def _raw_fftnd(a, s=None, axes=None, function=fft, norm=None):
a = asarray(a)
s, axes = _cook_nd_args(a, s, axes)
itl = list(range(len(axes)))
itl.reverse()
for ii in itl:
a = function(a, n=s[ii], axis=axes[ii], norm=norm)
return a
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., 4., -5., -4., -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)
def byte_bounds(a):
"""
Returns pointers to the end-points of an array.
Parameters
----------
a : ndarray
Input array. It must conform to the Python-side of the array
interface.
Returns
-------
(low, high) : tuple of 2 integers
The first integer is the first byte of the array, the second
integer is just past the last byte of the array. If `a` is not
contiguous it will not use every byte between the (`low`, `high`)
values.
Examples
--------
>>> I = np.eye(2, dtype='f'); I.dtype
dtype('float32')
>>> low, high = np.byte_bounds(I)
>>> high - low == I.size*I.itemsize
True
>>> I = np.eye(2, dtype='G'); I.dtype
dtype('complex192')
>>> low, high = np.byte_bounds(I)
>>> high - low == I.size*I.itemsize
True
"""
ai = a.__array_interface__
a_data = ai['data'][0]
astrides = ai['strides']
ashape = ai['shape']
bytes_a = asarray(a).dtype.itemsize
a_low = a_high = a_data
if astrides is None:
# contiguous case
a_high += a.size * bytes_a
else:
for shape, stride in zip(ashape, astrides):
if stride < 0:
a_low += (shape - 1) * stride
else:
a_high += (shape - 1) * stride
a_high += bytes_a
return a_low, a_high
def _raw_fft(a, n=None, axis=-1, init_function=fftpack.cffti,
work_function=fftpack.cfftf, fft_cache=_fft_cache):
a = asarray(a)
if n is None:
n = a.shape[axis]
if n < 1:
raise ValueError("Invalid number of FFT data points (%d) specified."
% n)
# We have to ensure that only a single thread can access a wsave array
# at any given time. Thus we remove it from the cache and insert it
# again after it has been used. Multiple threads might create multiple
# copies of the wsave array. This is intentional and a limitation of
# the current C code.
wsave = fft_cache.pop_twiddle_factors(n)
if wsave is None:
wsave = init_function(n)
if a.shape[axis] != n:
s = list(a.shape)
if s[axis] > n:
index = [slice(None)]*len(s)
index[axis] = slice(0, n)
a = a[tuple(index)]
else:
index = [slice(None)]*len(s)
index[axis] = slice(0, s[axis])
s[axis] = n
z = zeros(s, a.dtype.char)
z[tuple(index)] = a
a = z
if axis != -1:
a = swapaxes(a, axis, -1)
r = work_function(a, wsave)
if axis != -1:
r = swapaxes(r, axis, -1)
# As soon as we put wsave back into the cache, another thread could pick it
# up and start using it, so we must not do this until after we're
# completely done using it ourselves.
fft_cache.put_twiddle_factors(n, wsave)
return r
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)
def fft(a, n=None, axis=-1, norm=None):
"""
Compute the one-dimensional discrete Fourier Transform.
This function computes the one-dimensional *n*-point discrete Fourier
Transform (DFT) with the efficient Fast Fourier Transform (FFT)
algorithm [CT].
Parameters
----------
a : array_like
Input array, can be complex.
n : int, optional
Length of the transformed axis of the output.
If `n` is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If `n` is not given,
the length of the input along the axis specified by `axis` is used.
axis : int, optional
Axis over which to compute the FFT. If not given, the last axis is
used.
norm : {None, "ortho"}, optional
.. versionadded:: 1.10.0
Normalization mode (see `numpy.fft`). Default is None.
Returns
-------
out : complex ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
Raises
------
IndexError
if `axes` is larger than the last axis of `a`.
See Also
--------
numpy.fft : for definition of the DFT and conventions used.
ifft : The inverse of `fft`.
fft2 : The two-dimensional FFT.
fftn : The *n*-dimensional FFT.
rfftn : The *n*-dimensional FFT of real input.
fftfreq : Frequency bins for given FFT parameters.
Notes
-----
FFT (Fast Fourier Transform) refers to a way the discrete Fourier
Transform (DFT) can be calculated efficiently, by using symmetries in the
calculated terms. The symmetry is highest when `n` is a power of 2, and
the transform is therefore most efficient for these sizes.
The DFT is defined, with the conventions used in this implementation, in
the documentation for the `numpy.fft` module.
References
----------
.. [CT] Cooley, James W., and John W. Tukey, 1965, "An algorithm for the
machine calculation of complex Fourier series," *Math. Comput.*
19: 297-301.
Examples
--------
>>> np.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8))
array([ -3.44505240e-16 +1.14383329e-17j,
8.00000000e+00 -5.71092652e-15j,
2.33482938e-16 +1.22460635e-16j,
1.64863782e-15 +1.77635684e-15j,
9.95839695e-17 +2.33482938e-16j,
0.00000000e+00 +1.66837030e-15j,
1.14383329e-17 +1.22460635e-16j,
-1.64863782e-15 +1.77635684e-15j])
In this example, real input has an FFT which is Hermitian, i.e., symmetric
in the real part and anti-symmetric in the imaginary part, as described in
the `numpy.fft` documentation:
>>> import matplotlib.pyplot as plt
>>> t = np.arange(256)
>>> sp = np.fft.fft(np.sin(t))
>>> freq = np.fft.fftfreq(t.shape[-1])
>>> plt.plot(freq, sp.real, freq, sp.imag)
[<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
>>> plt.show()
"""
a = asarray(a).astype(complex, copy=False)
if n is None:
n = a.shape[axis]
output = _raw_fft(a, n, axis, fftpack.cffti, fftpack.cfftf, _fft_cache)
if _unitary(norm):
output *= 1 / sqrt(n)
return output
def __div__(self, other):
return self._rc(divide(self.array, asarray(other)))
def __rdiv__(self, other):
return self._rc(divide(asarray(other), self.array))
def __rsub__(self, other):
return self._rc(asarray(other) - self.array)
def __mul__(self, other):
return self._rc(multiply(self.array, asarray(other)))
def __sub__(self, other):
return self._rc(self.array - asarray(other))
def __add__(self, other):
return self._rc(self.array + asarray(other))
def __setitem__(self, index, value):
self.array[index] = asarray(value, self.dtype)
def __rpow__(self, other):
return self._rc(power(asarray(other), self.array))
def __pow__(self, other):
return self._rc(power(self.array, asarray(other)))
