Radio Interferometer Measurement Equation

Functions used to compute the terms of the Radio Interferometer Measurement Equation (RIME). It describes the response of an interferometer to a sky model.

\[V_{pq} = G_{p} \left( \sum_{s} E_{ps} L_{p} K_{ps} B_{s} K_{qs}^H L_{q}^H E_{qs}^H \right) G_{q}^H\]

where for antenna \(p\) and \(q\), and source \(s\):

  • \(G_{p}\) represents direction-independent effects.
  • \(E_{ps}\) represents direction-dependent effects.
  • \(L_{p}\) represents the feed rotation.
  • \(K_{ps}\) represents the phase delay term.
  • \(B_{s}\) represents the brightness matrix.

The RIME is more formally described in the following four papers:

Numpy

predict_vis(time_index, antenna1, antenna2) Multiply Jones terms together to form model visibilities according to the following formula:
phase_delay(lm, uvw, frequency) Computes the phase delay (K) term:
parallactic_angles(times, antenna_positions, …) Computes parallactic angles per timestep for the given reference antenna position and field centre.
feed_rotation(parallactic_angles[, feed_type]) Computes the 2x2 feed rotation (L) matrix from the parallactic_angles.
transform_sources(lm, parallactic_angles, …) Creates beam sampling coordinates suitable for use in beam_cube_dde() by:
beam_cube_dde(beam, coords, l_grid, m_grid, …) Computes Direction Dependent Effects (E) by sampling complex values in beam at the coordinates coords.
zernike_dde(coords, coeffs, noll_index) Computes Direction Dependent Effects by evaluating Zernicke Polynomials defined by coefficients coeffs and noll indexes noll_index at the specified coordinates coords.
africanus.rime.predict_vis(time_index, antenna1, antenna2, dde1_jones=None, source_coh=None, dde2_jones=None, die1_jones=None, base_vis=None, die2_jones=None)[source]

Multiply Jones terms together to form model visibilities according to the following formula:

\[V_{pq} = G_{p} \left( B_{pq} + \sum_{s} A_{ps} X_{pqs} A_{qs}^H \right) G_{q}^H\]

where for antenna \(p\) and \(q\), and source \(s\):

  • \(B_{{pq}}\) represent base coherencies.
  • \(E_{{ps}}\) represents Direction-Dependent Jones terms.
  • \(X_{{pqs}}\) represents a coherency matrix (per-source).
  • \(G_{{p}}\) represents Direction-Independent Jones terms.

Generally, \(E_{ps}\), \(G_{p}\), \(X_{pqs}\) should be formed by using the RIME API functions and combining them together with einsum().

Please read the Notes

Parameters:
time_index : numpy.ndarray

Time index used to look up the antenna Jones index for a particular baseline. shape (row,).

antenna1 : numpy.ndarray

Antenna 1 index used to look up the antenna Jones for a particular baseline. with shape (row,).

antenna2 : numpy.ndarray

Antenna 2 index used to look up the antenna Jones for a particular baseline. with shape (row,).

dde1_jones : numpy.ndarray, optional

\(A_{ps}\) Direction-Dependent Jones terms for the first antenna. shape (source,time,ant,chan,corr_1,corr_2)

source_coh : numpy.ndarray, optional

\(X_{pqs}\) Direction-Dependent Coherency matrix for the baseline. with shape (source,row,chan,corr_1,corr_2)

dde2_jones : numpy.ndarray, optional

\(A_{qs}\) Direction-Dependent Jones terms for the second antenna. shape (source,time,ant,chan,corr_1,corr_2)

die1_jones : numpy.ndarray, optional

\(G_{ps}\) Direction-Independent Jones terms for the first antenna of the baseline. with shape (time,ant,chan,corr_1,corr_2)

base_vis : numpy.ndarray, optional

\(B_{pq}\) base visibilities, added to source coherency summation before multiplication with die1_jones and die2_jones.

die2_jones : numpy.ndarray, optional

\(G_{ps}\) Direction-Independent Jones terms for the second antenna of the baseline. with shape (time,ant,chan,corr_1,corr_2)
Returns:
visibilities : numpy.ndarray

