This section develops a special case of Wave Field Synthesis (WFS) by spatially sampling simple plane waves. Sampling plane waves is much simpler than the traditional WFS formulation which begins with the classical KirchhoffHelmholtz integral (Firtha, 2018; Pierce, 1989). In return for this simplicity, we are restricted to virtual primary sources that are many wavelengths away from the speaker array, and on the other side of it from the listener. As we shall see, we can relax these restrictions in various ways, and the remaining sampling conditions are generally equally binding for WFS systems. In other words, spatial sampling theory is fundamental to all spatial audio systems using discrete drivers arranged in linear, planar, or even more general array geometries. What does not seem to be generally known, however, is that a samplingbased approach is sufficient (and much more to the point) for deriving and optimizing the system, as pursued in this paper.
Figure 2 shows a monochromatic plane wave traveling down and to the right at a 45 degree angle. The solid black line across the middle represents the microphonearray, ideally a uniformly spaced grid of tiny omnidirectional pressure microphones having no ``acoustic shadow'' at all; these microphones serve to sample the plane wave along the line. In the 3D case, the line represents one cut along a planar microphonearray. The sinusoid drawn along the microphone line indicates the pressure seen by each microphone. By the sampling theorem (applied now to spatial sampling using a microphonearray), we must have more than two microphones per wavelength along the line array. Thus, the required microphone density is determined by the minimum incident wavelength and the maximum angle of incidence , as derived below.

Figure 3 illustrates the geometry of the wavelengths involved. The wavelength of the incident sinusoidal plane wave is denoted , and denotes the wavelength of the sinusoidal pressure fluctuation seen by the microphone line array. As Fig.3 makes clear, from the angle of incidence and incident wavelength , we have
(1) 
(2)  
(3) 
For example, choosing kHz and (stage angle 90 degrees), and using m/s for sound speed, we obtain mm, or about halfinch spacing for the microphones. (The coincident speakerarray has the same samplingdensity requirement as the microphonearray.)
Reducing either or relaxes the spatial sampling density requirement. For example, if the ``stage width'' is reduced from 90 degrees ( ) down to 40 degrees ( ), then oneinch spacing of the microphones (and speakers) is allowed. If we bandlimit our reconstruction bandwidth to 5 kHz, then we get by with fourinch spacing, as pursued below in a practical PBAP design (§2.15).
If we don't bandlimit to below the spatial Nyquist limit, then we obtain ``spatial angle aliasing'' at very high frequencies for sources near the left or right edge of the ``stage viewing window''. That is, for sources near the left or right edges of the ``stage'', the highestfrequency components may not appear to come from the same direction as components below the cutoff frequency of 5 kHz. On the other hand, perception is such that the apparent angleofarrival typically may not alias at high frequencies because the desired angle remains a psychoacoustic choice and keeps the whole source spectrum in one place. Sources near the center of the stage are spatially oversampled by the microphonearray at all frequencies, so they are never a problem. In lowpassedwidebandnoise tests (see Appendix A), highfrequency spatial aliasing has been observed to break up a formerly coherent wideband virtual noise source, but not in a manner indicating ``folding'' as in aliasing of tones due to temporal sampling, but instead as the sound of a spurious new noise source somewhere along the array. The psychoacoustics of spatial aliasing perception is a fascinating topic for ongoing research.
http://arxiv.org/abs/1911.07575
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