412  Spatial Sound

For the (L − 1)-order far-field spatial Ambisonics, if P(r, Ω, f) is a plane wave with a unit amplitude given in Equation (9.3.16) and P′(r, Ω, f) is a truncation of Equation (9.3.17) up to the (L − 1) order, the mean square error caused by truncation is calculated from Equation (9.5.4) as

l 0

Errors are independent from the target plane wave direction. When the radius of a region and the wave number satisfy the condition of Equation (9.3.30) for ideal reconstruction, the mean square error given in Equation (9.5.5) is less than 0.04 or −14 dB (Ward and Abhayapala, 2001). Similar to the case of horizontal reproduction, discrete arrays with a finite number of secondary sources are used in practical spatial Ambisonics. The overall error in the reconstructed sound field is a mix of truncation and mirror spatial-spectral errors. This analysis is similar to that of horizontal Ambisonics (Poletti, 2005b) and thus omitted here.

9.5.2  Discrete secondary source array and spatial-spectral aliasing error in Ambisonics

Errors caused by the truncation of the spatial harmonic decomposition of Ambisonic sound field are discussed in Section 9.5.1, and a mix of truncation and mirror spatial-spectral errors is analyzed. In this section, mirror spatial-spectral error in Ambisonic-reconstructed sound field caused by discrete array of secondary sources is further analyzed (Spors and Rabenstain, 2006). In the case of digital signal processing, converting the time domain signal to the frequency domain signal is convenient for analyzing the mirror frequency spectra and frequency aliasing caused by the discrete sampling of a continuous time signal. Similarly, converting the sound pressure in the spatial domain to that in the spatial-spectral domain is convenient for the analysis in this section. For convenience in mathematical expression, complex-valued Fourier series is expanded here. In Section 9.2.2, the expansions of complexand real-valued Fourier series are equivalent.

For continuous and uniform secondary source arrays on a horizontal circle with radius r0, Equation (9.2.26) gives the relationship of the reconstructed sound pressures, the transfer function from secondary sources to receiver positions, and the driving signals in the spatialspectral (azimuthal spectrum) domain:

A discrete and uniform secondary source array on a horizontal circle is equivalent to applying spatial or azimuthal sampling to the continuous azimuthal function of the driving signal E(θ′, f) of the secondary source along θ′. For a discrete and uniform array of M secondary sources with an azimuthal interval of 2π/M, the azimuthal-sampled driving signal is

Analysis of multichannel sound field recording and reconstruction  413

The spatial-spectral domain representation of driving signals is given by the azimuthal Fourier coefficients in Equation (9.5.7):

v

The spatial or azimuthal spectrum of the sampled driving signal is an infinite repetition of the original azimuthal spectrum with displacements of the integral multiplication of M along q axis, or the q-order azimuthal spectrum component becomes a mix of all original q + vM order azimuthal spectrum components. The azimuthal–spectral domain representation of the reconstructed sound field by discrete secondary source array can be obtained by substituting Eq(f) in Equation (9.5.6) with Esamp, q(f) in Equation (9.5.8):

v

With the azimuthal Fourier representation given in Equation (9.5.9), the reconstructed sound pressure in spatial domain is expressed as

v q

In Equation (9.5.10), the term of v = 0 represents the ideal or target reconstructed sound pressure in Equation (9.5.6), and all terms of v ≠ 0 represent the contributions of mirror spatial or azimuthal spectrum caused by azimuthal sampling. If the azimuthal spectrum of driving signals is azimuthally bandlimited, i.e., for an odd number of discrete secondary sources, the azimuthal spectrum satisfies

for an even number of discrete secondary sources, the azimuthal spectrum satisfies

No overlap exists between target and mirror azimuthal spectra, i.e., no azimuthal spectrum aliasing.

414  Spatial Sound

Even if the target driving signal is azimuthally bandlimited and thus satisfies the anti-alias- ing condition given in Equation (9.5.11a) or (7.5.11b), the mirror azimuthal spectra caused by azimuthal sampling still appear in the reconstructed sound field and consequently lead to error. This observation is because Green’s or transfer functions from secondary sources to receiver positions are not always azimuthally bandlimited, so they may not satisfy the condition of anti-azimuthal-aliasing in some receiver positions. For example, the spatialor azimuthal-spectral domain representation of the transfer function from a secondary plane wave source to the receiver position is expressed in Equation (9.2.15). The reconstructed sound pressure in the spatial domain is given by substituting Equation (9.2.15) into Equation (9.5.10):

v q

If a driving signal is azimuthally bandlimited with azimuthal harmonics up to order Q = max(|q|), then similar to the case in Equation (9.3.14), Jq(kr) oscillates and decays when its order q is not less than [exp(1)kr / 2]. Under the condition given by Equation (9.5.13),

the function Jq+vM(kr) serves as anti-azimuthal-aliasing filtering to remove the influence of mirror azimuthal spectra effectively. Conversely, when the frequency and the distance of the receiver position increase so that Equation (9.5.13) is not satisfied, the mirror azimuthal spectra cause error in reconstructed sound pressure.

Similarly, the azimuthal spectrum representation of the transfer function from a straightline secondary source to the receiver position is given in Equation (9.2.5). The reconstructed sound pressure in the spatial domain is given by substituting Equation (9.2.20) into Equation (9.5.10):

The aforementioned discussion is suitable for reconstruction with straight-line secondary sources.

In Equation (9.5.10), the contribution of v = 0 term is denoted by P′ta(r, θ,r0, f), the contributions of all v ≠ 0 terms are denoted by P′er(r, θ, r0, f). The relative energy error in the reconstructed sound field caused by azimuthal sampling is evaluated by

The above analysis can be extended to spatial Ambisonics, but the spatial spectrum aliasing in spatial Ambisonics is more complicated than horizontal Ambisonics. In addition, the analysis of azimuthal spectrum aliasing in this section is suitable not only for horizontal Ambisonics but also for arbitrary sound field reconstruction with a uniform secondary source