def radar_cross_section(frequency, width, incident_angle, observation_angle):
"""
Calculate the bistatic radar cross section for a 2D strip.
:param frequency: The frequency of the incident energy (Hz).
:param width: The width of the strip (m).
:param incident_angle: The incident angle (deg).
:param observation_angle: The observation angle (deg).
:return: The bistatic radar cross section (m^2).
"""
# Wavelength
wavelength = c / frequency
# Wavenumber
k = 2.0 * pi / wavelength
phi_i = radians(incident_angle)
phi_o = radians(observation_angle)
rcs_tm = k * width**2 * sin(phi_i) * sinc(width / wavelength *
(cos(phi_o) + cos(phi_i)))**2
rcs_te = k * width**2 * sin(phi_o) * sinc(width / wavelength *
(cos(phi_o) + cos(phi_i)))**2
return rcs_tm, rcs_te
def radar_cross_section(frequency, width, length, incident_theta,
observation_theta, observation_phi):
"""
Calculate the bistatic radar cross section for a rectangular plate.
:param frequency: The frequency of the incident energy (Hz).
:param width: The width of the plate (m).
:param length: The length of the plate (m).
:param incident_theta: The incident angle theta (deg).
:param observation_theta: The observation angle theta (deg).
:param observation_phi: The observation angle phi (deg).
:return: The bistatic radar cross section (m^2).
"""
# Wavelength
wavelength = c / frequency
theta_i = radians(incident_theta)
theta_o = radians(observation_theta)
phi_o = radians(observation_phi)
x = width / wavelength * sin(theta_o) * cos(phi_o)
y = length / wavelength * (sin(theta_o) * sin(phi_o) - sin(theta_i))
rcs_tm = 4.0 * pi * (length * width / wavelength)**2 * (
cos(theta_i)**2 * (cos(theta_o)**2 * cos(phi_o)**2 +
sin(phi_o)**2)) * sinc(x)**2 * sinc(y)**2
rcs_te = 4.0 * pi * (length * width / wavelength)**2 * (
cos(theta_o)**2 * sin(phi_o)**2 +
cos(phi_o)**2) * sinc(x)**2 * sinc(y)**2
return rcs_tm, rcs_te
def getXiAuto(self,rp,rt,z,pk_lin,pars):
k = self.k
if not self.fit_aiso:
ap=pars["ap"]
at=pars["at"]
else:
ap=pars["aiso"]*pars["1+epsilon"]*pars["1+epsilon"]
at=pars["aiso"]/pars["1+epsilon"]
ar=np.sqrt(rt**2*at**2+rp**2*ap**2)
mur=rp*ap/ar
muk = model.muk
kp = k * muk
kt = k * np.sqrt(1-muk**2)
bias_lya = pars["bias_lya*(1+beta_lya)"]/(1.+pars["beta_lya"])
beta_lya = pars["beta_lya"]
if self.uv_fluct:
bias_gamma = pars["bias_gamma"]
bias_prim = pars["bias_prim"]
lambda_uv = pars["lambda_uv"]
W = sp.arctan(k*lambda_uv)/(k*lambda_uv)
bias_lya_prim = bias_lya + bias_gamma*W/(1+bias_prim*W)
beta_lya = bias_lya*beta_lya/bias_lya_prim
bias_lya = bias_lya_prim
if self.lls:
bias_lls = pars["bias_lls"]
beta_lls = pars["beta_lls"]
L0_lls = pars["L0_lls"]
F_lls = sp.sinc(kp*L0_lls/sp.pi)
beta_lya = (bias_lya*beta_lya + bias_lls*beta_lls*F_lls)/(bias_lya+bias_lls*F_lls)
bias_lya = bias_lya + bias_lls*F_lls
pk_full = pk_lin * (1+beta_lya*muk**2)**2*bias_lya**2
Lpar=pars["Lpar_auto"]
Lper=pars["Lper_auto"]
Gpar = sp.sinc(kp*Lpar/2/sp.pi)
Gper = sp.sinc(kt*Lper/2/sp.pi)
pk_full*=Gpar**2
pk_full*=Gper**2
sigmaNLper = pars["SigmaNL_perp"]
sigmaNLpar = sigmaNLper*pars["1+f"]
pk_nl = sp.exp(-(kp*sigmaNLpar)**2/2-(kt*sigmaNLper)**2/2)
pk_full *= pk_nl
pk_full *= self.DNL(k,muk,self.pk,self.q1_dnl,self.kv_dnl,self.av_dnl,self.bv_dnl,self.kp_dnl,self.dnl_model)
evol = self.evolution_Lya_bias(z,[pars["alpha_lya"]])*self.evolution_growth_factor(z)
evol /= self.evolution_Lya_bias(self.zref,[pars["alpha_lya"]])*self.evolution_growth_factor(self.zref)
evol = evol**2.
return self.Pk2Xi(ar,mur,k,pk_full,ell_max=self.ell_max)*evol
def getXiAuto2D(self, rp, rt, z, pk2d, pars):
if not self.fit_aiso:
ap = pars["ap"]
at = pars["at"]
else:
ap = pars["aiso"] * pars["1+epsilon"] * pars["1+epsilon"]
at = pars["aiso"] / pars["1+epsilon"]
art = at * rt
arp = ap * rp
bias_lya = pars["bias_lya*(1+beta_lya)"] / (1. + pars["beta_lya"])
beta_lya = pars["beta_lya"]
if self.uv_fluct:
bias_gamma = pars["bias_gamma"]
bias_prim = pars["bias_prim"]
lambda_uv = pars["lambda_uv"]
W = sp.arctan(self.k * lambda_uv) / (self.k * lambda_uv)
bias_lya_prim = bias_lya + bias_gamma * W / (1 + bias_prim * W)
beta_lya = bias_lya * beta_lya / bias_lya_prim
bias_lya = bias_lya_prim
if self.lls:
bias_lls = pars["bias_lls"]
beta_lls = pars["beta_lls"]
L0_lls = pars["L0_lls"]
F_lls = sp.sinc(self.kp * L0_lls / sp.pi)
beta_lya = (bias_lya * beta_lya + bias_lls * beta_lls * F_lls) / (
bias_lya + bias_lls * F_lls)
bias_lya = bias_lya + bias_lls * F_lls
sigmaNLper = pars["SigmaNL_perp"]
sigmaNLpar = sigmaNLper * pars["1+f"]
pk_full = pk2d * sp.exp(
-(sigmaNLper**2 * self.kt**2 + sigmaNLpar**2 * self.kp**2) / 2)
pk_full = pk_full * (1 + beta_lya * self.muk**2)**2 * bias_lya**2
Lpar = pars["Lpar_auto"]
Lper = pars["Lper_auto"]
pk_full *= sp.sinc(self.kp * Lpar / 2 / sp.pi)**2
pk_full *= sp.sinc(self.kt * Lper / 2 / sp.pi)**2
pk_full *= self.DNL(self.k, self.muk, self.pk_2d, self.q1_dnl,
self.kv_dnl, self.av_dnl, self.bv_dnl, self.kp_dnl,
self.dnl_model)
evol = self.evolution_Lya_bias(
z, [pars["alpha_lya"]]) * self.evolution_growth_factor(z)
evol /= self.evolution_Lya_bias(
self.zref, [pars["alpha_lya"]]) * self.evolution_growth_factor(
self.zref)
evol = evol**2.
return fftlog.Pk2XiA(self.k1d, pk_full, arp, art) * evol
def sincKer(width, scale):
'''
Return a sinc filter kernel with specified width and scale
width - total number of points in the filter (not radius!)
scale - zeros will be centered around 0 with spacing "scale"
'''
x = r_[0:width + 1] - width * 0.5
k = numpy.outer(scipy.sinc(x / scale), scipy.sinc(x / scale))
return k
def sincKer(width, scale):
'''
Return a sinc filter kernel with specified width and scale
width - total number of points in the filter (not radius!)
