Higher-order Ambisonics adds degree-n spherical harmonics to increase soundfield spatial resolution at the cost of (n+1)² channels

Spherical harmonics are mathematical functions on the sphere analogous to frequency bins in the Fourier transform: combining more of them allows more precise representation of an arbitrary soundfield. First-order Ambisonics (4 channels) uses the degree-0 omni and three degree-1 bidirectional patterns. Each additional order adds (2n+1) more components: 2nd order adds 5 (total 9 channels), 3rd order adds 7 (total 16 channels), and so on — always (n+1)² total. In the ATK, 3rd order is the most common practical choice, 5th order is the practical maximum. For 2D (horizontal-plane only) decoding, only the outer-edge spherical harmonics are used — those with maximum sensitivity on the horizontal plane — reducing channel count. HOA provides a larger sweet spot and more precise phantom images than FOA, especially with many loudspeakers.

Examples

Channel counts: order 0 = 1, order 1 = 4, order 2 = 9, order 3 = 16, order 5 = 36. HoaOrder.new(3).size returns 16 in SuperCollider.

Assessment

Explain why HOA is described as having a larger sweet spot than FOA. Calculate the number of channels in a 4th-order Ambisonic signal. State the practical processing cost tradeoff.