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Edit File: fftpack.py
""" Discrete Fourier Transforms Routines in this module: fft(a, n=None, axis=-1) ifft(a, n=None, axis=-1) rfft(a, n=None, axis=-1) irfft(a, n=None, axis=-1) hfft(a, n=None, axis=-1) ihfft(a, n=None, axis=-1) fftn(a, s=None, axes=None) ifftn(a, s=None, axes=None) rfftn(a, s=None, axes=None) irfftn(a, s=None, axes=None) fft2(a, s=None, axes=(-2,-1)) ifft2(a, s=None, axes=(-2, -1)) rfft2(a, s=None, axes=(-2,-1)) irfft2(a, s=None, axes=(-2, -1)) i = inverse transform r = transform of purely real data h = Hermite transform n = n-dimensional transform 2 = 2-dimensional transform (Note: 2D routines are just nD routines with different default behavior.) The underlying code for these functions is an f2c-translated and modified version of the FFTPACK routines. """ from __future__ import division, absolute_import, print_function __all__ = ['fft', 'ifft', 'rfft', 'irfft', 'hfft', 'ihfft', 'rfftn', 'irfftn', 'rfft2', 'irfft2', 'fft2', 'ifft2', 'fftn', 'ifftn'] from numpy.core import asarray, zeros, swapaxes, shape, conjugate, \ take from . import fftpack_lite as fftpack _fft_cache = {} _real_fft_cache = {} 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) try: # Thread-safety note: We rely on list.pop() here to atomically # retrieve-and-remove a wsave from the cache. This ensures that no # other thread can get the same wsave while we're using it. wsave = fft_cache.setdefault(n, []).pop() except (IndexError): 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[index] else: index = [slice(None)]*len(s) index[axis] = slice(0, s[axis]) s[axis] = n z = zeros(s, a.dtype.char) z[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[n].append(wsave) return r def fft(a, n=None, axis=-1): """ 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. 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]) >>> 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() 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. """ return _raw_fft(a, n, axis, fftpack.cffti, fftpack.cfftf, _fft_cache) def ifft(a, n=None, axis=-1): """ 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+1]`` should contain the positive-frequency terms, and ``a[n/2+1:]`` should contain the negative-frequency terms, in order of decreasingly negative frequency. 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. 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--') [<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>] >>> plt.legend(('real', 'imaginary')) <matplotlib.legend.Legend object at 0x...> >>> plt.show() """ a = asarray(a).astype(complex) if n is None: n = shape(a)[axis] return _raw_fft(a, n, axis, fftpack.cffti, fftpack.cfftb, _fft_cache) / n def rfft(a, n=None, axis=-1): """ 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. 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 Hermite-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. """ a = asarray(a).astype(float) return _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftf, _real_fft_cache) def irfft(a, n=None, axis=-1): """ 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 Hermite-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. 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 Hermite-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. """ a = asarray(a).astype(complex) if n is None: n = (shape(a)[axis] - 1) * 2 return _raw_fft(a, n, axis, fftpack.rffti, fftpack.rfftb, _real_fft_cache) / n def hfft(a, n=None, axis=-1): """ Compute the FFT of a signal whose spectrum has Hermitian symmetry. Parameters ---------- a : array_like The input array. n : int, optional The length of the FFT. axis : int, optional The axis over which to compute the FFT, assuming Hermitian symmetry of the spectrum. Default is the last axis. Returns ------- out : ndarray The transformed input. 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 is real in the frequency domain and has Hermite symmetry in the time domain. So here it's `hfft` for which you must supply the length of the result if it is to be odd: ``ihfft(hfft(a), len(a)) == a``, within numerical accuracy. Examples -------- >>> 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.]]) """ a = asarray(a).astype(complex) if n is None: n = (shape(a)[axis] - 1) * 2 return irfft(conjugate(a), n, axis) * n def ihfft(a, n=None, axis=-1): """ Compute the inverse FFT of a signal whose spectrum has Hermitian symmetry. Parameters ---------- a : array_like Input array. n : int, optional Length of the inverse FFT. axis : int, optional Axis over which to compute the inverse FFT, assuming Hermitian symmetry of the spectrum. Default is the last axis. Returns ------- out : ndarray The transformed input. See also -------- hfft, irfft Notes ----- `hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the opposite case: here the signal is real in the frequency domain and has Hermite symmetry in the time domain. So here it's `hfft` for which you must supply the length of the result if it is to be odd: ``ihfft(hfft(a), len(a)) == a``, within numerical accuracy. """ a = asarray(a).astype(float) if n is None: n = shape(a)[axis] return conjugate(rfft(a, n, axis))/n def _cook_nd_args(a, s=None, axes=None, invreal=0): if s is None: shapeless = 1 if axes is None: s = list(a.shape) else: s = take(a.shape, axes) else: shapeless = 0 s = list(s) if axes is None: axes = list(range(-len(s), 0)) if len(s) != len(axes): raise ValueError("Shape and axes have different lengths.") if invreal and shapeless: s[-1] = (a.shape[axes[-1]] - 1) * 2 return s, axes def _raw_fftnd(a, s=None, axes=None, function=fft): 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]) return a def fftn(a, s=None, axes=None): """ Compute the N-dimensional discrete Fourier Transform. This function computes the *N*-dimensional discrete Fourier Transform over any number of axes in an *M*-dimensional array by means of the Fast Fourier Transform (FFT). Parameters ---------- a : array_like Input array, can be complex. 