content: """Filter design.
  """
  from __future__ import division, print_function, absolute_import
  
  import warnings
  import math
  
  import numpy
  import numpy as np
  from numpy import (atleast_1d, poly, polyval, roots, real, asarray,
                     resize, pi, absolute, logspace, r_, sqrt, tan, log10,
                     arctan, arcsinh, sin, exp, cosh, arccosh, ceil, conjugate,
                     zeros, sinh, append, concatenate, prod, ones, array,
                     mintypecode)
  from numpy.polynomial.polynomial import polyval as npp_polyval
  
  from scipy import special, optimize
  from scipy.special import comb, factorial
  from scipy._lib._numpy_compat import polyvalfromroots
  
  
  __all__ = ['findfreqs', 'freqs', 'freqz', 'tf2zpk', 'zpk2tf', 'normalize',
             'lp2lp', 'lp2hp', 'lp2bp', 'lp2bs', 'bilinear', 'iirdesign',
             'iirfilter', 'butter', 'cheby1', 'cheby2', 'ellip', 'bessel',
             'band_stop_obj', 'buttord', 'cheb1ord', 'cheb2ord', 'ellipord',
             'buttap', 'cheb1ap', 'cheb2ap', 'ellipap', 'besselap',
             'BadCoefficients', 'freqs_zpk', 'freqz_zpk',
             'tf2sos', 'sos2tf', 'zpk2sos', 'sos2zpk', 'group_delay',
             'sosfreqz', 'iirnotch', 'iirpeak']
  
  
  class BadCoefficients(UserWarning):
      """Warning about badly conditioned filter coefficients"""
      pass
  
  abs = absolute
  
  
  def findfreqs(num, den, N, kind='ba'):
      """
      Find array of frequencies for computing the response of an analog filter.
  
      Parameters
      ----------
      num, den : array_like, 1-D
          The polynomial coefficients of the numerator and denominator of the
          transfer function of the filter or LTI system, where the coefficients
          are ordered from highest to lowest degree. Or, the roots  of the
          transfer function numerator and denominator (i.e. zeroes and poles).
      N : int
          The length of the array to be computed.
      kind : str {'ba', 'zp'}, optional
          Specifies whether the numerator and denominator are specified by their
          polynomial coefficients ('ba'), or their roots ('zp').
  
      Returns
      -------
      w : (N,) ndarray
          A 1-D array of frequencies, logarithmically spaced.
  
      Examples
      --------
      Find a set of nine frequencies that span the "interesting part" of the
      frequency response for the filter with the transfer function
  
          H(s) = s / (s^2 + 8s + 25)
  
      >>> from scipy import signal
      >>> signal.findfreqs([1, 0], [1, 8, 25], N=9)
      array([  1.00000000e-02,   3.16227766e-02,   1.00000000e-01,
               3.16227766e-01,   1.00000000e+00,   3.16227766e+00,
               1.00000000e+01,   3.16227766e+01,   1.00000000e+02])
      """
      if kind == 'ba':
          ep = atleast_1d(roots(den)) + 0j
          tz = atleast_1d(roots(num)) + 0j
      elif kind == 'zp':
          ep = atleast_1d(den) + 0j
          tz = atleast_1d(num) + 0j
      else:
          raise ValueError("input must be one of {'ba', 'zp'}")
  
      if len(ep) == 0:
          ep = atleast_1d(-1000) + 0j
  
      ez = r_['-1',
              numpy.compress(ep.imag >= 0, ep, axis=-1),
              numpy.compress((abs(tz) < 1e5) & (tz.imag >= 0), tz, axis=-1)]
  
      integ = abs(ez) < 1e-10
      hfreq = numpy.around(numpy.log10(numpy.max(3 * abs(ez.real + integ) +
                                                 1.5 * ez.imag)) + 0.5)
      lfreq = numpy.around(numpy.log10(0.1 * numpy.min(abs(real(ez + integ)) +
                                                       2 * ez.imag)) - 0.5)
  
      w = logspace(lfreq, hfreq, N)
      return w
  
  
  def freqs(b, a, worN=None, plot=None):
      """
      Compute frequency response of analog filter.
  
      Given the M-order numerator `b` and N-order denominator `a` of an analog
      filter, compute its frequency response::
  
               b[0]*(jw)**M + b[1]*(jw)**(M-1) + ... + b[M]
       H(w) = ----------------------------------------------
               a[0]*(jw)**N + a[1]*(jw)**(N-1) + ... + a[N]
  
      Parameters
      ----------
      b : array_like
          Numerator of a linear filter.
      a : array_like
          Denominator of a linear filter.
      worN : {None, int, array_like}, optional
          If None, then compute at 200 frequencies around the interesting parts
          of the response curve (determined by pole-zero locations).  If a single
          integer, then compute at that many frequencies.  Otherwise, compute the
          response at the angular frequencies (e.g. rad/s) given in `worN`.
      plot : callable, optional
          A callable that takes two arguments. If given, the return parameters
          `w` and `h` are passed to plot. Useful for plotting the frequency
          response inside `freqs`.
  
      Returns
      -------
      w : ndarray
          The angular frequencies at which `h` was computed.
      h : ndarray
          The frequency response.
  
      See Also
      --------
      freqz : Compute the frequency response of a digital filter.
  
      Notes
      -----
      Using Matplotlib's "plot" function as the callable for `plot` produces
      unexpected results,  this plots the real part of the complex transfer
      function, not the magnitude.  Try ``lambda w, h: plot(w, abs(h))``.
  
      Examples
      --------
      >>> from scipy.signal import freqs, iirfilter
  
      >>> b, a = iirfilter(4, [1, 10], 1, 60, analog=True, ftype='cheby1')
  
      >>> w, h = freqs(b, a, worN=np.logspace(-1, 2, 1000))
  
      >>> import matplotlib.pyplot as plt
      >>> plt.semilogx(w, 20 * np.log10(abs(h)))
      >>> plt.xlabel('Frequency')
      >>> plt.ylabel('Amplitude response [dB]')
      >>> plt.grid()
      >>> plt.show()
  
      """
      if worN is None:
          w = findfreqs(b, a, 200)
      elif isinstance(worN, int):
          N = worN
          w = findfreqs(b, a, N)
      else:
          w = worN
      w = atleast_1d(w)
      s = 1j * w
      h = polyval(b, s) / polyval(a, s)
      if plot is not None:
          plot(w, h)
  
      return w, h
  
  
  def freqs_zpk(z, p, k, worN=None):
      """
      Compute frequency response of analog filter.
  
