# Ambisonics

# 1.1 Order 0

• W (monophonic)

# 1.2 Order 1

Order 0 plus

• Y (positive left, negative right)
• Z (positive up, negative down)
• X (positive front, negative back)

# 1.3 Order N

Order N-1 plus

• 2N+1 degree N polynomials in x, y, z.

# 2.1 Encoder In C

Formulas copy/pasted from:

https://web.archive.org/web/20240420095322/http://www.angelofarina.it/Aurora/HOA_explicit_formulas.htm

May have transcription errors…

void encode(float *w, int order, float x, float y, float z)
{
  float x2, y2, z2, x4, y4, z4, x6, y6, z6;
  if (order >= 0)
  {
    w[0] = 1;
  }
  if (order >= 1)
  {
    w[1] = y;
    w[2] = z;
    w[3] = x;
  }
  if (order >= 2)
  {
    x2 = x * x;
    y2 = y * y;
    z2 = z * z;
    w[4] = sqrtf(3) * x * y;
    w[5] = sqrtf(3) * y * z;
    w[6] = 0.5 * (3 * z2 - 1);
    w[7] = sqrtf(3) * x * z;
    w[8] = 0.5 * sqrtf(3) * (x2 - y2);
  }
  if (order >= 3)
  {
    w[ 9] = sqrtf(5./8.) * y * (3 * x2 - y2);
    w[10] = sqrtf(15) * x * y * z;
    w[11] = sqrtf(3./8.) * y * (5 * z2 - 1);
    w[12] = 0.5 * z * (5 * z2 - 3);
    w[13] = sqrtf(3./8.) * x * (5 * z2 - 1);
    w[14] = sqrtf(15./4.) * z * (x2 - y2);
    w[15] = sqrtf(5./8.) * x * (x2 - 3 * y2);
  }
  if (order >= 4)
  {
    x4 = x2 * x2;
    y4 = y2 * y2;
    z4 = z2 * z2;
    w[16] = sqrtf(35./4.)*x*y*(x2-y2);
    w[17] = sqrtf(35./8.)*y*z*(3*x2-y2);
    w[18] = sqrtf(5./4.)*x*y*(7*z2-1);
    w[19] = sqrtf(5./8.)*y*z*(7*z2-3);
    w[20] = 1./8.*(35*z4-30*z2+3);
    w[21] = sqrtf(5./8.)*x*z*(7*z2-3);
    w[22] = sqrtf(5./16.)*(x2-y2)*(7*z2-1);
    w[23] = sqrtf(35./8.)*x*z*(x2-3*y2);
    w[24] = sqrtf(35./64.)*(x4-6*x2*y2+y4);
  }
  if (order >= 5)
  {
    w[25] = sqrtf(63./128.)*y*(5*x4-10*x2*y2+y4);
    w[26] = sqrtf(315./4.)*x*y*z*(x2-y2);
    w[27] = sqrtf(35./128.)*y*(y4-2*x2*y2-3*x4-8*y2*z2+24*x2*z2);
    w[28] = sqrtf(105./4.)*x*y*z*(2*z2-x2-y2);
    w[29] = sqrtf(15./64.)*y*(x4+2*x2*y2+y4-12*x2*z2-12*y2*z2+8*z4);
    w[30] = 1./8.*z*(63*z4-70*z2+15);
    w[31] = sqrtf(15./64.)*x*(x4+2*x2*y2+y4-12*x2*z2-12*y2*z2+8*z4);
    w[32] = sqrtf(105./16.)*z*(2*x2*z2-2*y2*z2-x4+y4);
    w[33] = sqrtf(35./128.)*x*(2*x2*y2+8*x2*z2-24*y2*z2-x4+3*y4);
    w[34] = sqrtf(315./64.)*z*(x4-6*x2*y2+y4);
    w[35] = sqrtf(63./128.)*x*(x4-10*x2*y2+5*y4);
  }
  if (order >= 6)
  {
    x6 = x2 * x4;
    y6 = y2 * y4;
    z6 = z2 * z4;
    w[36] = 1./8*sqrtf(231./2)*x*y*(3*x4-10*x2*y2+3*y4);
    w[37] = 3./8*sqrtf(77./2)*y*z*(5*x4-10*x2*y2+y4);
    w[38] = 3./4*sqrtf(7)*x*y*(-x4+y4 +10*x2*y2-10*y2*z2);
    w[39] = 1./8*sqrtf(105./2)*y*z*(-9*x4+3*y4-6*x2*y2+24*x2*z2-8*y2*z2);
    w[40] = 1./8*sqrtf(105./2)*x*y*(x4+y4+16*z4+2*x2*y2-16*x2*z2-16*y2*z2);
    w[41] = 1./8*sqrtf(21)*y*z*(5*x4+5*y4+8*z4+10*x2*y2-20*x2*z2-20*y2*z2);
    w[42] = 1./16*(231*z6-315*z4+105*z2-5);
    w[43] = 1./8*sqrtf(21)*x*z*(5*x4+5*y4+8*z4+10*x2*y2-20*x2*z2-20*y2*z2);
    w[44] = 1./16*sqrtf(105./2)*(x6-y6+x4*y2-x2*y4-16*x4*z2+16*x2*z4+16*y4*z2-16*y2*z4);
    w[45] = 1./8*sqrtf(105./2)*x*z*(-3*x4+6*x2*y2+8*x2*z2-24*y2*z2+9*y4);
    w[46] = 3./16*sqrtf(7)*(-x6+5*x4*y2+10*x4*z2+5*x2*y4+10*y4*z2-y6-60*x2*y2*z2);
    w[47] = 3./8*sqrtf(77./2)*x*z*(5*y4-10*x2*y2+x4);
    w[48] = 1./16*sqrtf(231./2)*(x6-15*x4*y2+15*x2*y4-y6);
  }
  if (order >= 7)
  {
    assert(! "high order ambisonics implemented");
    abort();
  }
}