| | """ |
| | Discrete Fourier Transforms |
| | |
| | Routines in this module: |
| | |
| | fft(a, n=None, axis=-1, norm="backward") |
| | ifft(a, n=None, axis=-1, norm="backward") |
| | rfft(a, n=None, axis=-1, norm="backward") |
| | irfft(a, n=None, axis=-1, norm="backward") |
| | hfft(a, n=None, axis=-1, norm="backward") |
| | ihfft(a, n=None, axis=-1, norm="backward") |
| | fftn(a, s=None, axes=None, norm="backward") |
| | ifftn(a, s=None, axes=None, norm="backward") |
| | rfftn(a, s=None, axes=None, norm="backward") |
| | irfftn(a, s=None, axes=None, norm="backward") |
| | fft2(a, s=None, axes=(-2,-1), norm="backward") |
| | ifft2(a, s=None, axes=(-2, -1), norm="backward") |
| | rfft2(a, s=None, axes=(-2,-1), norm="backward") |
| | irfft2(a, s=None, axes=(-2, -1), norm="backward") |
| | |
| | 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.) |
| | |
| | """ |
| | __all__ = ['fft', 'ifft', 'rfft', 'irfft', 'hfft', 'ihfft', 'rfftn', |
| | 'irfftn', 'rfft2', 'irfft2', 'fft2', 'ifft2', 'fftn', 'ifftn'] |
| |
|
| | import functools |
| |
|
| | from numpy.core import asarray, zeros, swapaxes, conjugate, take, sqrt |
| | from . import _pocketfft_internal as pfi |
| | from numpy.core.multiarray import normalize_axis_index |
| | from numpy.core import overrides |
| |
|
| |
|
| | array_function_dispatch = functools.partial( |
| | overrides.array_function_dispatch, module='numpy.fft') |
| |
|
| |
|
| | |
| | |
| | |
| | |
| | def _raw_fft(a, n, axis, is_real, is_forward, inv_norm): |
| | axis = normalize_axis_index(axis, a.ndim) |
| | if n is None: |
| | n = a.shape[axis] |
| |
|
| | fct = 1/inv_norm |
| |
|
| | if a.shape[axis] != n: |
| | s = list(a.shape) |
| | index = [slice(None)]*len(s) |
| | if s[axis] > n: |
| | index[axis] = slice(0, n) |
| | a = a[tuple(index)] |
| | else: |
| | index[axis] = slice(0, s[axis]) |
| | s[axis] = n |
| | z = zeros(s, a.dtype.char) |
| | z[tuple(index)] = a |
| | a = z |
| |
|
| | if axis == a.ndim-1: |
| | r = pfi.execute(a, is_real, is_forward, fct) |
| | else: |
| | a = swapaxes(a, axis, -1) |
| | r = pfi.execute(a, is_real, is_forward, fct) |
| | r = swapaxes(r, axis, -1) |
| | return r |
| |
|
| |
|
| | def _get_forward_norm(n, norm): |
| | if n < 1: |
| | raise ValueError(f"Invalid number of FFT data points ({n}) specified.") |
| |
|
| | if norm is None or norm == "backward": |
| | return 1 |
| | elif norm == "ortho": |
| | return sqrt(n) |
| | elif norm == "forward": |
| | return n |
| | raise ValueError(f'Invalid norm value {norm}; should be "backward",' |
| | '"ortho" or "forward".') |
| |
|
| |
|
| | def _get_backward_norm(n, norm): |
| | if n < 1: |
| | raise ValueError(f"Invalid number of FFT data points ({n}) specified.") |
| |
|
| | if norm is None or norm == "backward": |
| | return n |
| | elif norm == "ortho": |
| | return sqrt(n) |
| | elif norm == "forward": |
| | return 1 |
| | raise ValueError(f'Invalid norm value {norm}; should be "backward", ' |
| | '"ortho" or "forward".') |
| |
|
| |
|
| | _SWAP_DIRECTION_MAP = {"backward": "forward", None: "forward", |
| | "ortho": "ortho", "forward": "backward"} |
| |
|
| |
|
| | def _swap_direction(norm): |
| | try: |
| | return _SWAP_DIRECTION_MAP[norm] |
| | except KeyError: |
| | raise ValueError(f'Invalid norm value {norm}; should be "backward", ' |
| | '"ortho" or "forward".') from None |
| |
|
| |
|
| | def _fft_dispatcher(a, n=None, axis=None, norm=None): |
| | return (a,) |
| |
|
| |
|
| | @array_function_dispatch(_fft_dispatcher) |
| | def fft(a, n=None, axis=-1, norm=None): |
| | """ |
| | Compute the one-dimensional discrete Fourier Transform. |
| | |
| | This function computes the one-dimensional *n*-point discrete Fourier |
| | Transform (DFT) with the efficient Fast Fourier Transform (FFT) |
| | algorithm [CT]. |
| | |
| | Parameters |
| | ---------- |
| | a : array_like |
| | Input array, can be complex. |
| | n : int, optional |
| | Length of the transformed axis of the output. |
| | If `n` is smaller than the length of the input, the input is cropped. |
| | If it is larger, the input is padded with zeros. If `n` is not given, |
| | the length of the input along the axis specified by `axis` is used. |
