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utils

A set of utility functions and constants used for unit conversions and cosmology as well as some generically useful math functions.

K_CMB2K_RJ(freq)

Convert from K_CMB to K_RJ.

Parameters:

Name Type Description Default
freq float

The observing frequency in Hz.

 required

Returns:

Name Type Description
K_CMB2K_RJ float

Conversion factor from K_CMB to K_RJ.
Source code in witch/utils.py 
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@partial(jax.jit, static_argnums=(0,))
def K_CMB2K_RJ(freq: float) -> float:
    """
    Convert from K_CMB to K_RJ.

    Parameters
    ----------
    freq : float
        The observing frequency in Hz.

    Returns
    -------
    K_CMB2K_RJ : float
        Conversion factor from K_CMB to K_RJ.
    """
    x = freq * h / kb / Tcmb
    return jnp.exp(x) * x * x / jnp.expm1(x) ** 2

beam_double_gauss(dr, fwhm1, amp1, fwhm2, amp2)

Helper function to generate a double gaussian beam.

Parameters:

Name Type Description Default
dr float

Pixel size.

 required
fwhm1 float

Full width half max of the primary gaussian in the same units as dr.

 required
amp1 float

Amplitude of the primary gaussian.

 required
fwhm2 float

Full width half max of the secondairy gaussian in the same units as dr.

 required
amp2 float

Amplitude of the secondairy gaussian.

 required

Returns:

Type Description
beam: Double gaussian beam.
Source code in witch/utils.py 
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def beam_double_gauss(
    dr: float, fwhm1: float, amp1: float, fwhm2: float, amp2: float
) -> jax.Array:
    """
    Helper function to generate a double gaussian beam.

    Parameters
    ----------
    dr : float
        Pixel size.
    fwhm1 : float
        Full width half max of the primary gaussian in the same units as `dr`.
    amp1 : float
        Amplitude of the primary gaussian.
    fwhm2 : float
        Full width half max of the secondairy gaussian in the same units as `dr`.
    amp2 : float
        Amplitude of the secondairy gaussian.

    Returns
    -------
        beam: Double gaussian beam.
    """
    x = jnp.arange(-1.5 * fwhm1 // (dr), 1.5 * fwhm1 // (dr)) * (dr)
    beam_xx, beam_yy = jnp.meshgrid(x, x)
    beam_rr = jnp.sqrt(beam_xx**2 + beam_yy**2)
    beam = amp1 * jnp.exp(-4 * jnp.log(2) * beam_rr**2 / fwhm1**2) + amp2 * jnp.exp(
        -4 * jnp.log(2) * beam_rr**2 / fwhm2**2
    )
    return beam / jnp.sum(beam)

bilinear_interp(x, y, xp, yp, fp)

JAX implementation of bilinear interpolation. Out of bounds values are set to 0. Using the repeated linear interpolation method here, see https://en.wikipedia.org/wiki/Bilinear_interpolation#Repeated_linear_interpolation.

Parameters:

Name Type Description Default
x Array

X values to return interpolated values at.

 required
y Array

Y values to return interpolated values at.

 required
xp Array

X values to interpolate with, should be 1D. Assumed to be sorted.

 required
yp Array

Y values to interpolate with, should be 1D. Assumed to be sorted.

 required
fp Array

Functon values at (xp, yp), should have shape (len(xp), len(yp)). Note that if you are using meshgrid, we assume 'ij' indexing.

 required

Returns:

Name Type Description
f Array

The interpolated values.
Source code in witch/utils.py 
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@jax.jit
def bilinear_interp(
    x: jax.Array, y: jax.Array, xp: jax.Array, yp: jax.Array, fp: jax.Array
) -> jax.Array:
    """
    JAX implementation of bilinear interpolation.
    Out of bounds values are set to 0.
    Using the repeated linear interpolation method here,
    see https://en.wikipedia.org/wiki/Bilinear_interpolation#Repeated_linear_interpolation.

    Parameters
    ----------
    x : jax.Array
        X values to return interpolated values at.
    y : jax.Array
        Y values to return interpolated values at.
    xp : jax.Array
        X values to interpolate with, should be 1D.
        Assumed to be sorted.
    yp : jax.Array
        Y values to interpolate with, should be 1D.
        Assumed to be sorted.
    fp : jax.Array
        Functon values at `(xp, yp)`, should have shape `(len(xp), len(yp))`.
        Note that if you are using meshgrid, we assume `'ij'` indexing.

