| """ |
| Discrete Fourier Transforms - helper.py |
| |
| """ |
| from __future__ import division, absolute_import, print_function |
| |
| from numpy.compat import integer_types |
| from numpy.core import integer, empty, arange, asarray, roll |
| from numpy.core.overrides import array_function_dispatch, set_module |
| |
| # Created by Pearu Peterson, September 2002 |
| |
| __all__ = ['fftshift', 'ifftshift', 'fftfreq', 'rfftfreq'] |
| |
| integer_types = integer_types + (integer,) |
| |
| |
| def _fftshift_dispatcher(x, axes=None): |
| return (x,) |
| |
| |
| @array_function_dispatch(_fftshift_dispatcher, module='numpy.fft') |
| def fftshift(x, axes=None): |
| """ |
| Shift the zero-frequency component to the center of the spectrum. |
| |
| This function swaps half-spaces for all axes listed (defaults to all). |
| Note that ``y[0]`` is the Nyquist component only if ``len(x)`` is even. |
| |
| Parameters |
| ---------- |
| x : array_like |
| Input array. |
| axes : int or shape tuple, optional |
| Axes over which to shift. Default is None, which shifts all axes. |
| |
| Returns |
| ------- |
| y : ndarray |
| The shifted array. |
| |
| See Also |
| -------- |
| ifftshift : The inverse of `fftshift`. |
| |
| Examples |
| -------- |
| >>> freqs = np.fft.fftfreq(10, 0.1) |
| >>> freqs |
| array([ 0., 1., 2., ..., -3., -2., -1.]) |
| >>> np.fft.fftshift(freqs) |
| array([-5., -4., -3., -2., -1., 0., 1., 2., 3., 4.]) |
| |
| Shift the zero-frequency component only along the second axis: |
| |
| >>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3) |
| >>> freqs |
| array([[ 0., 1., 2.], |
| [ 3., 4., -4.], |
| [-3., -2., -1.]]) |
| >>> np.fft.fftshift(freqs, axes=(1,)) |
| array([[ 2., 0., 1.], |
| [-4., 3., 4.], |
| [-1., -3., -2.]]) |
| |
| """ |
| x = asarray(x) |
| if axes is None: |
| axes = tuple(range(x.ndim)) |
| shift = [dim // 2 for dim in x.shape] |
| elif isinstance(axes, integer_types): |
| shift = x.shape[axes] // 2 |
| else: |
| shift = [x.shape[ax] // 2 for ax in axes] |
| |
| return roll(x, shift, axes) |
| |
| |
| @array_function_dispatch(_fftshift_dispatcher, module='numpy.fft') |
| def ifftshift(x, axes=None): |
| """ |
| The inverse of `fftshift`. Although identical for even-length `x`, the |
| functions differ by one sample for odd-length `x`. |
| |
| Parameters |
| ---------- |
| x : array_like |
| Input array. |
| axes : int or shape tuple, optional |
| Axes over which to calculate. Defaults to None, which shifts all axes. |
| |
| Returns |
| ------- |
| y : ndarray |
| The shifted array. |
| |
| See Also |
| -------- |
| fftshift : Shift zero-frequency component to the center of the spectrum. |
| |
| Examples |
| -------- |
| >>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3) |
| >>> freqs |
| array([[ 0., 1., 2.], |
| [ 3., 4., -4.], |
| [-3., -2., -1.]]) |
| >>> np.fft.ifftshift(np.fft.fftshift(freqs)) |
| array([[ 0., 1., 2.], |
| [ 3., 4., -4.], |
| [-3., -2., -1.]]) |
| |
| """ |
| x = asarray(x) |
| if axes is None: |
| axes = tuple(range(x.ndim)) |
| shift = [-(dim // 2) for dim in x.shape] |
| elif isinstance(axes, integer_types): |
