def test_norm_int():
# simps(y, x) == 2.0
x = np.linspace(0, np.pi, 100)
y = np.sin(x)
for scale in [True, False]:
yy = norm_int(y, x, area=10.0, scale=scale)
assert np.allclose(simps(yy, x), 10.0)
def test_norm_int():
# simps(y, x) == 2.0
x = np.linspace(0, np.pi, 100)
y = np.sin(x)
for scale in [True, False]:
yy = norm_int(y, x, area=10.0, scale=scale)
assert np.allclose(simps(yy,x), 10.0)
def cut_norm(full_y, dt, area=1.0):
"""Cut out an FFT spectrum from scipy.fftpack.fft() (or numpy.fft.fft())
result and normalize the integral `int(f) y(f) df = area`.
full_y : 1d array
Result of fft(...)
dt : time step
area : integral area
"""
full_faxis = np.fft.fftfreq(full_y.shape[0], dt)
split_idx = full_faxis.shape[0] / 2
y_out = full_y[:split_idx]
faxis = full_faxis[:split_idx]
return faxis, num.norm_int(y_out, faxis, area=area)
def cut_norm(full_y, dt, area=1.0):
"""Cut out an FFT spectrum from scipy.fftpack.fft() (or numpy.fft.fft())
result and normalize the integral `int(f) y(f) df = area`.
full_y : 1d array
Result of fft(...)
dt : time step
area : integral area
"""
full_faxis = np.fft.fftfreq(full_y.shape[0], dt)
split_idx = full_faxis.shape[0]/2
y_out = full_y[:split_idx]
faxis = full_faxis[:split_idx]
return faxis, num.norm_int(y_out, faxis, area=area)
def pdos(vel,
dt=1.0,
m=None,
full_out=False,
area=1.0,
window=True,
npad=None,
tonext=False,
mirr=False,
method='direct'):
"""Phonon DOS by FFT of the VACF or direct FFT of atomic velocities.
Integral area is normalized to `area`. It is possible (and recommended) to
zero-padd the velocities (see `npad`).
Parameters
----------
vel : 3d array (nstep, natoms, 3)
atomic velocities
dt : time step
m : 1d array (natoms,),
atomic mass array, if None then mass=1.0 for all atoms is used
full_out : bool
area : float
normalize area under frequency-PDOS curve to this value
window : bool
use Welch windowing on data before FFT (reduces leaking effect,
recommended)
npad : {None, int}
method='direct' only: Length of zero padding along `axis`. `npad=None`
= no padding, `npad > 0` = pad by a length of ``(nstep-1)*npad``. `npad
> 5` usually results in sufficient interpolation.
tonext : bool
method='direct' only: Pad `vel` with zeros along `axis` up to the next
power of two after the array length determined by `npad`. This gives
you speed, but variable (better) frequency resolution.
mirr : bool
method='vacf' only: mirror one-sided VACF at t=0 before fft
Returns
-------
if full_out = False
| ``(faxis, pdos)``
| faxis : 1d array [1/unit(dt)]
| pdos : 1d array, the phonon DOS, normalized to `area`
if full_out = True
| if method == 'direct':
| ``(faxis, pdos, (full_faxis, full_pdos, split_idx))``
| if method == 'vavcf':
| ``(faxis, pdos, (full_faxis, full_pdos, split_idx, vacf, fft_vacf))``
| fft_vacf : 1d complex array, result of fft(vacf) or fft(mirror(vacf))
| vacf : 1d array, the VACF
Examples
--------
>>> from pwtools.constants import fs,rcm_to_Hz
>>> tr = Trajectory(...)
>>> # freq in [Hz] if timestep in [s]
>>> freq,dos = pdos(tr.velocity, m=tr.mass, dt=tr.timestep*fs,
>>> method='direct', npad=1)
>>> # frequency in [1/cm]
>>> plot(freq/rcm_to_Hz, dos)
Notes
-----
padding (only method='direct'): With `npad` we pad the velocities `vel`
with ``npad*(nstep-1)`` zeros along `axis` (the time axis) before FFT
b/c the signal is not periodic. For `npad=1`, this gives us the exact
same spectrum and frequency resolution as with ``pdos(...,
method='vacf',mirr=True)`` b/c the array to be fft'ed has length
``2*nstep-1`` along the time axis in both cases (remember that the
array length = length of the time axis influences the freq.
resolution). FFT is only fast for arrays with length = a power of two.
Therefore, you may get very different fft speeds depending on whether
``2*nstep-1`` is a power of two or not (in most cases it won't). Try
using `tonext` but remember that you get another (better) frequency
resolution.
References
----------
[1] Phys Rev B 47(9) 4863, 1993
See Also
--------
:func:`pwtools.signal.fftsample`
:func:`pwtools.signal.acorr`
:func:`direct_pdos`
:func:`vacf_pdos`
"""
mass = m
# assume vel.shape = (nstep,natoms,3)
axis = 0
assert vel.shape[-1] == 3
if mass is not None:
assert len(mass) == vel.shape[1], "len(mass) != vel.shape[1]"
# define here b/c may be used twice below
mass_bc = mass[None, :, None]
if window:
sl = [None] * vel.ndim
sl[axis] = slice(None) # ':'
vel2 = vel * (welch(vel.shape[axis])[sl])
else:
vel2 = vel
# handle options which are mutually exclusive
if method == 'vacf':
assert npad in [0, None], "use npad={0,None} for method='vacf'"
# padding
if npad is not None:
nadd = (vel2.shape[axis] - 1) * npad
if tonext:
vel2 = pad_zeros(vel2,
tonext=True,
tonext_min=vel2.shape[axis] + nadd,
axis=axis)
else:
vel2 = pad_zeros(vel2, tonext=False, nadd=nadd, axis=axis)
if method == 'direct':
full_fft_vel = np.abs(fft(vel2, axis=axis))**2.0
full_faxis = np.fft.fftfreq(vel2.shape[axis], dt)
split_idx = len(full_faxis) // 2
faxis = full_faxis[:split_idx]
