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703 lines
21 KiB
Python
703 lines
21 KiB
Python
"""
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Discrete Fourier Transforms - basic.py
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"""
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# Created by Pearu Peterson, August,September 2002
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from __future__ import division, print_function, absolute_import
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__all__ = ['fft','ifft','fftn','ifftn','rfft','irfft',
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'fft2','ifft2']
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from numpy import swapaxes, zeros
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import numpy
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from . import _fftpack
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from scipy.fftpack.helper import _init_nd_shape_and_axes_sorted
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import atexit
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atexit.register(_fftpack.destroy_zfft_cache)
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atexit.register(_fftpack.destroy_zfftnd_cache)
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atexit.register(_fftpack.destroy_drfft_cache)
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atexit.register(_fftpack.destroy_cfft_cache)
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atexit.register(_fftpack.destroy_cfftnd_cache)
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atexit.register(_fftpack.destroy_rfft_cache)
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del atexit
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def istype(arr, typeclass):
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return issubclass(arr.dtype.type, typeclass)
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def _datacopied(arr, original):
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"""
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Strict check for `arr` not sharing any data with `original`,
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under the assumption that arr = asarray(original)
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"""
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if arr is original:
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return False
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if not isinstance(original, numpy.ndarray) and hasattr(original, '__array__'):
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return False
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return arr.base is None
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# XXX: single precision FFTs partially disabled due to accuracy issues
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# for large prime-sized inputs.
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#
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# See http://permalink.gmane.org/gmane.comp.python.scientific.devel/13834
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# ("fftpack test failures for 0.8.0b1", Ralf Gommers, 17 Jun 2010,
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# @ scipy-dev)
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#
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# These should be re-enabled once the problems are resolved
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def _is_safe_size(n):
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"""
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Is the size of FFT such that FFTPACK can handle it in single precision
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with sufficient accuracy?
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Composite numbers of 2, 3, and 5 are accepted, as FFTPACK has those
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"""
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n = int(n)
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if n == 0:
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return True
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# Divide by 3 until you can't, then by 5 until you can't
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for c in (3, 5):
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while n % c == 0:
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n //= c
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# Return True if the remainder is a power of 2
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return not n & (n-1)
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def _fake_crfft(x, n, *a, **kw):
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if _is_safe_size(n):
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return _fftpack.crfft(x, n, *a, **kw)
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else:
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return _fftpack.zrfft(x, n, *a, **kw).astype(numpy.complex64)
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def _fake_cfft(x, n, *a, **kw):
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if _is_safe_size(n):
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return _fftpack.cfft(x, n, *a, **kw)
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else:
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return _fftpack.zfft(x, n, *a, **kw).astype(numpy.complex64)
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def _fake_rfft(x, n, *a, **kw):
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if _is_safe_size(n):
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return _fftpack.rfft(x, n, *a, **kw)
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else:
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return _fftpack.drfft(x, n, *a, **kw).astype(numpy.float32)
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def _fake_cfftnd(x, shape, *a, **kw):
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if numpy.all(list(map(_is_safe_size, shape))):
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return _fftpack.cfftnd(x, shape, *a, **kw)
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else:
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return _fftpack.zfftnd(x, shape, *a, **kw).astype(numpy.complex64)
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_DTYPE_TO_FFT = {
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# numpy.dtype(numpy.float32): _fftpack.crfft,
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numpy.dtype(numpy.float32): _fake_crfft,
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numpy.dtype(numpy.float64): _fftpack.zrfft,
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# numpy.dtype(numpy.complex64): _fftpack.cfft,
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numpy.dtype(numpy.complex64): _fake_cfft,
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numpy.dtype(numpy.complex128): _fftpack.zfft,
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}
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_DTYPE_TO_RFFT = {
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# numpy.dtype(numpy.float32): _fftpack.rfft,
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numpy.dtype(numpy.float32): _fake_rfft,
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numpy.dtype(numpy.float64): _fftpack.drfft,
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}
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_DTYPE_TO_FFTN = {
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# numpy.dtype(numpy.complex64): _fftpack.cfftnd,
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numpy.dtype(numpy.complex64): _fake_cfftnd,
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numpy.dtype(numpy.complex128): _fftpack.zfftnd,
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# numpy.dtype(numpy.float32): _fftpack.cfftnd,
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numpy.dtype(numpy.float32): _fake_cfftnd,
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numpy.dtype(numpy.float64): _fftpack.zfftnd,
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}
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def _asfarray(x):
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"""Like numpy asfarray, except that it does not modify x dtype if x is
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already an array with a float dtype, and do not cast complex types to
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real."""
