def _focused(omega, x0, n0, xs, ns, *, c=None):
r"""Driving function for 2/3-dimensional WFS for a focused source.
Parameters
----------
omega : float
Angular frequency of focused source.
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of normal vectors of secondary sources.
xs : (3,) array_like
Position of focused source.
ns : (3,) array_like
Direction of focused source.
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
-----
.. math::
D(\x_0,\w) = \i\wc \frac{\scalarprod{\x_0-\x_\text{s}}{\n_0}}
{|\x_0-\x_\text{s}|^\frac{3}{2}}
\e{\i\wc |\x_0-\x_\text{s}|}
Examples
--------
.. plot::
:context: close-figs
d, selection, secondary_source = sfs.fd.wfs.focused_3d(
omega, array.x, array.n, xs_focused, ns_focused)
plot(d, selection, secondary_source)
"""
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)
d = 1j * k * _inner1d(ds, n0) / r ** (3 / 2) * _np.exp(1j * k * r)
selection = _util.source_selection_focused(ns, x0, xs)
return d, selection, _secondary_source_point(omega, c)
def _focused(omega, x0, n0, xs, ns, *, c=None):
r"""Driving function for 2/3-dimensional WFS for a focused source.
Parameters
----------
omega : float
Angular frequency of focused source.
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of normal vectors of secondary sources.
xs : (3,) array_like
Position of focused source.
ns : (3,) array_like
Direction of focused source.
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
-----
.. math::
D(\x_0,\w) = \i\wc \frac{\scalarprod{\x_0-\x_\text{s}}{\n_0}}
{|\x_0-\x_\text{s}|^\frac{3}{2}}
\e{\i\wc |\x_0-\x_\text{s}|}
Examples
--------
.. plot::
:context: close-figs
d, selection, secondary_source = sfs.fd.wfs.focused_3d(
omega, array.x, array.n, xs_focused, ns_focused)
plot(d, selection, secondary_source)
"""
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)
d = 1j * k * _inner1d(ds, n0) / r**(3 / 2) * _np.exp(1j * k * r)
selection = _util.source_selection_focused(ns, x0, xs)
return d, selection, _secondary_source_point(omega, c)
def plane_25d(x0, n0, n=[0, 1, 0], xref=[0, 0, 0], c=None):
r"""Plane wave model by 2.5-dimensional WFS.
Parameters
----------
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of secondary source orientations.
n : (3,) array_like, optional
Normal vector (propagation direction) of synthesized plane wave.
xref : (3,) array_like, optional
Reference position
c : float, optional
Speed of sound
Returns
-------
delays : (N,) numpy.ndarray
Delays of secondary sources in seconds.
weights : (N,) numpy.ndarray
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.td.synthesize()`.
Notes
-----
2.5D correction factor
.. math::
g_0 = \sqrt{2 \pi |x_\mathrm{ref} - x_0|}
d using a plane wave as source model
.. math::
d_{2.5D}(x_0,t) = h(t)
2 g_0 \scalarprod{n}{n_0}
\dirac{t - \frac{1}{c} \scalarprod{n}{x_0}}
with wfs(2.5D) prefilter h(t), which is not implemented yet.
See :sfs:`d_wfs/#equation-td-wfs-plane-25d`
Examples
--------
.. plot::
:context: close-figs
delays, weights, selection, secondary_source = \
sfs.td.wfs.plane_25d(array.x, array.n, npw)
d = sfs.td.wfs.driving_signals(delays, weights, signal)
plot(d, selection, secondary_source)
"""
if c is None:
c = _default.c
x0 = _util.asarray_of_rows(x0)
n0 = _util.asarray_of_rows(n0)
n = _util.normalize_vector(n)
xref = _util.asarray_1d(xref)
g0 = _np.sqrt(2 * _np.pi * _np.linalg.norm(xref - x0, axis=1))
delays = _inner1d(n, x0) / c
weights = 2 * g0 * _inner1d(n, n0)
selection = _util.source_selection_plane(n0, n)
return delays, weights, selection, _secondary_source_point(c)
def focused_25d(x0, n0, xs, ns, xref=[0, 0, 0], c=None):
r"""Point source by 2.5-dimensional WFS.
