Module « scipy.linalg »

Fonction dft - module scipy.linalg

Signature de la fonction dft

def dft(n, scale=None) 

Description

dft.__doc__

    Discrete Fourier transform matrix.

    Create the matrix that computes the discrete Fourier transform of a
    sequence [1]_. The nth primitive root of unity used to generate the
    matrix is exp(-2*pi*i/n), where i = sqrt(-1).

    Parameters
    ----------
    n : int
        Size the matrix to create.
    scale : str, optional
        Must be None, 'sqrtn', or 'n'.
        If `scale` is 'sqrtn', the matrix is divided by `sqrt(n)`.
        If `scale` is 'n', the matrix is divided by `n`.
        If `scale` is None (the default), the matrix is not normalized, and the
        return value is simply the Vandermonde matrix of the roots of unity.

    Returns
    -------
    m : (n, n) ndarray
        The DFT matrix.

    Notes
    -----
    When `scale` is None, multiplying a vector by the matrix returned by
    `dft` is mathematically equivalent to (but much less efficient than)
    the calculation performed by `scipy.fft.fft`.

    .. versionadded:: 0.14.0

    References
    ----------
    .. [1] "DFT matrix", https://en.wikipedia.org/wiki/DFT_matrix

    Examples
    --------
    >>> from scipy.linalg import dft
    >>> np.set_printoptions(precision=2, suppress=True)  # for compact output
    >>> m = dft(5)
    >>> m
    array([[ 1.  +0.j  ,  1.  +0.j  ,  1.  +0.j  ,  1.  +0.j  ,  1.  +0.j  ],
           [ 1.  +0.j  ,  0.31-0.95j, -0.81-0.59j, -0.81+0.59j,  0.31+0.95j],
           [ 1.  +0.j  , -0.81-0.59j,  0.31+0.95j,  0.31-0.95j, -0.81+0.59j],
           [ 1.  +0.j  , -0.81+0.59j,  0.31-0.95j,  0.31+0.95j, -0.81-0.59j],
           [ 1.  +0.j  ,  0.31+0.95j, -0.81+0.59j, -0.81-0.59j,  0.31-0.95j]])
    >>> x = np.array([1, 2, 3, 0, 3])
    >>> m @ x  # Compute the DFT of x
    array([ 9.  +0.j  ,  0.12-0.81j, -2.12+3.44j, -2.12-3.44j,  0.12+0.81j])

    Verify that ``m @ x`` is the same as ``fft(x)``.

    >>> from scipy.fft import fft
    >>> fft(x)     # Same result as m @ x
    array([ 9.  +0.j  ,  0.12-0.81j, -2.12+3.44j, -2.12-3.44j,  0.12+0.81j])