      Given the zeros `z`, poles `p`, and gain `k` of a filter, compute its
      frequency response::
  
                  (jw-z[0]) * (jw-z[1]) * ... * (jw-z[-1])
       H(w) = k * ----------------------------------------
                  (jw-p[0]) * (jw-p[1]) * ... * (jw-p[-1])
  
      Parameters
      ----------
      z : array_like
          Zeroes of a linear filter
      p : array_like
          Poles of a linear filter
      k : scalar
          Gain of a linear filter
      worN : {None, int, array_like}, optional
          If None, then compute at 200 frequencies around the interesting parts
          of the response curve (determined by pole-zero locations).  If a single
          integer, then compute at that many frequencies.  Otherwise, compute the
          response at the angular frequencies (e.g. rad/s) given in `worN`.
  
      Returns
      -------
      w : ndarray
          The angular frequencies at which `h` was computed.
      h : ndarray
          The frequency response.
  
      See Also
      --------
      freqs : Compute the frequency response of an analog filter in TF form
      freqz : Compute the frequency response of a digital filter in TF form
      freqz_zpk : Compute the frequency response of a digital filter in ZPK form
  
      Notes
      -----
      .. versionadded: 0.19.0
  
      Examples
      --------
      >>> from scipy.signal import freqs_zpk, iirfilter
  
      >>> z, p, k = iirfilter(4, [1, 10], 1, 60, analog=True, ftype='cheby1',
      ...                     output='zpk')
  
      >>> w, h = freqs_zpk(z, p, k, worN=np.logspace(-1, 2, 1000))
  
      >>> import matplotlib.pyplot as plt
      >>> plt.semilogx(w, 20 * np.log10(abs(h)))
      >>> plt.xlabel('Frequency')
      >>> plt.ylabel('Amplitude response [dB]')
      >>> plt.grid()
      >>> plt.show()
  
      """
      k = np.asarray(k)
      if k.size > 1:
          raise ValueError('k must be a single scalar gain')
  
      if worN is None:
          w = findfreqs(z, p, 200, kind='zp')
      elif isinstance(worN, int):
          N = worN
          w = findfreqs(z, p, N, kind='zp')
      else:
          w = worN
  
      w = atleast_1d(w)
      s = 1j * w
      num = polyvalfromroots(s, z)
      den = polyvalfromroots(s, p)
      h = k * num/den
      return w, h
  
  
  def freqz(b, a=1, worN=None, whole=False, plot=None):
      """
      Compute the frequency response of a digital filter.
  
      Given the M-order numerator `b` and N-order denominator `a` of a digital
      filter, compute its frequency response::
  
                   jw               -jw               -jwM
          jw    B(e  )  b[0] + b[1]e    + .... + b[M]e
       H(e  ) = ---- = -----------------------------------
                   jw               -jw               -jwN
                A(e  )  a[0] + a[1]e    + .... + a[N]e
  
      Parameters
      ----------
      b : array_like
          numerator of a linear filter
      a : array_like
          denominator of a linear filter
      worN : {None, int, array_like}, optional
          If None (default), then compute at 512 frequencies equally spaced
          around the unit circle.
          If a single integer, then compute at that many frequencies.
          If an array_like, compute the response at the frequencies given (in
          radians/sample).
      whole : bool, optional
          Normally, frequencies are computed from 0 to the Nyquist frequency,
          pi radians/sample (upper-half of unit-circle).  If `whole` is True,
          compute frequencies from 0 to 2*pi radians/sample.
      plot : callable
          A callable that takes two arguments. If given, the return parameters
          `w` and `h` are passed to plot. Useful for plotting the frequency
          response inside `freqz`.
  
      Returns
      -------
      w : ndarray
          The normalized frequencies at which `h` was computed, in
          radians/sample.
      h : ndarray
          The frequency response, as complex numbers.
  
      See Also
      --------
      sosfreqz
  
      Notes
      -----
      Using Matplotlib's "plot" function as the callable for `plot` produces
      unexpected results,  this plots the real part of the complex transfer
      function, not the magnitude.  Try ``lambda w, h: plot(w, abs(h))``.
  
      Examples
      --------
      >>> from scipy import signal
      >>> b = signal.firwin(80, 0.5, window=('kaiser', 8))
      >>> w, h = signal.freqz(b)
  
      >>> import matplotlib.pyplot as plt
      >>> fig = plt.figure()
      >>> plt.title('Digital filter frequency response')
      >>> ax1 = fig.add_subplot(111)
  
      >>> plt.plot(w, 20 * np.log10(abs(h)), 'b')
      >>> plt.ylabel('Amplitude [dB]', color='b')
      >>> plt.xlabel('Frequency [rad/sample]')
  
      >>> ax2 = ax1.twinx()
      >>> angles = np.unwrap(np.angle(h))
      >>> plt.plot(w, angles, 'g')
      >>> plt.ylabel('Angle (radians)', color='g')
      >>> plt.grid()
      >>> plt.axis('tight')
      >>> plt.show()
  
      """
      b, a = map(atleast_1d, (b, a))
      if whole:
          lastpoint = 2 * pi
      else:
          lastpoint = pi
      if worN is None:
          N = 512
          w = numpy.linspace(0, lastpoint, N, endpoint=False)
      elif isinstance(worN, int):
          N = worN
          w = numpy.linspace(0, lastpoint, N, endpoint=False)
      else:
          w = worN
      w = atleast_1d(w)
      zm1 = exp(-1j * w)
      h = polyval(b[::-1], zm1) / polyval(a[::-1], zm1)
      if plot is not None:
          plot(w, h)
  
      return w, h
  
  
  def freqz_zpk(z, p, k, worN=None, whole=False):
      """
      Compute the frequency response of a digital filter in ZPK form.
  
      Given the Zeros, Poles and Gain of a digital filter, compute its frequency
      response::
  
      :math:`H(z)=k \prod_i (z - Z[i]) / \prod_j (z - P[j])`
  
      where :math:`k` is the `gain`, :math:`Z` are the `zeros` and :math:`P` are
      the `poles`.
  
      Parameters
      ----------
      z : array_like
          Zeroes of a linear filter
      p : array_like
          Poles of a linear filter
      k : scalar
          Gain of a linear filter
      worN : {None, int, array_like}, optional
          If None (default), then compute at 512 frequencies equally spaced
          around the unit circle.
          If a single integer, then compute at that many frequencies.
          If an array_like, compute the response at the frequencies given (in
          radians/sample).
      whole : bool, optional
          Normally, frequencies are computed from 0 to the Nyquist frequency,
          pi radians/sample (upper-half of unit-circle).  If `whole` is True,
          compute frequencies from 0 to 2*pi radians/sample.
  
      Returns
      -------
      w : ndarray
          The normalized frequencies at which `h` was computed, in
          radians/sample.
      h : ndarray
          The frequency response.
  
