Analysis of multichannel sound field recording and reconstruction  415

array in a horizontal circle. This analysis can also be extended to arbitrary three-dimensional sound field reconstruction technique with a uniform secondary source array on a spherical surface. Therefore, the analysis in this section is general.

9.6  MULTICHANNEL RECONSTRUCTED SOUND FIELD ANALYSIS IN THE SPATIAL DOMAIN

9.6.1  Basic method for analysis in the spatial domain

The general formulation for multichannel sound field reconstruction in the spatial domain, including the formulations for continuous secondary source array in Equation (9.2.1) and discrete and finite secondary source array in Equation (9.2.6), is presented in Section 9.2.1. The methods for analyzing a multichannel reconstructed sound field in two special spatialspectral domains are discussed in Sections 9.2.2 and 9.2.3, and the Ambisonic sound field is analyzed in Section 9.3. The analysis in the spatial-spectral domain is convenient and appropriate for some regular second source arrays, such as uniform or nearly uniform array in a circle or spherical surfaces.

Equation (9.2.1) or (9.2.6) can be directly analyzed and solved in the spatial domain. For continuous and uniform secondary source arrays in a horizontal circle or on a spherical surface, analyses in the spatial domain and the spatial-spectral domain are basically equivalent. However, for arbitrary discrete arrays with a finite number of secondary sources, analyses in the spatial domain often lead to significant results. Given the secondary source array and driving signals, reconstructed sound pressures can be directly evaluated from Equations (9.2.1) and (9.2.6) in the spatial domain. Conversely, under certain conditions, given secondary source array and target sound field, driving signals can be derived directly in the spatial domain.

In the case of discrete array with a finite number of secondary sources, M secondary sources are arranged at positions ri (i = 0, 1…M − 1). The receiver position or controlled point in the

sound field is specified by rcontro, then the sound pressures at a controlled point are calculated with Equation (9.2.6):

i 0

According to Equation (9.6.1) and under certain conditions, driving signals Ei(ri, f) can be derived by minimizing the error of the reconstructed sound pressures at a set of receiver positions or controlled points within a special spatial region. Indeed, the resultant driving signals depend on the chosen error criteria and controlled points. This phenomenon is the basic consideration and method of multichannel reconstructed sound field analysis in the spatial domain.

9.6.2  Minimizing error in reconstructed sound field and summing localization equation

M secondary sources are arranged in a horizontal circle with radius r0, the azimuth of the ith secondary source is θi, and the normalized amplitude of driving signals is Ai. For a unit signal waveform EA(f) = 1 in the frequency domain, the driving signals of secondary sources are Ei = Ai. Two controlled points rcontro,L and rcontro,R are located at a distance of r = a (radius

416  Spatial Sound

of head) and azimuth of ±90°, respectively. This simplified head model is used for binaural pressure analysis in Sections 2.1.1 and 3.2.1 in which the effect of head shadow is neglected. Secondary sources are located at a far-field distance with r0 a; according to Equation (9.6.1), pressures at two controlled points are the superposition of pressures caused by incident plane waves from secondary sources:

i 0

Given the target pressures PL and PR at two controlled points (two ears), two linear Equations for the normalized amplitudes Ai of driving signals can be obtained by matching the left sides of Equation (9.6.2) with the target pressures. However, in the case of more than two secondary sources (M > 2), the equations are underdetermined with more unknown Ai than equations exist. Therefore, a unique solution of driving signal amplitudes cannot be obtained. Conversely, given the driving signal amplitudes, the pressures at two controlled points can be calculated from Equation (9.6.2), and the direction of summing virtual sources can be evaluated.

If a target (virtual) source is located at the far-field distance rI a and at the azimuth of θI, according to Equation (1.2.6), the pressures of a plane wave at two controlled points created by the target sources are given as

PL P rcontro,L, I , f PA exp jka cos 90 I ,

(9.6.3)

PR P rcontro,R, I , f PA exp jka cos 90 I .

The overall square error of complex-valued pressures at two controlled points is evaluated using

Equations (9.6.2) and (9.6.3) are substituted into Equation (9.6.4) to evaluate the virtual source direction, and the amplitude PA and the incident azimuth θI of a target plane wave are chosen to minimize the error in Equation (9.6.4) or equally

The optimal matched target or virtual source direction is found by using the following equation:

Analysis of multichannel sound field recording and reconstruction  417

Equation (9.6.6) is the summing localization equation of multiple horizontal secondary sources (loudspeakers) for a fixed head in Equation (3.2.6). At low frequencies with ka 1, Equation (9.6.6) is simplified into Equation (3.2.7).

