"""Compute driving functions for various systems.
.. include:: math-definitions.rst
"""
import numpy as np
from numpy.core.umath_tests import inner1d # element-wise inner product
from scipy.special import jn, hankel2
from .. import util
from .. import defs
[docs]def wfs_2d_line(omega, x0, n0, xs, c=None):
"""Line source by 2-dimensional WFS.
::
D(x0,k) = j/2 k (x0-xs) n0 / |x0-xs| * H1(k |x0-xs|)
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
xs = util.asarray_1d(xs)
k = util.wavenumber(omega, c)
ds = x0 - xs
r = np.linalg.norm(ds, axis=1)
return -1j/2 * k * inner1d(ds, n0) / r * hankel2(1, k * r)
def _wfs_point(omega, x0, n0, xs, c=None):
"""Point source by two- or three-dimensional WFS.
::
(x0-xs) n0
D(x0,k) = j k ------------- e^(-j k |x0-xs|)
|x0-xs|^(3/2)
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
xs = util.asarray_1d(xs)
k = util.wavenumber(omega, c)
ds = x0 - xs
r = np.linalg.norm(ds, axis=1)
return 1j * k * inner1d(ds, n0) / r ** (3 / 2) * np.exp(-1j * k * r)
wfs_2d_point = _wfs_point
[docs]def wfs_25d_point(omega, x0, n0, xs, xref=[0, 0, 0], c=None, omalias=None):
"""Point source by 2.5-dimensional WFS.
::
____________ (x0-xs) n0
D(x0,k) = \|j k |xref-x0| ------------- e^(-j k |x0-xs|)
|x0-xs|^(3/2)
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
xs = util.asarray_1d(xs)
xref = util.asarray_1d(xref)
k = util.wavenumber(omega, c)
ds = x0 - xs
r = np.linalg.norm(ds, axis=1)
return wfs_25d_preeq(omega, omalias, c) * \
np.sqrt(np.linalg.norm(xref - x0)) * inner1d(ds, n0) / \
r ** (3 / 2) * np.exp(-1j * k * r)
wfs_3d_point = _wfs_point
def _wfs_plane(omega, x0, n0, n=[0, 1, 0], c=None):
"""Plane wave by two- or three-dimensional WFS.
Eq.(17) from :cite:`Spors2008`::
D(x0,k) = j k n n0 e^(-j k n x0)
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
n = util.normalize_vector(n)
k = util.wavenumber(omega, c)
return 2j * k * np.inner(n, n0) * np.exp(-1j * k * np.inner(n, x0))
wfs_2d_plane = _wfs_plane
[docs]def wfs_25d_plane(omega, x0, n0, n=[0, 1, 0], xref=[0, 0, 0], c=None,
omalias=None):
"""Plane wave by 2.5-dimensional WFS.
::
____________
D_2.5D(x0,w) = \|j k |xref-x0| n n0 e^(-j k n x0)
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
n = util.normalize_vector(n)
xref = util.asarray_1d(xref)
k = util.wavenumber(omega, c)
return wfs_25d_preeq(omega, omalias, c) * \
np.sqrt(2*np.pi * np.linalg.norm(xref - x0)) * \
np.inner(n, n0) * np.exp(-1j * k * np.inner(n, x0))
wfs_3d_plane = _wfs_plane
def _wfs_focused(omega, x0, n0, xs, c=None):
"""Focused source by two- or three-dimensional WFS.
::
(x0-xs) n0
D(x0,k) = j k ------------- e^(j k |x0-xs|)
|x0-xs|^(3/2)
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
xs = util.asarray_1d(xs)
k = util.wavenumber(omega, c)
ds = x0 - xs
r = np.linalg.norm(ds, axis=1)
return 1j * k * inner1d(ds, n0) / r ** (3 / 2) * np.exp(1j * k * r)
wfs_2d_focused = _wfs_focused
[docs]def wfs_25d_focused(omega, x0, n0, xs, xref=[0, 0, 0], c=None, omalias=None):
"""Focused source by 2.5-dimensional WFS.