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if hasattr(x, "dtype") and x.dtype.char in numpy.typecodes["AllFloat"]:
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# 'dtype' attribute does not ensure that the
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# object is an ndarray (e.g. Series class
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# from the pandas library)
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if x.dtype == numpy.half:
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# no half-precision routines, so convert to single precision
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return numpy.asarray(x, dtype=numpy.float32)
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return numpy.asarray(x, dtype=x.dtype)
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else:
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# We cannot use asfarray directly because it converts sequences of
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# complex to sequence of real
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ret = numpy.asarray(x)
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if ret.dtype == numpy.half:
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return numpy.asarray(ret, dtype=numpy.float32)
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elif ret.dtype.char not in numpy.typecodes["AllFloat"]:
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return numpy.asfarray(x)
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return ret
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def _fix_shape(x, n, axis):
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""" Internal auxiliary function for _raw_fft, _raw_fftnd."""
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s = list(x.shape)
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if s[axis] > n:
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index = [slice(None)]*len(s)
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index[axis] = slice(0,n)
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x = x[tuple(index)]
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return x, False
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else:
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index = [slice(None)]*len(s)
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index[axis] = slice(0,s[axis])
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s[axis] = n
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z = zeros(s,x.dtype.char)
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z[tuple(index)] = x
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return z, True
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def _raw_fft(x, n, axis, direction, overwrite_x, work_function):
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""" Internal auxiliary function for fft, ifft, rfft, irfft."""
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if n is None:
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n = x.shape[axis]
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elif n != x.shape[axis]:
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x, copy_made = _fix_shape(x,n,axis)
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overwrite_x = overwrite_x or copy_made
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if n < 1:
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raise ValueError("Invalid number of FFT data points "
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"(%d) specified." % n)
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if axis == -1 or axis == len(x.shape)-1:
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r = work_function(x,n,direction,overwrite_x=overwrite_x)
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else:
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x = swapaxes(x, axis, -1)
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r = work_function(x,n,direction,overwrite_x=overwrite_x)
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r = swapaxes(r, axis, -1)
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return r
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def fft(x, n=None, axis=-1, overwrite_x=False):
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"""
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Return discrete Fourier transform of real or complex sequence.
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The returned complex array contains ``y(0), y(1),..., y(n-1)`` where
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``y(j) = (x * exp(-2*pi*sqrt(-1)*j*np.arange(n)/n)).sum()``.
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Parameters
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----------
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x : array_like
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Array to Fourier transform.
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n : int, optional
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Length of the Fourier transform. If ``n < x.shape[axis]``, `x` is
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truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
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default results in ``n = x.shape[axis]``.
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axis : int, optional
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Axis along which the fft's are computed; the default is over the
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last axis (i.e., ``axis=-1``).
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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z : complex ndarray
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with the elements::
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[y(0),y(1),..,y(n/2),y(1-n/2),...,y(-1)] if n is even
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[y(0),y(1),..,y((n-1)/2),y(-(n-1)/2),...,y(-1)] if n is odd
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where::
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y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n), j = 0..n-1
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See Also
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--------
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ifft : Inverse FFT
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rfft : FFT of a real sequence
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Notes
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-----
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The packing of the result is "standard": If ``A = fft(a, n)``, then
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``A[0]`` contains the zero-frequency term, ``A[1:n/2]`` contains the
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positive-frequency terms, and ``A[n/2:]`` contains the negative-frequency
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terms, in order of decreasingly negative frequency. So for an 8-point
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transform, the frequencies of the result are [0, 1, 2, 3, -4, -3, -2, -1].
