def test_fix_with_subclass(self):
class MyArray(nx.ndarray):
def __new__(cls, data, metadata=None):
res = nx.array(data, copy=True).view(cls)
res.metadata = metadata
return res
def __array_wrap__(self, obj, context=None):
obj.metadata = self.metadata
return obj
def __array_finalize__(self, obj):
self.metadata = getattr(obj, 'metadata', None)
return self
a = nx.array([1.1, -1.1])
m = MyArray(a, metadata='foo')
f = ufl.fix(m)
assert_array_equal(f, nx.array([1, -1]))
assert_(isinstance(f, MyArray))
assert_equal(f.metadata, 'foo')
# check 0d arrays don't decay to scalars
m0d = m[0,...]
m0d.metadata = 'bar'
f0d = ufl.fix(m0d)
assert_(isinstance(f0d, MyArray))
assert_equal(f0d.metadata, 'bar')
def test_3D_array(self):
a = array([[1, 2], [1, 2]])
b = array([[2, 3], [2, 3]])
a = array([a, a])
b = array([b, b])
res = [atleast_2d(a), atleast_2d(b)]
desired = [a, b]
assert_array_equal(res, desired)
def test_bad_out_shape(self):
a = array([1, 2])
b = array([3, 4])
assert_raises(ValueError, concatenate, (a, b), out=np.empty(5))
assert_raises(ValueError, concatenate, (a, b), out=np.empty((4,1)))
assert_raises(ValueError, concatenate, (a, b), out=np.empty((1,4)))
concatenate((a, b), out=np.empty(4))
def test_isneginf(self):
a = nx.array([nx.inf, -nx.inf, nx.nan, 0.0, 3.0, -3.0])
out = nx.zeros(a.shape, bool)
tgt = nx.array([False, True, False, False, False, False])
res = ufl.isneginf(a)
assert_equal(res, tgt)
res = ufl.isneginf(a, out)
assert_equal(res, tgt)
assert_equal(out, tgt)
def test_fix(self):
a = nx.array([[1.0, 1.1, 1.5, 1.8], [-1.0, -1.1, -1.5, -1.8]])
out = nx.zeros(a.shape, float)
tgt = nx.array([[1., 1., 1., 1.], [-1., -1., -1., -1.]])
res = ufl.fix(a)
assert_equal(res, tgt)
res = ufl.fix(a, out)
assert_equal(res, tgt)
assert_equal(out, tgt)
assert_equal(ufl.fix(3.14), 3)
def test_stack():
# non-iterable input
assert_raises(TypeError, stack, 1)
# 0d input
for input_ in [(1, 2, 3),
[np.int32(1), np.int32(2), np.int32(3)],
[np.array(1), np.array(2), np.array(3)]]:
assert_array_equal(stack(input_), [1, 2, 3])
# 1d input examples
a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
r1 = array([[1, 2, 3], [4, 5, 6]])
assert_array_equal(np.stack((a, b)), r1)
assert_array_equal(np.stack((a, b), axis=1), r1.T)
# all input types
assert_array_equal(np.stack(list([a, b])), r1)
assert_array_equal(np.stack(array([a, b])), r1)
# all shapes for 1d input
arrays = [np.random.randn(3) for _ in range(10)]
axes = [0, 1, -1, -2]
expected_shapes = [(10, 3), (3, 10), (3, 10), (10, 3)]
for axis, expected_shape in zip(axes, expected_shapes):
assert_equal(np.stack(arrays, axis).shape, expected_shape)
assert_raises_regex(np.AxisError, 'out of bounds', stack, arrays, axis=2)
assert_raises_regex(np.AxisError, 'out of bounds', stack, arrays, axis=-3)
# all shapes for 2d input
arrays = [np.random.randn(3, 4) for _ in range(10)]
axes = [0, 1, 2, -1, -2, -3]
expected_shapes = [(10, 3, 4), (3, 10, 4), (3, 4, 10),
(3, 4, 10), (3, 10, 4), (10, 3, 4)]
for axis, expected_shape in zip(axes, expected_shapes):
assert_equal(np.stack(arrays, axis).shape, expected_shape)
# empty arrays
assert_(stack([[], [], []]).shape == (3, 0))
assert_(stack([[], [], []], axis=1).shape == (0, 3))
# edge cases
assert_raises_regex(ValueError, 'need at least one array', stack, [])
assert_raises_regex(ValueError, 'must have the same shape',
stack, [1, np.arange(3)])
assert_raises_regex(ValueError, 'must have the same shape',
stack, [np.arange(3), 1])
assert_raises_regex(ValueError, 'must have the same shape',
stack, [np.arange(3), 1], axis=1)
assert_raises_regex(ValueError, 'must have the same shape',
stack, [np.zeros((3, 3)), np.zeros(3)], axis=1)
assert_raises_regex(ValueError, 'must have the same shape',
stack, [np.arange(2), np.arange(3)])
def ihfft(a, n=None, axis=-1, norm=None):
"""
Compute the inverse FFT of a signal that has Hermitian symmetry.