::
____________ (x0-xs) n0
D(x0,w) = \|j k |xref-x0| ------------- e^(j k |x0-xs|)
|x0-xs|^(3/2)
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
xs = util.asarray_1d(xs)
xref = util.asarray_1d(xref)
k = util.wavenumber(omega, c)
ds = x0 - xs
r = np.linalg.norm(ds, axis=1)
return wfs_25d_preeq(omega, omalias, c) * \
np.sqrt(np.linalg.norm(xref - x0)) * inner1d(ds, n0) / \
r ** (3 / 2) * np.exp(1j * k * r)
wfs_3d_focused = _wfs_focused
[docs]def wfs_25d_preeq(omega, omalias, c):
"""Preqeualization for 2.5D WFS."""
if omalias is None:
return np.sqrt(1j * util.wavenumber(omega, c))
else:
if omega <= omalias:
return np.sqrt(1j * util.wavenumber(omega, c))
else:
return np.sqrt(1j * util.wavenumber(omalias, c))
[docs]def delay_3d_plane(omega, x0, n0, n=[0, 1, 0], c=None):
"""Plane wave by simple delay of secondary sources."""
x0 = util.asarray_of_rows(x0)
n = util.normalize_vector(n)
k = util.wavenumber(omega, c)
return np.exp(-1j * k * np.inner(n, x0))
[docs]def source_selection_plane(n0, n):
"""Secondary source selection for a plane wave.
Eq.(13) from :cite:`Spors2008`
"""
n0 = util.asarray_of_rows(n0)
n = util.normalize_vector(n)
return np.inner(n, n0) >= defs.selection_tolerance
[docs]def source_selection_point(n0, x0, xs):
"""Secondary source selection for a point source.
Eq.(15) from :cite:`Spors2008`
"""
n0 = util.asarray_of_rows(n0)
x0 = util.asarray_of_rows(x0)
xs = util.asarray_1d(xs)
ds = x0 - xs
return inner1d(ds, n0) >= defs.selection_tolerance
[docs]def source_selection_line(n0, x0, xs):
"""Secondary source selection for a line source.
compare Eq.(15) from :cite:`Spors2008`
"""
return source_selection_point(n0, x0, xs)
[docs]def source_selection_focused(ns, x0, xs):
"""Secondary source selection for a focused source.
Eq.(2.78) from :cite:`Wierstorf2014`
"""
x0 = util.asarray_of_rows(x0)
xs = util.asarray_1d(xs)
ns = util.normalize_vector(ns)
ds = xs - x0
return inner1d(ns, ds) >= defs.selection_tolerance
[docs]def source_selection_all(N):
"""Select all secondary sources."""
return np.ones(N, dtype=bool)
[docs]def nfchoa_2d_plane(omega, x0, r0, n=[0, 1, 0], max_order=None, c=None):
r"""Plane wave by two-dimensional NFC-HOA.
.. math::
D(\phi_0, \omega) =
-\frac{2\i}{\pi r_0}
\sum_{m=-M}^M
\frac{\i^{-m}}{\Hankel{2}{m}{\wc r_0}}
\e{\i m (\phi_0 - \phi_\text{pw})}
See http://sfstoolbox.org/#equation-D.nfchoa.pw.2D.
"""
x0 = util.asarray_of_rows(x0)
k = util.wavenumber(omega, c)
n = util.normalize_vector(n)
phi, _, r = util.cart2sph(*n)
phi0 = util.cart2sph(*x0.T)[0]
M = _max_order_circular_harmonics(len(x0), max_order)
d = 0
for m in range(-M, M + 1):
d += 1j**-m / hankel2(m, k * r0) * np.exp(1j * m * (phi0 - phi))
return -2j / (np.pi*r0) * d
[docs]def nfchoa_25d_point(omega, x0, r0, xs, max_order=None, c=None):
r"""Point source by 2.5-dimensional NFC-HOA.