      See Also
      --------
      freqs : Compute the frequency response of an analog filter in TF form
      freqs_zpk : Compute the frequency response of an analog filter in ZPK form
      freqz : Compute the frequency response of a digital filter in TF form
  
      Notes
      -----
      .. versionadded: 0.19.0
  
      Examples
      --------
      >>> from scipy import signal
      >>> z, p, k = signal.butter(4, 0.2, output='zpk')
      >>> w, h = signal.freqz_zpk(z, p, k)
  
      >>> import matplotlib.pyplot as plt
      >>> fig = plt.figure()
      >>> plt.title('Digital filter frequency response')
      >>> ax1 = fig.add_subplot(111)
  
      >>> plt.plot(w, 20 * np.log10(abs(h)), 'b')
      >>> plt.ylabel('Amplitude [dB]', color='b')
      >>> plt.xlabel('Frequency [rad/sample]')
  
      >>> ax2 = ax1.twinx()
      >>> angles = np.unwrap(np.angle(h))
      >>> plt.plot(w, angles, 'g')
      >>> plt.ylabel('Angle (radians)', color='g')
      >>> plt.grid()
      >>> plt.axis('tight')
      >>> plt.show()
  
      """
      z, p = map(atleast_1d, (z, p))
      if whole:
          lastpoint = 2 * pi
      else:
          lastpoint = pi
      if worN is None:
          N = 512
          w = numpy.linspace(0, lastpoint, N, endpoint=False)
      elif isinstance(worN, int):
          N = worN
          w = numpy.linspace(0, lastpoint, N, endpoint=False)
      else:
          w = worN
      w = atleast_1d(w)
      zm1 = exp(1j * w)
      h = k * polyvalfromroots(zm1, z) / polyvalfromroots(zm1, p)
  
      return w, h
  
  
  def group_delay(system, w=None, whole=False):
      r"""Compute the group delay of a digital filter.
  
      The group delay measures by how many samples amplitude envelopes of
      various spectral components of a signal are delayed by a filter.
      It is formally defined as the derivative of continuous (unwrapped) phase::
  
                 d        jw
       D(w) = - -- arg H(e)
                dw
  
      Parameters
      ----------
      system : tuple of array_like (b, a)
          Numerator and denominator coefficients of a filter transfer function.
      w : {None, int, array-like}, optional
          If None (default), then compute at 512 frequencies equally spaced
          around the unit circle.
          If a single integer, then compute at that many frequencies.
          If array, compute the delay at the frequencies given
          (in radians/sample).
      whole : bool, optional
          Normally, frequencies are computed from 0 to the Nyquist frequency,
          pi radians/sample (upper-half of unit-circle).  If `whole` is True,
          compute frequencies from 0 to ``2*pi`` radians/sample.
  
      Returns
      -------
      w : ndarray
          The normalized frequencies at which the group delay was computed,
          in radians/sample.
      gd : ndarray
          The group delay.
  
      Notes
      -----
      The similar function in MATLAB is called `grpdelay`.
  
      If the transfer function :math:`H(z)` has zeros or poles on the unit
      circle, the group delay at corresponding frequencies is undefined.
      When such a case arises the warning is raised and the group delay
      is set to 0 at those frequencies.
  
      For the details of numerical computation of the group delay refer to [1]_.
  
      .. versionadded: 0.16.0
  
      See Also
      --------
      freqz : Frequency response of a digital filter
  
      References
      ----------
      .. [1] Richard G. Lyons, "Understanding Digital Signal Processing,
             3rd edition", p. 830.
  
      Examples
      --------
      >>> from scipy import signal
      >>> b, a = signal.iirdesign(0.1, 0.3, 5, 50, ftype='cheby1')
      >>> w, gd = signal.group_delay((b, a))
  
      >>> import matplotlib.pyplot as plt
      >>> plt.title('Digital filter group delay')
      >>> plt.plot(w, gd)
      >>> plt.ylabel('Group delay [samples]')
      >>> plt.xlabel('Frequency [rad/sample]')
      >>> plt.show()
  
      """
      if w is None:
          w = 512
  
      if isinstance(w, int):
          if whole:
              w = np.linspace(0, 2 * pi, w, endpoint=False)
          else:
              w = np.linspace(0, pi, w, endpoint=False)
  
      w = np.atleast_1d(w)
      b, a = map(np.atleast_1d, system)
      c = np.convolve(b, a[::-1])
      cr = c * np.arange(c.size)
      z = np.exp(-1j * w)
      num = np.polyval(cr[::-1], z)
      den = np.polyval(c[::-1], z)
      singular = np.absolute(den) < 10 * EPSILON
      if np.any(singular):
          warnings.warn(
              "The group delay is singular at frequencies [{0}], setting to 0".
              format(", ".join("{0:.3f}".format(ws) for ws in w[singular]))
          )
  
      gd = np.zeros_like(w)
      gd[~singular] = np.real(num[~singular] / den[~singular]) - a.size + 1
      return w, gd
  
  
  def _validate_sos(sos):
      """Helper to validate a SOS input"""
      sos = np.atleast_2d(sos)
      if sos.ndim != 2:
          raise ValueError('sos array must be 2D')
      n_sections, m = sos.shape
      if m != 6:
          raise ValueError('sos array must be shape (n_sections, 6)')
      if not (sos[:, 3] == 1).all():
          raise ValueError('sos[:, 3] should be all ones')
      return sos, n_sections
  
  
  def sosfreqz(sos, worN=None, whole=False):
      """
      Compute the frequency response of a digital filter in SOS format.
  
      Given `sos`, an array with shape (n, 6) of second order sections of
      a digital filter, compute the frequency response of the system function::
  
                 B0(z)   B1(z)         B{n-1}(z)
          H(z) = ----- * ----- * ... * ---------
                 A0(z)   A1(z)         A{n-1}(z)
  
      for z = exp(omega*1j), where B{k}(z) and A{k}(z) are numerator and
      denominator of the transfer function of the k-th second order section.
  
      Parameters
      ----------
      sos : array_like
          Array of second-order filter coefficients, must have shape
          ``(n_sections, 6)``. Each row corresponds to a second-order
          section, with the first three columns providing the numerator
          coefficients and the last three providing the denominator
          coefficients.
      worN : {None, int, array_like}, optional
          If None (default), then compute at 512 frequencies equally spaced
          around the unit circle.
          If a single integer, then compute at that many frequencies.
          If an array_like, compute the response at the frequencies given (in
          radians/sample).
      whole : bool, optional
          Normally, frequencies are computed from 0 to the Nyquist frequency,
          pi radians/sample (upper-half of unit-circle).  If `whole` is True,
          compute frequencies from 0 to 2*pi radians/sample.
  
      Returns
      -------
      w : ndarray
          The normalized frequencies at which `h` was computed, in
          radians/sample.
      h : ndarray
          The frequency response, as complex numbers.
  
      See Also
      --------
      freqz, sosfilt
  
      Notes
      -----
  
      .. versionadded:: 0.19.0
  
      Examples
      --------
      Design a 15th-order bandpass filter in SOS format.
  