Parameters
----------
a : array_like
Input array.
n : int, optional
Length of the inverse FFT, the number of points along
transformation axis in the input to use. 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 inverse FFT. If not given, the last
axis is used.
norm : {None, "ortho"}, optional
Normalization mode (see `numpy.fft`). Default is None.
.. versionadded:: 1.10.0
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.
The length of the transformed axis is ``n//2 + 1``.
See also
--------
hfft, irfft
Notes
-----
`hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
opposite case: here the signal has Hermitian symmetry in the time
domain and is real in the frequency domain. So here it's `hfft` for
which you must supply the length of the result if it is to be odd:
* even: ``ihfft(hfft(a, 2*len(a) - 2) == a``, within roundoff error,
* odd: ``ihfft(hfft(a, 2*len(a) - 1) == a``, within roundoff error.
Examples
--------
>>> spectrum = np.array([ 15, -4, 0, -1, 0, -4])
>>> np.fft.ifft(spectrum)
array([ 1.+0.j, 2.-0.j, 3.+0.j, 4.+0.j, 3.+0.j, 2.-0.j])
>>> np.fft.ihfft(spectrum)
array([ 1.-0.j, 2.-0.j, 3.-0.j, 4.-0.j])
"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=float)
if n is None:
n = a.shape[axis]
unitary = _unitary(norm)
output = conjugate(rfft(a, n, axis))
return output * (1 / (sqrt(n) if unitary else n))
def test_3d(self):
a000 = np.ones((2, 2, 2), int) * 1
a100 = np.ones((3, 2, 2), int) * 2
a010 = np.ones((2, 3, 2), int) * 3
a001 = np.ones((2, 2, 3), int) * 4
a011 = np.ones((2, 3, 3), int) * 5
a101 = np.ones((3, 2, 3), int) * 6
a110 = np.ones((3, 3, 2), int) * 7
a111 = np.ones((3, 3, 3), int) * 8
result = np.block([
[
[a000, a001],
[a010, a011],
],
[
[a100, a101],
[a110, a111],
]
])
expected = array([[[1, 1, 4, 4, 4],
[1, 1, 4, 4, 4],
[3, 3, 5, 5, 5],
[3, 3, 5, 5, 5],
[3, 3, 5, 5, 5]],
[[1, 1, 4, 4, 4],
[1, 1, 4, 4, 4],
[3, 3, 5, 5, 5],
[3, 3, 5, 5, 5],
[3, 3, 5, 5, 5]],
[[2, 2, 6, 6, 6],
[2, 2, 6, 6, 6],
[7, 7, 8, 8, 8],
[7, 7, 8, 8, 8],
[7, 7, 8, 8, 8]],
[[2, 2, 6, 6, 6],
[2, 2, 6, 6, 6],
[7, 7, 8, 8, 8],
[7, 7, 8, 8, 8],
[7, 7, 8, 8, 8]],
[[2, 2, 6, 6, 6],
[2, 2, 6, 6, 6],
[7, 7, 8, 8, 8],
[7, 7, 8, 8, 8],
[7, 7, 8, 8, 8]]])
assert_array_equal(result, expected)
def test_concatenate(self):
# Test concatenate function
# One sequence returns unmodified (but as array)
r4 = list(range(4))
assert_array_equal(concatenate((r4,)), r4)
# Any sequence
assert_array_equal(concatenate((tuple(r4),)), r4)
assert_array_equal(concatenate((array(r4),)), r4)
# 1D default concatenation
r3 = list(range(3))
assert_array_equal(concatenate((r4, r3)), r4 + r3)
# Mixed sequence types
assert_array_equal(concatenate((tuple(r4), r3)), r4 + r3)
assert_array_equal(concatenate((array(r4), r3)), r4 + r3)
# Explicit axis specification
assert_array_equal(concatenate((r4, r3), 0), r4 + r3)