Model visibilities of shape (row,chan,corr_1,corr_2)

Notes

  • Direction-Dependent terms (dde{1,2}_jones) and Independent (die{1,2}_jones) are optional, but if one is present, the other must be present.
  • The inputs to this function involve row, time and ant (antenna) dimensions.
  • Each row is associated with a pair of antenna Jones matrices at a particular timestep via the time_index, antenna1 and antenna2 inputs.
  • The row dimension must be an increasing partial order in time.
africanus.rime.phase_delay(lm, uvw, frequency)[source]

Computes the phase delay (K) term:

\[ \begin{align}\begin{aligned}& {\Large e^{-2 \pi i (u l + v m + w (n - 1))} }\\& \textrm{where } n = \sqrt{1 - l^2 - m^2}\end{aligned}\end{align} \]
Parameters:
lm : numpy.ndarray

LM coordinates of shape (source, 2) with L and M components in the last dimension.

uvw : numpy.ndarray

UVW coordinates of shape (row, 3) with U, V and W components in the last dimension.

frequency : numpy.ndarray

frequencies of shape (chan,)
Returns:
complex_phase : numpy.ndarray

complex of shape (source, row, chan)

Notes

Corresponds to the complex exponential of the Van Cittert-Zernike Theorem.

MeqTrees uses a positive sign convention and so any UVW coordinates must be inverted in order for their phase delay terms (and therefore visibilities) to agree.

africanus.rime.parallactic_angles(times, antenna_positions, field_centre, backend='casa')[source]

Computes parallactic angles per timestep for the given reference antenna position and field centre.

Parameters:
times : numpy.ndarray

Array of Mean Julian Date times in seconds with shape (time,),

antenna_positions : numpy.ndarray

Antenna positions of shape (ant, 3) in metres in the ITRF frame.

field_centre : numpy.ndarray

Field centre of shape (2,) in radians

backend : {‘casa’, ‘test’}, optional

Backend to use for calculating the parallactic angles.

  • casa defers to an implementation depending on python-casacore. This backend should be used by default.
  • test creates parallactic angles by multiplying the times and antenna_position arrays. It exist solely for testing.
Returns:
parallactic_angles : numpy.ndarray

Parallactic angles of shape (time,ant)
africanus.rime.feed_rotation(parallactic_angles, feed_type='linear')[source]

Computes the 2x2 feed rotation (L) matrix from the parallactic_angles.

\[\begin{split}\textrm{linear} \begin{bmatrix} cos(pa) & sin(pa) \\ -sin(pa) & cos(pa) \end{bmatrix} \qquad \textrm{circular} \begin{bmatrix} e^{-i pa} & 0 \\ 0 & e^{i pa} \end{bmatrix}\end{split}\]
Parameters:
parallactic_angles : numpy.ndarray

floating point parallactic angles. Of shape (pa0, pa1, ..., pan).

feed_type : {‘linear’, ‘circular’}

The type of feed
Returns:
feed_matrix : numpy.ndarray

Feed rotation matrix of shape (pa0, pa1,...,pan,2,2)
africanus.rime.transform_sources(lm, parallactic_angles, pointing_errors, antenna_scaling, frequency, dtype=None)[source]

Creates beam sampling coordinates suitable for use in beam_cube_dde() by:

  1. Rotating lm coordinates by the parallactic_angles
  2. Adding pointing_errors
  3. Scaling by antenna_scaling
Parameters:
lm : numpy.ndarray

LM coordinates of shape (src,2) in radians offset from the phase centre.

parallactic_angles : numpy.ndarray

parallactic angles of shape (time, antenna) in radians.

pointing_errors : numpy.ndarray

LM pointing errors for each antenna at each timestep in radians. Has shape (time, antenna, 2)

antenna_scaling : numpy.ndarray

antenna scaling factor for each channel and each antenna. Has shape (antenna, chan)

frequency : numpy.ndarray

frequencies for each channel. Has shape (chan,)

dtype : numpy.dtype, optional

Numpy dtype of result array. Should be float32 or float64. Defaults to float64
Returns:
coords : numpy.ndarray

coordinates of shape (3, src, time, antenna, chan) where each coordinate component represents l, m and frequency, respectively.
africanus.rime.beam_cube_dde(beam, coords, l_grid, m_grid, freq_grid, spline_order=1, mode=u'nearest')[source]

Computes Direction Dependent Effects (E) by sampling complex values in beam at the coordinates coords.

