text: """Multitapering method.
  
  This module provides a multi-tapering method for spectral estimation.
  The computation of the tapering windows being computationally expensive, a C
  module is used to compute the tapering window (see :func:`spectrum.mtm.dpss`).
  
  The estimation itself can be performed with the :func:`spectrum.mtm.pmtm` function
  
  
  """
  import sys
  import platform
  import os
  
  import numpy as np
  import ctypes
  from ctypes import *
  import ctypes as C
  from ctypes import POINTER
  import os
  from os.path import join as pj
  from spectrum.tools import nextpow2
  from numpy.ctypeslib import load_library
  from spectrum.psd import Spectrum
  
  """
  
  from ctypes import *
  my_sum=CDLL('sum.so')
  a=numpy.array(range(10),dtype=float)
  my_sum.sum(a.ctypes.data_as(c_void_p),int(10))
  
  
  which you want to call from python. First make a shared object library by doing (at the command line)
  
  gcc -c sum.c
  gcc -shared -o sum.so sum.o
  
  Note that on OSX -shared should be replaced by -dynamiclib and sum.so should be called sum.dylib Then you can do
  
  """
  
  # Import shared mtspec library
  if hasattr(sys, 'frozen'):
      p = os.path.abspath(os.path.dirname(sys.executable))
  else:
      p = os.path.abspath(os.path.dirname(__file__))
  
  lib_name = 'mydpss'
  try:
      mtspeclib = load_library(lib_name, p)
  except:
      print("Library %s not found" % lib_name)
  
  
  class MultiTapering(Spectrum):
      """
  
      See :func:`pmtm` for details
  
      """
      def __init__(self, data, NW=None, k=None, NFFT=None, e=None, v=None,
                   method="adapt", scale_by_freq=True, sampling=1):
          super(MultiTapering, self).__init__(data, sampling=sampling, NFFT=NFFT,
              scale_by_freq=scale_by_freq)
  
          self.NW = NW
          self.k = k
          self.e = e
          self.v = v
          self.method = method
  
      def __call__(self):
          Sk_complex, weights, eigenvalues = pmtm(self.data, self.NW, self.k,
              NFFT=self.NFFT, e=self.e, v=self.v, method=self.method, show=False)
          Sk = abs(Sk_complex)**2
  
          if self.method == "adapt":
              Sk = Sk.transpose()
              Sk = np.mean(Sk * weights, axis=1)
          else:
              Sk = np.mean(Sk * weights, axis=0)
          self.Sk = Sk
          self.weights = weights
          self.eigenvalues = eigenvalues
  
          if self.datatype == "real":
              newpsd = self.Sk[0:int(self.NFFT//2)] * 2
              newpsd = np.append(newpsd, self.Sk[int(self.NFFT//2)]*2)
              self.psd = newpsd
          else:
              self.psd = self.Sk
  
      def __str_title(self):
          return "Multitapering estimate\n"
  
      def __str__(self):
          return super(MultiTapering, self).__str__()
  
  
  def pmtm(x, NW=None, k=None, NFFT=None, e=None, v=None, method='adapt', show=False):
      """Multitapering spectral estimation
  
      :param array x: the data
      :param float NW: The time half bandwidth parameter (typical values are
          2.5,3,3.5,4). Must be provided otherwise the tapering windows and
          eigen values (outputs of dpss) must be provided
      :param int k: uses the first k Slepian sequences. If *k* is not provided,
          *k* is set to *NW*2*.
      :param NW:
      :param e: the matrix containing the tapering windows
      :param v: the window concentrations (eigenvalues)
      :param str method: set how the eigenvalues are used. Must be
          in ['unity', 'adapt', 'eigen']
      :param bool show: plot results
      :return: Sk (complex), weights, eigenvalues
  
      Usually in spectral estimation the mean to reduce bias is to use tapering window.
      In order to reduce variance we need to average different spectrum. The problem
      is that we have only one set of data. Thus we need to decompose a set into several
      segments. Such method are well-known: simple daniell's periodogram, Welch's method
      and so on. The drawback of such methods is a loss of resolution since the segments
      used to compute the spectrum are smaller than the data set.
      The interest of multitapering method is to keep a good resolution while reducing
      bias and variance.
  