# Including negative
assert_array_equal(concatenate((r4, r3), -1), r4 + r3)
# 2D
a23 = array([[10, 11, 12], [13, 14, 15]])
a13 = array([[0, 1, 2]])
res = array([[10, 11, 12], [13, 14, 15], [0, 1, 2]])
assert_array_equal(concatenate((a23, a13)), res)
assert_array_equal(concatenate((a23, a13), 0), res)
assert_array_equal(concatenate((a23.T, a13.T), 1), res.T)
assert_array_equal(concatenate((a23.T, a13.T), -1), res.T)
# Arrays much match shape
assert_raises(ValueError, concatenate, (a23.T, a13.T), 0)
# 3D
res = arange(2 * 3 * 7).reshape((2, 3, 7))
a0 = res[..., :4]
a1 = res[..., 4:6]
a2 = res[..., 6:]
assert_array_equal(concatenate((a0, a1, a2), 2), res)
assert_array_equal(concatenate((a0, a1, a2), -1), res)
assert_array_equal(concatenate((a0.T, a1.T, a2.T), 0), res.T)
out = res.copy()
rout = concatenate((a0, a1, a2), 2, out=out)
assert_(out is rout)
assert_equal(res, rout)
def test_out_dtype(self):
out = np.empty(4, np.float32)
res = concatenate((array([1, 2]), array([3, 4])), out=out)
assert_(out is res)
out = np.empty(4, np.complex64)
res = concatenate((array([0.1, 0.2]), array([0.3, 0.4])), out=out)
assert_(out is res)
# invalid cast
out = np.empty(4, np.int32)
assert_raises(TypeError, concatenate,
(array([0.1, 0.2]), array([0.3, 0.4])), out=out)
def test_concatenate_axis_None(self):
a = np.arange(4, dtype=np.float64).reshape((2, 2))
b = list(range(3))
c = ['x']
r = np.concatenate((a, a), axis=None)
assert_equal(r.dtype, a.dtype)
assert_equal(r.ndim, 1)
r = np.concatenate((a, b), axis=None)
assert_equal(r.size, a.size + len(b))
assert_equal(r.dtype, a.dtype)
r = np.concatenate((a, b, c), axis=None)
d = array(['0.0', '1.0', '2.0', '3.0',
'0', '1', '2', 'x'])
assert_array_equal(r, d)
out = np.zeros(a.size + len(b))
r = np.concatenate((a, b), axis=None)
rout = np.concatenate((a, b), axis=None, out=out)
assert_(out is rout)
assert_equal(r, rout)
def test_2D_array(self):
a = array([[1, 2], [1, 2]])
b = array([[2, 3], [2, 3]])
res = [atleast_3d(a), atleast_3d(b)]
desired = [a[:,:, newaxis], b[:,:, newaxis]]
assert_array_equal(res, desired)
def rfftn(a, s=None, axes=None, norm=None):
"""
Compute the N-dimensional discrete Fourier Transform for real input.
This function computes the N-dimensional discrete Fourier Transform over
any number of axes in an M-dimensional real array by means of the Fast
Fourier Transform (FFT). By default, all axes are transformed, with the
real transform performed over the last axis, while the remaining
transforms are complex.
Parameters
----------
a : array_like
Input array, taken to be real.
s : sequence of ints, optional
Shape (length along each transformed axis) to use from the input.
(``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.).
The final element of `s` corresponds to `n` for ``rfft(x, n)``, while
for the remaining axes, it corresponds to `n` for ``fft(x, n)``.