The optimal matched target or virtual plane wave amplitude and the corresponding mini-

mal square error Err4,min can be evaluated from Equation (9.6.5). The general results are complicated and omitted here. At low frequencies with ka 1, the best-matched plane wave

amplitude is given as

i 0

It is the sum of the normalized amplitudes of the driving signals of secondary sources, if the normalized amplitudes of driving signals satisfy

i 0

The optimal matched target sound field is a plane wave with a unit amplitude and incident from azimuth θI,

If the controlled points are continuously and uniformly distributed in a circle centered at the origin and r = a, where a can be the head radius, but this condition is not limited to this phenomenon. Under the far-far-field condition, the superposed pressure at the controlled point (a, θ) caused by M secondary sources is given as

i 0

If a target (virtual) source is located at a far-field distance and at an azimuth of θI , the pressure of the plane wave at the controlled point (a, θ) created by the target sources is given as

The integral square error of complex-valued pressure (reconstructed wavefront) over the circle of controlled points is evaluated by

Analysis of multichannel sound field recording and reconstruction  419

the sound pressures at O controlled points specified by the position vector rcontro, o, o = 0, 1 … (O − 1) are given as

i 0

Equation (9.6.16) can be written as a matrix form:

where P′ = [P′(rcontro,0, f), P′(rcontro,1, f), P′(rcontro,O−1, f)]T is an O × 1 column vector or matrix composed of the sound pressures at O controlled points; E is an M × 1 column vector or

matrix composed of the driving signals of M secondary sources; and [G] is an O × M matrix composed of the complex-valued transfer functions from M secondary sources to O con-

trolled points, whose entries are Goi = G(rcontro,o, ri, f), o = 0, 1 … (O – 1), i = 0, 1 … (M – 1). Equation (9.6.17) is the general formulation for controlling the sound pressures at mul-

tiple receiver positions by multiple secondary sources. This formulation is suitable for various sound field reconstruction systems and secondary source arrays. From the point of signal processing, this problem occurs in a multi-input and multi-output (MIMO) system. If driving signals in Equation (9.6.17) are chosen so that the reconstructed sound pressures at the O controlled points match with the target sound pressures, then the O × 1 vector on the left side of Equation (9.6.17) becomes P′ = P. In this case, Equation (9.6.17) is a matrix equation or a set of linear equations with respect to vector E or M driving signals Ei(ri, f). Solving the vector E of driving signals is a multichannel inverse filtering problem (Nelson et al., 1996).

When the number of controlled points is equal to the number of secondary sources and matrix [G] is a full rank, i.e., rank [G] = O = M, a unique solution of Equation (9.6.17) for driving signals is given as

In this case, the errors in the reconstructed sound pressures at the O controlled points vanish.

Generally, the rank of matrix [G] is rank[G] = K ≤ min(O,M). When the number of controlled points is fewer than that of secondary sources, i.e., K ≤ O < M, Equation (9.6.17) is underdetermined, and infinite sets of the solution for driving signals exist. The pseudoinverse solution that minimizes the overall power of driving signals is given as

where superscript “+” denotes the transpose and conjugation of the matrix. Regularization can be applied to the solution to avoid the ill condition or instability in the pseudoinverse of matrix {[G][G]+} at some frequencies:

Analysis of multichannel sound field recording and reconstruction  421

where [ ] is an O × M singular value matrix, whose K non-zero left-diagonal entries are the singular values of [G] in descending order, i.e., δ0 ≥ δ1 ≥ … ≥ δK − 1 ≥ 0, then

[U] and [V] are O × O and M × M unitarity matrices, respectively, with [U]−1 = [U]+ and [V]−1 = [V]+. The preceding K columns of matrix [U] are constructed from the K orthonormal and normalized eigenvectors uκ, whereas the preceding K columns of matrix [V] are constructed from the K orthonormal and normalized eigenvectors vκ. Substituting Equation (9.6.26) into Equation (9.6.17) yields

If P′ = P at each given frequency, driving signals are solved from Equation (9.6.28):

where [1/ ] is an M × O diagonal matrix with K non-zero left-diagonal entries:

When δκ is small, the corresponding matrix entry 1 in Equation (9.6.30) is large, leading to the instability of driving signals in Equation (9.6.29). In this case, the δκ in Equation (9.6.27) that is larger than a certain threshold is retained, and other small δκ are discarded. Thus, the solution of driving signals given in Equations (9.6.29) and (9.6.30) becomes stable.