      How does it work? First we compute different simple periodogram with the whole data
      set (to keep good resolution) but each periodgram is computed with a different
      tapering windows. Then, we average all these spectrum. To avoid redundancy and bias
      due to the tapers mtm use special tapers.
  
      .. plot::
          :width: 80%
          :include-source:
  
          from spectrum import data_cosine, dpss, pmtm
  
          data = data_cosine(N=2048, A=0.1, sampling=1024, freq=200)
          # If you already have the DPSS windows
          [tapers, eigen] = dpss(2048, 2.5, 4)
          res = pmtm(data, e=tapers, v=eigen, show=False)
          # You do not need to compute the DPSS before end
          res = pmtm(data, NW=2.5, show=False)
          res = pmtm(data, NW=2.5, k=4, show=True)
  
  
      .. versionchanged:: 0.6.2
  
          APN modified method to return each Sk as complex values, the eigenvalues
          and the weights
  
      """
      assert method in ['adapt','eigen','unity']
  
      N = len(x)
  
      # if dpss not provided, compute them
      if e is None and v is None:
          if NW is not None:
              [tapers, eigenvalues] = dpss(N, NW, k=k)
          else:
              raise ValueError("NW must be provided (e.g. 2.5, 3, 3.5, 4")
      elif e is not None and v is not None:
          eigenvalues = e[:]
          tapers = v[:]
      else:
          raise ValueError("if e provided, v must be provided as well and viceversa.")
      nwin = len(eigenvalues) # length of the eigen values vector to be used later
  
      # set the NFFT
      if NFFT==None:
          NFFT = max(256, 2**nextpow2(N))
  
      Sk_complex = np.fft.fft(np.multiply(tapers.transpose(), x), NFFT)
      Sk = abs(Sk_complex)**2
  
      # si nfft smaller thqn N, cut otherwise add wero.
      # compute
      if method in ['eigen', 'unity']:
          if method == 'unity':
              weights = np.ones((nwin, 1))
          elif method == 'eigen':
              # The S_k spectrum can be weighted by the eigenvalues, as in Park et al.
              weights = np.array([_x/float(i+1) for i,_x in enumerate(eigenvalues)])
              weights = weights.reshape(nwin,1)
  
      elif method == 'adapt':
          # This version uses the equations from [2] (P&W pp 368-370).
  
          # Wrap the data modulo nfft if N > nfft
          sig2 = np.dot(x, x) / float(N)
          Sk = abs(np.fft.fft(np.multiply(tapers.transpose(), x), NFFT))**2
          Sk = Sk.transpose()
          S = (Sk[:,0] + Sk[:,1]) / 2    # Initial spectrum estimate
          S = S.reshape(NFFT, 1)
          Stemp = np.zeros((NFFT,1))
          S1 = np.zeros((NFFT,1))
          # Set tolerance for acceptance of spectral estimate:
          tol = 0.0005 * sig2 / float(NFFT)
          i = 0
          a = sig2 * (1 - eigenvalues)
  
          # converges very quickly but for safety; set i<100
          while sum(np.abs(S-S1))/NFFT > tol and i<100:
              i = i + 1
              # calculate weights
              b1 = np.multiply(S, np.ones((1,nwin)))
              b2 = np.multiply(S,eigenvalues.transpose()) + np.ones((NFFT,1))*a.transpose()
              b = b1/b2
  
              # calculate new spectral estimate
              wk=(b**2)*(np.ones((NFFT,1))*eigenvalues.transpose())
              S1 = sum(wk.transpose()*Sk.transpose())/ sum(wk.transpose())
              S1 = S1.reshape(NFFT, 1)
              Stemp = S1
              S1 = S
              S = Stemp   # swap S and S1
          weights=wk
  
      if show is True:
          from pylab import semilogy
          if method == "adapt":
              Sk = np.mean(Sk * weights, axis=1)
          else:
              Sk = np.mean(Sk * weights, axis=0)
          semilogy(Sk)
  
      return Sk_complex, weights, eigenvalues
  
  
  def dpss(N, NW=None, k=None):
      r"""Discrete prolate spheroidal (Slepian) sequences
  
      Calculation of the Discrete Prolate Spheroidal Sequences also known as the
      slepian sequences, and the corresponding eigenvalues.
  