      >>> from scipy import signal
      >>> sos = signal.ellip(15, 0.5, 60, (0.2, 0.4), btype='bandpass',
      ...                    output='sos')
  
      Compute the frequency response at 1500 points from DC to Nyquist.
  
      >>> w, h = signal.sosfreqz(sos, worN=1500)
  
      Plot the response.
  
      >>> import matplotlib.pyplot as plt
      >>> plt.subplot(2, 1, 1)
      >>> db = 20*np.log10(np.abs(h))
      >>> plt.plot(w/np.pi, db)
      >>> plt.ylim(-75, 5)
      >>> plt.grid(True)
      >>> plt.yticks([0, -20, -40, -60])
      >>> plt.ylabel('Gain [dB]')
      >>> plt.title('Frequency Response')
      >>> plt.subplot(2, 1, 2)
      >>> plt.plot(w/np.pi, np.angle(h))
      >>> plt.grid(True)
      >>> plt.yticks([-np.pi, -0.5*np.pi, 0, 0.5*np.pi, np.pi],
      ...            [r'$-\\pi$', r'$-\\pi/2$', '0', r'$\\pi/2$', r'$\\pi$'])
      >>> plt.ylabel('Phase [rad]')
      >>> plt.xlabel('Normalized frequency (1.0 = Nyquist)')
      >>> plt.show()
  
      If the same filter is implemented as a single transfer function,
      numerical error corrupts the frequency response:
  
      >>> b, a = signal.ellip(15, 0.5, 60, (0.2, 0.4), btype='bandpass',
      ...                    output='ba')
      >>> w, h = signal.freqz(b, a, worN=1500)
      >>> plt.subplot(2, 1, 1)
      >>> db = 20*np.log10(np.abs(h))
      >>> plt.plot(w/np.pi, db)
      >>> plt.subplot(2, 1, 2)
      >>> plt.plot(w/np.pi, np.angle(h))
      >>> plt.show()
  
      """
  
      sos, n_sections = _validate_sos(sos)
      if n_sections == 0:
          raise ValueError('Cannot compute frequencies with no sections')
      h = 1.
      for row in sos:
          w, rowh = freqz(row[:3], row[3:], worN=worN, whole=whole)
          h *= rowh
      return w, h
  
  
  def _cplxreal(z, tol=None):
      """
      Split into complex and real parts, combining conjugate pairs.
  
      The 1D input vector `z` is split up into its complex (`zc`) and real (`zr`)
      elements.  Every complex element must be part of a complex-conjugate pair,
      which are combined into a single number (with positive imaginary part) in
      the output.  Two complex numbers are considered a conjugate pair if their
      real and imaginary parts differ in magnitude by less than ``tol * abs(z)``.
  
      Parameters
      ----------
      z : array_like
          Vector of complex numbers to be sorted and split
      tol : float, optional
          Relative tolerance for testing realness and conjugate equality.
          Default is ``100 * spacing(1)`` of `z`'s data type (i.e. 2e-14 for
          float64)
  
      Returns
      -------
      zc : ndarray
          Complex elements of `z`, with each pair represented by a single value
          having positive imaginary part, sorted first by real part, and then
          by magnitude of imaginary part.  The pairs are averaged when combined
          to reduce error.
      zr : ndarray
          Real elements of `z` (those having imaginary part less than
          `tol` times their magnitude), sorted by value.
  
      Raises
      ------
      ValueError
          If there are any complex numbers in `z` for which a conjugate
          cannot be found.
  
      See Also
      --------
      _cplxpair
  
      Examples
      --------
      >>> a = [4, 3, 1, 2-2j, 2+2j, 2-1j, 2+1j, 2-1j, 2+1j, 1+1j, 1-1j]
      >>> zc, zr = _cplxreal(a)
      >>> print zc
      [ 1.+1.j  2.+1.j  2.+1.j  2.+2.j]
      >>> print zr
      [ 1.  3.  4.]
      """
  
      z = atleast_1d(z)
      if z.size == 0:
          return z, z
      elif z.ndim != 1:
          raise ValueError('_cplxreal only accepts 1D input')
  
      if tol is None:
          # Get tolerance from dtype of input
          tol = 100 * np.finfo((1.0 * z).dtype).eps
  
      # Sort by real part, magnitude of imaginary part (speed up further sorting)
      z = z[np.lexsort((abs(z.imag), z.real))]
  
      # Split reals from conjugate pairs
      real_indices = abs(z.imag) <= tol * abs(z)
      zr = z[real_indices].real
  
      if len(zr) == len(z):
          # Input is entirely real
          return array([]), zr
  
      # Split positive and negative halves of conjugates
      z = z[~real_indices]
      zp = z[z.imag > 0]
      zn = z[z.imag < 0]
  
      if len(zp) != len(zn):
          raise ValueError('Array contains complex value with no matching '
                           'conjugate.')
  
      # Find runs of (approximately) the same real part
      same_real = np.diff(zp.real) <= tol * abs(zp[:-1])
      diffs = numpy.diff(concatenate(([0], same_real, [0])))
      run_starts = numpy.where(diffs > 0)[0]
      run_stops = numpy.where(diffs < 0)[0]
  
      # Sort each run by their imaginary parts
      for i in range(len(run_starts)):
          start = run_starts[i]
          stop = run_stops[i] + 1
          for chunk in (zp[start:stop], zn[start:stop]):
              chunk[...] = chunk[np.lexsort([abs(chunk.imag)])]
  
      # Check that negatives match positives
      if any(abs(zp - zn.conj()) > tol * abs(zn)):
          raise ValueError('Array contains complex value with no matching '
                           'conjugate.')
  
      # Average out numerical inaccuracy in real vs imag parts of pairs
      zc = (zp + zn.conj()) / 2
  
      return zc, zr
  
  
  def _cplxpair(z, tol=None):
      """
      Sort into pairs of complex conjugates.
  
      Complex conjugates in `z` are sorted by increasing real part.  In each
      pair, the number with negative imaginary part appears first.
  
      If pairs have identical real parts, they are sorted by increasing
      imaginary magnitude.
  
      Two complex numbers are considered a conjugate pair if their real and
      imaginary parts differ in magnitude by less than ``tol * abs(z)``.  The
      pairs are forced to be exact complex conjugates by averaging the positive
      and negative values.
  
      Purely real numbers are also sorted, but placed after the complex
      conjugate pairs.  A number is considered real if its imaginary part is
      smaller than `tol` times the magnitude of the number.
  
      Parameters
      ----------
      z : array_like
          1-dimensional input array to be sorted.
      tol : float, optional
          Relative tolerance for testing realness and conjugate equality.
          Default is ``100 * spacing(1)`` of `z`'s data type (i.e. 2e-14 for
          float64)
  
      Returns
      -------
      y : ndarray
          Complex conjugate pairs followed by real numbers.
  
      Raises
      ------
      ValueError
          If there are any complex numbers in `z` for which a conjugate
          cannot be found.
  