Along any axis, if the given shape is smaller than that of the input,
the input is cropped. If it is larger, the input is padded with zeros.
if `s` is not given, the shape of the input along the axes specified
by `axes` is used.
axes : sequence of ints, optional
Axes over which to compute the FFT. If not given, the last ``len(s)``
axes are used, or all axes if `s` is also not specified.
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 axes
indicated by `axes`, or by a combination of `s` and `a`,
as explained in the parameters section above.
The length of the last axis transformed will be ``s[-1]//2+1``,
while the remaining transformed axes will have lengths according to
`s`, or unchanged from the input.
Raises
------
ValueError
If `s` and `axes` have different length.
IndexError
If an element of `axes` is larger than than the number of axes of `a`.
See Also
--------
irfftn : The inverse of `rfftn`, i.e. the inverse of the n-dimensional FFT
of real input.
fft : The one-dimensional FFT, with definitions and conventions used.
rfft : The one-dimensional FFT of real input.
fftn : The n-dimensional FFT.
rfft2 : The two-dimensional FFT of real input.
Notes
-----
The transform for real input is performed over the last transformation
axis, as by `rfft`, then the transform over the remaining axes is
performed as by `fftn`. The order of the output is as for `rfft` for the
final transformation axis, and as for `fftn` for the remaining
transformation axes.
See `fft` for details, definitions and conventions used.
Examples
--------
>>> a = np.ones((2, 2, 2))
>>> np.fft.rfftn(a)
array([[[ 8.+0.j, 0.+0.j],
[ 0.+0.j, 0.+0.j]],
[[ 0.+0.j, 0.+0.j],
[ 0.+0.j, 0.+0.j]]])
>>> np.fft.rfftn(a, axes=(2, 0))
array([[[ 4.+0.j, 0.+0.j],
[ 4.+0.j, 0.+0.j]],
[[ 0.+0.j, 0.+0.j],
[ 0.+0.j, 0.+0.j]]])
"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=float)
s, axes = _cook_nd_args(a, s, axes)
a = rfft(a, s[-1], axes[-1], norm)
for ii in range(len(axes)-1):
a = fft(a, s[ii], axes[ii], norm)
return a
def poly(seq_of_zeros):
"""
Find the coefficients of a polynomial with the given sequence of roots.
Returns the coefficients of the polynomial whose leading coefficient
is one for the given sequence of zeros (multiple roots must be included
in the sequence as many times as their multiplicity; see Examples).
A square matrix (or array, which will be treated as a matrix) can also
be given, in which case the coefficients of the characteristic polynomial
of the matrix are returned.
Parameters
----------
seq_of_zeros : array_like, shape (N,) or (N, N)
A sequence of polynomial roots, or a square array or matrix object.
Returns
-------
c : ndarray
1D array of polynomial coefficients from highest to lowest degree:
``c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]``
where c[0] always equals 1.
Raises
------
ValueError
If input is the wrong shape (the input must be a 1-D or square
2-D array).
See Also
--------
polyval : Compute polynomial values.
roots : Return the roots of a polynomial.
polyfit : Least squares polynomial fit.
poly1d : A one-dimensional polynomial class.
Notes
-----
Specifying the roots of a polynomial still leaves one degree of
freedom, typically represented by an undetermined leading
coefficient. [1]_ In the case of this function, that coefficient -
the first one in the returned array - is always taken as one. (If
for some reason you have one other point, the only automatic way
presently to leverage that information is to use ``polyfit``.)
The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n`
matrix **A** is given by
:math:`p_a(t) = \\mathrm{det}(t\\, \\mathbf{I} - \\mathbf{A})`,
where **I** is the `n`-by-`n` identity matrix. [2]_
References
----------
.. [1] M. Sullivan and M. Sullivan, III, "Algebra and Trignometry,
Enhanced With Graphing Utilities," Prentice-Hall, pg. 318, 1996.
.. [2] G. Strang, "Linear Algebra and Its Applications, 2nd Edition,"
Academic Press, pg. 182, 1980.
