import numpy as np
from numpy.fft import fft, ifft, fftshift, ifftshift
from .FFTBase import FFTBase
class T1FFT(FFTBase):
"""
The Fast Fourier Transform on the Circle / 1-Torus / 1-Sphere.
"""
@staticmethod
def analyze(f, axis=0):
"""
Compute the Fourier Transform of the discretely sampled function f : T^1 -> C.
Let f : T^1 -> C be a band-limited function on the circle.
The samples f(theta_k) correspond to points on a regular grid on the circle, as returned by spaces.T1.linspace:
theta_k = 2 pi k / N
for k = 0, ..., N - 1
This function computes
\hat{f}_n = (1/N) \sum_{k=0}^{N-1} f(theta_k) e^{-i n theta_k}
which, if f has band-limit less than N, is equal to:
\hat{f}_n = \int_0^{2pi} f(theta) e^{-i n theta} dtheta / 2pi,
= <f(theta), e^{i n theta}>
where dtheta / 2pi is the normalized Haar measure on T^1, and < , > denotes the inner product on Hilbert space,
with respect to which this transform is unitary.
The range of frequencies n is -floor(N/2) <= n <= ceil(N/2) - 1
:param f:
:param axis:
:return:
"""
# The numpy FFT returns coefficients in a different order than we want them,
# and using a different normalization.
fhat = fft(f, axis=axis)
fhat = fftshift(fhat, axes=axis)
return fhat / f.shape[axis]
@staticmethod
def synthesize(f_hat, axis=0):
"""
Compute the inverse / synthesis Fourier transform of the function f_hat : Z -> C.
The function f_hat(n) is sampled at points in a limited range -floor(N/2) <= n <= ceil(N/2) - 1
This function returns
f[k] = f(theta_k) = sum_{n=-floor(N/2)}^{ceil(N/2)-1} f_hat(n) exp(i n theta_k)
where theta_k = 2 pi k / N
for k = 0, ..., N - 1
:param f_hat:
:param axis:
:return:
"""
f_hat = ifftshift(f_hat * f_hat.shape[axis], axes=axis)
f = ifft(f_hat, axis=axis)
return f
@staticmethod
def analyze_naive(f):
f_hat = np.zeros_like(f)
for n in range(f.size):
for k in range(f.size):
theta_k = k * 2 * np.pi / f.size
f_hat[n] += f[k] * np.exp(-1j * n * theta_k)
return fftshift(f_hat / f.size, axes=0)