    Returns
    -------
    f : jax.Array
        The interpolated values.
    """
    if len(xp.shape) != 1:
        raise ValueError("xp must be 1D")
    if len(yp.shape) != 1:
        raise ValueError("yp must be 1D")
    if fp.shape != xp.shape + yp.shape:
        raise ValueError(
            "Incompatible shapes for fp, xp, yp: %s, %s, %s",
            fp.shape,
            xp.shape,
            yp.shape,
        )

    # Figure out bounds and mapping
    # This breaks if xp, yp is not sorted
    ix = jnp.clip(jnp.searchsorted(xp, x, side="right"), 1, len(xp) - 1)
    iy = jnp.clip(jnp.searchsorted(yp, y, side="right"), 1, len(yp) - 1)
    q_11 = fp[ix - 1, iy - 1]
    q_21 = fp[ix, iy - 1]
    q_12 = fp[ix - 1, iy]
    q_22 = fp[ix, iy]

    # Interpolate in x to start
    denom_x = xp[ix] - xp[ix - 1]
    dx_1 = x - xp[ix - 1]
    dx_2 = xp[ix] - x
    f_xy1 = (dx_2 * q_11 + dx_1 * q_21) / denom_x
    f_xy2 = (dx_2 * q_12 + dx_1 * q_22) / denom_x

    # Now do y as well
    denom_y = yp[iy] - yp[iy - 1]
    dy_1 = y - yp[iy - 1]
    dy_2 = yp[iy] - y
    f = (dy_2 * f_xy1 + dy_1 * f_xy2) / denom_y

    # Zero out the out of bounds values
    f = jnp.where((x < xp[0]) + (x > xp[-1]) + (y < yp[0]) + (y > yp[-1]), 0.0, f)

    return f

fft_conv(image, kernel)

Perform a convolution using FFTs for speed with jax.

Parameters:

Name Type Description Default
image ArrayLike

Data to be convolved.

 required
kernel ArrayLike

Convolution kernel.

 required

Returns:

Name Type Description
convolved_map Array

Image convolved with kernel.
Source code in witch/utils.py 
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@jax.jit
def fft_conv(image: ArrayLike, kernel: ArrayLike) -> jax.Array:
    """
    Perform a convolution using FFTs for speed with jax.

    Parameters
    ----------
    image : ArrayLike
        Data to be convolved.
    kernel : ArrayLike
        Convolution kernel.

    Returns
    -------
    convolved_map : jax.Array
        Image convolved with kernel.
    """
    Fmap = jnp.fft.fft2(jnp.fft.fftshift(image))
    Fkernel = jnp.fft.fft2(jnp.fft.fftshift(kernel))
    convolved_map = jnp.fft.fftshift(jnp.real(jnp.fft.ifft2(Fmap * Fkernel)))

    return convolved_map

get_da(z) cached

Get factor to convert from arcseconds to MPc.

Parameters:

Name Type Description Default
z ArrayLike

The redshift at which to compute the factor.

 required

Returns:

Name Type Description
da Array

Conversion factor from arcseconds to MPc.
Source code in witch/utils.py 
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@cache
def get_da(z: ArrayLike) -> jax.Array:
    """
    Get factor to convert from arcseconds to MPc.

    Parameters
    ----------
    z : ArrayLike
        The redshift at which to compute the factor.

    Returns
    -------
    da : jax.Array
        Conversion factor from arcseconds to MPc.
    """
    return jnp.interp(z, zline, daline)

get_hz(z) cached

Get the dimensionless hubble constant, h, at a given redshift.

Parameters:

Name Type Description Default
z ArrayLike

The redshift at which to compute h.

 required

Returns:

Name Type Description
hz Array

h at the given z.
Source code in witch/utils.py 
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@cache
def get_hz(z: ArrayLike) -> jax.Array:
    """
    Get the dimensionless hubble constant, h, at a given redshift.

    Parameters
    ----------
    z : ArrayLike
        The redshift at which to compute h.

    Returns
    -------
    hz : jax.Array
        h at the given z.
    """
    return jnp.interp(z, zline, hzline)

get_nz(z) cached

Get the critical density at a given redshift.

Parameters:

Name Type Description Default
z ArrayLike

The redshift at which to compute the critical density.

 required

Returns:

Name Type Description
nz Array

Critical density at the given z. This is in units of solar masses per cubic Mpc.
Source code in witch/utils.py 
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@cache
def get_nz(z: ArrayLike) -> jax.Array:
    """
    Get the critical density at a given redshift.

    Parameters
    ----------
    z : ArrayLike
        The redshift at which to compute the critical density.

    Returns
    -------
    nz : jax.Array
        Critical density at the given z.
        This is in units of solar masses per cubic Mpc.
    """
    return jnp.interp(z, zline, nzline)

tod_hi_pass(tod, N_filt)

High pass a tod with a tophat

Parameters:

Name Type Description Default
tod Array

TOD to high pass.

 required
N_filt int

N_filt of tophat.

 required

Returns:

Name Type Description
tod_filtered Array

High pass filtered TOD
Source code in witch/utils.py 
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@partial(jax.jit, static_argnums=(1,))
def tod_hi_pass(tod: jax.Array, N_filt: int) -> jax.Array:
    """
    High pass a tod with a tophat

    Parameters
    ----------
    tod : jax.Array
        TOD to high pass.
    N_filt : int
        N_filt of tophat.

    Returns
    -------
    tod_filtered : jax.Array
        High pass filtered TOD
    """
    mask = jnp.ones(tod.shape)
    mask = mask.at[..., :N_filt].set(0.0)

    ## apply the filter in fourier space
    Ftod = jnp.fft.fft(tod)
    Ftod_filtered = Ftod * mask
    tod_filtered = jnp.fft.ifft(Ftod_filtered).real
    return tod_filtered

y2K_CMB(freq, Te)

Convert from compton y to K_CMB.