scale - zeros will be centered around 0 with spacing "scale"
'''
x = r_[0:width + 1] - width * 0.5
k = numpy.outer(scipy.sinc(x / scale), scipy.sinc(x / scale))
return k
def getXiAutoQSO(self, rp, rt, z, pk_lin, pars):
k = self.k
if not self.fit_aiso:
ap = pars["ap"]
at = pars["at"]
else:
ap = pars["aiso"] * pars["1+epsilon"] * pars["1+epsilon"]
at = pars["aiso"] / pars["1+epsilon"]
ar = sp.sqrt(rt**2 * at**2 + rp**2 * ap**2)
mur = rp * ap / ar
muk = model.muk
kp = k * muk
kt = k * sp.sqrt(1 - muk**2)
### QSO-QSO auto correlation
bias_qso = pars["bias_qso"]
beta_qso = pars["growth_rate"] / bias_qso
pk_full = pk_lin * (bias_qso * (1. + beta_qso * muk**2))**2
### Velocity dispersion
if (self.velo_gauss):
pk_full *= sp.exp(-0.5 * (kp * pars['sigma_velo_gauss'])**2)
if (self.velo_lorentz):
pk_full /= 1. + (kp * pars['sigma_velo_lorentz'])**2
### Peak broadening
sigmaNLper = pars["SigmaNL_perp"]
sigmaNLpar = sigmaNLper * pars["1+f"]
pk_full *= sp.exp(-0.5 * ((sigmaNLper * kt)**2 + (sigmaNLpar * kp)**2))
### Pixel size
Lpar = pars["Lpar_autoQSO"]
Lper = pars["Lper_autoQSO"]
pk_full *= sp.sinc(kp * Lpar / 2. / sp.pi)**2
pk_full *= sp.sinc(kt * Lper / 2. / sp.pi)**2
### Redshift evolution
qso_evol = [pars['qso_evol_0'], pars['qso_evol_1']]
evol = sp.power(
self.evolution_growth_factor(z) /
self.evolution_growth_factor(self.zref), 2.)
evol *= sp.power(
self.evolution_QSO_bias(z, qso_evol) /
self.evolution_QSO_bias(self.zref, qso_evol), 2.)
return self.Pk2Xi(ar, mur, k, pk_full, ell_max=self.ell_max) * evol
def sinc_interp1d(x, s, r):
"""Interpolates `x`, sampled at times `s`
Output `y` is sampled at times `r`
inspired from from Matlab:
http://phaseportrait.blogspot.com/2008/06/sinc-interpolation-in-matlab.html
:param ndarray x: input data time series
:param ndarray s: input sampling time series (regular sample interval)
:param ndarray r: output sampling time series
:return ndarray: output data time series (regular sample interval)
"""
# init
s = sp.asarray(s)
r = sp.asarray(r)
x = sp.asarray(x)
if x.ndim == 1:
x = sp.atleast_2d(x)
else:
if x.shape[0] == len(s):
x = x.T
else:
if x.shape[1] != s.shape[0]:
raise ValueError('x and s must be same temporal extend')
if sp.allclose(s, r):
return x.T
T = s[1] - s[0]
# resample
sincM = sp.tile(r, (len(s), 1)) - sp.tile(s[:, sp.newaxis], (1, len(r)))
return sp.vstack([sp.dot(xx, sp.sinc(sincM / T)) for xx in x]).T
def shg_perfect_type_I(self, I1, lambd1, L, theta, phi):
"""
Calculate the amplitude of the sh with perfect type I phase matching. Effectively, there is only one incident
beam since both fundamental beams have the same intensity and wavelength.
Equation 2.2.19 on page 78 of Nonlinear Optics by Robert Boyd, Third Edition.
:param I1: Intensity of the fundamental.
:param lambd1: The wavelength of the fundamental.
:param L: Length of the crystal.
:param theta: Polar angle.
:param phi: Azimuthal angle.
:return: Intensity of the type I second harmonic.
"""
deff = 7.36e-12
n1 = self.outer_normal(lambd1, theta, phi)
n2 = self.outer_normal(lambd1, theta, phi)
n3 = self.inner_normal(2 * lambd1, theta, phi)
omega3 = pc.c * n1 / lambd1 + pc.c * n2 / lambd1
del_k = self.delta_k_I(lambd1, lambd1 / 2, theta, phi)
return 8 * deff**2 * omega3**2 * I1**2 / (n1 * n2 * n3 * pc.epsilon0 *
pc.c**2) * L**2 * sp.sinc(
del_k * L / 2)**2
def sinc_interp1d(x, s, r):
"""Interpolates `x`, sampled at times `s`
Output `y` is sampled at times `r`
inspired from from Matlab:
http://phaseportrait.blogspot.com/2008/06/sinc-interpolation-in-matlab.html
:param ndarray x: input data time series
:param ndarray s: input sampling time series (regular sample interval)
:param ndarray r: output sampling time series
:return ndarray: output data time series (regular sample interval)
"""
# init
s = sp.asarray(s)
r = sp.asarray(r)
x = sp.asarray(x)
if x.ndim == 1:
x = sp.atleast_2d(x)
else:
if x.shape[0] == len(s):
x = x.T
else:
if x.shape[1] != s.shape[0]:
raise ValueError('x and s must be same temporal extend')
if sp.allclose(s, r):
return x.T
T = s[1] - s[0]
# resample
sincM = sp.tile(r, (len(s), 1)) - sp.tile(s[:, sp.newaxis], (1, len(r)))
return sp.vstack([sp.dot(xx, sp.sinc(sincM / T)) for xx in x]).T
def radar_cross_section_3d(frequency, radius, incident_angle,
observation_angle, number_of_modes, length):
"""
Calculate the bistatic radar cross section for a finite length cylinder with oblique incidence.
:param frequency: The frequency of the incident energy (Hz).
:param radius: The radius of the cylinder (m).
:param incident_angle: The angle of incidence from z-axis (deg).
:param observation_angle: The observation angle (deg).
:param number_of_modes: The number of terms to take in the summation.
:param length: The length of the cylinder (m).
:return: The bistatic radar cross section for the infinite cylinder (m^2).
"""
# Wavelength
wavelength = c / frequency
theta_i = radians(incident_angle)
theta_o = theta_i
# Calculate the 2D RCS
rcs_te, rcs_tm = radar_cross_section(frequency, radius, incident_angle,
observation_angle, number_of_modes)
value = 2.0 * length ** 2 / wavelength * sin(theta_o) ** 2 * \
sinc(length / wavelength * (cos(theta_i) + cos(theta_o))) ** 2
return rcs_te * value, rcs_tm * value
def sum_frequenc_I(self, I1, I2, lambd1, lambd3, L, theta, phi):
"""
Calculate the intensity of the summed frequency pulse with fundamental pulse intensities I1 and I2 for Type I
phase matching.
Equation 2.2.19 on page 78 of Nonlinear Optics by Robert Boyd, Third Edition.
:param I1: Intensity of one of the fundamental beams.
:param I2: Intensity of the other fundamental beam.
:param lambd1: One of the fundamental wavelengths.
:param lambd3: The target sum frequency.
:param L: The length of the crystal.
:param theta: Polar angle.
:param phi: Azimuthal angle.
:return: Intensity of the type I summed frequency beam.
"""
omega3 = pc.c / lambd3
lambd2 = lambd1 * lambd3 / (lambd1 - lambd3
) # From w3 = w1 + w2, w=2pi*c/lambda
deff = 7.36e-12 # this is really the type II deff
n1 = self.outer_normal(lambd1, theta, phi)
n2 = self.outer_normal(lambd2, theta, phi)
n3 = self.inner_normal(lambd3, theta, phi)
del_k = self.delta_k_I(lambd1, lambd3, theta, phi)
return 8 * deff**2 * omega3**2 * I1 * I2 / (
n1 * n2 * n3 * pc.epsilon0 * pc.c**2) * L**2 * sp.sinc(
del_k * L / 2)**2
def far_fields(width, height, frequency, r, theta, phi):
"""
Calculate the far zone electric and magnetic fields for a rectangular aperture in a ground plane.
with a TE10 distribution of fields in the aperture.
:param r: The range to the field point (m).
:param theta: The theta angle to the field point (rad).
:param phi: The phi angle to the field point (rad).
:param width: The width of the aperture (m).
:param height: The height of the aperture (m).
:param frequency: The operating frequency (Hz).
:return: The far zone electric and magnetic fields (V/m), (A/m).
