sfs.fd.esa

Compute ESA driving functions for various systems.

ESA is abbreviation for equivalent scattering approach.

ESA driving functions for an edge-shaped SSD are provided below. Further ESA for different geometries might be added here.

Note that mode-matching (such as NFC-HOA, SDM) are equivalent to ESA in their specific geometries (spherical/circular, planar/linear).

import matplotlib.pyplot as plt
import numpy as np
import sfs

plt.rcParams['figure.figsize'] = 6, 6

f = 343  # Hz
omega = 2 * np.pi * f  # rad / s
k = omega / sfs.default.c  # rad / m

npw = sfs.util.direction_vector(np.radians(-45))
xs = np.array([-0.828427, 0.828427, 0])

grid = sfs.util.xyz_grid([-1, 5], [-5, 1], 0, spacing=0.02)
dx, L = 0.05, 4  # m
N = int(L / dx)
array = sfs.array.edge(N, dx, center=[0, 0, 0],
                       orientation=[0, -1, 0])

xref = np.array([2, -2, 0])
x_norm = np.linalg.norm(xs - xref)
norm_ls = (np.sqrt(8 * np.pi * k * x_norm) *
           np.exp(+1j * np.pi / 4) *
           np.exp(-1j * k * x_norm))
norm_pw = np.exp(+1j * 4*np.pi*np.sqrt(2))


def plot(d, selection, secondary_source, norm_ref):
    # the series expansion is numerically tricky, hence
    d = np.nan_to_num(d)
    # especially handle the origin loudspeaker
    d[N] = 0  # as it tends to nan/inf
    p = sfs.fd.synthesize(d, selection, array, secondary_source, grid=grid)
    sfs.plot2d.amplitude(p * norm_ref, grid)
    sfs.plot2d.loudspeakers(array.x, array.n,
                            selection * array.a, size=0.15)
    plt.xlim(-0.5, 4.5)
    plt.ylim(-4.5, 0.5)
    plt.grid(True)

Functions

line_2d_edge(omega, x0, xs, *[, alpha, Nc, c])

Driving function for 2-dimensional line source with edge ESA.

line_2d_edge_dipole_ssd(omega, x0, xs, *[, ...])

Driving function for 2-dimensional line source with edge dipole ESA.

plane_2d_edge(omega, x0[, n, alpha, Nc, c])

Driving function for 2-dimensional plane wave with edge ESA.

plane_2d_edge_dipole_ssd(omega, x0[, n, ...])

Driving function for 2-dimensional plane wave with edge dipole ESA.

point_25d_edge(omega, x0, xs, *[, xref, ...])

Driving function for 2.5-dimensional point source with edge ESA.
sfs.fd.esa.plane_2d_edge(omega, x0, n=[0, 1, 0], *, alpha=4.71238898038469, Nc=None, c=None)[source]

Driving function for 2-dimensional plane wave with edge ESA.

Driving function for a virtual plane wave using the 2-dimensional ESA for an edge-shaped secondary source distribution consisting of monopole line sources.

Parameters:

  • omega (float) – Angular frequency.

  • x0 (int(N, 3) array_like) – Sequence of secondary source positions.

  • n ((3,) array_like, optional) – Normal vector of synthesized plane wave.

  • alpha (float, optional) – Outer angle of edge.

  • Nc (int, optional) – Number of elements for series expansion of driving function. Estimated if not given.

  • c (float, optional) – Speed of sound

Returns:

  • d ((N,) numpy.ndarray) – Complex weights of secondary sources.

  • selection ((N,) numpy.ndarray) – Boolean array containing True or False depending on whether the corresponding secondary source is “active” or not.

  • secondary_source_function (callable) – A function that can be used to create the sound field of a single secondary source. See sfs.fd.synthesize().

Notes

One leg of the secondary sources has to be located on the x-axis (y0=0), the edge at the origin.

Derived from [SSR16]

Examples

d, selection, secondary_source = sfs.fd.esa.plane_2d_edge(
    omega, array.x, npw, alpha=np.pi*3/2)
plot(d, selection, secondary_source, norm_pw)
_images/sfs-fd-esa-2.svg
sfs.fd.esa.plane_2d_edge_dipole_ssd(omega, x0, n=[0, 1, 0], *, alpha=4.71238898038469, Nc=None, c=None)[source]

Driving function for 2-dimensional plane wave with edge dipole ESA.

Driving function for a virtual plane wave using the 2-dimensional ESA for an edge-shaped secondary source distribution consisting of dipole line sources.

Parameters:

  • omega (float) – Angular frequency.

  • x0 (int(N, 3) array_like) – Sequence of secondary source positions.

  • n ((3,) array_like, optional) – Normal vector of synthesized plane wave.

  • alpha (float, optional) – Outer angle of edge.

  • Nc (int, optional) – Number of elements for series expansion of driving function. Estimated if not given.

  • c (float, optional) – Speed of sound

Returns:

  • d ((N,) numpy.ndarray) – Complex weights of secondary sources.

