation formula of spherical harmonic functions in Equation (A.17) in Appendix A is used to obtain the second equality on the right side of Equation (9.3.26). From Equations (9.3.22) and (9.3.25), the pseudoinverse solution of the decoding matrix becomes

For nearly uniform secondary source arrays, all weights are approximately equal to λi = λ = 4π/M, Equation (9.3.27a) becomes

Similar to the case of horizontal sound field, an arbitrary spatial incident sound field can be decomposed by spherical harmonic functions in the form of the third equality on the right

side of Equation (9.3.17), but the coefficients B of decomposition are no longer the form

lm

given in Equation (9.3.18). Moreover, for a non-harmonic incident sound field, Blm depends

sources. If the array of M secondary sources satisfies the condition of discrete orthogonality, the spherical harmonic decomposition of reconstructed sound field matches with that of the target sound field up to the order (L − 1). Generally, if a target incident sound field is spatially bandlimited so that all the l > (L − 1)-order components in the spherical harmonic decom-

position of the directional distribution function PA in , f of the amplitude of the incident plane wave vanish, the preceding (L − 1) order spherical harmonic components of the reconstructed sound field match with those of the target sound field within the desired spherical region with radius r. The discrete orthogonality of spherical harmonic functions depends on M and configuration of secondary sources. The Shannon–Nyquist spatial sampling theorem requires the minimal number of secondary sources:

For most practical arrays (Section 9.4.1), the required number of secondary sources exceeds the lower limit given in Equation (9.3.28) and may be much higher than the lower limit in some instances.

The spatial sampling and recovery theorem are extended to the case of spatial Ambisonic sound field in the above discussion. In summary, L2 coincident microphones with different order directivities are needed to record the independent signals of (L − 1) order spatially