Both real and imaginary beam values are sampled at the given coordinates and normalised to form a mean of circular quantities.

l_grid, m_grid and freq_grid can be obtained from beam_grids().

Parameters:
beam : numpy.ndarray

complex beam cube of shape (beam_lw, beam_mh, beam_nud, corr_1, corr_2) where beam_lw is the grid width of the l dimension, beam_mh is the grid height of the m dimension and beam_nud is the grid depth of the frequency dimension. Either corr_1 or both corr_1 and corr_2 may be present, representing 1, 2 or 2x2 correlations respectively.

coords : numpy.ndarray

beam cube coordinates of shape (coords, dim_1, ..., dim_n) where coord always has size 3 and refers to (l,m,frequency).

l_grid : numpy.ndarray

Monotonically increasing or decreasing grid values for the l axis, with shape (beam_lw,). If decreasing, the

m_grid : numpy.ndarray

Monotonically increasing or decreasing grid values for the m axis, with shape (beam_mh,)

freq_grid : numpy.ndarray

Monotonically increasing grid values for the frequency axis, with shape (beam_nud,)

spline_order : int

Spline order to use in scipy.ndimage.interpolation.map_coordinates(). Defaults to 1 (‘linear’)

mode : str

Border mode to use in scipy.ndimage.interpolation.map_coordinates() Defaults to ‘nearest’
Returns:
ddes : numpy.ndarray

Sampled complex beam values at the specified coordinates with shape (dim_1, ..., dim_n, corr_1, corr_2)
africanus.rime.zernike_dde(coords, coeffs, noll_index)[source]

Computes Direction Dependent Effects by evaluating Zernicke Polynomials defined by coefficients coeffs and noll indexes noll_index at the specified coordinates coords.

Decomposition of a voxel beam cube into Zernicke polynomial coefficients can be achieved through the use of the eidos package.

Parameters:
coords : numpy.ndarray

Float coordinates at which to evaluate the zernike polynomials. Has shape (3, source, time, ant, chan). The three components in the first dimension represent l, m and frequency coordinates, respectively.

coeffs : numpy.ndarray

complex Zernicke polynomial coefficients. Has shape (ant, chan, corr_1, ..., corr_n, poly) where poly is the number of polynomial coefficients and corr_1, ..., corr_n are a variable number of correlation dimensions.

noll_index : numpy.ndarray

Noll index associated with each polynomial coefficient. Has shape (ant, chan, corr_1, ..., corr_n, poly).
Returns:
dde : numpy.ndarray

complex values with shape (source, time, ant, chan, corr_1, ..., corr_n)

Cuda

predict_vis(time_index, antenna1, antenna2, …) Multiply Jones terms together to form model visibilities according to the following formula:
phase_delay(lm, uvw, frequency) Computes the phase delay (K) term:
feed_rotation(parallactic_angles[, feed_type]) Computes the 2x2 feed rotation (L) matrix from the parallactic_angles.
africanus.rime.cuda.predict_vis(time_index, antenna1, antenna2, dde1_jones, source_coh, dde2_jones, die1_jones, base_vis, die2_jones)[source]

Multiply Jones terms together to form model visibilities according to the following formula:

\[V_{pq} = G_{p} \left( B_{pq} + \sum_{s} A_{ps} X_{pqs} A_{qs}^H \right) G_{q}^H\]

where for antenna \(p\) and \(q\), and source \(s\):

  • \(B_{{pq}}\) represent base coherencies.
  • \(E_{{ps}}\) represents Direction-Dependent Jones terms.
  • \(X_{{pqs}}\) represents a coherency matrix (per-source).
  • \(G_{{p}}\) represents Direction-Independent Jones terms.

