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and two examples of formulation in a spatial-spectral domain are presented. In Section 9.3, the reconstructed sound field of Ambisonics in the spatial-spectral domain is analyzed in detail; the decoding equation and deriving signals for arbitrary-order Ambisonics with various secondary source arrays or loudspeaker configurations are derived; the theorem of the spatial sampling and reconstruction of sound field are discussed; near-field compensated higher-order Ambisonics is addressed; and the applications of spatial-spectral analysis in some Ambisonic-like techniques are outlined. In Section 9.4, the secondary sources arrays and stability of Ambisonic sound field are analyzed, and some spatial transformations of Ambisonic sound field are discussed. In Section 9.5, errors in Ambisonic sound field are evaluated, and the problems of spatial aliasing caused by discrete secondary source arrays in a horizontal circle or on a spatial spherical surface are analyzed. In Section 9.6, the basic method of spatial domain analysis of multichannel sound field is introduced, and method of multiple matching receiver positions and their relations with the mode-matching method are outlined. In Section 9.7, the problem of active compensation for reflections in a listening room for multichannel sound reproduction is addressed. In Section 9.8, microphone array techniques for sound field recording, especially Ambisonic recording, are discussed on the basis of the spatial sampling and reconstruction theorem of a sound field.

9.1  EACH ORDER APPROXIMATION OF IDEAL REPRODUCTION AND AMBISONICS

9.1.1  Each order approximation of ideal horizontal reproduction

In this section, the recorded and reproduced signals of Ambisonics are derived from each order approximation of an ideal reproduction (Xie and Xie, 1996). The case of horizontal reproduction is discussed first. Similar to the discussion in Section 3.1, the discussion here starts with an ideal horizontal reproduction by arranging an infinite number of loudspeakers on a circle uniformly and continuously around a listener or receiver region. If the radius r0 of the circle is large enough, then the incident wave in a region near the center of the circle can be approximated as the superposition of plane waves from each loudspeaker. According to Equation (1.2.12), for an original or target plane wave with unit amplitude and incident from a horizontal azimuth θS, the azimuthal distribution function of the complex-valued amplitude of the incident wave is taken in the form of Dirac delta function PA in S in , where θin is the azimuthal coordinate of a sound field. For an ideal reproduction, the azimuthal distribution function A(θ′, θS) of the normalized amplitude of loudspeaker signals should match with that of the target sound field, where θ′ is the azimuth of continuous loudspeaker arrangement. Letting θ′ = θin, the normalized signal amplitude for loudspeakers at an arbitrary azimuth θ′ is given as

As in the case of the preceding chapters, a unit transfer coefficient from the loudspeaker signal to the pressure amplitude of the free-field plane wave at the origin is assumed in Equation (9.1.1), e.g., EA = PA. Therefore, in an ideal reproduction, only the loudspeaker at azimuth θ′ = θS is active, and the other loudspeakers are inactive.

A(θ′, θS) is a periodic function of azimuth θS or θ′ with a period of 2π (360°), so it can be expanded into a complexor real-valued Fourier series within the azimuthal region of (−π, π]:

Analysis of multichannel sound field recording and reconstruction  351

Therefore, the normalized amplitudes of loudspeaker signals in an ideal reproduction can be decomposed into a linear combination of azimuthal harmonics {exp(jqθS)} or {cos(qθS), sin(qθS)} with infinite orders. Complexand real-valued Fourier expansions are mathematically equivalent. The case of real-valued Fourier expansion is discussed in the following.