"""
# Calculate the wavenumber
k = 2.0 * pi * frequency / c
# Calculate the wave impedance
eta = sqrt(mu_0 / epsilon_0)
# Define the x and y wavenumber components
kx = k * width * 0.5 * sin(theta) * cos(phi)
ky = k * height * 0.5 * sin(theta) * sin(phi)
# Define the radial-component of the electric far field (V/m)
e_r = 0.0
# Define the theta-component of the electric far field (V/m)
e_theta = 1j * width * height * k / (2.0 * pi * r) * exp(-1j * k * r) * (-0.5 * pi * sin(phi)) * \
cos(kx) / (kx ** 2 - (0.5 * pi) ** 2) * sinc(ky)
# Define the phi-component of the electric far field (V/m)
e_phi = 1j * width * height * k / (2.0 * pi * r) * exp(-1j * k * r) * (-0.5 * pi * cos(theta) * cos(phi)) * \
cos(kx) / (kx ** 2 - (0.5 * pi) ** 2) * sinc(ky)
# Define the radial-component of the magnetic far field (A/m)
h_r = 0.0
# Define the theta-component of the magnetic far field (A/m)
h_theta = -1j * width * height * k / (2.0 * pi * eta * r) * exp(-1j * k * r) * \
(-0.5 * pi * cos(theta) * cos(phi)) * cos(kx) / (kx ** 2 - (0.5 * pi) ** 2) * sinc(ky)
# Define the phi-component of the magnetic far field (A/m)
h_phi = 1j * width * height * k / (pi * eta * r) * exp(-1j * k * r) * (-0.5 * pi * sin(phi)) * \
cos(kx) / (kx ** 2 - (0.5 * pi) ** 2) * sinc(ky)
# Return all six components of the far field
return e_r, e_theta, e_phi, h_r, h_theta, h_phi
def raised2(t, beta, T):
a = 1.0 / T
b = sinc(t / T)
c = cos((pi * beta * t) / T)
d = 1 - pow(2.0 * beta * t / T, 2)
#print "a:%s, b:%s, c:%s, d:%s" % (a,b,c,d)
return a * b * (c / d)
def far_fields(guide_height, horn_width, horn_effective_length, frequency, r,
theta, phi):
"""
Calculate the electric and magnetic fields in the far field of the horn.
:param r: The distance to the field point (m).
:param theta: The theta angle to the field point (rad).
:param phi: The phi angle to the field point (rad).
:param guide_height: The height of the waveguide feed (m).
:param horn_width: The width of the horn (m).
:param horn_effective_length: The effective length of the horn (m).
:param frequency: The operating frequency (Hz).
:return: The electric and magnetic fields radiated by the horn (V/m), (A/m).
"""
# Calculate the wavenumber
k = 2.0 * pi * frequency / c
# Calculate the wave impedance
eta = sqrt(mu_0 / epsilon_0)
# Define the radial-component of the electric field
e_r = 0.0
# Define the theta-component of the electric field
e_theta = sin(phi) * (1.0 + cos(theta)) * sinc(k * guide_height * 0.5 * sin(theta) * sin(phi)) * \
I(k, r, theta, phi, horn_width, guide_height, horn_effective_length)
# Define the phi-component of the electric field
e_phi = cos(phi) * (1.0 + cos(theta)) * sinc(k * guide_height * 0.5 * sin(theta) * sin(phi)) *\
I(k, r, theta, phi, horn_width, guide_height, horn_effective_length)
# Define the radial-component of the magnetic field
h_r = 0.0
# Define the theta-component of the magnetic field
h_theta = -cos(phi) / eta * (1.0 + cos(theta)) * sinc(k * guide_height * 0.5 * sin(theta) * sin(phi)) * \
I(k, r, theta, phi, horn_width, guide_height, horn_effective_length)
# Define the phi-component of the magnetic field
h_phi = sin(phi) / eta * (1.0 + cos(theta)) * sinc(k * guide_height * 0.5 * sin(theta) * sin(phi)) * \
I(k, r, theta, phi, horn_width, guide_height, horn_effective_length)
# Return all six components of the far field
return e_r, e_theta, e_phi, h_r, h_theta, h_phi
def single_pulse(time_delay, doppler_frequency, pulse_width):
"""
Calculate the ambiguity function for a continuous wave single pulse.
:param time_delay: The time delay for the ambiguity function (seconds).
:param doppler_frequency: The Doppler frequency for the ambiguity function (Hz).
:param pulse_width: The pulse width (seconds).
:return: The ambiguity function for a CW pulse.
"""
ambiguity = abs((1.0 - abs(time_delay / pulse_width)) *
sinc(doppler_frequency * (pulse_width - abs(time_delay))))**2
ambiguity[abs(time_delay) > pulse_width] = 0
return ambiguity
def sinc_seq(A = 1.0, tf = 1.0, delta = 0.1):
t = -tf
dt = 0.001
events = []
v_last = None
# Note use of round() to compensate for floating-point arithmetic errors
# that lead to inexact results.
while t <= tf:
v = delta * floor(round(A*sinc(t) / delta, 10))
if v != v_last:
events += [(t, v)]
v_last = v
t += dt
return array(events)
def _get_kernel(self,x1,func_name):
#evaluate pixel shift
if func_name in func_dic.keys():
func = func_dic[func_name]
else:
func = lambda kx: sp.sinc(kx)*get_window(func_name,kx.size)
i = sp.searchsorted(self.x,x1)
dpix = (x1 - self.x[i - 1])/(self.x[i] - self.x[i - 1])
if dpix == 1.0: dpix = 0
assert sp.absolute(dpix) < 1.0
kx = sp.r_[-self.kw:self.kw + 1] + dpix
k = func(kx)
k = k/sp.sum(k)
return k
def lfm_pulse(time_delay, doppler_frequency, pulse_width, bandwidth):
"""
Calculate the ambiguity function for a linear frequency modulated single pulse.
:param time_delay: The time delay for the ambiguity function (seconds).
:param doppler_frequency: The Doppler frequency for the ambiguity function (Hz).
:param pulse_width: The waveform pulse width (seconds).
:param bandwidth: The waveform band width (Hz).
:return: The ambiguity function for an LFM pulse.
"""
ambiguity = abs((1.0 - abs(time_delay) / pulse_width) *
sinc(pi * pulse_width * (bandwidth / pulse_width * time_delay + doppler_frequency) *
(1.0 - abs(time_delay) / pulse_width)))**2
ambiguity[abs(time_delay) > pulse_width] = 0
return ambiguity
def sincKer(width, even, scale = 2): ''' Return a sinc filter kernel with specified width and scale width - width of the filter (used by the window) even - true if the generated filter should have an even number of points. scale - zeros will be centred around 0 with spacing "scale" ''' if even: numPts = max(2*int((width+1)/2), 2) else: numPts = max(2*int(width/2) + 1, 3) x = r_[0:numPts]-(numPts-1)*0.5 k = scipy.sinc(x/scale) k *= cos(linspace(-pi/4,pi/4,len(k))) ker = numpy.outer(k,k) ker /= sum(ker.ravel()) return ker
def sincKer(width, even, scale=2):
'''
Return a sinc filter kernel with specified width and scale
width - width of the filter (used by the window)
even - true if the generated filter should have an even number of points.
scale - zeros will be centred around 0 with spacing "scale"
'''
if even:
numPts = max(2 * int((width + 1) / 2), 2)
else:
numPts = max(2 * int(width / 2) + 1, 3)
x = r_[0:numPts] - (numPts - 1) * 0.5
k = scipy.sinc(x / scale)
k *= cos(linspace(-pi / 4, pi / 4, len(k)))
ker = numpy.outer(k, k)
ker /= sum(ker.ravel())
return ker
def radial_transform(self, width) :
"""Return the radial beam Fourier transform function.
In the radial direction the beam is just top hat function, so the
Fourrier transform is a sinc function.
Parameters
----------
width : float
The frequency width of the beam function.
Returns
-------
transform : function
Call signiture : transform(k_rad). Vectorized radial beam transform
as a function of radial wave number. Accepts an array of wave
numbers (with units the reciprocal of those of `width`) and returns
an array of the same shape.