| shift = -(x.shape[axes] // 2) |
| else: |
| shift = [-(x.shape[ax] // 2) for ax in axes] |
| |
| return roll(x, shift, axes) |
| |
| |
| @set_module('numpy.fft') |
| def fftfreq(n, d=1.0): |
| """ |
| Return the Discrete Fourier Transform sample frequencies. |
| |
| The returned float array `f` contains the frequency bin centers in cycles |
| per unit of the sample spacing (with zero at the start). For instance, if |
| the sample spacing is in seconds, then the frequency unit is cycles/second. |
| |
| Given a window length `n` and a sample spacing `d`:: |
| |
| f = [0, 1, ..., n/2-1, -n/2, ..., -1] / (d*n) if n is even |
| f = [0, 1, ..., (n-1)/2, -(n-1)/2, ..., -1] / (d*n) if n is odd |
| |
| Parameters |
| ---------- |
| n : int |
| Window length. |
| d : scalar, optional |
| Sample spacing (inverse of the sampling rate). Defaults to 1. |
| |
| Returns |
| ------- |
| f : ndarray |
| Array of length `n` containing the sample frequencies. |
| |
| Examples |
| -------- |
| >>> signal = np.array([-2, 8, 6, 4, 1, 0, 3, 5], dtype=float) |
| >>> fourier = np.fft.fft(signal) |
| >>> n = signal.size |
| >>> timestep = 0.1 |
| >>> freq = np.fft.fftfreq(n, d=timestep) |
| >>> freq |
| array([ 0. , 1.25, 2.5 , ..., -3.75, -2.5 , -1.25]) |
| |
| """ |
| if not isinstance(n, integer_types): |
| raise ValueError("n should be an integer") |
| val = 1.0 / (n * d) |
| results = empty(n, int) |
| N = (n-1)//2 + 1 |
| p1 = arange(0, N, dtype=int) |
| results[:N] = p1 |
| p2 = arange(-(n//2), 0, dtype=int) |
| results[N:] = p2 |
| return results * val |
| |
| |
| @set_module('numpy.fft') |
| def rfftfreq(n, d=1.0): |
| """ |
| Return the Discrete Fourier Transform sample frequencies |
| (for usage with rfft, irfft). |
| |
| The returned float array `f` contains the frequency bin centers in cycles |
| per unit of the sample spacing (with zero at the start). For instance, if |
| the sample spacing is in seconds, then the frequency unit is cycles/second. |
| |
| Given a window length `n` and a sample spacing `d`:: |
| |
| f = [0, 1, ..., n/2-1, n/2] / (d*n) if n is even |
| f = [0, 1, ..., (n-1)/2-1, (n-1)/2] / (d*n) if n is odd |
| |
| Unlike `fftfreq` (but like `scipy.fftpack.rfftfreq`) |
| the Nyquist frequency component is considered to be positive. |
| |
| Parameters |
| ---------- |
| n : int |
| Window length. |
| d : scalar, optional |
| Sample spacing (inverse of the sampling rate). Defaults to 1. |
| |
| Returns |
| ------- |
| f : ndarray |
| Array of length ``n//2 + 1`` containing the sample frequencies. |
| |
| Examples |
| -------- |
| >>> signal = np.array([-2, 8, 6, 4, 1, 0, 3, 5, -3, 4], dtype=float) |
| >>> fourier = np.fft.rfft(signal) |
| >>> n = signal.size |
| >>> sample_rate = 100 |
| >>> freq = np.fft.fftfreq(n, d=1./sample_rate) |
| >>> freq |
| array([ 0., 10., 20., ..., -30., -20., -10.]) |
| >>> freq = np.fft.rfftfreq(n, d=1./sample_rate) |
| >>> freq |
| array([ 0., 10., 20., 30., 40., 50.]) |
| |
| """ |
| if not isinstance(n, integer_types): |
| raise ValueError("n should be an integer") |
| val = 1.0/(n*d) |
| N = n//2 + 1 |
| results = arange(0, N, dtype=int) |
| return results * val |