.. math::
D(\phi_0, \omega) =
\frac{1}{2 \pi r_0}
\sum_{m=-M}^M
\frac{\hankel{2}{|m|}{\wc r}}{\hankel{2}{|m|}{\wc r_0}}
\e{\i m (\phi_0 - \phi)}
See http://sfstoolbox.org/#equation-D.nfchoa.ps.2.5D.
"""
x0 = util.asarray_of_rows(x0)
k = util.wavenumber(omega, c)
xs = util.asarray_1d(xs)
phi, _, r = util.cart2sph(*xs)
phi0 = util.cart2sph(*x0.T)[0]
M = _max_order_circular_harmonics(len(x0), max_order)
hr = util.spherical_hn2(range(0, M + 1), k * r)
hr0 = util.spherical_hn2(range(0, M + 1), k * r0)
d = 0
for m in range(-M, M + 1):
d += hr[abs(m)] / hr0[abs(m)] * np.exp(1j * m * (phi0 - phi))
return d / (2 * np.pi * r0)
[docs]def nfchoa_25d_plane(omega, x0, r0, n=[0, 1, 0], max_order=None, c=None):
r"""Plane wave by 2.5-dimensional NFC-HOA.
.. math::
D(\phi_0, \omega) =
\frac{2\i}{r_0}
\sum_{m=-M}^M
\frac{\i^{-|m|}}{\wc \hankel{2}{|m|}{\wc r_0}}
\e{\i m (\phi_0 - \phi_\text{pw})}
See http://sfstoolbox.org/#equation-D.nfchoa.pw.2.5D.
"""
x0 = util.asarray_of_rows(x0)
k = util.wavenumber(omega, c)
n = util.normalize_vector(n)
phi, _, r = util.cart2sph(*n)
phi0 = util.cart2sph(*x0.T)[0]
M = _max_order_circular_harmonics(len(x0), max_order)
d = 0
hn2 = util.spherical_hn2(range(0, M + 1), k * r0)
for m in range(-M, M + 1):
d += 1j**-abs(m) / (k * hn2[abs(m)]) * np.exp(1j * m * (phi0 - phi))
return -2 / r0 * d
[docs]def sdm_2d_line(omega, x0, n0, xs, c=None):
"""Line source by two-dimensional SDM.
The secondary sources have to be located on the x-axis (y0=0).
Derived from :cite:`Spors2009`, Eq.(9), Eq.(4)::
D(x0,k) =
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
xs = util.asarray_1d(xs)
k = util.wavenumber(omega, c)
ds = x0 - xs
r = np.linalg.norm(ds, axis=1)
return - 1j/2 * k * xs[1] / r * hankel2(1, k * r)
[docs]def sdm_2d_plane(omega, x0, n0, n=[0, 1, 0], c=None):
"""Plane wave by two-dimensional SDM.
The secondary sources have to be located on the x-axis (y0=0).
Derived from :cite:`Ahrens2012`, Eq.(3.73), Eq.(C.5), Eq.(C.11)::
D(x0,k) = kpw,y * e^(-j*kpw,x*x)
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
n = util.normalize_vector(n)
k = util.wavenumber(omega, c)
return k * n[1] * np.exp(-1j * k * n[0] * x0[:, 0])
[docs]def sdm_25d_plane(omega, x0, n0, n=[0, 1, 0], xref=[0, 0, 0], c=None):
"""Plane wave by 2.5-dimensional SDM.
The secondary sources have to be located on the x-axis (y0=0).
Eq.(3.79) from :cite:`Ahrens2012`::
D_2.5D(x0,w) =
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
n = util.normalize_vector(n)
xref = util.asarray_1d(xref)
k = util.wavenumber(omega, c)
return 4j * np.exp(-1j*k*n[1]*xref[1]) / hankel2(0, k*n[1]*xref[1]) * \
np.exp(-1j*k*n[0]*x0[:, 0])
[docs]def sdm_25d_point(omega, x0, n0, xs, xref=[0, 0, 0], c=None):
"""Point source by 2.5-dimensional SDM.