"""
factor = width/2.0/sp.pi
return lambda k_rad : sp.sinc(k_rad * factor)
def radial_transform(self, width):
"""Return the radial beam Fourier transform function.
In the radial direction the beam is just top hat function, so the
Fourier transform is a sinc function.
Parameters
----------
width: float
The frequency width of the beam function.
Returns
-------
transform: function
Call signature: transform(k_rad). Vectorized radial beam transform
as a function of radial wave number. Accepts an array of wave
numbers (with units the reciprocal of those of `width`) and returns
an array of the same shape.
"""
factor = width / 2. / sp.pi
return lambda k_rad: sp.sinc(k_rad * factor)
def lanczos(N, tau=1):
'''
return N-point Lanczos window with width tau
'''
return scipy.sinc(linspace(-1, 1, N) / tau)
def make_amb(Fsorg,m_up,plen,nlags,nspec=128,winname = 'boxcar'):
""" Make the ambiguity function dictionary that holds the lag ambiguity and
range ambiguity. Uses a sinc function weighted by a blackman window. Currently
only set up for an uncoded pulse.
Inputs:
Fsorg: A scalar, the original sampling frequency in Hertz.
m_up: The upsampled ratio between the original sampling rate and the rate of
the ambiguity function up sampling.
plen: The length of the pulse in samples at the original sampling frequency.
nlags: The number of lags used.
Outputs:
Wttdict: A dictionary with the keys 'WttAll' which is the full ambiguity function
for each lag, 'Wtt' is the max for each lag for plotting, 'Wrange' is the
ambiguity in the range with the lag dimension summed, 'Wlag' The ambiguity
for the lag, 'Delay' the numpy array for the lag sampling, 'Range' the array
for the range sampling and 'WttMatrix' for a matrix that will impart the ambiguity
function on a pulses.
"""
# make the sinc
nsamps = sp.floor(8.5*m_up)
nsamps = nsamps-(1-sp.mod(nsamps,2))
nvec = sp.arange(-sp.floor(nsamps/2.0),sp.floor(nsamps/2.0)+1)
pos_windows = ['boxcar', 'triang', 'blackman', 'hamming', 'hann', 'bartlett', 'flattop', 'parzen', 'bohman', 'blackmanharris', 'nuttall', 'barthann']
curwin = scisig.get_window(winname,nsamps)
outsinc = curwin*sp.sinc(nvec/m_up)
outsinc = outsinc/sp.sum(outsinc)
dt = 1/(Fsorg*m_up)
Delay = sp.arange(-(len(nvec)-1),m_up*(nlags+5))*dt
t_rng = sp.arange(0,1.5*plen,dt)
numdiff = len(Delay)-len(outsinc)
outsincpad = sp.pad(outsinc,(0,numdiff),mode='constant',constant_values=(0.0,0.0))
(srng,d2d)=sp.meshgrid(t_rng,Delay)
# envelop function
envfunc = sp.zeros(d2d.shape)
envfunc[(d2d-srng+plen-Delay.min()>=0)&(d2d-srng+plen-Delay.min()<=plen)]=1
envfunc = envfunc/sp.sqrt(envfunc.sum(axis=0).max())
#create the ambiguity function for everything
Wtt = sp.zeros((nlags,d2d.shape[0],d2d.shape[1]))
cursincrep = sp.tile(outsincpad[:,sp.newaxis],(1,d2d.shape[1]))
Wt0 = Wta = cursincrep*envfunc
Wt0fft = sp.fft(Wt0,axis=0)
for ilag in sp.arange(nlags):
cursinc = sp.roll(outsincpad,ilag*m_up)
cursincrep = sp.tile(cursinc[:,sp.newaxis],(1,d2d.shape[1]))
Wta = cursincrep*envfunc
#do fft based convolution, probably best method given sizes
Wtafft = scfft.fft(Wta,axis=0)
if ilag==0:
nmove = len(nvec)-1
else:
nmove = len(nvec)
Wtt[ilag] = sp.roll(scfft.ifft(Wtafft*sp.conj(Wt0fft),axis=0).real,nmove,axis=0)
# make matrix to take
# imat = sp.eye(nspec)
# tau = sp.arange(-sp.floor(nspec/2.),sp.ceil(nspec/2.))/Fsorg
# tauint = Delay
# interpmat = spinterp.interp1d(tau,imat,bounds_error=0,axis=0)(tauint)
# lagmat = sp.dot(Wtt.sum(axis=2),interpmat)
# # triangle window
tau = sp.arange(-sp.floor(nspec/2.),sp.ceil(nspec/2.))/Fsorg
amb1d = plen-tau
amb1d[amb1d<0]=0.
amb1d[tau<0]=0.
amb1d=amb1d/plen
kp = sp.argwhere(amb1d>0).flatten()
lagmat = sp.zeros((Wtt.shape[0],nspec))
lagmat.flat[sp.ravel_multi_index((sp.arange(Wtt.shape[0]),kp),lagmat.shape)]=amb1d[kp]
Wttdict = {'WttAll':Wtt,'Wtt':Wtt.max(axis=0),'Wrange':Wtt.sum(axis=1),'Wlag':Wtt.sum(axis=2),
'Delay':Delay,'Range':v_C_0*t_rng/2.0,'WttMatrix':lagmat}
return Wttdict
def raised(x, beta, T):
return (pi / 4.0 * T) * sinc(1 / 2.0 * beta)
def periodic_sinc_deft(x, max_w):
"""
Return a sinc-based Fourier transform of a discrete event sequence x.
The sequence x is assumed to define a possibly non-uniformly sampled
periodic time-domain signal.
This approach is a semi-analytical technique that essentially works by
treating the DE sequence as defining a piecewise-constant signal, and
treating the FT as the sum of the FTs of the rectangular pulses that make up
that piecewise-constant signal.
This version of the DEFT assumes a periodic signal, and like the standard
DFT constructs the transform from a finite segment of the signal by
assuming a periodic extension of the signal. The result is essentially a
frequency sampling of the nonperiodic DEFT at frequencies that are multiples
of the fundamental frequency.
The rect function is defined as
rect(t/tau) = 0 if |t| > tau/2
1 if |t| < tau/2
and has Fourier transform
F(w) = tau * sinc(w*tau/(2*pi))
Given events e0 = (t0, v0) and e1 = (t1, v1) we treat them as defining
a rectangle rect((t - ((t1 - t0)/2 + t0))/|t1 - t0|) with FT
F(w) = v0 * exp(-j*w*(t0+t1)/2) * tau * sinc(w*tau/(2*pi))
where tau = |t1 - t0|
Parameters
----------
x : event array
array to transform, where each event in the array is a 2-tuple
(time, value)
max_w : maximum radian frequency over which to evaluate the transform
Returns
-------
z : complex array
z = sum[n=0..N] v0 * exp(-1j*w*(t0 + t1)/2) * tau * sinc(w*tau/(2*pi))
where
(t0, v0) = x[n]
(t1, v1) = x[n+1]
tau = t1 - t0
P = max(t) - min(t)
w = 2*pi*k/P for k in -max_w*P/(2*pi) .. max_w*P/(2*pi)
"""
ts = [t for (t,v) in x]
P = max(ts) - min(ts) # Period
w0 = (2*pi)/P # Fundamental radian frequency
T = max(ts) # Signal end point
# Construct frequency sample points
k_bound = floor(max_w/w0)
w = w0*arange(-k_bound, k_bound+1, dtype=complex)
z = zeros(len(w), dtype=complex)
xf = append(x, [(T, x[0,1])], axis=0) # manage the wraparound
for n in range(len(x)):
(t0, v0) = xf[n]
(t1, v1) = xf[n+1]
tau = abs(t1 - t0)
z = z + (v0 * exp(-1j*w*(t0 + tau/2)) * tau * sinc(w*tau/(2*pi)))
return asarray(zip(w,z/P))
def make_amb(Fsorg, m_up, plen, pulse, nspec=128, winname='boxcar'):
"""
Make the ambiguity function dictionary that holds the lag ambiguity and
range ambiguity. Uses a sinc function weighted by a blackman window. Currently
only set up for an uncoded pulse.