Equation (9.1.2) is analyzed from the points of multichannel sound recording and reproduction to obtain insights into physical significance. From the point of multichannel sound reproduction, Equation (9.1.2) represents the azimuthal distribution of the normalized amplitude of loudspeaker signals as a function of the azimuth θ′. It also represents the azimuthal distribution function of the normalized amplitude of the free-field plane wave incident to the center of the circle. As a zero-order approximation, the expansion in Equation (9.1.2) is truncated only to the term of q = 0, and other terms are omitted. Then, the normalized signal amplitude of the loudspeaker at arbitrary azimuth θ′ is given as

Equation (9.1.3) is equivalent to presenting the mono signal captured by an omnidirectional microphone to all loudspeakers in reproduction. The sound pressure at the center of the circle is a superposition of plane waves with equal amplitude and phase from the continuous azimuthal directions of all loudspeakers. Therefore, zero-order reproduction cannot recreate the spatial information of a target plane wave, or can create a perceived virtual source at the top direction similar to the case in Section 6.4.3.

As the first-order approximation, the expansion in Equation (9.1.2) is truncated up to the term of q = 1, and higher terms are omitted. Then, the normalized signal amplitude of loudspeakers at arbitrary azimuth θ′ is expressed as

Except for the difference in overall gain, Equation (9.1.4) is directly proportional to the conventional solution of the normalized amplitude of loudspeaker signals for first-order horizontal Ambisonics in Equations (4.3.15) and (4.3.25). Variations in the azimuthal distribution function or horizontal polar pattern of the normalized amplitude with the difference in θS − θ′ = θS − θi between the target and loudspeaker azimuths are illustrated in Section 4.1.3 and Figures 4.4, 4.5, and 4.17. For an arbitrary loudspeaker, the normalized signal magnitude maximizes when the loudspeaker azimuth coincides with the target source azimuth at θ′ = θS. When the loudspeaker azimuth deviates from the target source azimuth, the normalized signal magnitude reduces and gradually vanishes. However, as the loudspeaker azimuth further deviates from the target source azimuth, a weak crosstalk with a reversal phase occurs in the loudspeakers close to the direction opposite to the target source. As proven in

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Equation (4.3.27), at the central listening position and low frequencies, the perceived virtual source direction in the first-order Ambisonic reproduction matches with that of the target source for the fixed head and the head oriented to the virtual source.

As the second-order approximation, the expansion in Equation (9.1.2) is truncated up to the term of q = 2, and the higher terms are omitted. Then, the normalized signal amplitude of loudspeakers at the arbitrary azimuth θ′ is expressed as

A , S 21 1 2 cos cos S 2 sin sin S 2 cos 2 cos 2 S 2 sin 2 sin 2 S . (9.1.5)

Except the difference in the overall gain, Equation (9.1.5) is directly proportional to the conventional solution of the normalized amplitude of loudspeaker signals for the secondorder horizontal Ambisonics in Equations (4.3.53) and (4.3.62). The variation in azimuthal distribution function or horizontal polar pattern of normalized amplitude with the difference θS − θ′ = θS − θi between target and loudspeaker azimuths are illustrated in Figure 4.17. In comparison with the case of the first-order reproduction, the relative signal magnitude of the loudspeaker nearest the target source azimuth increases, and the relative signal magnitudes (crosstalk) of the other loudspeakers decrease. Therefore, the incident power in reproduction is more focused on the target azimuth of θ′ = θS, and the performance of directional information reproduction is improved.

When the expansion in Equation (9.1.2) is truncated up to the term of q = 3 or higher, the conventional solution of loudspeaker signals for the thirdor higher-order horizontal Ambisonics is achieved. As illustrated in Section 4.3.3 and Figure 4.17, as the order increases, the relative signal magnitude of the loudspeaker consistent with the target source direction increases, and the relative signal magnitudes (crosstalk) of the other loudspeakers decrease. Then, the incident power in reproduction is gradually focused on the target azimuth of θ′ = θS. In other words, as the order increases, the approximated reproduction approaches ideal reproduction, and the reproduction of spatial information gradually improves. When the order in Equation (9.1.2) tends to infinity, the approximated reproduction achieves the limitation of ideal reproduction. Generally, when the expansion in Equation (9.1.2) is truncated to an arbitrary-order Q, the normalized amplitude of loudspeaker signals is expressed as