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To rearrange the fft output so that the zero-frequency component is
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centered, like [-4, -3, -2, -1, 0, 1, 2, 3], use `fftshift`.
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Both single and double precision routines are implemented. Half precision
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inputs will be converted to single precision. Non floating-point inputs
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will be converted to double precision. Long-double precision inputs are
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not supported.
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This function is most efficient when `n` is a power of two, and least
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efficient when `n` is prime.
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Note that if ``x`` is real-valued then ``A[j] == A[n-j].conjugate()``.
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If ``x`` is real-valued and ``n`` is even then ``A[n/2]`` is real.
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If the data type of `x` is real, a "real FFT" algorithm is automatically
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used, which roughly halves the computation time. To increase efficiency
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a little further, use `rfft`, which does the same calculation, but only
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outputs half of the symmetrical spectrum. If the data is both real and
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symmetrical, the `dct` can again double the efficiency, by generating
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half of the spectrum from half of the signal.
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Examples
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--------
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>>> from scipy.fftpack import fft, ifft
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>>> x = np.arange(5)
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>>> np.allclose(fft(ifft(x)), x, atol=1e-15) # within numerical accuracy.
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True
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"""
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tmp = _asfarray(x)
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try:
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work_function = _DTYPE_TO_FFT[tmp.dtype]
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except KeyError:
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raise ValueError("type %s is not supported" % tmp.dtype)
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if not (istype(tmp, numpy.complex64) or istype(tmp, numpy.complex128)):
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overwrite_x = 1
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overwrite_x = overwrite_x or _datacopied(tmp, x)
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if n is None:
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n = tmp.shape[axis]
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elif n != tmp.shape[axis]:
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tmp, copy_made = _fix_shape(tmp,n,axis)
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overwrite_x = overwrite_x or copy_made
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if n < 1:
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raise ValueError("Invalid number of FFT data points "
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"(%d) specified." % n)
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if axis == -1 or axis == len(tmp.shape) - 1:
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return work_function(tmp,n,1,0,overwrite_x)
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tmp = swapaxes(tmp, axis, -1)
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tmp = work_function(tmp,n,1,0,overwrite_x)
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return swapaxes(tmp, axis, -1)
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def ifft(x, n=None, axis=-1, overwrite_x=False):
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"""
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Return discrete inverse Fourier transform of real or complex sequence.
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The returned complex array contains ``y(0), y(1),..., y(n-1)`` where
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``y(j) = (x * exp(2*pi*sqrt(-1)*j*np.arange(n)/n)).mean()``.
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Parameters
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----------
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x : array_like
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Transformed data to invert.
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n : int, optional
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Length of the inverse Fourier transform. If ``n < x.shape[axis]``,
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`x` is truncated. If ``n > x.shape[axis]``, `x` is zero-padded.
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The default results in ``n = x.shape[axis]``.
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axis : int, optional
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Axis along which the ifft's are computed; the default is over the
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last axis (i.e., ``axis=-1``).
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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ifft : ndarray of floats
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The inverse discrete Fourier transform.
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See Also
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--------
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fft : Forward FFT
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Notes
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-----
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Both single and double precision routines are implemented. Half precision
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inputs will be converted to single precision. Non floating-point inputs
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will be converted to double precision. Long-double precision inputs are
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not supported.
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This function is most efficient when `n` is a power of two, and least
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efficient when `n` is prime.
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If the data type of `x` is real, a "real IFFT" algorithm is automatically
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used, which roughly halves the computation time.
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Examples
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--------
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>>> from scipy.fftpack import fft, ifft
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>>> import numpy as np
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>>> x = np.arange(5)
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>>> np.allclose(ifft(fft(x)), x, atol=1e-15) # within numerical accuracy.