Args:
Fsorg (:obj:`float`): A scalar, the original sampling frequency in Hertz.
m_up (:obj:`int`): The upsampled ratio between the original sampling rate and the rate of
the ambiguity function up sampling.
plen (:obj:`int`): The length of the pulse in samples at the original sampling frequency.
nlags (:obj:`int`): The number of lags used.
Returns:
Wttdict (:obj:`dict`): A dictionary with the keys 'WttAll' which is the full ambiguity function
for each lag, 'Wtt' is the max for each lag for plotting, 'Wrange' is the
ambiguity in the range with the lag dimension summed, 'Wlag' The ambiguity
for the lag, 'Delay' the numpy array for the lag sampling, 'Range' the array
for the range sampling and 'WttMatrix' for a matrix that will impart the ambiguity
function on a pulses.
"""
nspec = int(nspec)
nlags = len(pulse)
# make the sinc
nsamps = sp.floor(8.5 * m_up)
nsamps = int(nsamps - (1 - sp.mod(nsamps, 2)))
# need to incorporate summation rule
vol = 1.
nvec = sp.arange(-sp.floor(nsamps / 2.0), sp.floor(nsamps / 2.0) + 1)
pos_windows = [
'boxcar', 'triang', 'blackman', 'hamming', 'hann', 'bartlett',
'flattop', 'parzen', 'bohman', 'blackmanharris', 'nuttall', 'barthann'
]
curwin = scisig.get_window(winname, nsamps)
# Apply window to the sinc function. This will act as the impulse respons of the filter
outsinc = curwin * sp.sinc(nvec / m_up)
outsinc = outsinc / sp.sum(outsinc)
dt = 1 / (Fsorg * m_up)
#make delay vector
Delay_num = sp.arange(-(len(nvec) - 1), m_up * (nlags + 5))
Delay = Delay_num * dt
t_rng = sp.arange(0, 1.5 * plen, dt)
if len(t_rng) > 2e4:
raise ValueError(
'The time array is way too large. plen should be in seconds.')
numdiff = len(Delay) - len(outsinc)
numback = int(nvec.min() / m_up - Delay_num.min())
numfront = numdiff - numback
# outsincpad = sp.pad(outsinc,(0,numdiff),mode='constant',constant_values=(0.0,0.0))
outsincpad = sp.pad(outsinc, (numback, numfront),
mode='constant',
constant_values=(0.0, 0.0))
(d2d, srng) = sp.meshgrid(Delay, t_rng)
# envelop function
t_p = sp.arange(nlags) / Fsorg
envfunc = sp.interp(sp.ravel(srng - d2d), t_p, pulse, left=0.,
right=0.).reshape(d2d.shape)
# envfunc = sp.zeros(d2d.shape)
# envfunc[(d2d-srng+plen-Delay.min()>=0)&(d2d-srng+plen-Delay.min()<=plen)]=1
envfunc = envfunc / sp.sqrt(envfunc.sum(axis=0).max())
#create the ambiguity function for everything
Wtt = sp.zeros((nlags, d2d.shape[0], d2d.shape[1]))
cursincrep = sp.tile(outsincpad[sp.newaxis, :], (len(t_rng), 1))
Wt0 = cursincrep * envfunc
Wt0fft = sp.fft(Wt0, axis=1)
for ilag in sp.arange(nlags):
cursinc = sp.roll(outsincpad, ilag * m_up)
cursincrep = sp.tile(cursinc[sp.newaxis, :], (len(t_rng), 1))
Wta = cursincrep * envfunc
#do fft based convolution, probably best method given sizes
Wtafft = scfft.fft(Wta, axis=1)
nmove = len(nvec) - 1
Wtt[ilag] = sp.roll(scfft.ifft(Wtafft * sp.conj(Wt0fft), axis=1).real,
nmove,
axis=1)
# make matrix to take
imat = sp.eye(nspec)
tau = sp.arange(-sp.floor(nspec / 2.), sp.ceil(nspec / 2.)) / Fsorg
tauint = Delay
interpmat = spinterp.interp1d(tau, imat, bounds_error=0, axis=0)(tauint)
lagmat = sp.dot(Wtt.sum(axis=1), interpmat)
W0 = lagmat[0].sum()
for ilag in range(nlags):
lagmat[ilag] = ((vol + ilag) / (vol * W0)) * lagmat[ilag]
Wttdict = {
'WttAll': Wtt,
'Wtt': Wtt.max(axis=0),
'Wrange': Wtt.sum(axis=1),
'Wlag': Wtt.sum(axis=2),
'Delay': Delay,
'Range': v_C_0 * t_rng / 2.0,
'WttMatrix': lagmat
}
return Wttdict
def trim_sinc(kx):
#equivalent to setting window to boxcar
return sp.sinc(kx)
def make_amb(Fsorg,m_up,plen,pulse,nspec=128,winname = 'boxcar'):
"""
Make the ambiguity function dictionary that holds the lag ambiguity and
range ambiguity. Uses a sinc function weighted by a blackman window. Currently
only set up for an uncoded pulse.
Args:
Fsorg (:obj:`float`): A scalar, the original sampling frequency in Hertz.
m_up (:obj:`int`): The upsampled ratio between the original sampling rate and the rate of
the ambiguity function up sampling.
plen (:obj:`int`): The length of the pulse in samples at the original sampling frequency.
nlags (:obj:`int`): The number of lags used.
Returns:
Wttdict (:obj:`dict`): A dictionary with the keys 'WttAll' which is the full ambiguity function
for each lag, 'Wtt' is the max for each lag for plotting, 'Wrange' is the
ambiguity in the range with the lag dimension summed, 'Wlag' The ambiguity
for the lag, 'Delay' the numpy array for the lag sampling, 'Range' the array
for the range sampling and 'WttMatrix' for a matrix that will impart the ambiguity
function on a pulses.
"""
nspec = int(nspec)
nlags = len(pulse)
# make the sinc
nsamps = sp.floor(8.5*m_up)
nsamps = int(nsamps-(1-sp.mod(nsamps, 2)))
# need to incorporate summation rule
vol = 1.
nvec = sp.arange(-sp.floor(nsamps/2.0), sp.floor(nsamps/2.0)+1)
pos_windows = ['boxcar', 'triang', 'blackman', 'hamming', 'hann',
'bartlett', 'flattop', 'parzen', 'bohman', 'blackmanharris',
'nuttall', 'barthann']
curwin = scisig.get_window(winname, nsamps)
# Apply window to the sinc function. This will act as the impulse respons of the filter
outsinc = curwin*sp.sinc(nvec/m_up)
outsinc = outsinc/sp.sum(outsinc)
dt = 1/(Fsorg*m_up)
#make delay vector
Delay_num = sp.arange(-(len(nvec)-1),m_up*(nlags+5))
Delay = Delay_num*dt
t_rng = sp.arange(0, 1.5*plen, dt)
if len(t_rng) > 2e4:
raise ValueError('The time array is way too large. plen should be in seconds.')