      See Also
      --------
      _cplxreal
  
      Examples
      --------
      >>> a = [4, 3, 1, 2-2j, 2+2j, 2-1j, 2+1j, 2-1j, 2+1j, 1+1j, 1-1j]
      >>> z = _cplxpair(a)
      >>> print(z)
      [ 1.-1.j  1.+1.j  2.-1.j  2.+1.j  2.-1.j  2.+1.j  2.-2.j  2.+2.j  1.+0.j
        3.+0.j  4.+0.j]
      """
  
      z = atleast_1d(z)
      if z.size == 0 or np.isrealobj(z):
          return np.sort(z)
  
      if z.ndim != 1:
          raise ValueError('z must be 1-dimensional')
  
      zc, zr = _cplxreal(z, tol)
  
      # Interleave complex values and their conjugates, with negative imaginary
      # parts first in each pair
      zc = np.dstack((zc.conj(), zc)).flatten()
      z = np.append(zc, zr)
      return z
  
  
  def tf2zpk(b, a):
      r"""Return zero, pole, gain (z, p, k) representation from a numerator,
      denominator representation of a linear filter.
  
      Parameters
      ----------
      b : array_like
          Numerator polynomial coefficients.
      a : array_like
          Denominator polynomial coefficients.
  
      Returns
      -------
      z : ndarray
          Zeros of the transfer function.
      p : ndarray
          Poles of the transfer function.
      k : float
          System gain.
  
      Notes
      -----
      If some values of `b` are too close to 0, they are removed. In that case,
      a BadCoefficients warning is emitted.
  
      The `b` and `a` arrays are interpreted as coefficients for positive,
      descending powers of the transfer function variable.  So the inputs
      :math:`b = [b_0, b_1, ..., b_M]` and :math:`a =[a_0, a_1, ..., a_N]`
      can represent an analog filter of the form:
  
      .. math::
  
          H(s) = \frac
          {b_0 s^M + b_1 s^{(M-1)} + \cdots + b_M}
          {a_0 s^N + a_1 s^{(N-1)} + \cdots + a_N}
  
      or a discrete-time filter of the form:
  
      .. math::
  
          H(z) = \frac
          {b_0 z^M + b_1 z^{(M-1)} + \cdots + b_M}
          {a_0 z^N + a_1 z^{(N-1)} + \cdots + a_N}
  
      This "positive powers" form is found more commonly in controls
      engineering.  If `M` and `N` are equal (which is true for all filters
      generated by the bilinear transform), then this happens to be equivalent
      to the "negative powers" discrete-time form preferred in DSP:
  
      .. math::
  
          H(z) = \frac
          {b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}}
          {a_0 + a_1 z^{-1} + \cdots + a_N z^{-N}}
  
      Although this is true for common filters, remember that this is not true
      in the general case.  If `M` and `N` are not equal, the discrete-time
      transfer function coefficients must first be converted to the "positive
      powers" form before finding the poles and zeros.
  
      """
      b, a = normalize(b, a)
      b = (b + 0.0) / a[0]
      a = (a + 0.0) / a[0]
      k = b[0]
      b /= b[0]
      z = roots(b)
      p = roots(a)
      return z, p, k
  
  
  def zpk2tf(z, p, k):
      """
      Return polynomial transfer function representation from zeros and poles
  
      Parameters
      ----------
      z : array_like
          Zeros of the transfer function.
      p : array_like
          Poles of the transfer function.
      k : float
          System gain.
  
      Returns
      -------
      b : ndarray
          Numerator polynomial coefficients.
      a : ndarray
          Denominator polynomial coefficients.
  
      """
      z = atleast_1d(z)
      k = atleast_1d(k)
      if len(z.shape) > 1:
          temp = poly(z[0])
          b = zeros((z.shape[0], z.shape[1] + 1), temp.dtype.char)
          if len(k) == 1:
              k = [k[0]] * z.shape[0]
          for i in range(z.shape[0]):
              b[i] = k[i] * poly(z[i])
      else:
          b = k * poly(z)
      a = atleast_1d(poly(p))
  
      # Use real output if possible.  Copied from numpy.poly, since
      # we can't depend on a specific version of numpy.
      if issubclass(b.dtype.type, numpy.complexfloating):
          # if complex roots are all complex conjugates, the roots are real.
          roots = numpy.asarray(z, complex)
          pos_roots = numpy.compress(roots.imag > 0, roots)
          neg_roots = numpy.conjugate(numpy.compress(roots.imag < 0, roots))
          if len(pos_roots) == len(neg_roots):
              if numpy.all(numpy.sort_complex(neg_roots) ==
                           numpy.sort_complex(pos_roots)):
                  b = b.real.copy()
  
      if issubclass(a.dtype.type, numpy.complexfloating):
          # if complex roots are all complex conjugates, the roots are real.
          roots = numpy.asarray(p, complex)
          pos_roots = numpy.compress(roots.imag > 0, roots)
          neg_roots = numpy.conjugate(numpy.compress(roots.imag < 0, roots))
          if len(pos_roots) == len(neg_roots):
              if numpy.all(numpy.sort_complex(neg_roots) ==
                           numpy.sort_complex(pos_roots)):
                  a = a.real.copy()
  
      return b, a
  
  
  def tf2sos(b, a, pairing='nearest'):
      """
      Return second-order sections from transfer function representation
  
      Parameters
      ----------
      b : array_like
          Numerator polynomial coefficients.
      a : array_like
          Denominator polynomial coefficients.
      pairing : {'nearest', 'keep_odd'}, optional
          The method to use to combine pairs of poles and zeros into sections.
          See `zpk2sos`.
  
      Returns
      -------
      sos : ndarray
          Array of second-order filter coefficients, with shape
          ``(n_sections, 6)``. See `sosfilt` for the SOS filter format
          specification.
  
      See Also
      --------
      zpk2sos, sosfilt
  
      Notes
      -----
      It is generally discouraged to convert from TF to SOS format, since doing
      so usually will not improve numerical precision errors. Instead, consider
      designing filters in ZPK format and converting directly to SOS. TF is
      converted to SOS by first converting to ZPK format, then converting
      ZPK to SOS.
  
      .. versionadded:: 0.16.0
      """
      return zpk2sos(*tf2zpk(b, a), pairing=pairing)
  
  
  def sos2tf(sos):
      """
      Return a single transfer function from a series of second-order sections
  
      Parameters
      ----------
      sos : array_like
          Array of second-order filter coefficients, must have shape
          ``(n_sections, 6)``. See `sosfilt` for the SOS filter format
          specification.
  
      Returns
      -------
      b : ndarray
          Numerator polynomial coefficients.
      a : ndarray
          Denominator polynomial coefficients.
  