The secondary sources have to be located on the x-axis (y0=0).
Driving funcnction from :cite:`Spors2010`, Eq.(24)::
D(x0,k) =
"""
x0 = util.asarray_of_rows(x0)
n0 = util.asarray_of_rows(n0)
xs = util.asarray_1d(xs)
xref = util.asarray_1d(xref)
k = util.wavenumber(omega, c)
ds = x0 - xs
r = np.linalg.norm(ds, axis=1)
return 1/2 * 1j * k * np.sqrt(xref[1] / (xref[1] - xs[1])) * \
xs[1] / r * hankel2(1, k * r)
[docs]def esa_edge_2d_plane(omega, x0, n=[0, 1, 0], alpha=3/2*np.pi, Nc=None, c=None):
"""Plane wave by two-dimensional ESA for an edge-shaped secondary source
distribution consisting of monopole line sources.
One leg of the secondary sources has to be located on the x-axis (y0=0),
the edge at the origin.
Derived from :cite:`Spors2016`
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
-------
(N,) numpy.ndarray
Complex weights of secondary sources.
"""
x0 = np.asarray(x0)
n = util.normalize_vector(n)
k = util.wavenumber(omega, c)
phi_s = np.arctan2(n[1], n[0]) + np.pi
L = x0.shape[0]
r = np.linalg.norm(x0, axis=1)
phi = np.arctan2(x0[:, 1], x0[:, 0])
phi = np.where(phi < 0, phi+2*np.pi, phi)
if Nc is None:
Nc = np.ceil(2 * k * np.max(r) * alpha/np.pi)
epsilon = np.ones(Nc) # weights for series expansion
epsilon[0] = 2
d = np.zeros(L, dtype=complex)
for m in np.arange(Nc):
nu = m*np.pi/alpha
d = d + 1/epsilon[m] * np.exp(1j*nu*np.pi/2) * np.sin(nu*phi_s) \
* np.cos(nu*phi) * nu/r * jn(nu, k*r)
d[phi > 0] = -d[phi > 0]
return 4*np.pi/alpha * d
[docs]def esa_edge_dipole_2d_plane(omega, x0, n=[0, 1, 0], alpha=3/2*np.pi, Nc=None, c=None):
"""Plane wave by two-dimensional ESA for an edge-shaped secondary source
distribution consisting of dipole line sources.
One leg of the secondary sources has to be located on the x-axis (y0=0),
the edge at the origin.
Derived from :cite:`Spors2016`
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
-------
(N,) numpy.ndarray
Complex weights of secondary sources.
"""
x0 = np.asarray(x0)
n = util.normalize_vector(n)
k = util.wavenumber(omega, c)
phi_s = np.arctan2(n[1], n[0]) + np.pi
L = x0.shape[0]
r = np.linalg.norm(x0, axis=1)
phi = np.arctan2(x0[:, 1], x0[:, 0])
phi = np.where(phi < 0, phi+2*np.pi, phi)
if Nc is None:
Nc = np.ceil(2 * k * np.max(r) * alpha/np.pi)
epsilon = np.ones(Nc) # weights for series expansion
epsilon[0] = 2
d = np.zeros(L, dtype=complex)
for m in np.arange(Nc):
nu = m*np.pi/alpha
d = d + 1/epsilon[m] * np.exp(1j*nu*np.pi/2) * np.cos(nu*phi_s) \
* np.cos(nu*phi) * jn(nu, k*r)
return 4*np.pi/alpha * d
[docs]def esa_edge_2d_line(omega, x0, xs, alpha=3/2*np.pi, Nc=None, c=None):
"""Line source by two-dimensional ESA for an edge-shaped secondary source
distribution constisting of monopole line sources.
One leg of the secondary sources have to be located on the x-axis (y0=0),
the edge at the origin.
Derived from :cite:`Spors2016`
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
-------
(N,) numpy.ndarray
Complex weights of secondary sources.
