def __fix_shape(x, n, axis, dct_or_dst):
tmp = _asfarray(x)
copy_made = _datacopied(tmp, x)
if n is None:
n = tmp.shape[axis]
elif n != tmp.shape[axis]:
tmp, copy_made2 = _fix_shape(tmp, n, axis)
copy_made = copy_made or copy_made2
if n < 1:
raise ValueError("Invalid number of %s data points "
"(%d) specified." % (dct_or_dst, n))
return tmp, n, copy_made
def __fix_shape(x, n, axis, dct_or_dst):
tmp = _asfarray(x)
copy_made = _datacopied(tmp, x)
if n is None:
n = tmp.shape[axis]
elif n != tmp.shape[axis]:
tmp, copy_made2 = _fix_shape(tmp, n, axis)
copy_made = copy_made or copy_made2
if n < 1:
raise ValueError("Invalid number of %s data points "
"(%d) specified." % (dct_or_dst, n))
return tmp, n, copy_made
def fft(x, n=None, axis=-1, overwrite_x=False):
"""
Return discrete Fourier transform of real or complex sequence.
The returned complex array contains ``y(0), y(1),..., y(n-1)`` where
``y(j) = (x * exp(-2*pi*sqrt(-1)*j*np.arange(n)/n)).sum()``.
Parameters
----------
x : array_like
Array to Fourier transform.
n : int, optional
Length of the Fourier transform. If ``n < x.shape[axis]``, `x` is
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
default results in ``n = x.shape[axis]``.
axis : int, optional
Axis along which the fft's are computed; the default is over the
last axis (i.e., ``axis=-1``).
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
Returns
-------
z : complex ndarray
with the elements::
[y(0),y(1),..,y(n/2),y(1-n/2),...,y(-1)] if n is even
[y(0),y(1),..,y((n-1)/2),y(-(n-1)/2),...,y(-1)] if n is odd
where::
y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n), j = 0..n-1
See Also
--------
ifft : Inverse FFT
rfft : FFT of a real sequence
Notes
-----
The packing of the result is "standard": If ``A = fft(a, n)``, then
``A[0]`` contains the zero-frequency term, ``A[1:n/2]`` contains the
positive-frequency terms, and ``A[n/2:]`` contains the negative-frequency
terms, in order of decreasingly negative frequency. So for an 8-point
transform, the frequencies of the result are [0, 1, 2, 3, -4, -3, -2, -1].
To rearrange the fft output so that the zero-frequency component is
centered, like [-4, -3, -2, -1, 0, 1, 2, 3], use `fftshift`.
Both single and double precision routines are implemented. Half precision
inputs will be converted to single precision. Non floating-point inputs
will be converted to double precision. Long-double precision inputs are
not supported.
This function is most efficient when `n` is a power of two, and least
efficient when `n` is prime.
Note that if ``x`` is real-valued then ``A[j] == A[n-j].conjugate()``.
If ``x`` is real-valued and ``n`` is even then ``A[n/2]`` is real.
If the data type of `x` is real, a "real FFT" algorithm is automatically
used, which roughly halves the computation time. To increase efficiency
a little further, use `rfft`, which does the same calculation, but only
outputs half of the symmetrical spectrum. If the data is both real and
symmetrical, the `dct` can again double the efficiency, by generating
half of the spectrum from half of the signal.
Examples
--------
# >>> from scipy.fftpack import fft, ifft
# >>> x = np.arange(5)
# >>> np.allclose(fft(ifft(x)), x, atol=1e-15) # within numerical accuracy.
True
"""
tmp = _asfarray(x)
try:
work_function = _DTYPE_TO_FFT[tmp.dtype]
except KeyError:
raise ValueError("type %s is not supported" % tmp.dtype)
if not (istype(tmp, numpy.complex64) or istype(tmp, numpy.complex128)):
overwrite_x = 1
overwrite_x = overwrite_x or _datacopied(tmp, x)
if n is None:
n = tmp.shape[axis]
elif n != tmp.shape[axis]:
tmp, copy_made = _fix_shape(tmp, n, axis)
overwrite_x = overwrite_x or copy_made
if n < 1:
raise ValueError("Invalid number of FFT data points "
"(%d) specified." % n)
if axis == -1 or axis == len(tmp.shape) - 1:
return work_function(tmp, n, 1, 0, overwrite_x)
tmp = swapaxes(tmp, axis, -1)
tmp = work_function(tmp, n, 1, 0, overwrite_x)
return swapaxes(tmp, axis, -1)