      Notes
      -----
      .. versionadded:: 0.16.0
      """
      sos = np.asarray(sos)
      b = [1.]
      a = [1.]
      n_sections = sos.shape[0]
      for section in range(n_sections):
          b = np.polymul(b, sos[section, :3])
          a = np.polymul(a, sos[section, 3:])
      return b, a
  
  
  def sos2zpk(sos):
      """
      Return zeros, poles, and gain of a series of second-order sections
  
      Parameters
      ----------
      sos : array_like
          Array of second-order filter coefficients, must have shape
          ``(n_sections, 6)``. See `sosfilt` for the SOS filter format
          specification.
  
      Returns
      -------
      z : ndarray
          Zeros of the transfer function.
      p : ndarray
          Poles of the transfer function.
      k : float
          System gain.
  
      Notes
      -----
      .. versionadded:: 0.16.0
      """
      sos = np.asarray(sos)
      n_sections = sos.shape[0]
      z = np.empty(n_sections*2, np.complex128)
      p = np.empty(n_sections*2, np.complex128)
      k = 1.
      for section in range(n_sections):
          zpk = tf2zpk(sos[section, :3], sos[section, 3:])
          z[2*section:2*(section+1)] = zpk[0]
          p[2*section:2*(section+1)] = zpk[1]
          k *= zpk[2]
      return z, p, k
  
  
  def _nearest_real_complex_idx(fro, to, which):
      """Get the next closest real or complex element based on distance"""
      assert which in ('real', 'complex')
      order = np.argsort(np.abs(fro - to))
      mask = np.isreal(fro[order])
      if which == 'complex':
          mask = ~mask
      return order[np.where(mask)[0][0]]
  
  
  def zpk2sos(z, p, k, pairing='nearest'):
      """
      Return second-order sections from zeros, poles, and gain of a system
  
      Parameters
      ----------
      z : array_like
          Zeros of the transfer function.
      p : array_like
          Poles of the transfer function.
      k : float
          System gain.
      pairing : {'nearest', 'keep_odd'}, optional
          The method to use to combine pairs of poles and zeros into sections.
          See Notes below.
  
      Returns
      -------
      sos : ndarray
          Array of second-order filter coefficients, with shape
          ``(n_sections, 6)``. See `sosfilt` for the SOS filter format
          specification.
  
      See Also
      --------
      sosfilt
  
      Notes
      -----
      The algorithm used to convert ZPK to SOS format is designed to
      minimize errors due to numerical precision issues. The pairing
      algorithm attempts to minimize the peak gain of each biquadratic
      section. This is done by pairing poles with the nearest zeros, starting
      with the poles closest to the unit circle.
  
      *Algorithms*
  
      The current algorithms are designed specifically for use with digital
      filters. (The output coefficents are not correct for analog filters.)
  
      The steps in the ``pairing='nearest'`` and ``pairing='keep_odd'``
      algorithms are mostly shared. The ``nearest`` algorithm attempts to
      minimize the peak gain, while ``'keep_odd'`` minimizes peak gain under
      the constraint that odd-order systems should retain one section
      as first order. The algorithm steps and are as follows:
  
      As a pre-processing step, add poles or zeros to the origin as
      necessary to obtain the same number of poles and zeros for pairing.
      If ``pairing == 'nearest'`` and there are an odd number of poles,
      add an additional pole and a zero at the origin.
  
      The following steps are then iterated over until no more poles or
      zeros remain:
  
      1. Take the (next remaining) pole (complex or real) closest to the
         unit circle to begin a new filter section.
  
      2. If the pole is real and there are no other remaining real poles [#]_,
         add the closest real zero to the section and leave it as a first
         order section. Note that after this step we are guaranteed to be
         left with an even number of real poles, complex poles, real zeros,
         and complex zeros for subsequent pairing iterations.
  
      3. Else:
  
          1. If the pole is complex and the zero is the only remaining real
             zero*, then pair the pole with the *next* closest zero
             (guaranteed to be complex). This is necessary to ensure that
             there will be a real zero remaining to eventually create a
             first-order section (thus keeping the odd order).
  
          2. Else pair the pole with the closest remaining zero (complex or
             real).
  
          3. Proceed to complete the second-order section by adding another
             pole and zero to the current pole and zero in the section:
  
              1. If the current pole and zero are both complex, add their
                 conjugates.
  
              2. Else if the pole is complex and the zero is real, add the
                 conjugate pole and the next closest real zero.
  
              3. Else if the pole is real and the zero is complex, add the
                 conjugate zero and the real pole closest to those zeros.
  
              4. Else (we must have a real pole and real zero) add the next
                 real pole closest to the unit circle, and then add the real
                 zero closest to that pole.
  
      .. [#] This conditional can only be met for specific odd-order inputs
             with the ``pairing == 'keep_odd'`` method.
  
      .. versionadded:: 0.16.0
  
      Examples
      --------
  
      Design a 6th order low-pass elliptic digital filter for a system with a
      sampling rate of 8000 Hz that has a pass-band corner frequency of
      1000 Hz.  The ripple in the pass-band should not exceed 0.087 dB, and
      the attenuation in the stop-band should be at least 90 dB.
  
      In the following call to `signal.ellip`, we could use ``output='sos'``,
      but for this example, we'll use ``output='zpk'``, and then convert to SOS
      format with `zpk2sos`:
  
      >>> from scipy import signal
      >>> z, p, k = signal.ellip(6, 0.087, 90, 1000/(0.5*8000), output='zpk')
  
      Now convert to SOS format.
  
      >>> sos = signal.zpk2sos(z, p, k)
  
      The coefficients of the numerators of the sections:
  
      >>> sos[:, :3]
      array([[ 0.0014154 ,  0.00248707,  0.0014154 ],
             [ 1.        ,  0.72965193,  1.        ],
             [ 1.        ,  0.17594966,  1.        ]])
  
      The symmetry in the coefficients occurs because all the zeros are on the
      unit circle.
  
      The coefficients of the denominators of the sections:
  
      >>> sos[:, 3:]
      array([[ 1.        , -1.32543251,  0.46989499],
             [ 1.        , -1.26117915,  0.6262586 ],
             [ 1.        , -1.25707217,  0.86199667]])
  
      The next example shows the effect of the `pairing` option.  We have a
      system with three poles and three zeros, so the SOS array will have
      shape (2, 6).  The means there is, in effect, an extra pole and an extra
      zero at the origin in the SOS representation.
  
      >>> z1 = np.array([-1, -0.5-0.5j, -0.5+0.5j])
      >>> p1 = np.array([0.75, 0.8+0.1j, 0.8-0.1j])
  
      With ``pairing='nearest'`` (the default), we obtain
  
      >>> signal.zpk2sos(z1, p1, 1)
      array([[ 1.  ,  1.  ,  0.5 ,  1.  , -0.75,  0.  ],
             [ 1.  ,  1.  ,  0.  ,  1.  , -1.6 ,  0.65]])
  
      The first section has the zeros {-0.5-0.05j, -0.5+0.5j} and the poles
      {0, 0.75}, and the second section has the zeros {-1, 0} and poles
      {0.8+0.1j, 0.8-0.1j}.  Note that the extra pole and zero at the origin
      have been assigned to different sections.
  