numdiff = len(Delay)-len(outsinc)
numback = int(nvec.min()/m_up-Delay_num.min())
numfront = numdiff-numback
# outsincpad = sp.pad(outsinc,(0,numdiff),mode='constant',constant_values=(0.0,0.0))
outsincpad = sp.pad(outsinc,(numback, numfront), mode='constant',
constant_values=(0.0, 0.0))
(d2d, srng)=sp.meshgrid(Delay, t_rng)
# envelop function
t_p = sp.arange(nlags)/Fsorg
envfunc = sp.interp(sp.ravel(srng-d2d), t_p,pulse, left=0., right=0.).reshape(d2d.shape)
# envfunc = sp.zeros(d2d.shape)
# envfunc[(d2d-srng+plen-Delay.min()>=0)&(d2d-srng+plen-Delay.min()<=plen)]=1
envfunc = envfunc/sp.sqrt(envfunc.sum(axis=0).max())
#create the ambiguity function for everything
Wtt = sp.zeros((nlags, d2d.shape[0], d2d.shape[1]))
cursincrep = sp.tile(outsincpad[sp.newaxis, :], (len(t_rng), 1))
Wt0 = cursincrep*envfunc
Wt0fft = sp.fft(Wt0, axis=1)
for ilag in sp.arange(nlags):
cursinc = sp.roll(outsincpad, ilag*m_up)
cursincrep = sp.tile(cursinc[sp.newaxis, :], (len(t_rng), 1))
Wta = cursincrep*envfunc
#do fft based convolution, probably best method given sizes
Wtafft = scfft.fft(Wta, axis=1)
nmove = len(nvec)-1
Wtt[ilag] = sp.roll(scfft.ifft(Wtafft*sp.conj(Wt0fft), axis=1).real,
nmove, axis=1)
# make matrix to take
imat = sp.eye(nspec)
tau = sp.arange(-sp.floor(nspec/2.), sp.ceil(nspec/2.))/Fsorg
tauint = Delay
interpmat = spinterp.interp1d(tau, imat, bounds_error=0, axis=0)(tauint)
lagmat = sp.dot(Wtt.sum(axis=1), interpmat)
W0 = lagmat[0].sum()
for ilag in range(nlags):
lagmat[ilag] = ((vol+ilag)/(vol*W0))*lagmat[ilag]
Wttdict = {'WttAll':Wtt, 'Wtt':Wtt.max(axis=0), 'Wrange':Wtt.sum(axis=1),
'Wlag':Wtt.sum(axis=2), 'Delay':Delay, 'Range':v_C_0*t_rng/2.0,
'WttMatrix':lagmat}
return Wttdict
def sinc_deft(x, T, w):
"""
Return a sinc-based Fourier transform of a discrete event sequence x.
The sequence x is assumed to define a possibly non-uniformly sampled
time-domain signal.
This approach is a semi-analytical technique that essentially works by
treating the DE sequence as defining a piecewise-constant signal, and
treating the FT as the sum of the FTs of the rectangular pulses that make up
that piecewise-constant signal.
This version of the DEFT assumes a finite signal duration,
with the signal assumed to return to its zero state at the end of that
duration (just as we assume that the signal starts at a zero state).
The rect function is defined as
rect(t/tau) = 0 if |t| > tau/2
1 if |t| < tau/2
and has Fourier transform
F(w) = tau * sinc(w*tau/(2*pi))
Given events e0 = (t0, v0) and e1 = (t1, v1) we treat them as defining
a rectangle v0 * rect((t - ((t1 - t0)/2 + t0))/|t1 - t0|) with FT
F(w) = v0 * exp(-j*w*(t0+t1)/2) * tau * sinc(w*tau/(2*pi))
where tau = |t1 - t0|
See McInnes, A. "A Discrete Event Fourier Transform", https://github.com/allanmcinnes/DEFT.
Parameters
----------
x : event array
array to transform, where each event in the array is a 2-tuple
(time, value)
T : signal termination time
w : array
array of radian frequencies over which the transform
should be evaluated
Returns
-------
z : complex array
z = sum[n=0..N] v0 * exp(-1j*w*(t0 + t1)/2) * tau * sinc(w*tau/(2*pi))
where
(t0, v0) = x[n]
(t1, v1) = x[n+1]
tau = t1 - t0
"""
assert T >= x[-1][0], \
"Termination tag T = {0:g} must be greater".format(T) + \
" than or equal to the tag of the final event in the signal, " + \
"t = {0:g}. ".format(x[-1][0]) + \
"However, T is {0:g} less than t.".format(x[-1][0] - T)
z = zeros(len(w), dtype=complex)
xf = append(x, [(T, 0.0)], axis=0) # Append a terminating zero event
for n in range(len(x)):
(t0, v0) = xf[n]
(t1, v1) = xf[n+1]
tau = abs(t1 - t0)
z = z + (v0 * exp(-1j*w*(t0 + tau/2)) * tau * sinc(w*tau/(2*pi)))
return asarray(zip(w,z))
def valueCross(self,pars):
qso_boost = pars["qso_metal_boost"]
qso_evol = [pars['qso_evol_0'],pars['qso_evol_1']]
bias_qso = pars["bias_qso"]
growth_rate = pars["growth_rate"]
beta_qso = growth_rate/bias_qso
bias_met = sp.array([pars['bias_'+met] for met in self.met_names])
beta_met = sp.array([pars['beta_'+met] for met in self.met_names])
Lpar = pars["Lpar_cross"]
Lper = pars["Lper_cross"]
### Scales
if (self.different_drp):
drp_met = sp.array([pars['drp_'+met] for met in self.met_names])
drp = sp.outer(sp.ones(self.nd_cross),drp_met)
else:
drp = pars["drp"]
if self.grid:
### Redshift evolution
z = self.grid_qso_met[:,:,2]
evol = sp.power( self.evolution_growth_factor(z)/self.evolution_growth_factor(self.zref),2. )
evol *= self.evolution_Lya_bias(z,[pars["alpha_lya"]])/self.evolution_Lya_bias(self.zref,[pars["alpha_lya"]])
evol *= self.evolution_QSO_bias(z,qso_evol)/self.evolution_QSO_bias(self.zref,qso_evol)
rp_shift = self.grid_qso_met[:,:,0]+drp
rt = self.grid_qso_met[:,:,1]
r = sp.sqrt(rp_shift**2 + rt**2)
mur = rp_shift/r
muk = cosmo_model.muk
kp = self.k * muk
kt = self.k * sp.sqrt(1.-muk**2)
### Correction to linear power-spectrum
pk_corr = (1.+0.*muk)*self.pk
pk_corr *= sp.sinc(kp*Lpar/2./sp.pi)**2
pk_corr *= sp.sinc(kt*Lper/2./sp.pi)**2
### Biases
b1b2 = qso_boost*bias_qso*bias_met
if self.grid:
xi_qso_met = sp.zeros(self.grid_qso_met[:,0,0].size)
for i in range(self.nmet):
pk_full = b1b2[i]*(1. + beta_met[i]*muk**2)*(1. + beta_qso*muk**2)*pk_corr
xi_qso_met += cosmo_model.Pk2Xi(r[:,i],mur[:,i],self.k,pk_full,ell_max=self.ell_max)*evol[:,i]
else:
nbins = list(self.xdmat.values())[0].shape[0]