| | axis : int, optional |
| | Axis over which to compute the FFT. If not given, the last axis is |
| | used. |
| | norm : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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 `axis` is not a valid 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([-2.33486982e-16+1.14423775e-17j, 8.00000000e+00-1.25557246e-15j, |
| | 2.33486982e-16+2.33486982e-16j, 0.00000000e+00+1.22464680e-16j, |
| | -1.14423775e-17+2.33486982e-16j, 0.00000000e+00+5.20784380e-16j, |
| | 1.14423775e-17+1.14423775e-17j, 0.00000000e+00+1.22464680e-16j]) |
| | |
| | In this example, real input has an FFT which is Hermitian, i.e., symmetric |
| | in the real part and anti-symmetric in the imaginary part, as described in |
| | the `numpy.fft` documentation: |
| | |
| | >>> import matplotlib.pyplot as plt |
| | >>> t = np.arange(256) |
| | >>> sp = np.fft.fft(np.sin(t)) |
| | >>> freq = np.fft.fftfreq(t.shape[-1]) |
| | >>> plt.plot(freq, sp.real, freq, sp.imag) |
| | [<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>] |
| | >>> plt.show() |
| | |
| | """ |
| | a = asarray(a) |
| | if n is None: |
| | n = a.shape[axis] |
| | inv_norm = _get_forward_norm(n, norm) |
| | output = _raw_fft(a, n, axis, False, True, inv_norm) |
| | return output |
| |
|
| |
|
| | @array_function_dispatch(_fft_dispatcher) |
| | 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 : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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 `axis` is not a valid 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]) # may vary |
| | |
| | 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, label='real') |
| | [<matplotlib.lines.Line2D object at ...>] |
| | >>> plt.plot(t, s.imag, '--', label='imaginary') |
| | [<matplotlib.lines.Line2D object at ...>] |
| | >>> plt.legend() |
| | <matplotlib.legend.Legend object at ...> |
| | >>> plt.show() |
| | |
| | """ |
| | a = asarray(a) |
| | if n is None: |
| | n = a.shape[axis] |
| | inv_norm = _get_backward_norm(n, norm) |
| | output = _raw_fft(a, n, axis, False, False, inv_norm) |
| | return output |
| |
|
| |
|
| | @array_function_dispatch(_fft_dispatcher) |
| | 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 : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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 not a valid 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]) # may vary |
| | >>> np.fft.rfft([0, 1, 0, 0]) |
| | array([ 1.+0.j, 0.-1.j, -1.+0.j]) # may vary |
| | |
| | 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) |
| | if n is None: |
| | n = a.shape[axis] |
| | inv_norm = _get_forward_norm(n, norm) |
| | output = _raw_fft(a, n, axis, True, True, inv_norm) |
| | return output |
| |
|
| |
|
| | @array_function_dispatch(_fft_dispatcher) |
| | def irfft(a, n=None, axis=-1, norm=None): |
| | """ |
| | Computes the inverse of `rfft`. |
| | |
| | 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 taken to be |
| | ``2*(m-1)`` where ``m`` is 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 : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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 not a valid 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)``. |
| | |
| | The correct interpretation of the hermitian input depends on the length of |
| | the original data, as given by `n`. This is because each input shape could |
| | correspond to either an odd or even length signal. By default, `irfft` |
| | assumes an even output length which puts the last entry at the Nyquist |
| | frequency; aliasing with its symmetric counterpart. By Hermitian symmetry, |
| | the value is thus treated as purely real. To avoid losing information, the |
| | correct length of the real input **must** be given. |
| | |
| | Examples |
| | -------- |
| | >>> np.fft.ifft([1, -1j, -1, 1j]) |
| | array([0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]) # may vary |
| | >>> 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) |
| | if n is None: |
| | n = (a.shape[axis] - 1) * 2 |
| | inv_norm = _get_backward_norm(n, norm) |
