function [A2B pv max_err] = A2B_matrix(Y, X, v_z_factor)
  % fit 4x4 matrix to measured mic data
  %
  % Y is the n x m matrix of measurments (’observations’),
  % n is number of measurement directions
  % m is number of microphones
  % X is the n x 3 matrix of are the direction cosines of the
  % measurment directions (’predictor variables’)
  % pv is the matrix of unknowns to be estimated
  % so Y = X pv
  if (nargin < 2 )
    X = measurement_dirs_z0(size(Y,1));
  end
  X1 = [ ones(size(Y,1), 1) X ];
  pv = X1 \ Y;
  max_err = max (Y - (X1 * pv));
  pv = fill_v_z( pv );
  A2B = [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1] / pv;
end

function X = measurement_dirs_z0(n)
  if (nargin < 1)
    n = 8;
  end
  x_range = 0 : 2*pi/n : 2*pi*(n-1)/n;
  X = [ cos(x_range)' sin(x_range)' zeros(n,1) ];
end

function pv = fill_v_z (pv, signs, elevation)
  if (nargin < 2)
    signs = [ +1 -1 -1 +1 ];
  end;
  if (nargin < 3)
    z_factor = sqrt(2)/2;
  else
    z_factor = tan(elevation);
  end;
  for i = 1:4
    pv(4,i) = signs(i) * sqrt( pv(2,i)^2 + pv(3,i)^2 ) * z_factor;
  end;
end