      :param int N: desired window length
      :param float NW: The time half bandwidth parameter (typical values are
          2.5,3,3.5,4).
      :param int k: returns the first k Slepian sequences. If *k* is not
          provided, *k* is set to *NW*2*.
      :return:
          * tapers, a matrix of tapering windows. Matrix is a N by *k* (k
            is the number of windows)
          * eigen, a vector of eigenvalues of length *k*
  
      The discrete prolate spheroidal or Slepian sequences derive from the following
      time-frequency concentration problem. For all finite-energy sequences index
      limited to some set , which sequence maximizes the following ratio:
  
      .. math::
  
          \lambda = \frac{\int_{-W}^{W}\left| X(f) \right|^2 df}
              {\int_{-F_s/2}^{F_s/2}\left| X(f) \right|^2 df}
  
      where :math:`F_s` is the sampling frequency and :math:`|W| < F_s/2`.
      This ratio determines which index-limited sequence has the largest proportion of its
      energy in the band :math:`[-W,W]` with :math:`0  < \lambda < 1`.
      The sequence maximizing the ratio is the first
      discrete prolate spheroidal or Slepian sequence. The second Slepian sequence
      maximizes the ratio and is orthogonal to the first Slepian sequence. The third
      Slepian sequence maximizes the ratio of integrals and is orthogonal to both
      the first and second Slepian sequences and so on.
  
      .. note:: Note about the implementation. Since the slepian generation is
          computationally expensive, we use a C implementation based on the C
          code written by Lees as published in:
  
              Lees, J. M. and J. Park (1995): Multiple-taper spectral analysis: A stand-alone
              C-subroutine: Computers & Geology: 21, 199-236.
  
          However, the original C code has been trimmed. Indeed, we only require the
          multitap function (that depends on jtridib, jtinvit functions only).
  
      .. plot::
          :width: 80%
          :include-source:
  
          from spectrum import *
          from pylab import *
          N = 512
          [w, eigens] = dpss(N, 2.5, 4)
          plot(w)
          title('Slepian Sequences N=%s, NW=2.5' % N)
          axis([0, N, -0.15, 0.15])
          legend(['1st window','2nd window','3rd window','4th window'])
  
      Windows are normalised:
  
      .. math::  \sum_k h_k h_k = 1
  
      :references: [Percival]_
  
          Slepian, D. Prolate spheroidal wave functions, Fourier analysis, and
          uncertainty V: The discrete case. Bell System Technical Journal,
          Volume 57 (1978), 1371430
  
      .. note:: the C code to create the slepian windows is extracted from original C code
          from Lees and Park (1995) and uses the conventions of Percival and Walden (1993).
          Functions that are not used here were removed.
  
      """
      assert NW < N/2 , "NW ({}) must be stricly less than N/2 ({}/2)".format(NW, N)
      if k is None:
          k = min(round(2*NW),N)
          k = int(max(k,1))
      from numpy import dot, zeros, arange, sqrt
      mtspeclib.multitap.restype = None
  
      lam = zeros(k, dtype=float)
      tapers = zeros(k*N, dtype=float)
      tapsum = zeros(k, dtype=float)
  
      res = mtspeclib.multitap(
          c_int(N),
          c_int(k),
          lam.ctypes.data_as(c_void_p),
          c_float(NW),
          tapers.ctypes.data_as(c_void_p),
          tapsum.ctypes.data_as(c_void_p),
          )
  
      # normalisation by sqtr(N). It is required to have normalised windows
      tapers = tapers.reshape(k,N).transpose() / sqrt(N)
  
      for i in range(k):
          # By convention (Percival and Walden, 1993 pg 379)
          # * symmetric tapers (k=0,2,4,...) should have a positive average.
          # * antisymmetric tapers should begin with a positive lobe
          if i%2 == 0:
              if tapsum[i]<0:
                  tapsum[i] *= -1
                  tapers[:,i] *= -1
          else:
              if tapers[0,i] < 0:
                  tapsum[i] *= -1
                  tapers[:,i] *= -1
  
      # Now find the eigenvalues of the original
      # Use the autocovariance sequence technique from Percival and Walden, 1993
      # pg 390 to get the eigenvalues more precisely (same as matlab output)
  
      # The values returned in lam are not exacly the same as in the following methods.
      acvs = _autocov(tapers.transpose(), debias=False) * N
      nidx = arange(N)
      W = float(NW)/N
      r = 4*W*np.sinc(2*W*nidx)
      r[0] = 2*W
      eigvals = dot(acvs, r)
  
      #return (tapers, lam)
      return [tapers, eigvals]
  
  
  def _other_dpss_method(N, NW, Kmax):
      """Returns the Discrete Prolate Spheroidal Sequences of orders [0,Kmax-1]
      for a given frequency-spacing multiple NW and sequence length N.
  