Examples
--------
Given a sequence of a polynomial's zeros:
>>> np.poly((0, 0, 0)) # Multiple root example
array([1, 0, 0, 0])
The line above represents z**3 + 0*z**2 + 0*z + 0.
>>> np.poly((-1./2, 0, 1./2))
array([ 1. , 0. , -0.25, 0. ])
The line above represents z**3 - z/4
>>> np.poly((np.random.random(1.)[0], 0, np.random.random(1.)[0]))
array([ 1. , -0.77086955, 0.08618131, 0. ]) #random
Given a square array object:
>>> P = np.array([[0, 1./3], [-1./2, 0]])
>>> np.poly(P)
array([ 1. , 0. , 0.16666667])
Note how in all cases the leading coefficient is always 1.
"""
seq_of_zeros = atleast_1d(seq_of_zeros)
sh = seq_of_zeros.shape
if len(sh) == 2 and sh[0] == sh[1] and sh[0] != 0:
seq_of_zeros = eigvals(seq_of_zeros)
elif len(sh) == 1:
dt = seq_of_zeros.dtype
# Let object arrays slip through, e.g. for arbitrary precision
if dt != object:
seq_of_zeros = seq_of_zeros.astype(mintypecode(dt.char))
else:
raise ValueError("input must be 1d or non-empty square 2d array.")
if len(seq_of_zeros) == 0:
return 1.0
dt = seq_of_zeros.dtype
a = ones((1, ), dtype=dt)
for k in range(len(seq_of_zeros)):
a = NX.convolve(a, array([1, -seq_of_zeros[k]], dtype=dt), mode='full')
if issubclass(a.dtype.type, NX.complexfloating):
# if complex roots are all complex conjugates, the roots are real.
roots = NX.asarray(seq_of_zeros, complex)
if NX.all(NX.sort(roots) == NX.sort(roots.conjugate())):
a = a.real.copy()
return a
def irfftn(a, s=None, axes=None, norm=None):
"""
Compute the inverse of the N-dimensional FFT of real input.
This function computes the inverse of the N-dimensional discrete
Fourier Transform for real input over any number of axes in an
M-dimensional array by means of the Fast Fourier Transform (FFT). In
other words, ``irfftn(rfftn(a), a.shape) == a`` to within numerical
accuracy. (The ``a.shape`` is necessary like ``len(a)`` is for `irfft`,
and for the same reason.)
The input should be ordered in the same way as is returned by `rfftn`,
i.e. as for `irfft` for the final transformation axis, and as for `ifftn`
along all the other axes.
Parameters
----------
a : array_like
Input array.
s : sequence of ints, optional
Shape (length of each transformed axis) of the output
(``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.). `s` is also the
number of input points used along this axis, except for the last axis,
where ``s[-1]//2+1`` points of the input are used.
Along any axis, if the shape indicated by `s` is smaller than that of
the input, the input is cropped. If it is larger, the input is padded
with zeros. If `s` is not given, the shape of the input along the
axes specified by `axes` is used.
axes : sequence of ints, optional
Axes over which to compute the inverse FFT. If not given, the last
`len(s)` axes are used, or all axes if `s` is also not specified.
Repeated indices in `axes` means that the inverse transform over that
axis is performed multiple times.
norm : {None, "ortho"}, optional
.. versionadded:: 1.10.0
Normalization mode (see `numpy.fft`). Default is None.
Returns
-------
out : ndarray
The truncated or zero-padded input, transformed along the axes
indicated by `axes`, or by a combination of `s` or `a`,
as explained in the parameters section above.
The length of each transformed axis is as given by the corresponding
element of `s`, or the length of the input in every axis except for the
last one if `s` is not given. In the final transformed axis the length
of the output when `s` is not given is ``2*(m-1)`` where ``m`` is the
length of the final transformed axis of the input. To get an odd
number of output points in the final axis, `s` must be specified.
Raises
------
ValueError
If `s` and `axes` have different length.
IndexError
If an element of `axes` is larger than than the number of axes of `a`.
See Also
--------
rfftn : The forward n-dimensional FFT of real input,
of which `ifftn` is the inverse.
fft : The one-dimensional FFT, with definitions and conventions used.
irfft : The inverse of the one-dimensional FFT of real input.
irfft2 : The inverse of the two-dimensional FFT of real input.