Parameters
----------
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of secondary source orientations.
xs : (3,) array_like
Virtual source position.
ns : (3,) array_like
Normal vector (propagation direction) of focused source.
This is used for secondary source selection,
see `sfs.util.source_selection_focused()`.
xref : (3,) array_like, optional
Reference position
c : float, optional
Speed of sound
Returns
-------
delays : (N,) numpy.ndarray
Delays of secondary sources in seconds.
weights: (N,) numpy.ndarray
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.td.synthesize()`.
Notes
-----
2.5D correction factor
.. math::
g_0 = \sqrt{\frac{|x_\mathrm{ref} - x_0|}
{|x_0-x_s| + |x_\mathrm{ref}-x_0|}}
d using a point source as source model
.. math::
d_{2.5D}(x_0,t) = h(t)
\frac{g_0 \scalarprod{(x_0 - x_s)}{n_0}}
{|x_0 - x_s|^{3/2}}
\dirac{t + \frac{|x_0 - x_s|}{c}}
with wfs(2.5D) prefilter h(t), which is not implemented yet.
See :sfs:`d_wfs/#equation-td-wfs-focused-25d`
Examples
--------
.. plot::
:context: close-figs
delays, weights, selection, secondary_source = \
sfs.td.wfs.focused_25d(array.x, array.n, xf, nf)
d = sfs.td.wfs.driving_signals(delays, weights, signal)
plot(d, selection, secondary_source, t=tf)
"""
if c is None:
c = _default.c
x0 = _util.asarray_of_rows(x0)
n0 = _util.asarray_of_rows(n0)
xs = _util.asarray_1d(xs)
xref = _util.asarray_1d(xref)
ds = x0 - xs
r = _np.linalg.norm(ds, axis=1)
g0 = _np.sqrt(_np.linalg.norm(xref - x0, axis=1)
/ (_np.linalg.norm(xref - x0, axis=1) + r))
delays = -r/c
weights = g0 * _inner1d(ds, n0) / (2 * _np.pi * r**(3/2))
selection = _util.source_selection_focused(ns, x0, xs)
return delays, weights, selection, _secondary_source_point(c)
def point_25d_legacy(omega, x0, n0, xs, xref=[0, 0, 0], c=None, omalias=None):
r"""Driving function for 2.5-dimensional WFS for a virtual point source.
.. versionadded:: 0.5
`point_25d()` was renamed to `point_25d_legacy()` (and a new
function with the name `point_25d()` was introduced). See notes for
further details.
Parameters
----------
omega : float
Angular frequency of point source.
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of normal vectors of secondary sources.
xs : (3,) array_like
Position of virtual point source.
xref : (3,) array_like, optional
Reference point for synthesized sound field.
c : float, optional
Speed of sound.
omalias: float, optional
Angular frequency where spatial aliasing becomes prominent.
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
-----
`point_25d_legacy()` derives 2.5D WFS from the 2D
Neumann-Rayleigh integral (i.e. the approach by Rabenstein & Spors), cf.
:cite:`Spors2008`.
.. math::
D(\x_0,\w) = \sqrt{\i\wc |\x_\text{ref}-\x_0|}
\frac{\scalarprod{\x_0-\x_\text{s}}{\n_0}}
{|\x_0-\x_\text{s}|^\frac{3}{2}}
\e{-\i\wc |\x_0-\x_\text{s}|}
The theoretical link of `point_25d()` and `point_25d_legacy()` was
introduced as *unified WFS framework* in :cite:`Firtha2017`.
Also cf. Eq. (2.145)-(2.147) :cite:`Schultz2016`.
Examples
--------
.. plot::
:context: close-figs
d, selection, secondary_source = sfs.fd.wfs.point_25d_legacy(
omega, array.x, array.n, xs)
normalize_gain = np.linalg.norm(xs)
plot(normalize_gain * d, selection, secondary_source)
"""
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)
d = (
preeq_25d(omega, omalias, c) *
_np.sqrt(_np.linalg.norm(xref - x0)) * _inner1d(ds, n0) /
r ** (3 / 2) * _np.exp(-1j * k * r))
selection = _util.source_selection_point(n0, x0, xs)
return d, selection, _secondary_source_point(omega, c)
def point_25d(omega, x0, n0, xs, xref=[0, 0, 0], c=None, omalias=None):
r"""Driving function for 2.5-dimensional WFS of a virtual point source.
