def test_pdos_1d():
pad=lambda x: pad_zeros(x, nadd=len(x)-1)
n=500; w=welch(n)
# 1 second signal
t=np.linspace(0,1,n); dt=t[1]-t[0]
# sum of sin()s with random freq and phase shift, 10 frequencies from
# f=0...100 Hz
v=np.array([np.sin(2*np.pi*f*t + rand()*2*np.pi) for f in rand(10)*100]).sum(0)
f=np.fft.fftfreq(2*n-1, dt)[:n]
c1=mirror(ifft(abs(fft(pad(v)))**2.0)[:n].real)
c2=correlate(v,v,'full')
c3=mirror(acorr(v,norm=False))
assert np.allclose(c1, c2)
assert np.allclose(c1, c3)
p1=(abs(fft(pad(v)))**2.0)[:n]
p2=(abs(fft(mirror(acorr(v,norm=False)))))[:n]
assert np.allclose(p1, p2)
p1=(abs(fft(pad(v*w)))**2.0)[:n]
p2=(abs(fft(mirror(acorr(v*w,norm=False)))))[:n]
assert np.allclose(p1, p2)
def test_pdos_1d():
pad=lambda x: pad_zeros(x, nadd=len(x)-1)
n=500; w=welch(n)
# 1 second signal
t=np.linspace(0,1,n); dt=t[1]-t[0]
# sum of sin()s with random freq and phase shift, 10 frequencies from
# f=0...100 Hz
v=np.array([np.sin(2*np.pi*f*t + rand()*2*np.pi) for f in rand(10)*100]).sum(0)
f=np.fft.fftfreq(2*n-1, dt)[:n]
c1=mirror(ifft(abs(fft(pad(v)))**2.0)[:n].real)
c2=correlate(v,v,'full')
c3=mirror(acorr(v,norm=False))
assert np.allclose(c1, c2)
assert np.allclose(c1, c3)
p1=(abs(fft(pad(v)))**2.0)[:n]
p2=(abs(fft(mirror(acorr(v,norm=False)))))[:n]
assert np.allclose(p1, p2)
p1=(abs(fft(pad(v*w)))**2.0)[:n]
p2=(abs(fft(mirror(acorr(v*w,norm=False)))))[:n]
assert np.allclose(p1, p2)
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
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)
See Also
--------
:func:`direct_pdos`
:func:`vacf_pdos`
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
--------
pwtools.signal.fftsample
pwtools.signal.acorr
"""
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