"""
x0 = np.asarray(x0)
k = util.wavenumber(omega, c)
phi_s = np.arctan2(xs[1], xs[0])
if phi_s < 0:
phi_s = phi_s + 2*np.pi
r_s = np.linalg.norm(xs)
L = x0.shape[0]
r = np.linalg.norm(x0, axis=1)
phi = np.arctan2(x0[:, 1], x0[:, 0])
phi = np.where(phi < 0, phi+2*np.pi, phi)
if Nc is None:
Nc = np.ceil(2 * k * np.max(r) * alpha/np.pi)
epsilon = np.ones(Nc) # weights for series expansion
epsilon[0] = 2
d = np.zeros(L, dtype=complex)
idx = (r <= r_s)
for m in np.arange(Nc):
nu = m*np.pi/alpha
f = 1/epsilon[m] * np.sin(nu*phi_s) * np.cos(nu*phi) * nu/r
d[idx] = d[idx] + f[idx] * jn(nu, k*r[idx]) * hankel2(nu, k*r_s)
d[~idx] = d[~idx] + f[~idx] * jn(nu, k*r_s) * hankel2(nu, k*r[~idx])
d[phi > 0] = -d[phi > 0]
return -1j*np.pi/alpha * d
[docs]def esa_edge_25d_point(omega, x0, xs, xref=[2, -2, 0], alpha=3/2*np.pi, Nc=None, c=None):
"""Point source by 2.5-dimensional ESA for an edge-shaped secondary source
distribution constisting of monopole line sources.
One leg of the secondary sources have to be located on the x-axis (y0=0),
the edge at the origin.
Derived from :cite:`Spors2016`
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
-------
(N,) numpy.ndarray
Complex weights of secondary sources.
"""
x0 = np.asarray(x0)
xs = np.asarray(xs)
xref = np.asarray(xref)
if np.isscalar(xref):
a = np.linalg.norm(xref)/np.linalg.norm(xref-xs)
else:
a = np.linalg.norm(xref-x0, axis=1)/np.linalg.norm(xref-xs)
return 1j*np.sqrt(a) * esa_edge_2d_line(omega, x0, xs, alpha=alpha, Nc=Nc, c=c)
[docs]def esa_edge_dipole_2d_line(omega, x0, xs, alpha=3/2*np.pi, Nc=None, c=None):
"""Line source by two-dimensional ESA for an edge-shaped secondary source
distribution constisting of dipole line sources.
One leg of the secondary sources have to be located on the x-axis (y0=0),
the edge at the origin.
Derived from :cite:`Spors2016`
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
-------
(N,) numpy.ndarray
Complex weights of secondary sources.
"""
x0 = np.asarray(x0)
k = util.wavenumber(omega, c)
phi_s = np.arctan2(xs[1], xs[0])
if phi_s < 0:
phi_s = phi_s + 2*np.pi
r_s = np.linalg.norm(xs)
L = x0.shape[0]
r = np.linalg.norm(x0, axis=1)
phi = np.arctan2(x0[:, 1], x0[:, 0])
phi = np.where(phi < 0, phi+2*np.pi, phi)
if Nc is None:
Nc = np.ceil(2 * k * np.max(r) * alpha/np.pi)
epsilon = np.ones(Nc) # weights for series expansion
epsilon[0] = 2
d = np.zeros(L, dtype=complex)
idx = (r <= r_s)
for m in np.arange(Nc):
nu = m*np.pi/alpha
f = 1/epsilon[m] * np.cos(nu*phi_s) * np.cos(nu*phi)
d[idx] = d[idx] + f[idx] * jn(nu, k*r[idx]) * hankel2(nu, k*r_s)
d[~idx] = d[~idx] + f[~idx] * jn(nu, k*r_s) * hankel2(nu, k*r[~idx])
return -1j*np.pi/alpha * d
def _max_order_circular_harmonics(N, max_order):
"""Compute order of 2D HOA."""
return N // 2 if max_order is None else max_order