.. versionchanged:: 0.5
see notes, old handling of `point_25d()` is now `point_25d_legacy()`
Parameters
----------
omega : float
Angular frequency of point source.
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of normal vectors of secondary sources.
xs : (3,) array_like
Position of virtual point source.
xref : (3,) array_like, optional
Reference point xref or contour xref(x0) for amplitude correct
synthesis.
c : float, optional
Speed of sound in m/s.
omalias: float, optional
Angular frequency where spatial aliasing becomes prominent.
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
-----
`point_25d()` derives 2.5D WFS from the 3D
Neumann-Rayleigh integral (i.e. the TU Delft approach).
The eq. (3.10), (3.11) in :cite:`Start1997`, equivalent to
Eq. (2.137) in :cite:`Schultz2016`
.. math::
D(\x_0,\w) = \sqrt{8 \pi \, \i\wc}
\sqrt{\frac{|\x_\text{ref}-\x_0| \cdot
|\x_0-\x_\text{s}|}{|\x_\text{ref}-\x_0| + |\x_0-\x_\text{s}|}}
\scalarprod{\frac{\x_0-\x_\text{s}}{|\x_0-\x_\text{s}|}}{\n_0}
\frac{\e{-\i\wc |\x_0-\x_\text{s}|}}{4\pi\,|\x_0-\x_\text{s}|}
is implemented.
The theoretical link of `point_25d()` and `point_25d_legacy()` was
introduced as *unified WFS framework* in :cite:`Firtha2017`.
Examples
--------
.. plot::
:context: close-figs
d, selection, secondary_source = sfs.fd.wfs.point_25d(
omega, array.x, array.n, xs)
normalize_gain = 4 * np.pi * np.linalg.norm(xs)
plot(normalize_gain * d, selection, secondary_source)
"""
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
dr = xref - x0
s = _np.linalg.norm(ds, axis=1)
r = _np.linalg.norm(dr, axis=1)
d = (
preeq_25d(omega, omalias, c) *
_np.sqrt(8 * _np.pi) *
_np.sqrt((r * s) / (r + s)) *
_inner1d(n0, ds) / s *
_np.exp(-1j * k * s) / (4 * _np.pi * s))
selection = _util.source_selection_point(n0, x0, xs)
return d, selection, _secondary_source_point(omega, c)
def point_25d_legacy(omega, x0, n0, xs, xref=[0, 0, 0], c=None, omalias=None):
r"""Driving function for 2.5-dimensional WFS for a virtual point source.
.. versionadded:: 0.5
`point_25d()` was renamed to `point_25d_legacy()` (and a new
function with the name `point_25d()` was introduced). See notes for
further details.
Parameters
----------
omega : float
Angular frequency of point source.
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of normal vectors of secondary sources.
xs : (3,) array_like
Position of virtual point source.
xref : (3,) array_like, optional
Reference point for synthesized sound field.
c : float, optional
Speed of sound.
omalias: float, optional
Angular frequency where spatial aliasing becomes prominent.
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
-----
`point_25d_legacy()` derives 2.5D WFS from the 2D
Neumann-Rayleigh integral (i.e. the approach by Rabenstein & Spors), cf.
:cite:`Spors2008`.
.. math::
D(\x_0,\w) = \sqrt{\i\wc |\x_\text{ref}-\x_0|}
\frac{\scalarprod{\x_0-\x_\text{s}}{\n_0}}
{|\x_0-\x_\text{s}|^\frac{3}{2}}
\e{-\i\wc |\x_0-\x_\text{s}|}
The theoretical link of `point_25d()` and `point_25d_legacy()` was
introduced as *unified WFS framework* in :cite:`Firtha2017`.
Also cf. Eq. (2.145)-(2.147) :cite:`Schultz2016`.