Notes
-----
See `fft` for definitions and conventions used.
See `rfft` for definitions and conventions used for real input.
Examples
--------
>>> a = np.zeros((3, 2, 2))
>>> a[0, 0, 0] = 3 * 2 * 2
>>> np.fft.irfftn(a)
array([[[ 1., 1.],
[ 1., 1.]],
[[ 1., 1.],
[ 1., 1.]],
[[ 1., 1.],
[ 1., 1.]]])
"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=complex)
s, axes = _cook_nd_args(a, s, axes, invreal=1)
for ii in range(len(axes)-1):
a = ifft(a, s[ii], axes[ii], norm)
a = irfft(a, s[-1], axes[-1], norm)
return a
def ifft(a, n=None, axis=-1, norm=None):
"""
Compute the one-dimensional inverse discrete Fourier Transform.
This function computes the inverse of the one-dimensional *n*-point
discrete Fourier transform computed by `fft`. In other words,
``ifft(fft(a)) == a`` to within numerical accuracy.
For a general description of the algorithm and definitions,
see `numpy.fft`.
The input should be ordered in the same way as is returned by `fft`,
i.e.,
* ``a[0]`` should contain the zero frequency term,
* ``a[1:n//2]`` should contain the positive-frequency terms,
* ``a[n//2 + 1:]`` should contain the negative-frequency terms, in
increasing order starting from the most negative frequency.
For an even number of input points, ``A[n//2]`` represents the sum of
the values at the positive and negative Nyquist frequencies, as the two
are aliased together. See `numpy.fft` for details.
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.
See notes about padding issues.
axis : int, optional
Axis over which to compute the inverse DFT. 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 : An introduction, with definitions and general explanations.
fft : The one-dimensional (forward) FFT, of which `ifft` is the inverse
ifft2 : The two-dimensional inverse FFT.
ifftn : The n-dimensional inverse FFT.
Notes
-----
If the input parameter `n` is larger than the size of the input, the input
is padded by appending zeros at the end. Even though this is the common
approach, it might lead to surprising results. If a different padding is
desired, it must be performed before calling `ifft`.
Examples
--------
>>> np.fft.ifft([0, 4, 0, 0])
array([ 1.+0.j, 0.+1.j, -1.+0.j, 0.-1.j])
Create and plot a band-limited signal with random phases:
>>> import matplotlib.pyplot as plt
>>> t = np.arange(400)
>>> n = np.zeros((400,), dtype=complex)
>>> n[40:60] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20,)))
>>> s = np.fft.ifft(n)
>>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--')
...
>>> plt.legend(('real', 'imaginary'))
...
>>> plt.show()
"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=complex)
if n is None:
n = a.shape[axis]
unitary = _unitary(norm)
output = _raw_fft(a, n, axis, fftpack.cffti, fftpack.cfftb, _fft_cache)
return output * (1 / (sqrt(n) if unitary else n))
def rfft(a, n=None, axis=-1, norm=None):
"""
Compute the one-dimensional discrete Fourier Transform for real input.
This function computes the one-dimensional *n*-point discrete Fourier
Transform (DFT) of a real-valued array by means of an efficient algorithm
called the Fast Fourier Transform (FFT).
Parameters
----------
a : array_like
Input array
n : int, optional
Number of points along transformation axis in the input to use.
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.
If `n` is even, the length of the transformed axis is ``(n/2)+1``.
If `n` is odd, the length is ``(n+1)/2``.
Raises
------
IndexError
If `axis` is larger than the last axis of `a`.
See Also
--------
numpy.fft : For definition of the DFT and conventions used.
irfft : The inverse of `rfft`.
fft : The one-dimensional FFT of general (complex) input.
fftn : The *n*-dimensional FFT.
rfftn : The *n*-dimensional FFT of real input.
Notes
-----
When the DFT is computed for purely real input, the output is
Hermitian-symmetric, i.e. the negative frequency terms are just the complex
conjugates of the corresponding positive-frequency terms, and the
negative-frequency terms are therefore redundant. This function does not
compute the negative frequency terms, and the length of the transformed
axis of the output is therefore ``n//2 + 1``.
