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scipy.fft.

# ifft

scipy.fft.ifft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, *, plan=None)

  Compute the 1-D inverse discrete Fourier Transform.

  This function computes the inverse of the 1-D n-point discrete Fourier transform computed by `fft`. In other words, `ifft(fft(x)) == x` to within numerical accuracy.

  The input should be ordered in the same way as is returned by `fft`, i.e.,

  - `x[0]` should contain the zero frequency term,

  - `x[1:n//2]` should contain the positive-frequency terms,

  - `x[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, `x[n//2]` represents the sum of the values at the positive and negative Nyquist frequencies, as the two are aliased together. See `fft` for details.

  Parameters:
    x 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

      Normalization mode (see `fft`). Default is “backward”.

    overwrite_x bool, optional

      If True, the contents of x can be destroyed; the default is False. See `fft` for more details.

    workers int, optional

      Maximum number of workers to use for parallel computation. If negative, the value wraps around from `os.cpu_count()`. See `fft` for more details.

    plan object, optional

      This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

      Added in version 1.5.0.

  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 x.

  See also

  `fft`

    The 1-D (forward) FFT, of which `ifft` is the inverse.

  `ifft2`

    The 2-D inverse FFT.

  `ifftn`

    The N-D 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`.

  If `x` is a 1-D array, then the `ifft` is equivalent to

      y[k] = np.sum(x * np.exp(2j * np.pi * k * np.arange(n)/n)) / len(x)

  As with `fft`, `ifft` has support for all floating point types and is optimized for real input.

  Array API Standard Support

  `ifft` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ✅ | ✅
  Dask | ⚠️ computes graph | n/a

  See Support for the array API standard for more information.

  Examples

      >>> import scipy.fft
      >>> import numpy as np
      >>> scipy.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
      >>> rng = np.random.default_rng()
      >>> t = np.arange(400)
      >>> n = np.zeros((400,), dtype=complex)
      >>> n[40:60] = np.exp(1j*rng.uniform(0, 2*np.pi, (20,)))
      >>> s = scipy.fft.ifft(n)
      >>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--')
      [<matplotlib.lines.Line2D object at ...>, <matplotlib.lines.Line2D object at ...>]
      >>> plt.legend(('real', 'imaginary'))
      <matplotlib.legend.Legend object at ...>
      >>> plt.show()