Phase conventions for spherical harmonics

[mberz]

After checking for consistency, the current implementation of spharpy supports the following phase conventions for real and complex valued spherical harmonics:

Complex-valued including the Condon-Shortley phase

same as in B. Rafaely, Fundamentals of Spherical Array Processing, 2015

import spharpy
import numpy as np

n_max = 3
sampling = spharpy.samplings.equal_area(0, n_points=700)
Y_nm = spharpy.spherical.spherical_harmonic_basis(n_max, sampling)

Real-valued without Condon-Shortley phase

which is the same as the implementation in F. Zotter and M. Frank, Ambisonics, 2019, 10.1007/978-3-030-17207-7

import spharpy
import numpy as np

n_max = 3
sampling = spharpy.samplings.equal_area(0, n_points=700)
Y_nm = spharpy.spherical.spherical_harmonic_basis_real(n_max, sampling)

Currently not implemented

Real-valued including the Condon-Shortley Phase

import spharpy
import numpy as np
from scipy.special import sph_harm

def sph_harm_real(m, n, theta, phi):
    """With Condon-Shortley phase convention"""
    # careful here, scipy uses phi as the elevation angle and
    # theta as the azimuth angle
    Y_nm_cplx = sph_harm(np.abs(m), n, theta, phi)

    if m >= 0:
        Y_nm = np.real(Y_nm_cplx)
        if m != 0:
            Y_nm *= np.sqrt(2)
    else:
        Y_nm = np.imag(Y_nm_cplx)

    return Y_nm

n_max = 3
sampling = spharpy.samplings.equal_area(0, n_points=700)
Y_nm = np.zeros((sampling.cshape[0], (n_max+1)**2), dtype=complex)
for acn in np.arange((n_max+1)**2):
    n, m = spharpy.spherical.acn2nm(acn)
    Y_nm[:, acn] = sph_harm_real(m, n, sampling.azimuth, sampling.colatitude)