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.). This 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. Repeated indices in `axes` means that the transform over that axis is performed multiple times. 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. 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 -------- numpy.fft : Overall view of discrete Fourier transforms, with definitions and conventions used. ifftn : The inverse of `fftn`, the inverse *n*-dimensional FFT. fft : The one-dimensional FFT, with definitions and conventions used. rfftn : The *n*-dimensional FFT of real input. fft2 : The two-dimensional FFT. fftshift : Shifts zero-frequency terms to centre of array Notes ----- The output, analogously to `fft`, contains the term for zero frequency in the low-order corner of all axes, the positive frequency terms in the first half of all axes, the term for the Nyquist frequency in the middle of all axes and the negative frequency terms in the second half of all axes, in order of decreasingly negative frequency. See `numpy.fft` for details, definitions and conventions used. Examples -------- >>> a = np.mgrid[:3, :3, :3][0] >>> np.fft.fftn(a, axes=(1, 2)) array([[[ 0.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j]], [[ 9.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j]], [[ 18.+0.j, 0.+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.fftn(a, (2, 2), axes=(0, 1)) array([[[ 2.+0.j, 2.+0.j, 2.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j]], [[-2.+0.j, -2.+0.j, -2.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j]]]) >>> import matplotlib.pyplot as plt >>> [X, Y] = np.meshgrid(2 * np.pi * np.arange(200) / 12, ... 2 * np.pi * np.arange(200) / 34) >>> S = np.sin(X) + np.cos(Y) + np.random.uniform(0, 1, X.shape) >>> FS = np.fft.fftn(S) >>> plt.imshow(np.log(np.abs(np.fft.fftshift(FS))**2)) <matplotlib.image.AxesImage object at 0x...> >>> plt.show() """ return _raw_fftnd(a, s, axes, fft) def ifftn(a, s=None, axes=None): """ Compute the N-dimensional inverse discrete Fourier Transform. This function computes the inverse of the N-dimensional discrete Fourier Transform over any number of axes in an M-dimensional array by means of the Fast Fourier Transform (FFT). In other words, ``ifftn(fftn(a)) == a`` to within numerical accuracy. For a description of the definitions and conventions used, see `numpy.fft`. The input, analogously to `ifft`, should be ordered in the same way as is returned by `fftn`, i.e. it should have the term for zero frequency in all axes in the low-order corner, the positive frequency terms in the first half of all axes, the term for the Nyquist frequency in the middle of all axes and the negative frequency terms in the second half of all axes, in order of decreasingly negative frequency. Parameters ---------- a : array_like Input array, can be complex. 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.). This corresponds to ``n`` for ``ifft(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. See notes for issue on `ifft` zero padding. axes : sequence of ints, optional Axes over which to compute the IFFT. 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. Returns ------- out : complex 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. 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 -------- numpy.fft : Overall view of discrete Fourier transforms, with definitions and conventions used. fftn : The forward *n*-dimensional FFT, of which `ifftn` is the inverse. ifft : The one-dimensional inverse FFT. ifft2 : The two-dimensional inverse FFT. ifftshift : Undoes `fftshift`, shifts zero-frequency terms to beginning of array. Notes ----- See `numpy.fft` for definitions and conventions used. Zero-padding, analogously with `ifft`, is performed by appending zeros to the input along the specified dimension. Although this is the common approach, it might lead to surprising results. If another form of zero padding is desired, it must be performed before `ifftn` is called. Examples -------- >>> a = np.eye(4) >>> np.fft.ifftn(np.fft.fftn(a, axes=(0,)), axes=(1,)) array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]]) Create and plot an image with band-limited frequency content: >>> import matplotlib.pyplot as plt >>> n = np.zeros((200,200), dtype=complex) >>> n[60:80, 20:40] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20, 20))) >>> im = np.fft.ifftn(n).real >>> plt.imshow(im) <matplotlib.image.AxesImage object at 0x...> >>> plt.show() """ return _raw_fftnd(a, s, axes, ifft) def fft2(a, s=None, axes=(-2, -1)): """ Compute the 2-dimensional discrete Fourier Transform This function computes the *n*-dimensional discrete Fourier Transform over any axes in an *M*-dimensional array by means of the Fast Fourier Transform (FFT). By default, the transform is computed over the last two axes of the input array, i.e., a 2-dimensional FFT. Parameters ---------- a : array_like Input array, can be complex 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.). This corresponds to `n` for `fft(x, n)`. Along each 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 two axes are used. A repeated index in `axes` means the transform over that axis is performed multiple times. A one-element sequence means that a one-dimensional FFT is performed. Returns ------- out : complex ndarray The truncated or zero-padded input, transformed along the axes indicated by `axes`, or the last two axes if `axes` is not given. Raises ------ ValueError If `s` and `axes` have different length, or `axes` not given and ``len(s) != 2``. IndexError If an element of `axes` is larger than than the number of axes of `a`. See Also -------- numpy.fft : Overall view of