| | output = _raw_fft(a, n, axis, True, False, inv_norm) |
| | return output |
| |
|
| |
|
| | @array_function_dispatch(_fft_dispatcher) |
| | 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 taken to be ``2*(m-1)`` |
| | where ``m`` is 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 : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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 not a valid 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. |
| | |
| | The correct interpretation of the hermitian input depends on the length of |
| | the original data, as given by `n`. This is because each input shape could |
| | correspond to either an odd or even length signal. By default, `hfft` |
| | assumes an even output length which puts the last entry at the Nyquist |
| | frequency; aliasing with its symmetric counterpart. By Hermitian symmetry, |
| | the value is thus treated as purely real. To avoid losing information, the |
| | shape of the full signal **must** be given. |
| | |
| | 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]) # may vary |
| | >>> 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], # may vary |
| | [ 0.+0.j, 0.-0.j]]) |
| | >>> freq_spectrum = np.fft.hfft(signal) |
| | >>> freq_spectrum |
| | array([[ 1., 1.], |
| | [ 2., -2.]]) |
| | |
| | """ |
| | a = asarray(a) |
| | if n is None: |
| | n = (a.shape[axis] - 1) * 2 |
| | new_norm = _swap_direction(norm) |
| | output = irfft(conjugate(a), n, axis, norm=new_norm) |
| | return output |
| |
|
| |
|
| | @array_function_dispatch(_fft_dispatcher) |
| | 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 : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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]) # may vary |
| | >>> np.fft.ihfft(spectrum) |
| | array([ 1.-0.j, 2.-0.j, 3.-0.j, 4.-0.j]) # may vary |
| | |
| | """ |
| | a = asarray(a) |
| | if n is None: |
| | n = a.shape[axis] |
| | new_norm = _swap_direction(norm) |
| | output = conjugate(rfft(a, n, axis, norm=new_norm)) |
| | return output |
| |
|
| |
|
| | 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, norm=None): |
| | a = asarray(a) |
| | s, axes = _cook_nd_args(a, s, axes) |
| | itl = list(range(len(axes))) |
| | itl.reverse() |
| | for ii in itl: |
| | a = function(a, n=s[ii], axis=axes[ii], norm=norm) |
| | return a |
| |
|
| |
|
| | def _fftn_dispatcher(a, s=None, axes=None, norm=None): |
| | return (a,) |
| |
|
| |
|
| | @array_function_dispatch(_fftn_dispatcher) |
| | def fftn(a, s=None, axes=None, norm=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. |
| | norm : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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], # may vary |
| | [ 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], # may vary |
| | [ 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, norm) |
| |
|
| |
|
| | @array_function_dispatch(_fftn_dispatcher) |
| | def ifftn(a, s=None, axes=None, norm=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. |
| | norm : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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], # may vary |
| | [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, norm) |
| |
|
| |
|
| | @array_function_dispatch(_fftn_dispatcher) |
| | def fft2(a, s=None, axes=(-2, -1), norm=None): |
| | """ |
| | 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. |
| | norm : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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([[ 50. +0.j , 0. +0.j , 0. +0.j , # may vary |
| | 0. +0.j , 0. +0.j ], |
| | [-12.5+17.20477401j, 0. +0.j , 0. +0.j , |
| | 0. +0.j , 0. +0.j ], |
| | [-12.5 +4.0614962j , 0. +0.j , 0. +0.j , |
| | 0. +0.j , 0. +0.j ], |
| | [-12.5 -4.0614962j , 0. +0.j , 0. +0.j , |
| | 0. +0.j , 0. +0.j ], |
| | [-12.5-17.20477401j, 0. +0.j , 0. +0.j , |
| | 0. +0.j , 0. +0.j ]]) |
| | |
| | """ |
| | return _raw_fftnd(a, s, axes, fft, norm) |
| |
|
| |
|
| | @array_function_dispatch(_fftn_dispatcher) |
| | def ifft2(a, s=None, axes=(-2, -1), norm=None): |
| | """ |
| | 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. |