When ``A = rfft(a)`` and fs is the sampling frequency, ``A[0]`` contains
the zero-frequency term 0*fs, which is real due to Hermitian symmetry.
If `n` is even, ``A[-1]`` contains the term representing both positive
and negative Nyquist frequency (+fs/2 and -fs/2), and must also be purely
real. If `n` is odd, there is no term at fs/2; ``A[-1]`` contains
the largest positive frequency (fs/2*(n-1)/n), and is complex in the
general case.
If the input `a` contains an imaginary part, it is silently discarded.
Examples
--------
>>> np.fft.fft([0, 1, 0, 0])
array([ 1.+0.j, 0.-1.j, -1.+0.j, 0.+1.j])
>>> np.fft.rfft([0, 1, 0, 0])
array([ 1.+0.j, 0.-1.j, -1.+0.j])
Notice how the final element of the `fft` output is the complex conjugate
of the second element, for real input. For `rfft`, this symmetry is
exploited to compute only the non-negative frequency terms.
"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=float)
output = _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftf,
_real_fft_cache)
if _unitary(norm):
if n is None:
n = a.shape[axis]
output *= 1 / sqrt(n)
return output
def hfft(a, n=None, axis=-1, norm=None):
"""
Compute the FFT of a signal that has Hermitian symmetry, i.e., a real
spectrum.
Parameters
----------
a : array_like
The input array.
n : int, optional
Length of the transformed axis of the output. For `n` output
points, ``n//2 + 1`` input points are necessary. If the input is
longer than this, it is cropped. If it is shorter than this, it is
padded with zeros. If `n` is not given, it is determined from the
length of the input along the axis specified by `axis`.
axis : int, optional
Axis over which to compute the FFT. If not given, the last
axis is used.
norm : {None, "ortho"}, optional
Normalization mode (see `numpy.fft`). Default is None.
.. versionadded:: 1.10.0
Returns
-------
out : ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
The length of the transformed axis is `n`, or, if `n` is not given,
``2*m - 2`` where ``m`` is the length of the transformed axis of
the input. To get an odd number of output points, `n` must be
specified, for instance as ``2*m - 1`` in the typical case,
Raises
------
IndexError
If `axis` is larger than the last axis of `a`.
See also
--------
rfft : Compute the one-dimensional FFT for real input.
ihfft : The inverse of `hfft`.
Notes
-----
`hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
opposite case: here the signal has Hermitian symmetry in the time
domain and is real in the frequency domain. So here it's `hfft` for
which you must supply the length of the result if it is to be odd.
* even: ``ihfft(hfft(a, 2*len(a) - 2) == a``, within roundoff error,
* odd: ``ihfft(hfft(a, 2*len(a) - 1) == a``, within roundoff error.
Examples
--------
>>> signal = np.array([1, 2, 3, 4, 3, 2])
>>> np.fft.fft(signal)
array([ 15.+0.j, -4.+0.j, 0.+0.j, -1.-0.j, 0.+0.j, -4.+0.j])
>>> np.fft.hfft(signal[:4]) # Input first half of signal
array([ 15., -4., 0., -1., 0., -4.])
>>> np.fft.hfft(signal, 6) # Input entire signal and truncate
array([ 15., -4., 0., -1., 0., -4.])
>>> signal = np.array([[1, 1.j], [-1.j, 2]])
>>> np.conj(signal.T) - signal # check Hermitian symmetry
array([[ 0.-0.j, 0.+0.j],
[ 0.+0.j, 0.-0.j]])
>>> freq_spectrum = np.fft.hfft(signal)
>>> freq_spectrum
array([[ 1., 1.],
[ 2., -2.]])
"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=complex)
if n is None:
n = (a.shape[axis] - 1) * 2
unitary = _unitary(norm)
return irfft(conjugate(a), n, axis) * (sqrt(n) if unitary else n)
def irfft(a, n=None, axis=-1, norm=None):
"""
Compute the inverse of the n-point DFT for real input.
This function computes the inverse of the one-dimensional *n*-point
discrete Fourier Transform of real input computed by `rfft`.