      See dpss function that is the official version. This version is indepedant
      of the C code and relies on Scipy function. However, it is slower by a factor 3
  
      Tridiagonal form of DPSS calculation from:
  
      """
      # here we want to set up an optimization problem to find a sequence
      # whose energy is maximally concentrated within band [-W,W].
      # Thus, the measure lambda(T,W) is the ratio between the energy within
      # that band, and the total energy. This leads to the eigen-system
      # (A - (l1)I)v = 0, where the eigenvector corresponding to the largest
      # eigenvalue is the sequence with maximally concentrated energy. The
      # collection of eigenvectors of this system are called Slepian sequences,
      # or discrete prolate spheroidal sequences (DPSS). Only the first K,
      # K = 2NW/dt orders of DPSS will exhibit good spectral concentration
      # [see http://en.wikipedia.org/wiki/Spectral_concentration_problem]
  
      # Here I set up an alternative symmetric tri-diagonal eigenvalue problem
      # such that
      # (B - (l2)I)v = 0, and v are our DPSS (but eigenvalues l2 != l1)
      # the main diagonal = ([N-1-2*t]/2)**2 cos(2PIW), t=[0,1,2,...,N-1]
      # and the first off-diangonal = t(N-t)/2, t=[1,2,...,N-1]
      # [see Percival and Walden, 1993]
      from scipy import linalg as la
      Kmax = int(Kmax)
      W = float(NW)/N
      ab = np.zeros((2,N), 'd')
      nidx = np.arange(N)
      ab[0,1:] = nidx[1:]*(N-nidx[1:])/2.
      ab[1] = ((N-1-2*nidx)/2.)**2 * np.cos(2*np.pi*W)
      # only calculate the highest Kmax-1 eigenvectors
      l,v = la.eig_banded(ab, select='i', select_range=(N-Kmax, N-1))
      dpss = v.transpose()[::-1]
  
      # By convention (Percival and Walden, 1993 pg 379)
      # * symmetric tapers (k=0,2,4,...) should have a positive average.
      # * antisymmetric tapers should begin with a positive lobe
      fix_symmetric = (dpss[0::2].sum(axis=1) < 0)
      for i, f in enumerate(fix_symmetric):
          if f:
              dpss[2*i] *= -1
      fix_skew = (dpss[1::2,1] < 0)
      for i, f in enumerate(fix_skew):
          if f:
              dpss[2*i+1] *= -1
  
      # Now find the eigenvalues of the original
      # Use the autocovariance sequence technique from Percival and Walden, 1993
      # pg 390
      # XXX : why debias false? it's all messed up o.w., even with means
      # on the order of 1e-2
      acvs = _autocov(dpss, debias=False) * N
      r = 4*W*np.sinc(2*W*nidx)
      r[0] = 2*W
      eigvals = np.dot(acvs, r)
      return dpss, eigvals
  
  
  def _autocov(s, **kwargs):
      """Returns the autocovariance of signal s at all lags.
  
      Adheres to the definition
      sxx[k] = E{S[n]S[n+k]} = cov{S[n],S[n+k]}
      where E{} is the expectation operator, and S is a zero mean process
      """
      # only remove the mean once, if needed
      debias = kwargs.pop('debias', True)
      axis = kwargs.get('axis', -1)
      if debias:
          s = _remove_bias(s, axis)
      kwargs['debias'] = False
      return _crosscov(s, s, **kwargs)
  
  
  def _crosscov(x, y, axis=-1, all_lags=False, debias=True):
      """Returns the crosscovariance sequence between two ndarrays.
      This is performed by calling fftconvolve on x, y[::-1]
  