Examples
--------
.. plot::
:context: close-figs
d, selection, secondary_source = sfs.fd.wfs.point_25d_legacy(
omega, array.x, array.n, xs)
normalize_gain = np.linalg.norm(xs)
plot(normalize_gain * d, selection, secondary_source)
"""
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)
d = (preeq_25d(omega, omalias, c) * _np.sqrt(_np.linalg.norm(xref - x0)) *
_inner1d(ds, n0) / r**(3 / 2) * _np.exp(-1j * k * r))
selection = _util.source_selection_point(n0, x0, xs)
return d, selection, _secondary_source_point(omega, c)
def point_25d(omega, x0, n0, xs, xref=[0, 0, 0], c=None, omalias=None):
r"""Driving function for 2.5-dimensional WFS of a virtual point source.
.. versionchanged:: 0.5
see notes, old handling of `point_25d()` is now `point_25d_legacy()`
Parameters
----------
omega : float
Angular frequency of point source.
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of normal vectors of secondary sources.
xs : (3,) array_like
Position of virtual point source.
xref : (3,) array_like, optional
Reference point xref or contour xref(x0) for amplitude correct
synthesis.
c : float, optional
Speed of sound in m/s.
omalias: float, optional
Angular frequency where spatial aliasing becomes prominent.
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
-----
`point_25d()` derives 2.5D WFS from the 3D
Neumann-Rayleigh integral (i.e. the TU Delft approach).
The eq. (3.10), (3.11) in :cite:`Start1997`, equivalent to
Eq. (2.137) in :cite:`Schultz2016`
.. math::
D(\x_0,\w) = \sqrt{8 \pi \, \i\wc}
\sqrt{\frac{|\x_\text{ref}-\x_0| \cdot
|\x_0-\x_\text{s}|}{|\x_\text{ref}-\x_0| + |\x_0-\x_\text{s}|}}
\scalarprod{\frac{\x_0-\x_\text{s}}{|\x_0-\x_\text{s}|}}{\n_0}
\frac{\e{-\i\wc |\x_0-\x_\text{s}|}}{4\pi\,|\x_0-\x_\text{s}|}
is implemented.
The theoretical link of `point_25d()` and `point_25d_legacy()` was
introduced as *unified WFS framework* in :cite:`Firtha2017`.
Examples
--------
.. plot::
:context: close-figs
d, selection, secondary_source = sfs.fd.wfs.point_25d(
omega, array.x, array.n, xs)
normalize_gain = 4 * np.pi * np.linalg.norm(xs)
plot(normalize_gain * d, selection, secondary_source)
"""
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
dr = xref - x0
s = _np.linalg.norm(ds, axis=1)
r = _np.linalg.norm(dr, axis=1)
d = (preeq_25d(omega, omalias, c) * _np.sqrt(8 * _np.pi) * _np.sqrt(
(r * s) / (r + s)) * _inner1d(n0, ds) / s * _np.exp(-1j * k * s) /
(4 * _np.pi * s))
selection = _util.source_selection_point(n0, x0, xs)
return d, selection, _secondary_source_point(omega, c)
def point_25d_legacy(x0, n0, xs, xref=[0, 0, 0], c=None):
r"""Driving function for 2.5-dimensional WFS of a virtual point source.
.. versionadded:: 0.61
`point_25d()` was renamed to `point_25d_legacy()` (and a new
function with the name `point_25d()` was introduced). See notes below
for further details.
Parameters
----------
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of secondary source orientations.
xs : (3,) array_like
Virtual source position.
xref : (3,) array_like, optional
Reference position
c : float, optional
Speed of sound
Returns
-------
delays : (N,) numpy.ndarray
Delays of secondary sources in seconds.
weights: (N,) numpy.ndarray
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.td.synthesize()`.
Notes
-----
2.5D correction factor
.. math::
g_0 = \sqrt{2 \pi |x_\mathrm{ref} - x_0|}
d using a point source as source model
.. math::
d_{2.5D}(x_0,t) =
\frac{g_0 \scalarprod{(x_0 - x_s)}{n_0}}
{2\pi |x_0 - x_s|^{3/2}}
\dirac{t - \frac{|x_0 - x_s|}{c}} \ast_t h(t)
with wfs(2.5D) prefilter h(t), which is not implemented yet.
