windows.cpp

#include <math.h>
#include "i0.h"

/*
   Translated from Python code at https://github.com/scipy/scipy/blob/master/scipy/signal/windows/windows.py

   Copyright (c) 2001, 2002 Enthought, Inc.
   All rights reserved.
  
   Copyright (c) 2003-2017 SciPy Developers.
   All rights reserved.
   

*/

/**
  def _len_guards(M):
    """Handle small or incorrect window lengths"""
    if int(M) != M or M < 0:
        raise ValueError('Window length M must be a non-negative integer')
    return M <= 1


  def _extend(M, sym):
    """Extend window by 1 sample if needed for DFT-even symmetry"""
    if not sym:
        return M + 1, True
    else:
        return M, False


  def _truncate(w, needed):
    """Truncate window by 1 sample if needed for DFT-even symmetry"""
    if needed:
        return w[:-1]
    else:
        return w
*/


/**
    def kaiser(M, beta, sym=True):
        r"""Return a Kaiser window.

        The Kaiser window is a taper formed by using a Bessel function.

        Parameters
        ----------
        M : int
            Number of points in the output window. If zero or less, an empty
            array is returned.
        beta : float
            Shape parameter, determines trade-off between main-lobe width and
            side lobe level. As beta gets large, the window narrows.
        sym : bool, optional
            When True (default), generates a symmetric window, for use in filter
            design.
            When False, generates a periodic window, for use in spectral analysis.

        Returns
        -------
        w : ndarray
            The window, with the maximum value normalized to 1 (though the value 1
            does not appear if `M` is even and `sym` is True).

        Notes
        -----
        The Kaiser window is defined as

        .. math::  w(n) = I_0\left( \beta \sqrt{1-\frac{4n^2}{(M-1)^2}}
                   \right)/I_0(\beta)

        with

        .. math:: \quad -\frac{M-1}{2} \leq n \leq \frac{M-1}{2},

        where :math:`I_0` is the modified zeroth-order Bessel function.

        The Kaiser was named for Jim Kaiser, who discovered a simple approximation
        to the DPSS window based on Bessel functions.
        The Kaiser window is a very good approximation to the Digital Prolate
        Spheroidal Sequence, or Slepian window, which is the transform which
        maximizes the energy in the main lobe of the window relative to total
        energy.

        The Kaiser can approximate other windows by varying the beta parameter.
        (Some literature uses alpha = beta/pi.) [4]_

        ====  =======================
        beta  Window shape
        ====  =======================
        0     Rectangular
        5     Similar to a Hamming
        6     Similar to a Hann
        8.6   Similar to a Blackman
        ====  =======================

        A beta value of 14 is probably a good starting point. Note that as beta
        gets large, the window narrows, and so the number of samples needs to be
        large enough to sample the increasingly narrow spike, otherwise NaNs will
        be returned.

        Most references to the Kaiser window come from the signal processing
        literature, where it is used as one of many windowing functions for
        smoothing values.  It is also known as an apodization (which means
        "removing the foot", i.e. smoothing discontinuities at the beginning
        and end of the sampled signal) or tapering function.

        References
        ----------
        .. [1] J. F. Kaiser, "Digital Filters" - Ch 7 in "Systems analysis by
               digital computer", Editors: F.F. Kuo and J.F. Kaiser, p 218-285.
               John Wiley and Sons, New York, (1966).
        .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The
               University of Alberta Press, 1975, pp. 177-178.
        .. [3] Wikipedia, "Window function",
               https://en.wikipedia.org/wiki/Window_function
        .. [4] F. J. Harris, "On the use of windows for harmonic analysis with the
               discrete Fourier transform," Proceedings of the IEEE, vol. 66,
               no. 1, pp. 51-83, Jan. 1978. :doi:`10.1109/PROC.1978.10837`.

        Examples
        --------
        Plot the window and its frequency response:

        >>> from scipy import signal
        >>> from scipy.fftpack import fft, fftshift
        >>> import matplotlib.pyplot as plt

        >>> window = signal.kaiser(51, beta=14)
        >>> plt.plot(window)
        >>> plt.title(r"Kaiser window ($\beta$=14)")
        >>> plt.ylabel("Amplitude")
        >>> plt.xlabel("Sample")

        >>> plt.figure()
        >>> A = fft(window, 2048) / (len(window)/2.0)
        >>> freq = np.linspace(-0.5, 0.5, len(A))
        >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
        >>> plt.plot(freq, response)
        >>> plt.axis([-0.5, 0.5, -120, 0])
        >>> plt.title(r"Frequency response of the Kaiser window ($\beta$=14)")
        >>> plt.ylabel("Normalized magnitude [dB]")
        >>> plt.xlabel("Normalized frequency [cycles per sample]")

        """
*/
int kaiser( float beta, int M, float *window )
{
  // # Docstring adapted from NumPy's kaiser function
  // if _len_guards(M):
  //     return np.ones(M)
  // M, needs_trunc = _extend(M, sym)

  int result = true;

  // n = np.arange(0, M)
  float n[M];
  for ( int i = 0; i < M; i++ )
    n[i] = i;
  // alpha = (M - 1) / 2.0
  float alpha = (M - 1) / 2.0;
  // w = (special.i0(beta * np.sqrt(1 - ((n - alpha) / alpha) ** 2.0)) /
  //      special.i0(beta))
  for ( int i = 0; i < M; i++ ) {
    float p = pow( (n[i] - alpha) / alpha, 2 );
    window[i] = i0( beta * sqrt(1 - p) ) / i0( beta );
  }

  return result;
}