Generally, \(E_{ps}\), \(G_{p}\), \(X_{pqs}\) should be formed by using the RIME API functions and combining them together with einsum().

Please read the Notes

Parameters:
time_index : cupy.ndarray

Time index used to look up the antenna Jones index for a particular baseline. shape (row,).

antenna1 : cupy.ndarray

Antenna 1 index used to look up the antenna Jones for a particular baseline. with shape (row,).

antenna2 : cupy.ndarray

Antenna 2 index used to look up the antenna Jones for a particular baseline. with shape (row,).

dde1_jones : cupy.ndarray, optional

\(A_{ps}\) Direction-Dependent Jones terms for the first antenna. shape (source,time,ant,chan,corr_1,corr_2)

source_coh : cupy.ndarray, optional

\(X_{pqs}\) Direction-Dependent Coherency matrix for the baseline. with shape (source,row,chan,corr_1,corr_2)

dde2_jones : cupy.ndarray, optional

\(A_{qs}\) Direction-Dependent Jones terms for the second antenna. shape (source,time,ant,chan,corr_1,corr_2)

die1_jones : cupy.ndarray, optional

\(G_{ps}\) Direction-Independent Jones terms for the first antenna of the baseline. with shape (time,ant,chan,corr_1,corr_2)

base_vis : cupy.ndarray, optional

\(B_{pq}\) base visibilities, added to source coherency summation before multiplication with die1_jones and die2_jones.

die2_jones : cupy.ndarray, optional

\(G_{ps}\) Direction-Independent Jones terms for the second antenna of the baseline. with shape (time,ant,chan,corr_1,corr_2)
Returns:
visibilities : cupy.ndarray

Model visibilities of shape (row,chan,corr_1,corr_2)

Notes

  • Direction-Dependent terms (dde{1,2}_jones) and Independent (die{1,2}_jones) are optional, but if one is present, the other must be present.
  • The inputs to this function involve row, time and ant (antenna) dimensions.
  • Each row is associated with a pair of antenna Jones matrices at a particular timestep via the time_index, antenna1 and antenna2 inputs.
  • The row dimension must be an increasing partial order in time.
africanus.rime.cuda.phase_delay(lm, uvw, frequency)[source]

Computes the phase delay (K) term:

\[ \begin{align}\begin{aligned}& {\Large e^{-2 \pi i (u l + v m + w (n - 1))} }\\& \textrm{where } n = \sqrt{1 - l^2 - m^2}\end{aligned}\end{align} \]
Parameters:
lm : cupy.ndarray

LM coordinates of shape (source, 2) with L and M components in the last dimension.

uvw : cupy.ndarray

UVW coordinates of shape (row, 3) with U, V and W components in the last dimension.

frequency : cupy.ndarray

frequencies of shape (chan,)
Returns:
complex_phase : cupy.ndarray

complex of shape (source, row, chan)

Notes

Corresponds to the complex exponential of the Van Cittert-Zernike Theorem.

MeqTrees uses a positive sign convention and so any UVW coordinates must be inverted in order for their phase delay terms (and therefore visibilities) to agree.

africanus.rime.cuda.feed_rotation(parallactic_angles, feed_type='linear')[source]

Computes the 2x2 feed rotation (L) matrix from the parallactic_angles.

\[\begin{split}\textrm{linear} \begin{bmatrix} cos(pa) & sin(pa) \\ -sin(pa) & cos(pa) \end{bmatrix} \qquad \textrm{circular} \begin{bmatrix} e^{-i pa} & 0 \\ 0 & e^{i pa} \end{bmatrix}\end{split}\]
Parameters:
parallactic_angles : cupy.ndarray

floating point parallactic angles. Of shape (pa0, pa1, ..., pan).

feed_type : {‘linear’, ‘circular’}

The type of feed
Returns:
feed_matrix : cupy.ndarray

Feed rotation matrix of shape (pa0, pa1,...,pan,2,2)