In other words, ``irfft(rfft(a), len(a)) == a`` to within numerical
accuracy. (See Notes below for why ``len(a)`` is necessary here.)
The input is expected to be in the form returned by `rfft`, i.e. the
real zero-frequency term followed by the complex positive frequency terms
in order of increasing frequency. Since the discrete Fourier Transform of
real input is Hermitian-symmetric, the negative frequency terms are taken
to be the complex conjugates of the corresponding positive frequency terms.
Parameters
----------
a : array_like
The input array.
n : int, optional
Length of the transformed axis of the output.
For `n` output points, ``n//2+1`` input points are necessary. If the
input is longer than this, it is cropped. If it is shorter than this,
it is padded with zeros. If `n` is not given, it is determined from
the length of the input along the axis specified by `axis`.
axis : int, optional
Axis over which to compute the inverse 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 : ndarray
The truncated or zero-padded input, transformed along the axis
indicated by `axis`, or the last one if `axis` is not specified.
The length of the transformed axis is `n`, or, if `n` is not given,
``2*(m-1)`` where ``m`` is the length of the transformed axis of the
input. To get an odd number of output points, `n` must be specified.
Raises
------
IndexError
If `axis` is larger than the last axis of `a`.
See Also
--------
numpy.fft : For definition of the DFT and conventions used.
rfft : The one-dimensional FFT of real input, of which `irfft` is inverse.
fft : The one-dimensional FFT.
irfft2 : The inverse of the two-dimensional FFT of real input.
irfftn : The inverse of the *n*-dimensional FFT of real input.
Notes
-----
Returns the real valued `n`-point inverse discrete Fourier transform
of `a`, where `a` contains the non-negative frequency terms of a
Hermitian-symmetric sequence. `n` is the length of the result, not the
input.
If you specify an `n` such that `a` must be zero-padded or truncated, the
extra/removed values will be added/removed at high frequencies. One can
thus resample a series to `m` points via Fourier interpolation by:
``a_resamp = irfft(rfft(a), m)``.
Examples
--------
>>> np.fft.ifft([1, -1j, -1, 1j])
array([ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j])
>>> np.fft.irfft([1, -1j, -1])
array([ 0., 1., 0., 0.])
Notice how the last term in the input to the ordinary `ifft` is the
complex conjugate of the second term, and the output has zero imaginary
part everywhere. When calling `irfft`, the negative frequencies are not
specified, and the output array is purely real.
"""
# The copy may be required for multithreading.
a = array(a, copy=True, dtype=complex)
if n is None:
n = (a.shape[axis] - 1) * 2
unitary = _unitary(norm)
output = _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftb,
_real_fft_cache)
return output * (1 / (sqrt(n) if unitary else n))
def test_1D_array(self):
a = array([1, 2])
b = array([2, 3])
res = [atleast_2d(a), atleast_2d(b)]
desired = [array([[1, 2]]), array([[2, 3]])]
assert_array_equal(res, desired)
def test_0D_array(self):
a = array(1)
b = array(2)
res = hstack([a, b])
desired = array([1, 2])
assert_array_equal(res, desired)
def test_0D_array(self):
a = array(1)
b = array(2)
res = [atleast_3d(a), atleast_3d(b)]
desired = [array([[[1]]]), array([[[2]]])]
assert_array_equal(res, desired)
def test_2D_array(self):
a = array([[1], [2]])
b = array([[1], [2]])
res = hstack([a, b])
desired = array([[1, 1], [2, 2]])
assert_array_equal(res, desired)
def test_1D_array(self):
a = array([1])
b = array([2])
res = vstack([a, b])
desired = array([[1], [2]])
assert_array_equal(res, desired)
def test_2D_array2(self):
a = array([1, 2])
b = array([1, 2])
res = vstack([a, b])
desired = array([[1, 2], [1, 2]])
assert_array_equal(res, desired)
def __new__(cls, data, metadata=None):
res = nx.array(data, copy=True).view(cls)
res.metadata = metadata
return res
def __init__(self, data, dtype=None, copy=True):
self.array = array(data, dtype, copy=copy)
