
    Return Hilbert transform of a periodic sequence x.

    If x_j and y_j are Fourier coefficients of periodic functions x
    and y, respectively, then::

      y_j = sqrt(-1)*sign(j) * x_j
      y_0 = 0

    Parameters
    ----------
    x : array_like
        The input array, should be periodic.
    _cache : dict, optional
        Dictionary that contains the kernel used to do a convolution with.

    Returns
    -------
    y : ndarray
        The transformed input.

    See Also
    --------
    scipy.signal.hilbert : Compute the analytic signal, using the Hilbert
                           transform.

    Notes
    -----
    If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``.

    For even len(x), the Nyquist mode of x is taken zero.

    The sign of the returned transform does not have a factor -1 that is more
    often than not found in the definition of the Hilbert transform.  Note also
    that `scipy.signal.hilbert` does have an extra -1 factor compared to this
    function.

    