Source code for TRXASprefitpack.mathfun.irf

'''
exp_conv_irf:
submodule for the mathematical functions for
irf (instrumental response function)

:copyright: 2022 by pistack (Junho Lee).
:license: LGPL3.
'''

from typing import Union, Tuple
import numpy as np

def gau_irf(t: Union[float, np.ndarray], fwhm: float) -> Union[float, np.ndarray]:

    '''
    Compute gaussian shape irf function

    Args:
      t: time
      fwhm: full width at half maximum of gaussian distribution

    Returns:
     normalized gaussian function.
    '''

    sigma = fwhm/(2*np.sqrt(2*np.log(2)))
    return np.exp(-t**2/(2*sigma**2))/(sigma*np.sqrt(2*np.pi))

def cauchy_irf(t: Union[float, np.ndarray], fwhm: float) -> Union[float, np.ndarray]:

    '''
    Compute lorenzian shape irf function

    Args:
      t: time
      fwhm: full width at half maximum of cauchy distribution

    Returns:
     normalized lorenzian function.
    '''

    gamma = fwhm/2
    return gamma/np.pi/(t**2+gamma**2)

def calc_fwhm(fwhm_G: float, fwhm_L: float) -> float:
  '''
  Calculate uniform fwhm of both gaussian and cauchy component of 
  pseudo voigt profile with fwhm_G, fwhm_L based on 
  Journal of Applied Crystallography. 33 (6): 1311–1316.

  Args:
    fwhm_G: full width at half maximum of gaussian part
    fwhm_L: full width at half maximum of lorenzian part
  Returns:
   uniform fwhm parameter of both gaussian and cauchy component of
   pseudo voigt profile
  '''
  f = fwhm_G**5+2.69269*fwhm_G**4*fwhm_L + \
    2.42843*fwhm_G**3*fwhm_L**2 + \
    4.47163*fwhm_G**2*fwhm_L**3 + \
    0.07842*fwhm_G*fwhm_L**4 + \
    fwhm_L**5
  return f**(1/5)
  
def calc_eta(fwhm_G: float, fwhm_L: float) -> float:
  '''
  Calculate eta of pseudo voigt profile with fwhm_G, fwhm_L based on 
  Journal of Applied Crystallography. 33 (6): 1311–1316.

  Args:
    fwhm_G: full width at half maximum of gaussian part
    fwhm_L: full width at half maximum of lorenzian part
  Returns:
   maxing parameter eta
  '''
  f = fwhm_G**5+2.69269*fwhm_G**4*fwhm_L + \
    2.42843*fwhm_G**3*fwhm_L**2 + \
    4.47163*fwhm_G**2*fwhm_L**3 + \
    0.07842*fwhm_G*fwhm_L**4 + \
    fwhm_L**5
  f = f**(1/5)
  x = fwhm_L/f
  eta = 1.36603*x-0.47719*x**2+0.11116*x**3
  return eta

def pvoigt_irf(t: Union[float, np.ndarray], fwhm: float, eta: float) -> Union[float, np.ndarray]:
    '''
    Compute pseudo voight shape irf function
    (i.e. linear combination of gaussian and lorenzian function)

    Args:
      t: time
      fwhm: uniform full width at half paramter
      eta: mixing paramter
    Returns:
     pseudo voigt profile
    '''
    u = gau_irf(t, fwhm)
    v = cauchy_irf(t, fwhm)

    return u + eta*(v-u)

def deriv_fwhm(fwhm_G: float, fwhm_L: float) -> Tuple[float, float]:
  '''
  Computes gradient of uniform fwhm parameter of pseudo voigt approximation
  based on Journal of Applied Crystallography. 33 (6): 1311-1316.

  Args:
   fwhm_G: full width at half maximum of gaussian part
   fwhm_L: full width at half maximum of lorenzian part
  
  Returns:
   gradient of fwhm(fwhm_G, fwhm_L)
  '''
  f = fwhm_G**5+2.69269*fwhm_G**4*fwhm_L + \
    2.42843*fwhm_G**3*fwhm_L**2 + \
    4.47163*fwhm_G**2*fwhm_L**3 + \
    0.07842*fwhm_G*fwhm_L**4 + \
    fwhm_L**5
  df_fwhm_G = 5*fwhm_G**4+10.77076*fwhm_G**3*fwhm_L + \
    7.28529*fwhm_G**2*fwhm_L**2+8.94326*fwhm_G*fwhm_L**3 + \
      0.07842*fwhm_L**4
  df_fwhm_L = 5*fwhm_L**4 + 0.31368*fwhm_L**3*fwhm_G + \
    13.41489*fwhm_G**2*fwhm_L**2 + 4.85686*fwhm_L*fwhm_G**3 + \
      2.69269*fwhm_G**4
  return df_fwhm_G/f**(4/5)/5, df_fwhm_L/f**(4/5)/5



def deriv_eta(fwhm_G: float, fwhm_L: float) -> Tuple[float, float]:
  '''
  Calculate gradient of eta of pseudo voigt profile with fwhm_G, fwhm_L based on 
  Journal of Applied Crystallography. 33 (6): 1311–1316.

  Args:
    fwhm_G: full width at half maximum of gaussian part
    fwhm_L: full width at half maximum of lorenzian part
  Returns:
   gradient of eta(fwhm_G, fwhm_L)
  '''
  f = fwhm_G**5+2.69269*fwhm_G**4*fwhm_L + \
    2.42843*fwhm_G**3*fwhm_L**2 + \
    4.47163*fwhm_G**2*fwhm_L**3 + \
    0.07842*fwhm_G*fwhm_L**4 + \
    fwhm_L**5
  g = f**(-1/5); x = fwhm_L*g
  df_fwhm_G = 5*fwhm_G**4+10.77076*fwhm_G**3*fwhm_L + \
    7.28529*fwhm_G**2*fwhm_L**2+8.94326*fwhm_G*fwhm_L**3 + \
      0.07842*fwhm_L**4
  df_fwhm_L = 5*fwhm_L**4 + 0.31368*fwhm_L**3*fwhm_G + \
    13.41489*fwhm_G**2*fwhm_L**2 + 4.85686*fwhm_L*fwhm_G**3 + \
      2.69269*fwhm_G**4
  dx_fwhm_G = -fwhm_L*df_fwhm_G*g/f/5
  dx_fwhm_L = g - fwhm_L*df_fwhm_L*g/f/5
  deta_x = 0.33348*x**2-0.95438*x+1.36603
  return deta_x*dx_fwhm_G, deta_x*dx_fwhm_L