      With ``pairing='keep_odd'``, we obtain:
  
      >>> signal.zpk2sos(z1, p1, 1, pairing='keep_odd')
      array([[ 1.  ,  1.  ,  0.  ,  1.  , -0.75,  0.  ],
             [ 1.  ,  1.  ,  0.5 ,  1.  , -1.6 ,  0.65]])
  
      The extra pole and zero at the origin are in the same section.
      The first section is, in effect, a first-order section.
  
      """
      # TODO in the near future:
      # 1. Add SOS capability to `filtfilt`, `freqz`, etc. somehow (#3259).
      # 2. Make `decimate` use `sosfilt` instead of `lfilter`.
      # 3. Make sosfilt automatically simplify sections to first order
      #    when possible. Note this might make `sosfiltfilt` a bit harder (ICs).
      # 4. Further optimizations of the section ordering / pole-zero pairing.
      # See the wiki for other potential issues.
  
      valid_pairings = ['nearest', 'keep_odd']
      if pairing not in valid_pairings:
          raise ValueError('pairing must be one of %s, not %s'
                           % (valid_pairings, pairing))
      if len(z) == len(p) == 0:
          return array([[k, 0., 0., 1., 0., 0.]])
  
      # ensure we have the same number of poles and zeros, and make copies
      p = np.concatenate((p, np.zeros(max(len(z) - len(p), 0))))
      z = np.concatenate((z, np.zeros(max(len(p) - len(z), 0))))
      n_sections = (max(len(p), len(z)) + 1) // 2
      sos = zeros((n_sections, 6))
  
      if len(p) % 2 == 1 and pairing == 'nearest':
          p = np.concatenate((p, [0.]))
          z = np.concatenate((z, [0.]))
      assert len(p) == len(z)
  
      # Ensure we have complex conjugate pairs
      # (note that _cplxreal only gives us one element of each complex pair):
      z = np.concatenate(_cplxreal(z))
      p = np.concatenate(_cplxreal(p))
  
      p_sos = np.zeros((n_sections, 2), np.complex128)
      z_sos = np.zeros_like(p_sos)
      for si in range(n_sections):
          # Select the next "worst" pole
          p1_idx = np.argmin(np.abs(1 - np.abs(p)))
          p1 = p[p1_idx]
          p = np.delete(p, p1_idx)
  
          # Pair that pole with a zero
  
          if np.isreal(p1) and np.isreal(p).sum() == 0:
              # Special case to set a first-order section
              z1_idx = _nearest_real_complex_idx(z, p1, 'real')
              z1 = z[z1_idx]
              z = np.delete(z, z1_idx)
              p2 = z2 = 0
          else:
              if not np.isreal(p1) and np.isreal(z).sum() == 1:
                  # Special case to ensure we choose a complex zero to pair
                  # with so later (setting up a first-order section)
                  z1_idx = _nearest_real_complex_idx(z, p1, 'complex')
                  assert not np.isreal(z[z1_idx])
              else:
                  # Pair the pole with the closest zero (real or complex)
                  z1_idx = np.argmin(np.abs(p1 - z))
              z1 = z[z1_idx]
              z = np.delete(z, z1_idx)
  
              # Now that we have p1 and z1, figure out what p2 and z2 need to be
              if not np.isreal(p1):
                  if not np.isreal(z1):  # complex pole, complex zero
                      p2 = p1.conj()
                      z2 = z1.conj()
                  else:  # complex pole, real zero
                      p2 = p1.conj()
                      z2_idx = _nearest_real_complex_idx(z, p1, 'real')
                      z2 = z[z2_idx]
                      assert np.isreal(z2)
                      z = np.delete(z, z2_idx)
              else:
                  if not np.isreal(z1):  # real pole, complex zero
                      z2 = z1.conj()
                      p2_idx = _nearest_real_complex_idx(p, z1, 'real')
                      p2 = p[p2_idx]
                      assert np.isreal(p2)
                  else:  # real pole, real zero
                      # pick the next "worst" pole to use
                      idx = np.where(np.isreal(p))[0]
                      assert len(idx) > 0
                      p2_idx = idx[np.argmin(np.abs(np.abs(p[idx]) - 1))]
                      p2 = p[p2_idx]
                      # find a real zero to match the added pole
                      assert np.isreal(p2)
                      z2_idx = _nearest_real_complex_idx(z, p2, 'real')
                      z2 = z[z2_idx]
                      assert np.isreal(z2)
                      z = np.delete(z, z2_idx)
                  p = np.delete(p, p2_idx)
          p_sos[si] = [p1, p2]
          z_sos[si] = [z1, z2]
      assert len(p) == len(z) == 0  # we've consumed all poles and zeros
      del p, z
  
      # Construct the system, reversing order so the "worst" are last
      p_sos = np.reshape(p_sos[::-1], (n_sections, 2))
      z_sos = np.reshape(z_sos[::-1], (n_sections, 2))
      gains = np.ones(n_sections)
      gains[0] = k
      for si in range(n_sections):
          x = zpk2tf(z_sos[si], p_sos[si], gains[si])
          sos[si] = np.concatenate(x)
      return sos
  
  
  def _align_nums(nums):
      """Aligns the shapes of multiple numerators.
  
      Given an array of numerator coefficient arrays [[a_1, a_2,...,
      a_n],..., [b_1, b_2,..., b_m]], this function pads shorter numerator
      arrays with zero's so that all numerators have the same length. Such
      alignment is necessary for functions like 'tf2ss', which needs the
      alignment when dealing with SIMO transfer functions.
  
      Parameters
      ----------
      nums: array_like
          Numerator or list of numerators. Not necessarily with same length.
  
      Returns
      -------
      nums: array
          The numerator. If `nums` input was a list of numerators then a 2d
          array with padded zeros for shorter numerators is returned. Otherwise
          returns ``np.asarray(nums)``.
      """
      try:
          # The statement can throw a ValueError if one
          # of the numerators is a single digit and another
          # is array-like e.g. if nums = [5, [1, 2, 3]]
          nums = asarray(nums)
  
          if not np.issubdtype(nums.dtype, np.number):
              raise ValueError("dtype of numerator is non-numeric")
  
          return nums
  
      except ValueError:
          nums = [np.atleast_1d(num) for num in nums]
          max_width = max(num.size for num in nums)
  
          # pre-allocate
          aligned_nums = np.zeros((len(nums), max_width))
  
          # Create numerators with padded zeros
          for index, num in enumerate(nums):
              aligned_nums[index, -num.size:] = num
  
          return aligned_nums
  
  
  def normalize(b, a):
      """Normalize numerator/denominator of a continuous-time transfer function.
  
