Sparse $\ell _{1}$-Optimal Multiloudspeaker Panning and Its Relation to Vector Base Amplitude Panning

Panning techniques, such as vector base amplitude panning (VBAP), are a widely used practical approach for spatial sound reproduction using multiple loudspeakers. Although limited to a relatively small listening area, they are very efficient and offer good localization accuracy, timbral quality, as well as a graceful degradation of quality outside the sweet spot. The aim of this paper is to investigate optimal sound reproduction techniques that adopt some of the advantageous properties of VBAP, such as the sparsity and the locality of the active loudspeakers for the reproduction of a single audio object. To this end, we state the task of multiloudspeaker panning as an ℓ
₁
optimization problem. We demonstrate and prove that the resulting solutions are exactly sparse. Moreover, we show the effect of adding a nonnegativity constraint on the loudspeaker gains in order to preserve the locality of the panning solution. Adding this constraint, ℓ
₁
-optimal panning can be formulated as a linear program. Using this representation, we prove that unique ℓ
₁
-optimal panning solutions incorporating a nonnegativity constraint are identical to VBAP using a Delaunay triangulation for the loudspeaker setup. Using results from linear programming and duality theory, we describe properties and special cases, such as solution ambiguity, of the VBAP solution.