| """ |
| Discrete Fourier Transforms - _basic.py |
| """ |
| |
| __all__ = ['fft','ifft','fftn','ifftn','rfft','irfft', |
| 'fft2','ifft2'] |
|
|
| from scipy.fft import _pocketfft |
| from ._helper import _good_shape |
|
|
|
|
| def fft(x, n=None, axis=-1, overwrite_x=False): |
| """ |
| Return discrete Fourier transform of real or complex sequence. |
| |
| The returned complex array contains ``y(0), y(1),..., y(n-1)``, where |
| |
| ``y(j) = (x * exp(-2*pi*sqrt(-1)*j*np.arange(n)/n)).sum()``. |
| |
| Parameters |
| ---------- |
| x : array_like |
| Array to Fourier transform. |
| n : int, optional |
| Length of the Fourier transform. If ``n < x.shape[axis]``, `x` is |
| truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The |
| default results in ``n = x.shape[axis]``. |
| axis : int, optional |
| Axis along which the fft's are computed; the default is over the |
| last axis (i.e., ``axis=-1``). |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| |
| Returns |
| ------- |
| z : complex ndarray |
| with the elements:: |
| |
| [y(0),y(1),..,y(n/2),y(1-n/2),...,y(-1)] if n is even |
| [y(0),y(1),..,y((n-1)/2),y(-(n-1)/2),...,y(-1)] if n is odd |
| |
| where:: |
| |
| y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n), j = 0..n-1 |
| |
| See Also |
| -------- |
| ifft : Inverse FFT |
| rfft : FFT of a real sequence |
| |
| Notes |
| ----- |
| The packing of the result is "standard": If ``A = fft(a, n)``, then |
| ``A[0]`` contains the zero-frequency term, ``A[1:n/2]`` contains the |
| positive-frequency terms, and ``A[n/2:]`` contains the negative-frequency |
| terms, in order of decreasingly negative frequency. So ,for an 8-point |
| transform, the frequencies of the result are [0, 1, 2, 3, -4, -3, -2, -1]. |
| To rearrange the fft output so that the zero-frequency component is |
| centered, like [-4, -3, -2, -1, 0, 1, 2, 3], use `fftshift`. |
| |
| Both single and double precision routines are implemented. Half precision |
| inputs will be converted to single precision. Non-floating-point inputs |
| will be converted to double precision. Long-double precision inputs are |
| not supported. |
| |
| This function is most efficient when `n` is a power of two, and least |
| efficient when `n` is prime. |
| |
| Note that if ``x`` is real-valued, then ``A[j] == A[n-j].conjugate()``. |
| If ``x`` is real-valued and ``n`` is even, then ``A[n/2]`` is real. |
| |
| If the data type of `x` is real, a "real FFT" algorithm is automatically |
| used, which roughly halves the computation time. To increase efficiency |
| a little further, use `rfft`, which does the same calculation, but only |
| outputs half of the symmetrical spectrum. If the data is both real and |
| symmetrical, the `dct` can again double the efficiency by generating |
| half of the spectrum from half of the signal. |
| |
| Examples |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fftpack import fft, ifft |
| >>> x = np.arange(5) |
| >>> np.allclose(fft(ifft(x)), x, atol=1e-15) # within numerical accuracy. |
| True |
| |
| """ |
| return _pocketfft.fft(x, n, axis, None, overwrite_x) |
|
|
|
|
| def ifft(x, n=None, axis=-1, overwrite_x=False): |
| """ |
| Return discrete inverse Fourier transform of real or complex sequence. |
| |
| The returned complex array contains ``y(0), y(1),..., y(n-1)``, where |
| |
| ``y(j) = (x * exp(2*pi*sqrt(-1)*j*np.arange(n)/n)).mean()``. |
| |
| Parameters |
| ---------- |
| x : array_like |
| Transformed data to invert. |
| n : int, optional |
| Length of the inverse Fourier transform. If ``n < x.shape[axis]``, |
| `x` is truncated. If ``n > x.shape[axis]``, `x` is zero-padded. |
| The default results in ``n = x.shape[axis]``. |
| axis : int, optional |
| Axis along which the ifft's are computed; the default is over the |
| last axis (i.e., ``axis=-1``). |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| |
| Returns |
| ------- |
| ifft : ndarray of floats |
| The inverse discrete Fourier transform. |
| |
| See Also |
| -------- |
| fft : Forward FFT |
| |
| Notes |
| ----- |
| Both single and double precision routines are implemented. Half precision |
| inputs will be converted to single precision. Non-floating-point inputs |
| will be converted to double precision. Long-double precision inputs are |
| not supported. |
| |
| This function is most efficient when `n` is a power of two, and least |
| efficient when `n` is prime. |
| |
| If the data type of `x` is real, a "real IFFT" algorithm is automatically |
| used, which roughly halves the computation time. |
| |
| Examples |
| -------- |
| >>> from scipy.fftpack import fft, ifft |
| >>> import numpy as np |
| >>> x = np.arange(5) |
| >>> np.allclose(ifft(fft(x)), x, atol=1e-15) # within numerical accuracy. |
| True |
| |
| """ |
| return _pocketfft.ifft(x, n, axis, None, overwrite_x) |
|
|
|
|
| def rfft(x, n=None, axis=-1, overwrite_x=False): |
| """ |
