Soundfield Control with Distributed Modal Constraints Dylan Menzies 1 1 Department of Media Technology, De Montfort University, LE1 9BH, Leicester, UK rdmg@dmu.ac.uk Abstract A previous paper investigated various methods for controlling an interior soundfield with a boundary array of a general shape. Some new proposals were made including one called Distributed Modal Con- straints, which is presented here in more detail. Subjects considered include the control of the com- plete interior of convex and concave boundaries, the control of subregions with optimal source distribu- tions, open boundaries, independent regions, point source targets, and representation and encoding. 1 Introduction We begin by reviewing existing sound-field con- trol theory and methods, with comments on the strengths and weaknesses of each. The method of Distributed Modal Constraints is introduced as a way to combine some of the best aspects. A series of simulations are then described in order to vali- date the method. All the functions and expansions used are provided in detail. Octave code used to generate the simulations can be obtained from the author. 1.1 Ambisonics High Order Ambisonics is based on the Fourier- Bessel expansion (FBE) of a soundfield about a point, either in two or three dimensions [1, 2]. The effectiveness of the expansion is greatly increased by the radial properties of the basis functions. For order N , they are highly suppressed for radius r < N/k in either 2 or 3 dimensions, where k is the wavenumber. This implies that the spherical volume r < N/k is accurately described by N basis functions. There exist, of course, fields in which the orders greater than N are significant compared to the lower orders for r < N/k, and these fields will be referred to as ill-conditioned. A natural ex- ample is the field near to a monopole, however ill- conditioning can occur in many ways. In the decoding process the sum of the FBEs of the contributing loudspeaker sources is matched to the desired FBE, a process generally known as mode-matching. Provided there are enough loud- speakers the accuracy of the reproduced FBE can be controlled accurately, with small interference from unwanted higher orders. Usually the speak- ers are arranged in a circular or spherical pattern, which simplifies and optimizes the decoding pro- cess. If the speaker boundary is non-spherical, it might be desirable to reproduce a soundfield over all parts of the interior. This is not possible how- ever, because the FBE expansion of each source is valid only as far as the radius extending from the FBE centre to the source, necessarily because any FBE field is sourceless. Beyond this the expansion field converges to a very different field compared to the actual field produced by a real source, and so mode matching in this region fails. Fig. 1 shows a (20λ, 5λ) dimension rectangular array, with source spacing λ/2, accurately repro- ducing a 2D FBE of a planewave travelling from the top with N = 20. The first plot shows the real part of the field, the second shows the abso- lute relative error, both with fixed colour keys that will be used for comparisons across all the figures in this article. The error plot clearly shows where the error is above and below 0.1, which is a use- ful practical error threshold. The expansion order limit based on a limited radius equal to half the width is N = (2.5)(2π) ≈ 15, however in this case the order can be pushed a little higher to N = 20, squashing the FBE region against the boundary. The error transition at the FBE edge is swift, as ex- pected. Fig. 2 shows that by N = 26 the solution has deteriorated, and cannot be improved or ex- tended to a wider region by change of order, source 1