Dask

predict_vis(time_index, antenna1, antenna2) Multiply Jones terms together to form model visibilities according to the following formula:
phase_delay(lm, uvw, frequency) Computes the phase delay (K) term:
parallactic_angles(times, antenna_positions, …) Computes parallactic angles per timestep for the given reference antenna position and field centre.
feed_rotation(parallactic_angles, feed_type) Computes the 2x2 feed rotation (L) matrix from the parallactic_angles.
transform_sources(lm, parallactic_angles, …) Creates beam sampling coordinates suitable for use in beam_cube_dde() by:
beam_cube_dde(beam, coords, l_grid, m_grid, …) Computes Direction Dependent Effects (E) by sampling complex values in beam at the coordinates coords.
zernike_dde(coords, coeffs, noll_index) Computes Direction Dependent Effects by evaluating Zernicke Polynomials defined by coefficients coeffs and noll indexes noll_index at the specified coordinates coords.
africanus.rime.dask.predict_vis(time_index, antenna1, antenna2, dde1_jones=None, source_coh=None, dde2_jones=None, die1_jones=None, base_vis=None, die2_jones=None)[source]

Multiply Jones terms together to form model visibilities according to the following formula:

\[V_{pq} = G_{p} \left( B_{pq} + \sum_{s} A_{ps} X_{pqs} A_{qs}^H \right) G_{q}^H\]

where for antenna \(p\) and \(q\), and source \(s\):

  • \(B_{{pq}}\) represent base coherencies.
  • \(E_{{ps}}\) represents Direction-Dependent Jones terms.
  • \(X_{{pqs}}\) represents a coherency matrix (per-source).
  • \(G_{{p}}\) represents Direction-Independent Jones terms.

Generally, \(E_{ps}\), \(G_{p}\), \(X_{pqs}\) should be formed by using the RIME API functions and combining them together with einsum().

Please read the Notes

Parameters:
time_index : dask.array.Array

Time index used to look up the antenna Jones index for a particular baseline. shape (row,).

antenna1 : dask.array.Array

Antenna 1 index used to look up the antenna Jones for a particular baseline. with shape (row,).

antenna2 : dask.array.Array

Antenna 2 index used to look up the antenna Jones for a particular baseline. with shape (row,).

dde1_jones : dask.array.Array, optional

\(A_{ps}\) Direction-Dependent Jones terms for the first antenna. shape (source,time,ant,chan,corr_1,corr_2)

source_coh : dask.array.Array, optional

\(X_{pqs}\) Direction-Dependent Coherency matrix for the baseline. with shape (source,row,chan,corr_1,corr_2)

dde2_jones : dask.array.Array, optional

\(A_{qs}\) Direction-Dependent Jones terms for the second antenna. shape (source,time,ant,chan,corr_1,corr_2)

die1_jones : dask.array.Array, optional

\(G_{ps}\) Direction-Independent Jones terms for the first antenna of the baseline. with shape (time,ant,chan,corr_1,corr_2)

base_vis : dask.array.Array, optional

\(B_{pq}\) base visibilities, added to source coherency summation before multiplication with die1_jones and die2_jones.

die2_jones : dask.array.Array, optional

\(G_{ps}\) Direction-Independent Jones terms for the second antenna of the baseline. with shape (time,ant,chan,corr_1,corr_2)
Returns:
visibilities : dask.array.Array

Model visibilities of shape (row,chan,corr_1,corr_2)

Notes

  • Direction-Dependent terms (dde{1,2}_jones) and Independent (die{1,2}_jones) are optional, but if one is present, the other must be present.

  • The inputs to this function involve row, time and ant (antenna) dimensions.

  • Each row is associated with a pair of antenna Jones matrices at a particular timestep via the time_index, antenna1 and antenna2 inputs.

  • The row dimension must be an increasing partial order in time.

    • The ant dimension should only contain a single chunk equal to the number of antenna. Since each row can contain any antenna, random access must be preserved along this dimension.