| Discrete Fourier transform of a real sequence. |
| |
| Parameters |
| ---------- |
| x : array_like, real-valued |
| The data to transform. |
| n : int, optional |
| Defines the length of the Fourier transform. If `n` is not specified |
| (the default) then ``n = x.shape[axis]``. If ``n < x.shape[axis]``, |
| `x` is truncated, if ``n > x.shape[axis]``, `x` is zero-padded. |
| axis : int, optional |
| The axis along which the transform is applied. The default is the |
| last axis. |
| overwrite_x : bool, optional |
| If set to true, the contents of `x` can be overwritten. Default is |
| False. |
| |
| Returns |
| ------- |
| z : real ndarray |
| The returned real array contains:: |
| |
| [y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2))] if n is even |
| [y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2)),Im(y(n/2))] if n is odd |
| |
| where:: |
| |
| y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k*2*pi/n) |
| j = 0..n-1 |
| |
| See Also |
| -------- |
| fft, irfft, scipy.fft.rfft |
| |
| Notes |
| ----- |
| Within numerical accuracy, ``y == rfft(irfft(y))``. |
| |
| 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. |
| |
| To get an output with a complex datatype, consider using the newer |
| function `scipy.fft.rfft`. |
| |
| Examples |
| -------- |
| >>> from scipy.fftpack import fft, rfft |
| >>> a = [9, -9, 1, 3] |
| >>> fft(a) |
| array([ 4. +0.j, 8.+12.j, 16. +0.j, 8.-12.j]) |
| >>> rfft(a) |
| array([ 4., 8., 12., 16.]) |
| |
| """ |
| return _pocketfft.rfft_fftpack(x, n, axis, None, overwrite_x) |
|
|
|
|
| def irfft(x, n=None, axis=-1, overwrite_x=False): |
| """ |
| Return inverse discrete Fourier transform of real sequence x. |
| |
| The contents of `x` are interpreted as the output of the `rfft` |
| function. |
| |
| Parameters |
| ---------- |
| x : array_like |
| Transformed data to invert. |
| n : int, optional |
| Length of the inverse Fourier transform. |
| If n < x.shape[axis], x is truncated. |
| If n > x.shape[axis], x is zero-padded. |
| The default results in n = x.shape[axis]. |
| axis : int, optional |
| Axis along which the ifft's are computed; the default is over |
| the last axis (i.e., axis=-1). |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| |
| Returns |
| ------- |
| irfft : ndarray of floats |
| The inverse discrete Fourier transform. |
| |
| See Also |
| -------- |
| rfft, ifft, scipy.fft.irfft |
| |
| Notes |
| ----- |
| The returned real array contains:: |
| |
| [y(0),y(1),...,y(n-1)] |
| |
| where for n is even:: |
| |
| y(j) = 1/n (sum[k=1..n/2-1] (x[2*k-1]+sqrt(-1)*x[2*k]) |
| * exp(sqrt(-1)*j*k* 2*pi/n) |
| + c.c. + x[0] + (-1)**(j) x[n-1]) |
| |
| 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 newer function `scipy.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.]) |
| |
| """ |
| return _pocketfft.irfft_fftpack(x, n, axis, None, overwrite_x) |
|
|
|
|
| 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-D) 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 ``numpy.take(x.shape, axes, axis=0)``. |
| If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. |
| If ``shape[i] < x.shape[i]``, the ith 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-D NumPy array |
| The (N-D) 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 |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fftpack import fftn, ifftn |
| >>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16)) |
| >>> np.allclose(y, fftn(ifftn(y))) |
| True |
| |
| """ |
| shape = _good_shape(x, shape, axes) |
| return _pocketfft.fftn(x, shape, axes, None, overwrite_x) |
|
|
|
|
| def ifftn(x, shape=None, axes=None, overwrite_x=False): |
| """ |
| Return inverse multidimensional 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 |
| |
| """ |
| shape = _good_shape(x, shape, axes) |
| return _pocketfft.ifftn(x, shape, axes, None, overwrite_x) |
|
|
|
|
| def fft2(x, shape=None, axes=(-2,-1), overwrite_x=False): |
| """ |
| 2-D discrete Fourier transform. |
| |
| Return the 2-D discrete Fourier transform of the 2-D argument |
| `x`. |
| |
| See Also |
| -------- |
| fftn : for detailed information. |
| |
| Examples |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fftpack import fft2, ifft2 |
| >>> y = np.mgrid[:5, :5][0] |
| >>> y |
| 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]]) |
| >>> np.allclose(y, ifft2(fft2(y))) |
| True |
| """ |
| 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 2-D discrete Fourier transform of |
| arbitrary type sequence x. |
| |
| See `ifft` for more information. |
| |
| See Also |
| -------- |
| fft2, ifft |
| |
| Examples |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fftpack import fft2, ifft2 |
| >>> y = np.mgrid[:5, :5][0] |
| >>> y |
| 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]]) |
| >>> np.allclose(y, fft2(ifft2(y))) |
| True |
| |
| """ |
| return ifftn(x,shape,axes,overwrite_x) |
|
|
