import numpy as np
from numpy.fft import fftn, ifftn
from ..base_channel import Channel
from tramp.utils.misc import complex2array, array2complex
[docs]class DFTChannel(Channel):
"""Discrete fourier transform x = FFT z.
Parameters
----------
- real: bool
If z supposed to be real
Notes
-----
The fft and ifft are scaled by sqrt(N) so that both are unitary.
For message passing it is more convenient to represent a complex array x
as a real array X where X[0] = x.real and X[1] = x.imag
In particular:
- output of sample(): X array of shape (2, x.shape)
- message bx, posterior rx: real arrays of shape (2, x.shape)
And if real=False (z complex):
- input of sample(): Z array of shape (2, z.shape)
- message bz, posterior rz: real arrays of shape (2, z.shape)
"""
def __init__(self, real=True):
self.real = real
self.repr_init()
def sample(self, Z):
"When real=False we assume Z[0] = Z.real and Z[1] = Z.imag"
if not self.real:
Z = array2complex(Z)
X = fftn(Z, norm="ortho")
X = complex2array(X)
return X
def math(self):
return r"$\mathcal{F}$"
def second_moment(self, tau_z):
return tau_z
def compute_forward_message(self, az, bz, ax, bx):
# x = FFT z
ax_new = az
if not self.real:
bz = array2complex(bz)
bx_new = fftn(bz, norm="ortho")
bx_new = complex2array(bx_new)
return ax_new, bx_new
def compute_backward_message(self, az, bz, ax, bx):
# z = IFFT x
az_new = ax
bx = array2complex(bx)
bz_new = ifftn(bx, norm="ortho")
if self.real:
bz_new = np.real(bz_new)
else:
bz_new = complex2array(bz_new)
return az_new, bz_new
def compute_forward_state_evolution(self, az, ax, tau_z):
ax_new = az
return ax_new
def compute_backward_state_evolution(self, az, ax, tau_z):
az_new = ax
return az_new
def compute_log_partition(self, az, bz, ax, bx):
_, bz_new = self.compute_backward_mean(az, bz, ax, bx)
b = bz + bz_new
a = az + ax
coef = 0.5 if self.real else 1
logZ = 0.5 * np.sum(b**2 / a) + coef * self.N * np.log(2 * np.pi / a)
return logZ
def compute_mutual_information(self, az, ax, tau_z):
a = ax + az
I = 0.5*np.log(a*tau_z)
return I
def compute_free_energy(self, az, ax, tau_z):
tau_x = self.second_moment(tau_z)
I = self.compute_mutual_information(az, ax, tau_z)
A = 0.5*(az*tau_z + ax*tau_x) - I + 0.5*np.log(2*np.pi*tau_z/np.e)
return A