From the points of multichannel sound recording, arbitrary Q ≥ 1 order Ambisonic signals A(θ′, θS) in Equation (9.1.6) are the linear combinations of (2Q + 1) independent signals or azimuthal harmonics. These independent signals can be theoretically recorded using (2Q + 1) coincident directional microphones. As stated in Sections 4.3.2 and 4.3.3, 1, cosθS, and sinθS are three normalized signals recorded with an omnidirectional microphone and two bidirectional microphones with their main axes pointing to the front and left directions, respectively; cosqθS and sinqθS (q ≥ 2) are normalized signals recorded with higher-directional microphones. In the polar patterns of the preceding three-order normalized signal amplitude

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distribution function PA in , f of the unnormalized amplitude of the incident plane wave. The coefficients of the azimuthal Fourier expansion are given as

In Equation (9.1.8), for an arbitrary target sound field, the Q-order Ambisonic-independent or encoding signals can be theoretically recorded by (2Q + 1) coincident microphones with different order directional characteristics. The preceding Q-order azimuthal harmonic components of the target sound field can be recovered from these microphone outputs after decoding. If the target or incident sound field is spatially bandlimited, i.e., all the q > Q-order azimuthal harmonics in the azimuthal Fourier expansion of the azimuthal distribution function PA in , f of the incident plane wave amplitude vanishes, the target sound field can be recovered exactly by using the Q-order Ambisonic signals from decoding outputs. The corresponding equations in the time domain can be obtained by applying an inverse Fourier transform similar to that in Equation (1.2.13) and the above equations in the frequency domain.

The Q-order circular sinc function is defined as

The magnitude of the circular sinc function maximizes at θ′ = θin and spreads to two sides around the center of θ′ = θin. Except a constant gain, the polar patterns of Q = 1, 2, and 3 order circular sinc function are identical to those in Figure 4.17. The first equality on the right side of Equation (9.1.8) can be written as

Equation (9.1.11) indicates that Q-order-reproduced signals are obtained by multiplying a Q-order circular sinc function (Q-order azimuthal sampling function) to PA in , f of the incident plane wave amplitude and then superposing (taking an integral) over all the azimuths. When Q tends to infinite, the circular sinc function reaches the Dirac delta function

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The normalized amplitude of the actual signal of the ith loudspeaker is Ai(θS). For a target plane wave with a unit amplitude, the reproduced sound pressure at the origin should satisfy the following equation:

i 0

Comparing Equation (9.1.16) and (9.1.17) yields

Equation (9.1.18) is the amplitude of the reproduced signals for the Q-order Ambisonics with constant-amplitude normalization given by Equations (4.3.63) and (4.3.80a).Therefore, for reproduction involving a finite number of horizontal loudspeakers with uniform configuration, the Q-order approximation of ideal reproduction leads to the Ambisonic decoding equation, the amplitude of the reproduced signals, and the normalized factor of the overall amplitude. The normalized signal amplitude of the ith loudspeaker in discrete configuration cannot be obtained directly by letting θ′ = θi in A(θ′, θS) for continuous configuration expressed in Equation (9.1.16). An overall gain and normalized factor should be supplemented.

In conclusion, the following statements have been mathematically proven:

1.An ideal horizontal reproduction requires an infinite number of loudspeakers arranged continuously and uniformly on a circle with a far-field radius.

2.The loudspeaker signals for an ideal reproduction can be expanded into an azimuthal Fourier series. The Q-order approximation of the azimuthal Fourier expansion is equivalent to the Q-order horizontal Ambisonic reproduced signals, which are the linear combination of (2Q + 1) independent or encoded signals. The decoding equation and reproduced signals of the conventional solution of Ambisonics is a natural consequence of each order approximation of ideal reproduced signals.

3.As the order Q of Ambisonics increases, the approximated reproduction gradually approaches ideal reproduction, but higher-order reproduction requires more independent signals and becomes complicated.

4.The independent signals of the Q-order Ambisonics, which involve the preceding Q-order azimuthal harmonic components of the target sound field, can be theoretically recorded by (2Q + 1) coincident microphones with appropriate directivities. Decoding can be regarded as beamforming processing.