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The aforementioned analysis indicates that spatial aliasing occurs when the frequency exceeds the upper limit of anti-spatial-aliasing reconstruction given by the interval of adjacent secondary sources. For the practical interval of adjacent secondary sources, the upper frequency limit is far from the desired value of 20 kHz. Fortunately, as stated in Section 1.6.5, the ITD below 1.5 kHz is a dominant lateral localization cue as long as the wideband stimuli include low–frequency components. Therefore, reproduction yields appropriate localization perception providing that target sound field can be reconstructed within a frequency range up to 1.5 kHz. However, spatial aliasing at high frequency cause timbre coloration in reproduction (Wittek, 2007; Wierstorf et al., 2014; Xie et al., 2015b). Specifically, driving signals are frequency dependent because high-pass filtering is included in Equation (10.1.19) to equalize them. Below the upper frequency limit for anti-spatial-aliasing reproduction, the spectra of the superposed sound pressure caused by all secondary sources are similar to those caused by target sources. Above the upper frequency limit, the spectra of the superposed sound pressure change, and timbre coloration occurs. In practice, a filter can be used to equalize the driving signals of secondary sources below the upper frequency limit for anti-spatial-aliasing reproduction, and a flat response of the filter is chosen above the upper frequency limit. Wittek et al. (2007) also suggested a hybrid reproduction method through which sound components below the frequency of anti-spatial aliasing are reproduced by WFS, and components above that frequency limit are reproduced by conventional stereophonic sound. Psychoacoustic experimental results reveal that the hybrid reproduction method yields a localization performance similar to that of reproduction by WFS only but reduces timbre coloration. The influence of spatial aliasing on WFS is further addressed in Section 10.3.2.
10.1.6 Some issues and related problems on WFS implementation
For an infinite linear array of secondary sources in Section 10.1.3, the target source and receiver region are restricted in half-horizontal planes on the two sides of the array. For a finite linear array of secondary sources in Section 10.1.4, the positions of the target source and the receiver are further restricted. A closed polygonal array of secondary sources composed of multiple finite sub-linear arrays can be used to reconstruct the sound field of the target source at various horizontal azimuths. Figure 10.8 illustrates the example of a rectangular array containing four finite sub-linear arrays. A finite sub-linear array is chosen to reconstruct the sound field according to the position of the target source. When the target source is located at the direction close to a vertex angle of the rectangular array, two adjacent sub-linear arrays may be used to reconstruct a sound field (Section 10.2.2). The curved array of secondary sources may also be used for WFS (Start, 1996), which is addressed in Sections 10.2.2 and 10.2.3.
In practical uses, the driving signals of secondary sources are often created with the modelbased method. For example, in practical music recording, instruments on a stage are divided into some groups according to their positions. The signals of each group of instruments are captured by a spot microphone at a close distance, and the driving signals of secondary sources are synthesized from the output signal of the spot microphones according to the position of the group of instruments. The overall driving signals are obtained from a mixture of these driving signals from each group.
In addition to reconstructing the direct sound caused by target sources, reconstructing a reflected sound field in a target room or hall, not a receiver or listening room, can be performed through WFS. As stated in Section 7.5.5, the information of reflections can be obtained via physical simulation. Under the approximation of geometrical acoustics, reflections in a hall can be modeled by a number of image sources. Therefore, a reflected sound
Spatial sound reproduction by wave field synthesis 453
Figure 10.8 Rectangular array composed of four finite sub-linear arrays.
field can be theoretically reconstructed by simulating all image sources in WFS. However, as the order of reflection increases, the number of image sources increases quickly, thereby causing difficulty in simulation. Nevertheless, this problem can be solved by simulating discrete early reflections through an image source method and simulating the late diffused reverberation via artificial reverberation algorithms (Vries et al., 1994b). Sonke and Vries (1997) proposed succeeding WFS processing. Actually, some considerations and methods in sound field approximation and psychoacoustics have been incorporated into practical WFS.
