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Stereo As Two Spatial Samples

It is interesting to consider the woofer upper frequency $ f_1$ in this example from a spatial sampling point of view. Suppose for simplicity that stereo will be used immediately above $ f_1$ . The array width is $ S$ , and for simplicity, consider the effective stereo speaker separation to be $ \Delta = S/2$ when half the speakers comprise the left channel and the other half comprise the right, or $ \Delta \approx S$ when using only the extreme left and right drivers, or various choices in between. Considering the stereo pair as two soundfield samples, stereo should take over for the subwoofer before the wavelength shrinks to $ \lambda_1=2\Delta$ (critical spatial sampling). The formula for $ f_1$ is then

$\displaystyle f_1 = \frac{c}{2\Delta}, \quad \Delta\in\left[\frac{S}{2},S\right].

In the above example with $ S\approx 3.5$ , we obtain $ f_1 = 343/(2
\Delta) \in [49,98]$ Hz, which includes the desirable 80 Hz setting. In the much smaller four-way Huygens Array of Fig.11 on page [*], the stereo speaker separation (center to center) is close to $ S=1/2$ meter, where we obtain the range $ [343,686]$ Hz, which is quite high compared to normal subwoofer crossovers. It is apparent that in typical listening geometries the subwoofer cutoff due to perceptual limits is comparable to that obtained by considering the stereo speakers as a pair of spatial samples needing to be outside each other's ``high-correlation zone.''32

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``A Spatial Sampling Approach to Wave Field Synthesis: PBAP and Huygens Arrays'', by Julius O. Smith III, Published 2019-11-18:
Copyright © 2020-05-15 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University