A formula for low-frequency interaural level difference (LF-ILD) as a function of source distance and direction is derived from rigid-sphere head-related transfer function (RS-HRTF) theory. Since ILD at low frequencies (typically ƒ < 700Hz) is a dominant cue for near-field distance perception in the free-field, a simple formula akin to the Woodworth formula for high-frequency interaural time difference, may be beneficial to easily and efficiently implement distance- and direction-dependent LF-ILDs for real-time spatial audio applications on devices with limited computational resources. By evaluating the limit as ƒ → 0 of the infinite series representation of the RS-HRTF for finite source distances, an exact, closed-form expression for the “DC gain” as a function of source distance and direction is derived. For a given source location, using this expression to compute the DC gain at the left and right “ears,” and then taking the ratio of the two quantities gives the desired LF-ILD. As an example, it is shown that the derived formula may be used to extrapolate far-field ILD spectra of a rigid sphere to the near-field exactly for low frequencies and with sufficiently high accuracy for higher frequencies. Furthermore, the derived formula, like the Woodworth formula, is well-suited to individualization.
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