xi_qso_met = sp.zeros(nbins)
for i in self.met_names:
bias_met = pars["bias_"+i]
beta_met = pars["beta_"+i]
recalc = beta_met != self.prev_pmet['beta_'+i] or\
growth_rate != self.prev_pmet['growth_rate'] or\
not sp.allclose(qso_evol,self.prev_pmet['qso_evol']) or\
self.prev_pmet['drp'] != drp
if recalc:
if self.verbose:
print("recalculating metal {}".format(i))
self.prev_pmet['beta_'+i] = beta_met
self.prev_pmet['growth_rate'] = growth_rate
self.prev_pmet['qso_evol'] = qso_evol
self.prev_pmet['drp'] = drp
z = self.xzeff[i]
evol = sp.power( self.evolution_growth_factor(z)/self.evolution_growth_factor(self.zref),2. )
evol *= self.evolution_Lya_bias(z,[pars["alpha_"+i]])/self.evolution_Lya_bias(self.zref,[pars["alpha_"+i]])
evol *= self.evolution_QSO_bias(z,qso_evol)/self.evolution_QSO_bias(self.zref,qso_evol)
rp = self.xrp[i] + drp
rt = self.xrt[i]
r = sp.sqrt(rp**2+rt**2)
w=r==0
r[w]=1e-6
mur = rp/r
pk_full = (1. + beta_met*muk**2)*(1. + beta_qso*muk**2)*pk_corr
self.prev_xi_qso_met[i] = cosmo_model.Pk2Xi(r,mur,self.k,pk_full,ell_max=self.ell_max)
self.prev_xi_qso_met[i] = self.xdmat[i].dot(self.prev_xi_qso_met[i]*evol)
xi_qso_met += qso_boost*bias_qso*bias_met*self.prev_xi_qso_met[i]
return xi_qso_met
def valueAuto(self,pars):
bias_lya=pars["bias_lya*(1+beta_lya)"]
beta_lya=pars["beta_lya"]
bias_lya /= 1+beta_lya
if self.templates:
bias_met=sp.array([pars['bias_'+met] for met in self.met_names])
beta_met=sp.array([pars['beta_'+met] for met in self.met_names])
amp=sp.zeros([self.nmet,3])
amp[:,0] = bias_met*(1 + (beta_lya+beta_met)/3 + beta_lya*beta_met/5)
amp[:,1] = bias_met*(2*(beta_lya+beta_met)/3 + 4*beta_lya*beta_met/7)
amp[:,2] = bias_met*8*beta_met*beta_lya/35
amp*=bias_lya
xi_lya_met=amp*self.temp_lya_met
xi_lya_met=sp.sum(xi_lya_met,axis=(1,2))
amp=sp.zeros([self.nmet,self.nmet,3])
bias_met2 = bias_met*bias_met[None,:]
amp[:,:,0] = bias_met2*(1+(beta_met+beta_met[None,:])/3+beta_met*beta_met[None,:]/5)
amp[:,:,1] = bias_met2*(2*(beta_met+beta_met[None,:])/3+4*beta_met*beta_met[None,:]/7)
amp[:,:,2] = bias_met2*8*beta_met*beta_met[None,:]/35
xi_met_met=amp*self.temp_met_met
xi_met_met=sp.sum(xi_met_met,axis=(1,2,3))
else:
muk = cosmo_model.muk
k = self.k
kp = k*muk
kt = k*sp.sqrt(1-muk**2)
nbins = self.dmat["LYA_"+self.met_names[0]].shape[0]
if self.hcds_mets:
bias_lls = pars["bias_lls"]
beta_lls = pars["beta_lls"]
L0_lls = pars["L0_lls"]
Flls = sp.sin(kp*L0_lls)/(kp*L0_lls)
Lpar_auto = pars["Lpar_auto"]
Lper_auto = pars["Lper_auto"]
alpha_lya = pars["alpha_lya"]
Gpar = sp.sinc(kp*Lpar_auto/2/sp.pi)**2
Gper = sp.sinc(kt*Lper_auto/2/sp.pi)**2
xi_lya_met = sp.zeros(nbins)
for met in self.met_names:
bias_met = pars['bias_'+met]
beta_met = pars['beta_'+met]
alpha_met = pars["alpha_"+met]
dm = self.dmat["LYA_"+met]
recalc = beta_met != self.prev_pmet["beta_"+met]\
or beta_lya != self.prev_pmet["beta_lya"]\
or alpha_lya != self.prev_pmet["alpha_lya"]\
or alpha_met != self.prev_pmet["alpha_"+met]
rt = self.auto_rt["LYA_"+met]
rp = self.auto_rp["LYA_"+met]
zeff = self.auto_zeff["LYA_"+met]
r = sp.sqrt(rt**2+rp**2)
w = (r==0)
r[w] = 1e-6
mur = rp/r
if recalc:
if self.verbose:
print("recalculating ",met)
pk = (1+beta_lya*muk**2)*(1+beta_met*muk**2)*self.pk
pk *= Gpar*Gper
xi = cosmo_model.Pk2Xi(r,mur,self.k,pk,ell_max=self.ell_max)
xi *= ((1+zeff)/(1+self.zref))**((alpha_lya-1)*(alpha_met-1))
self.prev_xi_lya_met["LYA_"+met] = self.dmat["LYA_"+met].dot(xi)
if self.hcds_mets:
recalc = self.prev_pmet["beta_lls"] != beta_lls\
or self.prev_pmet["L0_lls"] != L0_lls
if recalc:
pk = (1+beta_lls*muk**2)*(1+beta_met*muk**2)*self.pk*Flls
pk *= Gpar*Gper
xi = cosmo_model.Pk2Xi(r,mur,self.k,pk,ell_max=self.ell_max)
xi *= ((1+zeff)/(1+self.zref))**((alpha_lya-1)*(alpha_met-1))
self.prev_xi_dla_met[met] = xi
xi_lya_met += bias_lya*bias_met*self.prev_xi_lya_met["LYA_"+met]
if self.hcds_mets:
xi_lya_met += bias_lls*bias_met*self.prev_xi_dla_met[met]
xi_met_met = sp.zeros(nbins)
for i,met1 in enumerate(self.met_names):
bias_met1 = pars['bias_'+met1]
beta_met1 = pars['beta_'+met1]
alpha_met1 = pars["alpha_"+met1]
for met2 in self.met_names[i:]:
rt = self.auto_rt[met1+"_"+met2]
rp = self.auto_rp[met1+"_"+met2]
zeff = self.auto_zeff[met1+"_"+met2]
bias_met2 = pars['bias_'+met2]
beta_met2 = pars['beta_'+met2]
alpha_met2 = pars["alpha_"+met2]
dm = self.dmat[met1+"_"+met2]
recalc = beta_met1 != self.prev_pmet["beta_"+met1]\
or beta_met2 != self.prev_pmet["beta_"+met2]
if recalc:
if self.verbose:
print("recalculating ",met1,met2)
r = sp.sqrt(rt**2+rp**2)
w=r==0
r[w]=1e-6
mur = rp/r
pk = (1+beta_met1*muk**2)*(1+beta_met2*muk**2)*self.pk
pk *= Gpar*Gper
xi = cosmo_model.Pk2Xi(r,mur,self.k,pk,ell_max=self.ell_max)
xi *= ((1+zeff)/(1+self.zref))**((alpha_met1-1)*(alpha_met2-1))
self.prev_xi_met_met[met1+"_"+met2] = self.dmat[met1+"_"+met2].dot(xi)
xi_met_met += bias_met1*bias_met2*self.prev_xi_met_met[met1+"_"+met2]
for i in self.prev_pmet:
self.prev_pmet[i] = pars[i]
return xi_lya_met + xi_met_met
def distribution3(x, y, x_avr, y_avr, x_scale, y_scale):
return sinc((x - x_avr) / x_scale) * sinc((y - y_avr) / y_scale)
def compute_sinc_kernel(acquisition_time, user_time,
symmetricization_trick=True,
L=None,
slice_index=None,
n_slices=None,
):
"""Computes the sinc kernel around given user times (user_time).