    • The chunks in the row and time dimension must align. This subtle point must be understood otherwise invalid results will be produced by the chunking scheme. In the example below we have four unique time indices [0,1,2,3], and four unique antenna [0,1,2,3] indexing 10 rows.

      #  Row indices into the time/antenna indexed arrays
time_idx = np.asarray([0,0,1,1,2,2,2,2,3,3])
ant1 = np.asarray(    [0,0,0,0,1,1,1,2,2,3]
ant2 = np.asarray(    [0,1,2,3,1,2,3,2,3,3])

      A reasonable chunking scheme for the row and time dimension would be (4,4,2) and (2,1,1) respectively. Another way of explaining this is that the first four rows contain two unique timesteps, the second four rows contain one unique timestep and the last two rows contain one unique timestep.

      Some rules of thumb:

      1. The number chunks in row and time must match although the individual chunk sizes need not.

      2. Unique timesteps should not be split across row chunks.

      3. For a Measurement Set whose rows are ordered on the TIME column, the following is a good way of obtaining the row chunking strategy:

        import numpy as np
import pyrap.tables as pt

ms = pt.table("data.ms")
times = ms.getcol("TIME")
unique_times, chunks = np.unique(times, return_counts=True)

      4. Use aggregate_chunks() to aggregate multiple row and time chunks into chunks large enough such that functions operating on the resulting data can drop the GIL and spend time processing the data. Expanding the previous example:

        # Aggregate row
utimes = unique_times.size
# Single chunk for each unique time
time_chunks = (1,)*utimes
# Aggregate row chunks into chunks <= 10000
aggregate_chunks((chunks, time_chunks), (10000, utimes))
africanus.rime.dask.phase_delay(lm, uvw, frequency)[source]

Computes the phase delay (K) term:

\[ \begin{align}\begin{aligned}& {\Large e^{-2 \pi i (u l + v m + w (n - 1))} }\\& \textrm{where } n = \sqrt{1 - l^2 - m^2}\end{aligned}\end{align} \]
Parameters:
lm : dask.array.Array

LM coordinates of shape (source, 2) with L and M components in the last dimension.

uvw : dask.array.Array

UVW coordinates of shape (row, 3) with U, V and W components in the last dimension.

frequency : dask.array.Array

frequencies of shape (chan,)
Returns:
complex_phase : dask.array.Array

complex of shape (source, row, chan)

Notes

Corresponds to the complex exponential of the Van Cittert-Zernike Theorem.

MeqTrees uses a positive sign convention and so any UVW coordinates must be inverted in order for their phase delay terms (and therefore visibilities) to agree.

africanus.rime.dask.parallactic_angles(times, antenna_positions, field_centre, **kwargs)[source]

Computes parallactic angles per timestep for the given reference antenna position and field centre.

Parameters:
times : dask.array.Array

Array of Mean Julian Date times in seconds with shape (time,),

antenna_positions : dask.array.Array

Antenna positions of shape (ant, 3) in metres in the ITRF frame.

field_centre : dask.array.Array

Field centre of shape (2,) in radians

backend : {‘casa’, ‘test’}, optional

Backend to use for calculating the parallactic angles.

  • casa defers to an implementation depending on python-casacore. This backend should be used by default.
  • test creates parallactic angles by multiplying the times and antenna_position arrays. It exist solely for testing.
Returns:
parallactic_angles : dask.array.Array

Parallactic angles of shape (time,ant)
africanus.rime.dask.feed_rotation(parallactic_angles, feed_type)[source]

Computes the 2x2 feed rotation (L) matrix from the parallactic_angles.

\[\begin{split}\textrm{linear} \begin{bmatrix} cos(pa) & sin(pa) \\ -sin(pa) & cos(pa) \end{bmatrix} \qquad \textrm{circular} \begin{bmatrix} e^{-i pa} & 0 \\ 0 & e^{i pa} \end{bmatrix}\end{split}\]
Parameters:
parallactic_angles : numpy.ndarray

floating point parallactic angles. Of shape (pa0, pa1, ..., pan).