discrete Fourier transforms, with definitions and conventions used. ifft2 : The inverse two-dimensional FFT. fft : The one-dimensional FFT. fftn : The *n*-dimensional FFT. fftshift : Shifts zero-frequency terms to the center of the array. For two-dimensional input, swaps first and third quadrants, and second and fourth quadrants. Notes ----- `fft2` is just `fftn` with a different default for `axes`. The output, analogously to `fft`, contains the term for zero frequency in the low-order corner of the transformed axes, the positive frequency terms in the first half of these axes, the term for the Nyquist frequency in the middle of the axes and the negative frequency terms in the second half of the axes, in order of decreasingly negative frequency. See `fftn` for details and a plotting example, and `numpy.fft` for definitions and conventions used. Examples -------- >>> a = np.mgrid[:5, :5][0] >>> np.fft.fft2(a) array([[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], [ 5.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], [ 10.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], [ 15.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], [ 20.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]]) """ return _raw_fftnd(a, s, axes, fft) def ifft2(a, s=None, axes=(-2, -1)): """ Compute the 2-dimensional inverse discrete Fourier Transform. This function computes the inverse of the 2-dimensional discrete Fourier Transform over any number of axes in an M-dimensional array by means of the Fast Fourier Transform (FFT). In other words, ``ifft2(fft2(a)) == a`` to within numerical accuracy. By default, the inverse transform is computed over the last two axes of the input array. The input, analogously to `ifft`, should be ordered in the same way as is returned by `fft2`, i.e. it should have the term for zero frequency in the low-order corner of the two axes, the positive frequency terms in the first half of these axes, the term for the Nyquist frequency in the middle of the axes and the negative frequency terms in the second half of both axes, in order of decreasingly negative frequency. Parameters ---------- a : array_like Input array, can be complex. s : sequence of ints, optional Shape (length of each axis) of the output (``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.). This corresponds to `n` for ``ifft(x, n)``. Along each 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. See notes for issue on `ifft` zero padding. axes : sequence of ints, optional Axes over which to compute the FFT. If not given, the last two axes are used. A repeated index in `axes` means the transform over that axis is performed multiple times. A one-element sequence means that a one-dimensional FFT is performed. Returns ------- out : complex ndarray The truncated or zero-padded input, transformed along the axes indicated by `axes`, or the last two axes if `axes` is not given. Raises ------ ValueError If `s` and `axes` have different length, or `axes` not given and ``len(s) != 2``. IndexError If an element of `axes` is larger than than the number of axes of `a`. See Also -------- numpy.fft : Overall view of discrete Fourier transforms, with definitions and conventions used. fft2 : The forward 2-dimensional FFT, of which `ifft2` is the inverse. ifftn : The inverse of the *n*-dimensional FFT. fft : The one-dimensional FFT. ifft : The one-dimensional inverse FFT. Notes ----- `ifft2` is just `ifftn` with a different default for `axes`. See `ifftn` for details and a plotting example, and `numpy.fft` for definition and conventions used. Zero-padding, analogously with `ifft`, is performed by appending zeros to the input along the specified dimension. Although this is the common approach, it might lead to surprising results. If another form of zero padding is desired, it must be performed before `ifft2` is called. Examples -------- >>> a = 4 * np.eye(4) >>> np.fft.ifft2(a) array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j], [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j], [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]]) """ return _raw_fftnd(a, s, axes, ifft) def rfftn(a, s=None, axes=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. 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]]]) """ a = asarray(a).astype(float) s, axes = _cook_nd_args(a, s, axes) a = rfft(a, s[-1], axes[-1]) for ii in range(len(axes)-1): a = fft(a, s[ii], axes[ii]) return a def rfft2(a, s=None, axes=(-2, -1)): """ Compute the 2-dimensional FFT of a real array. Parameters ---------- a : array Input array, taken to be real. s : sequence of ints, optional Shape of the FFT. axes : sequence of ints, optional Axes over which to compute the FFT. Returns ------- out : ndarray The result of the real 2-D FFT. See Also -------- rfftn : Compute the N-dimensional discrete Fourier Transform for real input. Notes ----- This is really just `rfftn` with different default behavior. For more details see `rfftn`. """ return rfftn(a, s, axes) def irfftn(a, s=None, axes=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. 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.]]]) """ a = asarray(a).astype(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]) a = irfft(a, s[-1], axes[-1]) return a def irfft2(a, s=None, axes=(-2, -1)): """ Compute the 2-dimensional inverse FFT of a real array. Parameters ---------- a : array_like The input array s : sequence of ints, optional Shape of the inverse FFT. axes : sequence of ints, optional The axes over which to compute the inverse fft. Default is the last two axes. Returns ------- out : ndarray The result of the inverse real 2-D FFT. See Also -------- irfftn : Compute the inverse of the N-dimensional FFT of real input. Notes ----- This is really `irfftn` with different defaults. For more details see `irfftn`. """ return irfftn(a, s, axes)