      Parameters
  
  
      x: ndarray
      y: ndarray
      axis: time axis
  
      all_lags: {True/False}
      whether to return all nonzero lags, or to clip the length of s_xy
      to be the length of x and y. If False, then the zero lag covariance
      is at index 0. Otherwise, it is found at (len(x) + len(y) - 1)/2
  
      debias: {True/False}
      Always removes an estimate of the mean along the axis, unless
      told not to.
  
  
      cross covariance is defined as
      sxy[k] := E{X[t]*Y[t+k]}, where X,Y are zero mean random processes
      """
      if x.shape[axis] != y.shape[axis]:
          raise ValueError(
              'crosscov() only works on same-length sequences for now'
              )
      if debias:
          x = _remove_bias(x, axis)
          y = _remove_bias(y, axis)
      slicing = [slice(d) for d in x.shape]
      slicing[axis] = slice(None,None,-1)
      sxy = _fftconvolve(x, y[tuple(slicing)], axis=axis, mode='full')
      N = x.shape[axis]
      sxy /= N
      if all_lags:
          return sxy
      slicing[axis] = slice(N-1,2*N-1)
      return sxy[tuple(slicing)]
  
  
  def _crosscorr(x, y, **kwargs):
      """
      Returns the crosscorrelation sequence between two ndarrays.
      This is performed by calling fftconvolve on x, y[::-1]
  
      Parameters
  
  
      x: ndarray
      y: ndarray
      axis: time axis
      all_lags: {True/False}
      whether to return all nonzero lags, or to clip the length of r_xy
      to be the length of x and y. If False, then the zero lag correlation
      is at index 0. Otherwise, it is found at (len(x) + len(y) - 1)/2
  
      Notes
  
  
      cross correlation is defined as
      rxy[k] := E{X[t]*Y[t+k]}/(E{X*X}E{Y*Y})**.5,
      where X,Y are zero mean random processes. It is the noramlized cross
      covariance.
      """
      sxy = _crosscov(x, y, **kwargs)
      # estimate sigma_x, sigma_y to normalize
      sx = np.std(x)
      sy = np.std(y)
      return sxy/(sx*sy)
  
  
  def _remove_bias(x, axis):
      "Subtracts an estimate of the mean from signal x at axis"
      padded_slice = [slice(d) for d in x.shape]
      padded_slice[axis] = np.newaxis
      mn = np.mean(x, axis=axis)
      return x - mn[tuple(padded_slice)]
  
  
  def _fftconvolve(in1, in2, mode="full", axis=None):
      """ Convolve two N-dimensional arrays using FFT. See convolve.
  
      This is a fix of scipy.signal.fftconvolve, adding an axis argument and
      importing locally the stuff only needed for this function
  
      """
      #Locally import stuff only required for this:
      from scipy.fftpack import fftn, fft, ifftn, ifft
      from scipy.signal.signaltools import _centered
      from numpy import array, product
  
  
      s1 = array(in1.shape)
      s2 = array(in2.shape)
      complex_result = (np.issubdtype(in1.dtype, np.complex) or
                        np.issubdtype(in2.dtype, np.complex))
  
      if axis is None:
          size = s1+s2-1
          fslice = tuple([slice(0, int(sz)) for sz in size])
      else:
          equal_shapes = s1==s2
          # allow equal_shapes[axis] to be False
          equal_shapes[axis] = True
          assert equal_shapes.all(), 'Shape mismatch on non-convolving axes'
          size = s1[axis]+s2[axis]-1
          fslice = [slice(l) for l in s1]
          fslice[axis] = slice(0, int(size))
          fslice = tuple(fslice)
  
      # Always use 2**n-sized FFT
      fsize = (2**np.ceil(np.log2(size))).astype(np.int64)
      if axis is None:
          IN1 = fftn(in1,fsize)
          IN1 *= fftn(in2,fsize)
          ret = ifftn(IN1)[fslice].copy()
      else:
          IN1 = fft(in1,fsize,axis=axis)
          IN1 *= fft(in2,fsize,axis=axis)
          ret = ifft(IN1,axis=axis)[fslice].copy()
      if not complex_result:
          del IN1
          ret = ret.real
      if mode == "full":
          return ret
      elif mode == "same":
          if product(s1,axis=0) > product(s2,axis=0):
              osize = s1
          else:
              osize = s2
          return _centered(ret,osize)
      elif mode == "valid":
          return _centered(ret,abs(s2-s1)+1)
  