See :sfs:`d_wfs/#equation-td-wfs-point-25d`
`point_25d_legacy()` derives 2.5D WFS from the 2D
Neumann-Rayleigh integral (i.e. the approach by Rabenstein & Spors), cf.
:cite:`Spors2008`.
The theoretical link of `point_25d()` and `point_25d_legacy()` was
introduced as *unified WFS framework* in :cite:`Firtha2017`.
Examples
--------
.. plot::
:context: close-figs
delays, weights, selection, secondary_source = \
sfs.td.wfs.point_25d(array.x, array.n, xs)
d = sfs.td.wfs.driving_signals(delays, weights, signal)
plot(d, selection, secondary_source, t=ts)
"""
if c is None:
c = _default.c
x0 = _util.asarray_of_rows(x0)
n0 = _util.asarray_of_rows(n0)
xs = _util.asarray_1d(xs)
xref = _util.asarray_1d(xref)
g0 = _np.sqrt(2 * _np.pi * _np.linalg.norm(xref - x0, axis=1))
ds = x0 - xs
r = _np.linalg.norm(ds, axis=1)
delays = r / c
weights = g0 * _inner1d(ds, n0) / (2 * _np.pi * r**(3 / 2))
selection = _util.source_selection_point(n0, x0, xs)
return delays, weights, selection, _secondary_source_point(c)
def point_25d(x0, n0, xs, xref=[0, 0, 0], c=None):
r"""Driving function for 2.5-dimensional WFS of a virtual point source.
.. versionchanged:: 0.61
see notes, old handling of `point_25d()` is now `point_25d_legacy()`
Parameters
----------
x0 : (N, 3) array_like
Sequence of secondary source positions.
n0 : (N, 3) array_like
Sequence of secondary source orientations.
xs : (3,) array_like
Virtual source position.
xref : (N, 3) array_like or (3,) array_like
Reference curve of correct amplitude xref(x0)
c : float, optional
Speed of sound
Returns
-------
delays : (N,) numpy.ndarray
Delays of secondary sources in seconds.
weights: (N,) numpy.ndarray
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.td.synthesize()`.
Notes
-----
Eq. (2.138) in :cite:`Schultz2016`:
.. math::
d_{2.5D}(x_0, x_{ref}, t) =
\sqrt{8\pi}
\frac{\scalarprod{(x_0 - x_s)}{n_0}}{|x_0 - x_s|}
\sqrt{\frac{|x_0 - x_s||x_0 - x_{ref}|}{|x_0 - x_s|+|x_0 - x_{ref}|}}
\cdot
\frac{\dirac{t - \frac{|x_0 - x_s|}{c}}}{4\pi |x_0 - x_s|} \ast_t h(t)
.. math::
h(t) = F^{-1}(\sqrt{\frac{j \omega}{c}})
with wfs(2.5D) prefilter h(t), which is not implemented yet.
`point_25d()` derives WFS from 3D to 2.5D via the stationary phase
approximation approach (i.e. the Delft approach).
The theoretical link of `point_25d()` and `point_25d_legacy()` was
introduced as *unified WFS framework* in :cite:`Firtha2017`.
Examples
--------
.. plot::
:context: close-figs
delays, weights, selection, secondary_source = \
sfs.td.wfs.point_25d(array.x, array.n, xs)
d = sfs.td.wfs.driving_signals(delays, weights, signal)
plot(d, selection, secondary_source, t=ts)
"""
if c is None:
c = _default.c
x0 = _util.asarray_of_rows(x0)
n0 = _util.asarray_of_rows(n0)
xs = _util.asarray_1d(xs)
xref = _util.asarray_of_rows(xref)
x0xs = x0 - xs
x0xref = x0 - xref
x0xs_n = _np.linalg.norm(x0xs, axis=1)
x0xref_n = _np.linalg.norm(x0xref, axis=1)
g0 = 1 / (_np.sqrt(2 * _np.pi) * x0xs_n**2)
g0 *= _np.sqrt((x0xs_n * x0xref_n) / (x0xs_n + x0xref_n))
delays = x0xs_n / c
weights = g0 * _inner1d(x0xs, n0)
selection = _util.source_selection_point(n0, x0, xs)
return delays, weights, selection, _secondary_source_point(c)