The driving signals of secondary sources in WFS can also be recorded with an appropriate microphone array, and the original sound field, including direct and reflected sound fields, can be physically reconstructed. This is a data-based method to derive driving signals. According to the basic principle of WFS, a microphone array whose configuration is identical to that of a secondary source array can be used to capture the signals of pressure or medium velocity in the original sound field. A microphone array with a configuration that differs from that of a secondary source array can also be used. In this case, the outputs of the microphone array should be converted to the driving signals of the secondary source array by a signal processing matrix to simulate acoustical transmissions from the positions of microphones to the positions of secondary sources (Berkhout et al., 1993). Or alternatively, the driving signals of secondary sources can be directly derived from the outputs of microphone arrays by signal processing because of the flexibility of WFS. For example, an arbitrary horizontal sound field in a source-free region can be decomposed as a linear superposition of the plane wave from various azimuths. Therefore, the azimuthal–frequency distribution function of the complexvalued amplitude of the incident plane wave can be first recorded and analyzed by a microphone array, and driving signals in WFS are derived from the resultant azimuthal–frequency distribution function to reconstruct incident plane waves from various directions. Hulsebos and Vries (2002) and Hulsebos et al. (2002) analyzed the reconstruction sound field in WFS with driving signals derived from three different microphone arrays, namely, linear, cross, and circular arrays. They found that a linear array with omnidirectional or bidirectional microphones cannot discriminate front and rear incident sound waves. Only the linear array with hypercardioid microphones can discriminate the front and suppress the rear incident sound
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waves. A linear array is invalid when a sound wave is incident from the direction parallel to the array, but this problem can be solved by a cross array with hypercardioid microphones. In addition, finite linear and cross microphone arrays cause an edge effect. In comparison with a linear and cross array, a circular microphone array discussed in Section 9.8.1 is more appropriate. If three-dimensional information is recorded, the spherical microphone array described in Section 9.8.2 is appropriate.
In addition to onsite recording, reflections of a target sound field can be simulated by convoluting with spatial room impulse responses similar to that in Section 7.5.5. The spatial room impulse responses of a target hall or room are initially measured by a microphone array or obtained through calculation and subsequently converted to impulse responses for secondary sources. The driving signals of secondary sources are obtained by convoluting the input stimulus with the impulse responses for secondary sources. A method similar to DiRAC in Section 7.6 is also applicable to WFS (Gauthier et al., 2014a, 2014b). The spatial information of the original sound field is recorded and analyzed by an appropriate microphone array. Then, the driving signals in WFS are simulated according to the parameters of the original sound field obtained from the analysis.
Similar to the case of a multichannel sound in Section 7.4.3, WFS is applicable to recreating a moving virtual source. As a direct method, piecewise static simulation is applied through which a moving virtual source is simulated as a series of static virtual sources in each short period. The driving signals in different short periods change according to the temporary position of the target source. However, piecewise static simulation causes some problems (Franck et al., 2007). One of the problems is time-variant coloration. According to Equation (10.1.29), the upper frequency limit of anti-spatial aliasing depends on the target source direction. Variations in the target source direction lead to a time-variant upper frequency limit and thus time-variant spatial aliasing. Nevertheless, this problem is solved by reducing the interval between adjacent secondary sources. Other problems include the need for a fractional delay in signals to simulate moving virtual sources, errors in the simulation of a Doppler frequency shift, and spectral broadening in a source signal, which is difficult to be overcome. Some methods, such as those focusing on the Doppler frequency shift and simulating a target source with complex radiation characteristics (Ahrens and Spors, 2008a, 2011), deriving signals via the spatial spectral division method, and applying the stationary phase method in the time domain (Firtha and Fiala, 2015a, 2015b), have been proposed to improve the simulation of a moving virtual source in WFS.
Similar to the upmixing of multichannel sound signals in Section 8.3, signal blind separation and extraction have been suggested separating source signals in stereophonic signals and create the driving signals of WFS (Cobos and Lopez, 2009).
In foregoing discussions, secondary sources are supposed to be ideal monopole point sources. Although practical loudspeaker systems possess a certain directivity, the influence of directivity can be compensated by the inverse filtering 1/ΓS(Φ, f) to the driving signals in Equation (10.1.22), where ΓS(Φ, f) is the frequency-dependent directivity of the loudspeaker system, and Φ is the angle between the secondary source-to-receiver connection line and the inward-normal direction of the array (Vries, 1996). However, such a compensation is valid for receiver positions at a special direction. Another study has recommended using multiactuator panels (MAPs) as the secondary source array of WFS (Boone, 2004). The advantage of MAPs is that they can create uniform sound radiation within wide frequency range and spatial region. They also satisfy the visual requirement (Pueo et al., 2010).
The group at the Delft University of Technology explored possible WFS applications (Boone and Verheijen, 1998), including commercial cinema, virtual reality theaters, and teleconference systems. WFS can also be applied to sound reinforcement (Vries et al., 1994a). Some applications of WFS are described in Chapter 16.