| | norm : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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], # may vary |
| | [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, norm) |
| |
|
| |
|
| | @array_function_dispatch(_fftn_dispatcher) |
| | 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 : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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], # may vary |
| | [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], # may vary |
| | [4.+0.j, 0.+0.j]], |
| | [[0.+0.j, 0.+0.j], |
| | [0.+0.j, 0.+0.j]]]) |
| | |
| | """ |
| | a = asarray(a) |
| | 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 |
| |
|
| |
|
| | @array_function_dispatch(_fftn_dispatcher) |
| | def rfft2(a, s=None, axes=(-2, -1), norm=None): |
| | """ |
| | 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. |
| | norm : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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`. |
| | |
| | Examples |
| | -------- |
| | >>> a = np.mgrid[:5, :5][0] |
| | >>> np.fft.rfft2(a) |
| | array([[ 50. +0.j , 0. +0.j , 0. +0.j ], |
| | [-12.5+17.20477401j, 0. +0.j , 0. +0.j ], |
| | [-12.5 +4.0614962j , 0. +0.j , 0. +0.j ], |
| | [-12.5 -4.0614962j , 0. +0.j , 0. +0.j ], |
| | [-12.5-17.20477401j, 0. +0.j , 0. +0.j ]]) |
| | """ |
| | return rfftn(a, s, axes, norm) |
| |
|
| |
|
| | @array_function_dispatch(_fftn_dispatcher) |
| | def irfftn(a, s=None, axes=None, norm=None): |
| | """ |
| | Computes the inverse of `rfftn`. |
| | |
| | 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. Except for the last axis which is taken to |
| | be ``2*(m-1)`` where ``m`` is the length of the input along that axis. |
| | 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 : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | 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. |
| | |
| | The correct interpretation of the hermitian input depends on the shape of |
| | the original data, as given by `s`. This is because each input shape could |
| | correspond to either an odd or even length signal. By default, `irfftn` |
| | assumes an even output length which puts the last entry at the Nyquist |
| | frequency; aliasing with its symmetric counterpart. When performing the |
| | final complex to real transform, the last value is thus treated as purely |
| | real. To avoid losing information, the correct shape of the real input |
| | **must** be given. |
| | |
| | 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) |
| | 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 |
| |
|
| |
|
| | @array_function_dispatch(_fftn_dispatcher) |
| | def irfft2(a, s=None, axes=(-2, -1), norm=None): |
| | """ |
| | Computes the inverse of `rfft2`. |
| | |
| | Parameters |
| | ---------- |
| | a : array_like |
| | The input array |
| | s : sequence of ints, optional |
| | Shape of the real output to the inverse FFT. |
| | axes : sequence of ints, optional |
| | The axes over which to compute the inverse fft. |
| | Default is the last two axes. |
| | norm : {"backward", "ortho", "forward"}, optional |
| | .. versionadded:: 1.10.0 |
| | |
| | Normalization mode (see `numpy.fft`). Default is "backward". |
| | Indicates which direction of the forward/backward pair of transforms |
| | is scaled and with what normalization factor. |
| | |
| | .. versionadded:: 1.20.0 |
| | |
| | The "backward", "forward" values were added. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | The result of the inverse real 2-D FFT. |
| | |
| | See Also |
| | -------- |
| | rfft2 : The forward two-dimensional FFT of real input, |
| | of which `irfft2` is the inverse. |
| | rfft : The one-dimensional FFT for real input. |
| | irfft : The inverse of the one-dimensional FFT of real input. |
| | 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`. |
| | |
| | Examples |
| | -------- |
| | >>> a = np.mgrid[:5, :5][0] |
| | >>> A = np.fft.rfft2(a) |
| | >>> np.fft.irfft2(A, s=a.shape) |
| | array([[0., 0., 0., 0., 0.], |
| | [1., 1., 1., 1., 1.], |
| | [2., 2., 2., 2., 2.], |
| | [3., 3., 3., 3., 3.], |
| | [4., 4., 4., 4., 4.]]) |
| | """ |
| | return irfftn(a, s, axes, norm) |
| |
|