# First split the array, then multiply by `mass` and average. If
# full_out, then we need full_fft_vel below, so copy before slicing.
arr = full_fft_vel.copy() if full_out else full_fft_vel
fft_vel = num.slicetake(arr,
slice(0, split_idx),
axis=axis,
copy=False)
if mass is not None:
fft_vel *= mass_bc
# average remaining axes, summing is enough b/c normalization is done below
# sums: (nstep, natoms, 3) -> (nstep, natoms) -> (nstep,)
pdos = num.sum(fft_vel, axis=axis, keepdims=True)
default_out = (faxis, num.norm_int(pdos, faxis, area=area))
if full_out:
# have to re-calculate this here b/c we never calculate the full_pdos
# normally
if mass is not None:
full_fft_vel *= mass_bc
full_pdos = num.sum(full_fft_vel, axis=axis, keepdims=True)
extra_out = (full_faxis, full_pdos, split_idx)
return default_out + extra_out
else:
return default_out
elif method == 'vacf':
vacf = fvacf(vel2, m=mass)
if mirr:
fft_vacf = fft(mirror(vacf))
else:
fft_vacf = fft(vacf)
full_faxis = np.fft.fftfreq(fft_vacf.shape[axis], dt)
full_pdos = np.abs(fft_vacf)
split_idx = len(full_faxis) // 2
faxis = full_faxis[:split_idx]
pdos = full_pdos[:split_idx]
default_out = (faxis, num.norm_int(pdos, faxis, area=area))
extra_out = (full_faxis, full_pdos, split_idx, vacf, fft_vacf)
if full_out:
return default_out + extra_out
else:
return default_out
fourier_out_data_fn = pj(fourier_dir, 'fourier_out_data_1d.txt')
fourier_in_fn = pj(fourier_dir, 'fourier_1d.in')
fourier_out_fn = pj(fourier_dir, 'fourier_1d.log')
fourier_in_txt = '%s\n%s\n%e\n%e\n%e\n%i' % (
fourier_in_data_fn, fourier_out_data_fn, dt / constants.th, 0,
fmax * fmax_extend_fac / (constants.c0 * 100), 1)
common.file_write(fourier_in_fn, fourier_in_txt)
# In order to make picky gfortrans happy, we need to use savetxt(...,
# fmt="%g") such that the first column is an integer (1 instead of
# 1.0000e+00).
np.savetxt(fourier_in_data_fn, fourier_in_data, fmt='%g')
common.system("%s < %s > %s" %
(fourier_exe, fourier_in_fn, fourier_out_fn))
fourier_out_data = np.loadtxt(fourier_out_data_fn)
f3 = fourier_out_data[:, 0] * (constants.c0 * 100) # 1/cm -> Hz
y3n = num.norm_int(fourier_out_data[:, 1], f3)
f1, y1n = cut_norm(y1, dt)
f2, y2n = cut_norm(y2, dt)
figs = []
axs = []
figs.append(plt.figure())
axs.append(figs[-1].add_subplot(111))
axs[-1].set_title('1d arr')
axs[-1].plot(f1, y1n, label='1d |fft(arr)|^2, direct')
axs[-1].plot(f2, y2n, label='1d |fft(acorr(arr))|, vacf')
if use_fourier:
axs[-1].plot(f3, y3n, label='1d fourier.x')
axs[-1].legend()
fourier_out_fn = pj(fourier_dir, 'fourier_1d.log')
fourier_in_txt = '%s\n%s\n%e\n%e\n%e\n%i' %(fourier_in_data_fn,
fourier_out_data_fn,
dt/constants.th,
0,
fmax*fmax_extend_fac/(constants.c0*100),
1)
common.file_write(fourier_in_fn, fourier_in_txt)
# In order to make picky gfortrans happy, we need to use savetxt(...,
# fmt="%g") such that the first column is an integer (1 instead of
# 1.0000e+00).
np.savetxt(fourier_in_data_fn, fourier_in_data, fmt='%g')
common.system("%s < %s > %s" %(fourier_exe, fourier_in_fn, fourier_out_fn))
fourier_out_data = np.loadtxt(fourier_out_data_fn)
f3 = fourier_out_data[:,0]*(constants.c0*100) # 1/cm -> Hz
y3n = num.norm_int(fourier_out_data[:,1], f3)
f1, y1n = cut_norm(y1, dt)
f2, y2n = cut_norm(y2, dt)
figs = []
axs = []
figs.append(plt.figure())
axs.append(figs[-1].add_subplot(111))
axs[-1].set_title('1d arr')
axs[-1].plot(f1, y1n, label='1d |fft(arr)|^2, direct')
axs[-1].plot(f2, y2n, label='1d |fft(acorr(arr))|, vacf')
if use_fourier:
axs[-1].plot(f3, y3n, label='1d fourier.x')
axs[-1].legend()