Parameters
----------
acquisition_time: 1D array of size n_scans
acquisition times for the TRs in the underlying experiment
(typically 0, TR, 2TR, ..., (n_scans - 1) TR or 0, 1, 2, ...,
(n_scans - 1)
user_time: 1D array of shape n_user_times
the times around which the kernels will be centered (this
is the times your want to predict response for)
symmetricization_trick: boolean (optional, default True)
if true symmetricization trick will be used to reflect the
acquisition times about the ordinate axis (this helps the
subsequenc sinc-based interpolation)
L: int (optional, default None)
width of Hanning Window to use in windowing the sinc kernel
(this should help preventing the 'teleportation' of artefacts across
different TRs, and also make the kernel sparse, thus speeding up
the ensuing linear algebra)
Returns
-------
sinc_kernel: 2D array of shape (len(user_time), 2n_scans - 1) if
symmetricization trick has been used or (len(user_time), n_scans)
otherwise
Raises
------
AssertionError
Examples
--------
>>> import slice_timing as st
>>> from numpy import *
>>> at = st.get_acquisition_time(10)
>>> ut = st.get_user_time(21, 10, slice_index=9)
>>> k = st.compute_sinc_kernel(at, ut)
"""
# sanitize the times
assert len(user_time.shape) == 1
assert len(acquisition_time.shape) == 1
# brag
if not slice_index is None:
if not n_slices is None:
print ("Estimating STC transform (sinc kernel) for slice "
"%i/%i...") % (slice_index + 1, n_slices)
else:
print ("Estimating STC transform (sinc kernel) for slice "
"%i") % (slice_index + 1)
# symmetricize the acq time
if symmetricization_trick:
acquisition_time = symmetricized(acquisition_time)
# compute time shifts
time_shift = np.array([
t - acquisition_time
for t in user_time])
# compute kernel
sinc_kernel = scipy.sinc(time_shift)
# modify the kernel with a Hanning window of width L
# around the user times (user_time)
if not L is None and L != INFINITY:
assert L > 0
sinc_kernel *= hanning_window(
time_shift, L)
# return computed kernel
return scipy.sparse.csr_matrix(sinc_kernel)
def sicifunc(z, a, b, c):
si, ci = scipy.special.sici(2.0 * b * (z - a))
#return c*((-1.0+np.cos(2.0*b*(a-z))+2.0*b*(a-z)*si)/(2.0*(b*b)*(z-a))+np.pi/(2.0*b))*b
return c * (-scipy.sinc(b * (z - a) / np.pi) * scipy.sin(b * (z - a)) +
si + np.pi / 2.0) / np.pi
def Fonction(x, y):
z = fabs(sinc(x) * sinc(y))
return z
Lcoh_E = sp.zeros(points) # coherence length as the DC field is varied
I_phi = sp.zeros(points) # second harmonic as phi is varied
I_theta = sp.zeros(points) # second harmonic as theta is varied
I_E = sp.zeros(points) # second harmonic as the DC field is varied
# 7.)___________________________________________________________________________________________________________________
for i in range(points):
n1_phi = n1_normal.n2(theta, phi_var[i])
n3_phi = n3_normal.n1(theta, phi_var[i])
n1_theta = n1_normal.n2(theta_var[i], phi)
n3_theta = n3_normal.n1(theta_var[i], phi)
Lcoh_phi[i] = lambd1 * lambd3 / (2 * ((n3_phi - n1_phi) * lambd1))
Lcoh_theta[i] = lambd1 * lambd3 / (2 * ((n3_theta - n1_theta) * lambd1))
I_phi[i] = 8 * (chi3 * E)**2 * omega3**2 * I1**2 * L**2 / (
n**3 * pc.epsilon0 * pc.c**2) * sp.sinc(L / Lcoh_phi[i])**2
I_theta[i] = 8 * (chi3 * E)**2 * omega3**2 * I1**2 * L**2 / (n**3 * pc.epsilon0 * pc.c**2) * \
sp.sinc(L / Lcoh_theta[i])**2
# 8.)___________________________________________________________________________________________________________________
fig = plt.figure()
ax1 = fig.add_subplot(2, 1, 1)
ax2 = fig.add_subplot(2, 1, 2)
ax1.plot(phi_var * 180 / sp.pi, I_phi)
ax2.plot(theta_var * 180 / sp.pi, I_theta)
# ax1.set_ylim([0, 15])
plt.show()
def lanczos(N, tau=1):
'''
return N-point Lanczos window with width tau
'''
return scipy.sinc(linspace(-1, 1, N) / tau)
def getXiCross(self,rp,rt,z,pk_lin,pars):
k = self.k
if not self.fit_aiso:
ap=pars["ap"]
at=pars["at"]
else:
ap=pars["aiso"]*pars["1+epsilon"]*pars["1+epsilon"]
at=pars["aiso"]/pars["1+epsilon"]
drp=pars["drp"]
Lpar=pars["Lpar_cross"]
Lper=pars["Lper_cross"]
qso_evol = [pars['qso_evol_0'],pars['qso_evol_1']]
rp_shift=rp+drp
ar=np.sqrt(rt**2*at**2+rp_shift**2*ap**2)
mur=rp_shift*ap/ar
muk = model.muk
kp = k * muk
kt = k * np.sqrt(1-muk**2)
bias_lya = pars["bias_lya*(1+beta_lya)"]/(1.+pars["beta_lya"])
beta_lya = pars["beta_lya"]
### UV fluctuation
if self.uv_fluct:
bias_gamma = pars["bias_gamma"]
bias_prim = pars["bias_prim"]
lambda_uv = pars["lambda_uv"]
W = sp.arctan(k*lambda_uv)/(k*lambda_uv)
bias_lya_prim = bias_lya + bias_gamma*W/(1+bias_prim*W)
beta_lya = bias_lya*beta_lya/bias_lya_prim
bias_lya = bias_lya_prim
### LYA-QSO cross correlation
bias_qso = pars["bias_qso"]
beta_qso = pars["growth_rate"]/bias_qso
pk_full = bias_lya*bias_qso*(1+beta_lya*muk**2)*(1+beta_qso*muk**2)*pk_lin
### HCDS-QSO cross correlation
if self.lls:
bias_lls = pars["bias_lls"]
beta_lls = pars["beta_lls"]
L0_lls = pars["L0_lls"]
F_lls = sp.sinc(kp*L0_lls/sp.pi)
pk_full+=bias_lls*F_lls*bias_qso*(1+beta_lls*muk**2)*(1+beta_qso*muk**2)*pk_lin
### Velocity dispersion
if (self.velo_gauss):
pk_full *= sp.exp( -0.25*(kp*pars['sigma_velo_gauss'])**2 )
if (self.velo_lorentz):
pk_full /= np.sqrt(1.+(kp*pars['sigma_velo_lorentz'])**2)
### Peak broadening
sigmaNLper = pars["SigmaNL_perp"]
sigmaNLpar = sigmaNLper*pars["1+f"]
pk_full *= sp.exp( -0.5*( (sigmaNLper*kt)**2 + (sigmaNLpar*kp)**2 ) )
### Pixel size
pk_full *= sp.sinc(kp*Lpar/2./sp.pi)**2
pk_full *= sp.sinc(kt*Lper/2./sp.pi)**2
### Non-linear correction
pk_full *= np.sqrt(self.DNL(self.k,self.muk,self.pk,self.q1_dnl,self.kv_dnl,self.av_dnl,self.bv_dnl,self.kp_dnl,self.dnl_model))
### Redshift evolution
evol = np.power( self.evolution_growth_factor(z)/self.evolution_growth_factor(self.zref),2. )
evol *= self.evolution_Lya_bias(z,[pars["alpha_lya"]])/self.evolution_Lya_bias(self.zref,[pars["alpha_lya"]])
evol *= self.evolution_QSO_bias(z,qso_evol)/self.evolution_QSO_bias(self.zref,qso_evol)
return self.Pk2Xi(ar,mur,k,pk_full,ell_max=self.ell_max)*evol
def lanczos(kx):
w = 5.
out = sp.sinc(kx/w)
m = abs(kx) > w
out[m] = 0
return out*sp.sinc(kx)
dx = 0.1
t0 = 0.0
tmax = 500.0
dt = 0.1
x = arange(xmin, xmax, dx)
temps = arange(t0, tmax, dt)
# calcul d'une superposition de deux OPPH en milieu non dispersif
omega0 = 1000.0
domega = 1.0
k0 = 0.5
dk = 0.01
#mescourbes=[[2*Psi0*cos(domega*t - dk*xi)*cos(omega0*t - k0*xi) for xi in x] for t in temps]
# calcul d'un paquet d'ondes
mescourbes = [[
2 * Psi0 * sinc(domega * t - dk * xi) * cos(omega0 * t - k0 * xi)
for xi in x
] for t in temps]
# tracé de l'animation
fig = pyplot.figure()
ax = pyplot.axes(xlim=(0, xmax), ylim=(-1.0, 1.0))
courbe, = ax.plot(x, mescourbes[0])
line_ani = animation.FuncAnimation(fig,
traceframe,
100,
interval=50,
repeat=True)
pyplot.show()
def sincSquared(x,A,B,tau,x0):
return A*(scipy.sinc(tau * 2*scipy.pi * (x-x0) ))**2 / (2*scipy.pi) + B
epsilon = 1e-5
numPrime0 = (function(x0+epsilon)-function(x0))/epsilon
xold = x0
xnew = xold - function(xold)/numPrime0
while abs((xold- xnew)) > tol:
xold = xnew
numPrimeNew = (function(xnew+epsilon)-function(xnew))/epsilon
xnew = xold - function(xold)/numPrimeNew
return xnew
func = lambda x: x**2 -1
fPrime = lambda x: 2*x
func1 = lambda x: sp.cos(x)
func2 = lambda x: sp.sin(1/x)*x**2
func3 = lambda x: sp.sinc(x) -x
func1Prime = lambda x: -sp.sin(x)
func2Prime = lambda x: 2*x*sp.sin(1/x) - sp.cos(1/x)
func3Prime = lambda x: -sp.sin(x)/x**2 + sp.cos(x)/x -1
# Problem 2
funcProb2 = lambda x: x**(1/3)
# print myNewNewton(funcProb2, .5)
# This will diverge as can easily be seen with this analtical derivation:
# x_n+1 = x_n - (x_n**1/3)/(x_n**(-2/3)/3)
# x_n+1 = x_n - 3*x_n
# x_n+1 = -2*x_n // This diverges!