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True
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"""
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tmp = _asfarray(x)
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try:
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work_function = _DTYPE_TO_FFT[tmp.dtype]
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except KeyError:
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raise ValueError("type %s is not supported" % tmp.dtype)
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if not (istype(tmp, numpy.complex64) or istype(tmp, numpy.complex128)):
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overwrite_x = 1
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overwrite_x = overwrite_x or _datacopied(tmp, x)
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if n is None:
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n = tmp.shape[axis]
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elif n != tmp.shape[axis]:
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tmp, copy_made = _fix_shape(tmp,n,axis)
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overwrite_x = overwrite_x or copy_made
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if n < 1:
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raise ValueError("Invalid number of FFT data points "
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"(%d) specified." % n)
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if axis == -1 or axis == len(tmp.shape) - 1:
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return work_function(tmp,n,-1,1,overwrite_x)
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tmp = swapaxes(tmp, axis, -1)
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tmp = work_function(tmp,n,-1,1,overwrite_x)
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return swapaxes(tmp, axis, -1)
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def rfft(x, n=None, axis=-1, overwrite_x=False):
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"""
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Discrete Fourier transform of a real sequence.
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Parameters
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----------
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x : array_like, real-valued
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The data to transform.
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n : int, optional
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Defines the length of the Fourier transform. If `n` is not specified
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(the default) then ``n = x.shape[axis]``. If ``n < x.shape[axis]``,
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`x` is truncated, if ``n > x.shape[axis]``, `x` is zero-padded.
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axis : int, optional
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The axis along which the transform is applied. The default is the
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last axis.
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overwrite_x : bool, optional
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If set to true, the contents of `x` can be overwritten. Default is
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False.
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Returns
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-------
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z : real ndarray
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The returned real array contains::
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[y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2))] if n is even
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[y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2)),Im(y(n/2))] if n is odd
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where::
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y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k*2*pi/n)
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j = 0..n-1
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See Also
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--------
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fft, irfft, numpy.fft.rfft
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Notes
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-----
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Within numerical accuracy, ``y == rfft(irfft(y))``.
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Both single and double precision routines are implemented. Half precision
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inputs will be converted to single precision. Non floating-point inputs
|
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will be converted to double precision. Long-double precision inputs are
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not supported.
|
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To get an output with a complex datatype, consider using the related
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function `numpy.fft.rfft`.
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Examples
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--------
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>>> from scipy.fftpack import fft, rfft
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>>> a = [9, -9, 1, 3]
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>>> fft(a)
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array([ 4. +0.j, 8.+12.j, 16. +0.j, 8.-12.j])
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>>> rfft(a)
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array([ 4., 8., 12., 16.])
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"""
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tmp = _asfarray(x)
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if not numpy.isrealobj(tmp):
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raise TypeError("1st argument must be real sequence")
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try:
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work_function = _DTYPE_TO_RFFT[tmp.dtype]
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except KeyError:
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raise ValueError("type %s is not supported" % tmp.dtype)
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overwrite_x = overwrite_x or _datacopied(tmp, x)
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return _raw_fft(tmp,n,axis,1,overwrite_x,work_function)
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def irfft(x, n=None, axis=-1, overwrite_x=False):
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"""
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Return inverse discrete Fourier transform of real sequence x.
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The contents of `x` are interpreted as the output of the `rfft`
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function.
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Parameters
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----------
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x : array_like
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Transformed data to invert.
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n : int, optional
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Length of the inverse Fourier transform.
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If n < x.shape[axis], x is truncated.
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If n > x.shape[axis], x is zero-padded.
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The default results in n = x.shape[axis].
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axis : int, optional
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Axis along which the ifft's are computed; the default is over
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the last axis (i.e., axis=-1).
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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irfft : ndarray of floats
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The inverse discrete Fourier transform.