The aforementioned method is essentially a multiple receiver position matching method through which sound fields are sampled spatially, and sound pressure at O receiver positions is controlled to match with those of target sound pressures as far as possible. If the target sound field is spatially bandlimited, the receiver region can be sampled by a grid of controlled points and the distance between adjacent controlled points does not exceed the minimal half wavelength. According to Shannon–Nyquist spatial sampling theorem, a match of the reconstructed sound pressures at all the controlled points means an accurate reconstruction of target sound field in the concerned region (i.e., the Gibbs effect on the boundary is neglected). Otherwise, spatial aliasing errors occur in the reconstructed sound

422  Spatial Sound

field. Similar to the case of Ambisonics in Section 9.4.1, the controlled points and secondary source array should be chosen appropriately to obtain stable driving signals by solving Equation (9.6.17) so that the transfer matrix [G] is well-conditioned within the concerned frequency range.

To increase the frequency limit of anti-spatial aliasing, Kolundžija et al. (2011) suggested using secondary sources only that contribute mostly to the reconstructed sound field to control the sound pressures at receiver positions. Active secondary sources in array are selected according to the positions of the target source and reconstructed region on the basis of appropriate geometrical acoustic criteria. In addition, equalization can be introduced to the filters for driving signals; in this way, the overall sound power at the controlled points can be constant.

As an example of multiple receiver positions matching method, horizontal far-field Ambisonics is considered. The target sound field is a plane wave with a unit amplitude and incident from θS. O controlled points are located uniformly in a circle with radius r, and the azimuth of the oth controlled point is θo, o = 0, 1 … (O − 1). Target sound pressures at controlled points are calculated with Equations (9.3.1) and (9.3.2). M secondary sources are arranged in a circle with radius r0, azimuth of the ith secondary source is θi, the corresponding normalized amplitude of driving signal is Ai, i = 0, 1 … (M − 1). For secondary sources at a far-field distance so that they can be approximated as plane wave sources, the reconstructed sound pressures at O controlled points are calculated using Equation (9.2.14). Matching the reconstructed sound pressures with the target sound pressures at the O controlled points yields a set of O equations:

According to the discussion in Equations (9.3.14) and (9.3.15), the summation of azimuthal harmonics in Equation (9.6.31) can be truncated up to order Q = integer (kr), which is equivalent to the sampling of the sound field along a circle with radius r at an interval of half wavelength. Moreover, driving signals satisfy Equation (9.3.5) if the number of the controlled points or azimuthal sampling points satisfies the condition of O = M ≥ (2Q + 1). This result can be proven by multiplying cosqθo or sinqθo to both sides of Equation (9.6.31), thereby summing over θo and using the discrete orthogonality of trigonometric functions given in Equations (4.3.16) to (4.3.18). The above example indicates that a match of sound pressures at discrete azimuthal sampling points yields results identical to those obtained by a match of sound pressure in a whole continuous circle if the condition of Shannon–Nyquist spatial sampling theorem is satisfied.

Multiple receiver position-matching methods are closely related to the mode-matching method in Section 9.2.2 (Nelson and Kahana, 2001). By substituting P of target sound pressures with P′ of arbitrary reconstructed sound pressure and using [U]+ = [U]−1, Equation (9.6.28) becomes

424  Spatial Sound

point sources, the following equation is obtained from Equations (9.2.37) and (9.2.41) or directly from Equation (9.3.35):

where [g] is an L2 × L2 diagonal matrix, whose diagonal entries associated with the l-order spherical harmonic functions are denoted by gl. Substituting Equation (9.6.38) into Equation (9.6.17) yields

Multiplying [Y3D(Ωo)] to both sides of Equation (9.6.39) and using Equation (9.6.36) yields

The left side of Equation (9.6.40) represents an L2 × 1 column vector P of the preceding

lm

(L − 1) order spherical harmonic coefficients (spectrum) of the sound pressures at O discrete

controlled points. This result can be derived by using the discrete orthogonality of the spheri-

Equation (9.6.42) indicates that a special spatial mode component of the sound field is also created by the corresponding spatial mode component of driving signals in spherical harmonic representation. Equation (9.6.42) is equivalent to Equation (9.2.45) except for a scale caused by the discrete secondary source array.

Two matrices, namely, an L2 × O matrix [TU] and an L2 × M matrix [TV] are introduced,

and they satisfy [TU] [TU]+ = [I] and [TV] [TV]+ = [I], where [I] is an L2 × L2 identity matrix. Inserting these two matrices into Equation (9.6.39) yields