feed_type : {‘linear’, ‘circular’}

The type of feed
Returns:
feed_matrix : numpy.ndarray

Feed rotation matrix of shape (pa0, pa1,...,pan,2,2)
africanus.rime.dask.transform_sources(lm, parallactic_angles, pointing_errors, antenna_scaling, frequency, dtype=None)[source]

Creates beam sampling coordinates suitable for use in beam_cube_dde() by:

  1. Rotating lm coordinates by the parallactic_angles
  2. Adding pointing_errors
  3. Scaling by antenna_scaling
Parameters:
lm : dask.array.Array

LM coordinates of shape (src,2) in radians offset from the phase centre.

parallactic_angles : dask.array.Array

parallactic angles of shape (time, antenna) in radians.

pointing_errors : dask.array.Array

LM pointing errors for each antenna at each timestep in radians. Has shape (time, antenna, 2)

antenna_scaling : dask.array.Array

antenna scaling factor for each channel and each antenna. Has shape (antenna, chan)

frequency : dask.array.Array

frequencies for each channel. Has shape (chan,)

dtype : numpy.dtype, optional

Numpy dtype of result array. Should be float32 or float64. Defaults to float64
Returns:
coords : dask.array.Array

coordinates of shape (3, src, time, antenna, chan) where each coordinate component represents l, m and frequency, respectively.
africanus.rime.dask.beam_cube_dde(beam, coords, l_grid, m_grid, freq_grid, spline_order=1, mode='nearest')[source]

Computes Direction Dependent Effects (E) by sampling complex values in beam at the coordinates coords.

Both real and imaginary beam values are sampled at the given coordinates and normalised to form a mean of circular quantities.

l_grid, m_grid and freq_grid can be obtained from beam_grids().

Parameters:
beam : dask.array.Array

complex beam cube of shape (beam_lw, beam_mh, beam_nud, corr_1, corr_2) where beam_lw is the grid width of the l dimension, beam_mh is the grid height of the m dimension and beam_nud is the grid depth of the frequency dimension. Either corr_1 or both corr_1 and corr_2 may be present, representing 1, 2 or 2x2 correlations respectively.

coords : dask.array.Array

beam cube coordinates of shape (coords, dim_1, ..., dim_n) where coord always has size 3 and refers to (l,m,frequency).

l_grid : dask.array.Array

Monotonically increasing or decreasing grid values for the l axis, with shape (beam_lw,). If decreasing, the

m_grid : dask.array.Array

Monotonically increasing or decreasing grid values for the m axis, with shape (beam_mh,)

freq_grid : dask.array.Array

Monotonically increasing grid values for the frequency axis, with shape (beam_nud,)

spline_order : int

Spline order to use in scipy.ndimage.interpolation.map_coordinates(). Defaults to 1 (‘linear’)

mode : str

Border mode to use in scipy.ndimage.interpolation.map_coordinates() Defaults to ‘nearest’
Returns:
ddes : dask.array.Array

Sampled complex beam values at the specified coordinates with shape (dim_1, ..., dim_n, corr_1, corr_2)
africanus.rime.dask.zernike_dde(coords, coeffs, noll_index)[source]

Computes Direction Dependent Effects by evaluating Zernicke Polynomials defined by coefficients coeffs and noll indexes noll_index at the specified coordinates coords.

Decomposition of a voxel beam cube into Zernicke polynomial coefficients can be achieved through the use of the eidos package.

Parameters:
coords : dask.array.Array

Float coordinates at which to evaluate the zernike polynomials. Has shape (3, source, time, ant, chan). The three components in the first dimension represent l, m and frequency coordinates, respectively.

coeffs : dask.array.Array

complex Zernicke polynomial coefficients. Has shape (ant, chan, corr_1, ..., corr_n, poly) where poly is the number of polynomial coefficients and corr_1, ..., corr_n are a variable number of correlation dimensions.

noll_index : dask.array.Array

Noll index associated with each polynomial coefficient. Has shape (ant, chan, corr_1, ..., corr_n, poly).
Returns:
dde : dask.array.Array

complex values with shape (source, time, ant, chan, corr_1, ..., corr_n)