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See Also
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--------
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rfft, ifft, numpy.fft.irfft
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Notes
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-----
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The returned real array contains::
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[y(0),y(1),...,y(n-1)]
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where for n is even::
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y(j) = 1/n (sum[k=1..n/2-1] (x[2*k-1]+sqrt(-1)*x[2*k])
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* exp(sqrt(-1)*j*k* 2*pi/n)
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+ c.c. + x[0] + (-1)**(j) x[n-1])
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|
|
|
and for n is odd::
|
|
|
|
y(j) = 1/n (sum[k=1..(n-1)/2] (x[2*k-1]+sqrt(-1)*x[2*k])
|
|
* exp(sqrt(-1)*j*k* 2*pi/n)
|
|
+ c.c. + x[0])
|
|
|
|
c.c. denotes complex conjugate of preceding expression.
|
|
|
|
For details on input parameters, see `rfft`.
|
|
|
|
To process (conjugate-symmetric) frequency-domain data with a complex
|
|
datatype, consider using the related function `numpy.fft.irfft`.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.fftpack import rfft, irfft
|
|
>>> a = [1.0, 2.0, 3.0, 4.0, 5.0]
|
|
>>> irfft(a)
|
|
array([ 2.6 , -3.16405192, 1.24398433, -1.14955713, 1.46962473])
|
|
>>> irfft(rfft(a))
|
|
array([1., 2., 3., 4., 5.])
|
|
|
|
"""
|
|
tmp = _asfarray(x)
|
|
if not numpy.isrealobj(tmp):
|
|
raise TypeError("1st argument must be real sequence")
|
|
|
|
try:
|
|
work_function = _DTYPE_TO_RFFT[tmp.dtype]
|
|
except KeyError:
|
|
raise ValueError("type %s is not supported" % tmp.dtype)
|
|
|
|
overwrite_x = overwrite_x or _datacopied(tmp, x)
|
|
|
|
return _raw_fft(tmp,n,axis,-1,overwrite_x,work_function)
|
|
|
|
|
|
def _raw_fftnd(x, s, axes, direction, overwrite_x, work_function):
|
|
"""Internal auxiliary function for fftnd, ifftnd."""
|
|
noaxes = axes is None
|
|
s, axes = _init_nd_shape_and_axes_sorted(x, s, axes)
|
|
|
|
# No need to swap axes, array is in C order
|
|
if noaxes:
|
|
for ax in axes:
|
|
x, copy_made = _fix_shape(x, s[ax], ax)
|
|
overwrite_x = overwrite_x or copy_made
|
|
return work_function(x, s, direction, overwrite_x=overwrite_x)
|
|
|
|
# Swap the request axes, last first (i.e. First swap the axis which ends up
|
|
# at -1, then at -2, etc...), such as the request axes on which the
|
|
# operation is carried become the last ones
|
|
for i in range(1, axes.size+1):
|
|
x = numpy.swapaxes(x, axes[-i], -i)
|
|
|
|
# We can now operate on the axes waxes, the p last axes (p = len(axes)), by
|
|
# fixing the shape of the input array to 1 for any axis the fft is not
|
|
# carried upon.
|
|
waxes = list(range(x.ndim - axes.size, x.ndim))
|
|
shape = numpy.ones(x.ndim)
|
|
shape[waxes] = s
|
|
|
|
for i in range(len(waxes)):
|
|
x, copy_made = _fix_shape(x, s[i], waxes[i])
|
|
overwrite_x = overwrite_x or copy_made
|
|
|
|
r = work_function(x, shape, direction, overwrite_x=overwrite_x)
|
|
|
|
# reswap in the reverse order (first axis first, etc...) to get original
|
|
# order
|
|
for i in range(len(axes), 0, -1):
|
|
r = numpy.swapaxes(r, -i, axes[-i])
|
|
|
|
return r
|
|
|
|
|
|
def fftn(x, shape=None, axes=None, overwrite_x=False):
|
|
"""
|
|
Return multidimensional discrete Fourier transform.
|
|
|
|
The returned array contains::
|
|
|
|
y[j_1,..,j_d] = sum[k_1=0..n_1-1, ..., k_d=0..n_d-1]
|
|
x[k_1,..,k_d] * prod[i=1..d] exp(-sqrt(-1)*2*pi/n_i * j_i * k_i)
|
|
|
|
where d = len(x.shape) and n = x.shape.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
The (n-dimensional) array to transform.
