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Sampling Spreading Loss

In addition to spatial audio oscillations from a point source, there is amplitude change due to ``spreading loss'' away from the source. To look at this, consider that a unit pressure point-source at the origin can be expressed as the real part of (Morse and Ingard, 1968)

$\displaystyle p(r,t) = \frac{e^{j(\omega t - k r)}}{r}

where $ r=\sqrt{x^2+y^2+z^2}$ is the radial coordinate axis. There is no change in the complex amplitude along directions with constant radius $ r$ . Along $ r$ we observe the maximum amplitude change-rate. This rate of change is approached asymptotically along the array in both directions. The pressure gradient is given by

$\displaystyle \nabla p(r,t) \eqsp \frac{\partial}{\partial r} p(r,t)
\eqsp -\left(\frac{1}{r}+jk\right)\,p(r,t)

Intuitively, a sampling grid that is adequate for sampling spatial frequencies $ k$ should be adequate for sampling spherical spreading (decay by $ 1/r$ ) when

$\displaystyle \frac{1}{r} \ll k_{\mbox{max}}
\eqsp \frac{2\pi}{\lambda_{\mbox{min}}}


$\displaystyle r \gg \lambda_{\mbox{min}}. \protect$ (8)

That is, to keep the rate of amplitude-change due to spreading loss much less than that due to acoustic vibration, we can keep all sources a few minimum-wavelengths or more away from the line array. Note that this strategy only provides approximately valid sampling of the $ 1/r$ spreading-loss decay, because $ 1/r$ is not a bandlimited function and therefore cannot be sampled without some error in the reconstruction.20Fortunately, the error can be made zero psychoacoustically at a reasonable sampling density. Amplitude error perception generally requires at least a quarter-dB difference, and that's in the most demanding case of comparing alternating amplitude levels.

Since the speaker spacing $ X$ must be smaller than half the minimum wavelength $ \lambda_{\mbox{min}}$ , we can stipulate that all sources should stay at least several speaker-spacings away from the array.

An alternative strategy to Eq.(8) is to double the linear sampling density of the array and allow amplitude-change due to spreading loss become comparable to that due to vibration. In this case, the minimum approach distance becomes ( $ r \ge 1/k_{\mbox{max}} =
X/\pi$ ), allowing sources get to within $ \approx
\lambda_{\mbox{min}}/6.28$ of the minimum wavelength from the array, or about a third of the center-to-center speaker spacing. Intuitively, thinking of the speaker array as a sampling grid, it makes sense to keep virtual point-sources on the order of a sample away or more.

We learn in a first course on digital signal processing that a signal must be bandlimited to less than half the sampling rate in order to avoid aliasing (Smith, 2007b). Setting a minimum on how close a virtual source may approach the sampling array effectively spatially bandlimits the wavefront geometry. This enables soundfield sampling to work as intended. Analogous bandlimiting happens along the time dimension when we apply an A/D lowpass filter prior to sampling in time.

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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