Parameters:

Name Type Description Default
freq float

The observing frequency in Hz.

 required
Te float

Electron temperature

 required

Returns:

Name Type Description
y2K_CMB float

Conversion factor from compton y to K_CMB.
Source code in witch/utils.py 
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@partial(jax.jit, static_argnums=(0, 1))
def y2K_CMB(freq: float, Te: float) -> float:
    """
    Convert from compton y to K_CMB.

    Parameters
    ----------
    freq : float
        The observing frequency in Hz.
    Te : float
        Electron temperature

    Returns
    -------
    y2K_CMB : float
        Conversion factor from compton y to K_CMB.
    """
    x = freq * h / kb / Tcmb
    xt = x / jnp.tanh(0.5 * x)
    st = x / jnp.sinh(0.5 * x)
    # fmt:off
    Y0 = -4.0 + xt
    Y1 = (-10.0
        + ((47.0 / 2.0) + (-(42.0 / 5.0) + (7.0 / 10.0) * xt) * xt) * xt
        + st * st * (-(21.0 / 5.0) + (7.0 / 5.0) * xt)
    )
    Y2 = ((-15.0 / 2.0)
        + ((1023.0 / 8.0) + ((-868.0 / 5.0) + ((329.0 / 5.0) + ((-44.0 / 5.0) + (11.0 / 30.0) * xt) * xt) * xt) * xt) * xt
        + ((-434.0 / 5.0) + ((658.0 / 5.0) + ((-242.0 / 5.0) + (143.0 / 30.0) * xt) * xt) * xt
        + (-(44.0 / 5.0) + (187.0 / 60.0) * xt) * (st * st)) * st * st
    )
    Y3 = ((15.0 / 2.0)
        + ((2505.0 / 8.0) + ((-7098.0 / 5.0) + ((14253.0 / 10.0) + ((-18594.0 / 35.0) 
         + ((12059.0 / 140.0) + ((-128.0 / 21.0) + (16.0 / 105.0) * xt) * xt) * xt) * xt) * xt) * xt) * xt
        + (((-7098.0 / 10.0) + ((14253.0 / 5.0) + ((-102267.0 / 35.0) + ((156767.0 / 140.0)
         + ((-1216.0 / 7.0) + (64.0 / 7.0) * xt) * xt) * xt) * xt) * xt)
         + (((-18594.0 / 35.0) + ((205003.0 / 280.0) + ((-1920.0 / 7.0) + (1024.0 / 35.0) * xt) * xt) * xt)
          + ((-544.0 / 21.0) + (992.0 / 105.0) * xt) * st * st) * st * st) * st * st
    )
    Y4 = ((-135.0 / 32.0)
        + ((30375.0 / 128.0) + ((-62391.0 / 10.0) + ((614727.0 / 40.0) + ((-124389.0 / 10.0) + ((355703.0 / 80.0) + ((-16568.0 / 21.0)
         + ((7516.0 / 105.0) + ((-22.0 / 7.0) + (11.0 / 210.0) * xt) * xt) * xt) * xt) * xt) * xt) * xt) * xt) * xt
        + ((-62391.0 / 20.0) + ((614727.0 / 20.0) + ((-1368279.0 / 20.0) + ((4624139.0 / 80.0) + ((-157396.0 / 7.0) + ((30064.0 / 7.0)
         + ((-2717.0 / 7.0) + (2761.0 / 210.0) * xt) * xt) * xt) * xt) * xt) * xt) * xt
         + ((-124389.0 / 10.0)
          + ((6046951.0 / 160.0) + ((-248520.0 / 7.0) + ((481024.0 / 35.0) + ((-15972.0 / 7.0) + (18689.0 / 140.0) * xt) * xt) * xt) * xt) * xt
          + ((-70414.0 / 21.0) + ((465992.0 / 105.0) + ((-11792.0 / 7.0) + (19778.0 / 105.0) * xt) * xt) * xt
           + ((-682.0 / 7.0) + (7601.0 / 210.0) * xt) * st * st) * st * st) * st * st) * st * st
    )
    # fmt:on
    factor = Y0 + (Te / me) * (
        Y1 + (Te / me) * (Y2 + (Te / me) * (Y3 + (Te / me) * Y4))
    )
    return factor * Tcmb

y2K_RJ(freq, Te)

Convert from compton y to K_RJ.

Parameters:

Name Type Description Default
freq float

The observing frequency in Hz.

 required
Te float

Electron temperature

 required

Returns:

Name Type Description
y2K_RJ float

Conversion factor from compton y to K_RJ.
Source code in witch/utils.py 
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@partial(jax.jit, static_argnums=(0, 1))
def y2K_RJ(freq: float, Te: float) -> float:
    """
    Convert from compton y to K_RJ.

    Parameters
    ----------
    freq : float
        The observing frequency in Hz.
    Te : float
        Electron temperature

    Returns
    -------
    y2K_RJ : float
        Conversion factor from compton y to K_RJ.
    """
    factor = y2K_CMB(freq, Te)
    return factor * K_CMB2K_RJ(freq)