|
|
shape : int or array_like of ints or None, optional
|
|
The shape of the result. If both `shape` and `axes` (see below) are
|
|
None, `shape` is ``x.shape``; if `shape` is None but `axes` is
|
|
not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
|
|
If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros.
|
|
If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to
|
|
length ``shape[i]``.
|
|
If any element of `shape` is -1, the size of the corresponding
|
|
dimension of `x` is used.
|
|
axes : int or array_like of ints or None, optional
|
|
The axes of `x` (`y` if `shape` is not None) along which the
|
|
transform is applied.
|
|
The default is over all axes.
|
|
overwrite_x : bool, optional
|
|
If True, the contents of `x` can be destroyed. Default is False.
|
|
|
|
Returns
|
|
-------
|
|
y : complex-valued n-dimensional numpy array
|
|
The (n-dimensional) DFT of the input array.
|
|
|
|
See Also
|
|
--------
|
|
ifftn
|
|
|
|
Notes
|
|
-----
|
|
If ``x`` is real-valued, then
|
|
``y[..., j_i, ...] == y[..., n_i-j_i, ...].conjugate()``.
|
|
|
|
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.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.fftpack import fftn, ifftn
|
|
>>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16))
|
|
>>> np.allclose(y, fftn(ifftn(y)))
|
|
True
|
|
|
|
"""
|
|
return _raw_fftn_dispatch(x, shape, axes, overwrite_x, 1)
|
|
|
|
|
|
def _raw_fftn_dispatch(x, shape, axes, overwrite_x, direction):
|
|
tmp = _asfarray(x)
|
|
|
|
try:
|
|
work_function = _DTYPE_TO_FFTN[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)
|
|
return _raw_fftnd(tmp, shape, axes, direction, overwrite_x, work_function)
|
|
|
|
|
|
def ifftn(x, shape=None, axes=None, overwrite_x=False):
|
|
"""
|
|
Return inverse multi-dimensional discrete Fourier transform.
|
|
|
|
The sequence can be of an arbitrary type.
|
|
|
|
The returned array contains::
|
|
|
|
y[j_1,..,j_d] = 1/p * sum[k_1=0..n_1-1, ..., k_d=0..n_d-1]
|
|
x[k_1,..,k_d] * prod[i=1..d] exp(sqrt(-1)*2*pi/n_i * j_i * k_i)
|
|
|
|
where ``d = len(x.shape)``, ``n = x.shape``, and ``p = prod[i=1..d] n_i``.
|
|
|
|
For description of parameters see `fftn`.
|
|
|
|
See Also
|
|
--------
|
|
fftn : for detailed information.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.fftpack import fftn, ifftn
|
|
>>> import numpy as np
|
|
>>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16))
|
|
>>> np.allclose(y, ifftn(fftn(y)))
|
|
True
|
|
|
|
"""
|
|
return _raw_fftn_dispatch(x, shape, axes, overwrite_x, -1)
|
|
|
|
|
|
def fft2(x, shape=None, axes=(-2,-1), overwrite_x=False):
|
|
"""
|
|
2-D discrete Fourier transform.
|
|
|
|
Return the two-dimensional discrete Fourier transform of the 2-D argument
|
|
`x`.
|
|
|
|
See Also
|
|
--------
|
|
fftn : for detailed information.
|
|
|
|
"""
|
|
return fftn(x,shape,axes,overwrite_x)
|
|
|
|
|
|
def ifft2(x, shape=None, axes=(-2,-1), overwrite_x=False):
|
|
"""
|
|
2-D discrete inverse Fourier transform of real or complex sequence.
|
|
|
|
Return inverse two-dimensional discrete Fourier transform of
|
|
arbitrary type sequence x.
|
|
|
|
See `ifft` for more information.
|
|
|
|
See also
|
|
--------
|
|
fft2, ifft
|
|
|
|
"""
|
|
return ifftn(x,shape,axes,overwrite_x)
|