  • selection ((N,) numpy.ndarray) – Boolean array containing True or False depending on whether the corresponding secondary source is “active” or not.

  • secondary_source_function (callable) – A function that can be used to create the sound field of a single secondary source. See sfs.fd.synthesize().

Notes

One leg of the secondary sources has to be located on the x-axis (y0=0), the edge at the origin.

Derived from [SSR16]

Examples

d, selection, secondary_source = sfs.fd.esa.plane_2d_edge_dipole_ssd(
    omega, array.x, npw, alpha=np.pi*3/2)
plot(d, selection, secondary_source, norm_ref=1)
_images/sfs-fd-esa-3.svg
sfs.fd.esa.line_2d_edge(omega, x0, xs, *, alpha=4.71238898038469, Nc=None, c=None)[source]

Driving function for 2-dimensional line source with edge ESA.

Driving function for a virtual line source using the 2-dimensional ESA for an edge-shaped secondary source distribution consisting of line sources.

Parameters:

  • omega (float) – Angular frequency.

  • x0 (int(N, 3) array_like) – Sequence of secondary source positions.

  • xs ((3,) array_like) – Position of synthesized line source.

  • alpha (float, optional) – Outer angle of edge.

  • Nc (int, optional) – Number of elements for series expansion of driving function. Estimated if not given.

  • c (float, optional) – Speed of sound

Returns:

  • d ((N,) numpy.ndarray) – Complex weights of secondary sources.

  • selection ((N,) numpy.ndarray) – Boolean array containing True or False depending on whether the corresponding secondary source is “active” or not.

  • secondary_source_function (callable) – A function that can be used to create the sound field of a single secondary source. See sfs.fd.synthesize().

Notes

One leg of the secondary sources has to be located on the x-axis (y0=0), the edge at the origin.

Derived from [SSR16]

Examples

d, selection, secondary_source = sfs.fd.esa.line_2d_edge(
    omega, array.x, xs, alpha=np.pi*3/2)
plot(d, selection, secondary_source, norm_ls)
_images/sfs-fd-esa-4.svg
sfs.fd.esa.line_2d_edge_dipole_ssd(omega, x0, xs, *, alpha=4.71238898038469, Nc=None, c=None)[source]

Driving function for 2-dimensional line source with edge dipole ESA.

Driving function for a virtual line source using the 2-dimensional ESA for an edge-shaped secondary source distribution consisting of dipole line sources.

Parameters:

  • omega (float) – Angular frequency.

  • x0 ((N, 3) array_like) – Sequence of secondary source positions.

  • xs ((3,) array_like) – Position of synthesized line source.

  • alpha (float, optional) – Outer angle of edge.

  • Nc (int, optional) – Number of elements for series expansion of driving function. Estimated if not given.

  • c (float, optional) – Speed of sound

Returns:

  • d ((N,) numpy.ndarray) – Complex weights of secondary sources.

  • selection ((N,) numpy.ndarray) – Boolean array containing True or False depending on whether the corresponding secondary source is “active” or not.

  • secondary_source_function (callable) – A function that can be used to create the sound field of a single secondary source. See sfs.fd.synthesize().

Notes

One leg of the secondary sources has to be located on the x-axis (y0=0), the edge at the origin.

Derived from [SSR16]

Examples

d, selection, secondary_source = sfs.fd.esa.line_2d_edge_dipole_ssd(
    omega, array.x, xs, alpha=np.pi*3/2)
plot(d, selection, secondary_source, norm_ref=1)
_images/sfs-fd-esa-5.svg
sfs.fd.esa.point_25d_edge(omega, x0, xs, *, xref=[2, -2, 0], alpha=4.71238898038469, Nc=None, c=None)[source]

Driving function for 2.5-dimensional point source with edge ESA.

Driving function for a virtual point source using the 2.5-dimensional ESA for an edge-shaped secondary source distribution consisting of point sources.

Parameters:

  • omega (float) – Angular frequency.

  • x0 (int(N, 3) array_like) – Sequence of secondary source positions.

  • xs ((3,) array_like) – Position of synthesized line source.

  • xref ((3,) array_like or float) – Reference position or reference distance

  • alpha (float, optional) – Outer angle of edge.

  • Nc (int, optional) – Number of elements for series expansion of driving function. Estimated if not given.

  • c (float, optional) – Speed of sound

Returns:

  • d ((N,) numpy.ndarray) – Complex weights of secondary sources.

  • selection ((N,) numpy.ndarray) – Boolean array containing True or False depending on whether the corresponding secondary source is “active” or not.

  • secondary_source_function (callable) – A function that can be used to create the sound field of a single secondary source. See sfs.fd.synthesize().

Notes

One leg of the secondary sources has to be located on the x-axis (y0=0), the edge at the origin.

Derived from [SSR16]

Examples

d, selection, secondary_source = sfs.fd.esa.point_25d_edge(
    omega, array.x, xs, xref=xref, alpha=np.pi*3/2)
plot(d, selection, secondary_source, norm_ref=1)
_images/sfs-fd-esa-6.svg