      If values of `b` are too close to 0, they are removed. In that case, a
      BadCoefficients warning is emitted.
  
      Parameters
      ----------
      b: array_like
          Numerator of the transfer function. Can be a 2d array to normalize
          multiple transfer functions.
      a: array_like
          Denominator of the transfer function. At most 1d.
  
      Returns
      -------
      num: array
          The numerator of the normalized transfer function. At least a 1d
          array. A 2d-array if the input `num` is a 2d array.
      den: 1d-array
          The denominator of the normalized transfer function.
  
      Notes
      -----
      Coefficients for both the numerator and denominator should be specified in
      descending exponent order (e.g., ``s^2 + 3s + 5`` would be represented as
      ``[1, 3, 5]``).
      """
      num, den = b, a
  
      den = np.atleast_1d(den)
      num = np.atleast_2d(_align_nums(num))
  
      if den.ndim != 1:
          raise ValueError("Denominator polynomial must be rank-1 array.")
      if num.ndim > 2:
          raise ValueError("Numerator polynomial must be rank-1 or"
                           " rank-2 array.")
      if np.all(den == 0):
          raise ValueError("Denominator must have at least on nonzero element.")
  
      # Trim leading zeros in denominator, leave at least one.
      den = np.trim_zeros(den, 'f')
  
      # Normalize transfer function
      num, den = num / den[0], den / den[0]
  
      # Count numerator columns that are all zero
      leading_zeros = 0
      for col in num.T:
          if np.allclose(col, 0, atol=1e-14):
              leading_zeros += 1
          else:
              break
  
      # Trim leading zeros of numerator
      if leading_zeros > 0:
          warnings.warn("Badly conditioned filter coefficients (numerator): the "
                        "results may be meaningless", BadCoefficients)
          # Make sure at least one column remains
          if leading_zeros == num.shape[1]:
              leading_zeros -= 1
          num = num[:, leading_zeros:]
  
      # Squeeze first dimension if singular
      if num.shape[0] == 1:
          num = num[0, :]
  
      return num, den
  
  
  def lp2lp(b, a, wo=1.0):
      """
      Transform a lowpass filter prototype to a different frequency.
  
      Return an analog low-pass filter with cutoff frequency `wo`
      from an analog low-pass filter prototype with unity cutoff frequency, in
      transfer function ('ba') representation.
  
      """
      a, b = map(atleast_1d, (a, b))
      try:
          wo = float(wo)
      except TypeError:
          wo = float(wo[0])
      d = len(a)
      n = len(b)
      M = max((d, n))
      pwo = pow(wo, numpy.arange(M - 1, -1, -1))
      start1 = max((n - d, 0))
      start2 = max((d - n, 0))
      b = b * pwo[start1] / pwo[start2:]
      a = a * pwo[start1] / pwo[start1:]
      return normalize(b, a)
  
  
  def lp2hp(b, a, wo=1.0):
      """
      Transform a lowpass filter prototype to a highpass filter.
  
      Return an analog high-pass filter with cutoff frequency `wo`
      from an analog low-pass filter prototype with unity cutoff frequency, in
      transfer function ('ba') representation.
  
      """
      a, b = map(atleast_1d, (a, b))
      try:
          wo = float(wo)
      except TypeError:
          wo = float(wo[0])
      d = len(a)
      n = len(b)
      if wo != 1:
          pwo = pow(wo, numpy.arange(max((d, n))))
      else:
          pwo = numpy.ones(max((d, n)), b.dtype.char)
      if d >= n:
          outa = a[::-1] * pwo
          outb = resize(b, (d,))
          outb[n:] = 0.0
          outb[:n] = b[::-1] * pwo[:n]
      else:
          outb = b[::-1] * pwo
          outa = resize(a, (n,))
          outa[d:] = 0.0
          outa[:d] = a[::-1] * pwo[:d]
  
      return normalize(outb, outa)
  
  
  def lp2bp(b, a, wo=1.0, bw=1.0):
      """
      Transform a lowpass filter prototype to a bandpass filter.
  
      Return an analog band-pass filter with center frequency `wo` and
      bandwidth `bw` from an analog low-pass filter prototype with unity
      cutoff frequency, in transfer function ('ba') representation.
  
      """
      a, b = map(atleast_1d, (a, b))
      D = len(a) - 1
      N = len(b) - 1
      artype = mintypecode((a, b))
      ma = max([N, D])
      Np = N + ma
      Dp = D + ma
      bprime = numpy.zeros(Np + 1, artype)
      aprime = numpy.zeros(Dp + 1, artype)
      wosq = wo * wo
      for j in range(Np + 1):
          val = 0.0
          for i in range(0, N + 1):
              for k in range(0, i + 1):
                  if ma - i + 2 * k == j:
                      val += comb(i, k) * b[N - i] * (wosq) ** (i - k) / bw ** i
          bprime[Np - j] = val
      for j in range(Dp + 1):
          val = 0.0
          for i in range(0, D + 1):
              for k in range(0, i + 1):
                  if ma - i + 2 * k == j:
                      val += comb(i, k) * a[D - i] * (wosq) ** (i - k) / bw ** i
          aprime[Dp - j] = val
  
      return normalize(bprime, aprime)
  
  
  def lp2bs(b, a, wo=1.0, bw=1.0):
      """
      Transform a lowpass filter prototype to a bandstop filter.
  
      Return an analog band-stop filter with center frequency `wo` and
      bandwidth `bw` from an analog low-pass filter prototype with unity
      cutoff frequency, in transfer function ('ba') representation.
  
      """
      a, b = map(atleast_1d, (a, b))
      D = len(a) - 1
      N = len(b) - 1
      artype = mintypecode((a, b))
      M = max([N, D])
      Np = M + M
      Dp = M + M
      bprime = numpy.zeros(Np + 1, artype)
      aprime = numpy.zeros(Dp + 1, artype)
      wosq = wo * wo
      for j in range(Np + 1):
          val = 0.0
          for i in range(0, N + 1):
              for k in range(0, M - i + 1):
                  if i + 2 * k == j:
                      val += (comb(M - i, k) * b[N - i] *
                              (wosq) ** (M - i - k) * bw ** i)
          bprime[Np - j] = val
      for j in range(Dp + 1):
          val = 0.0
          for i in range(0, D + 1):
              for k in range(0, M - i + 1):
                  if i + 2 * k == j:
                      val += (comb(M - i, k) * a[D - i] *
                              (wosq) ** (M - i - k) * bw ** i)
          aprime[Dp - j] = val
  
      return normalize(bprime, aprime)
  
  
  def bilinear(b, a, fs=1.0):
      """Return a digital filter from an analog one using a bilinear transform.
  
      The bilinear transform substitutes ``(z-1) / (z+1)`` for ``s``.
      """
      fs = float(fs)
      a, b = map(atleast_1d, (a, b))
      D = len(a) - 1
      N = len(b) - 1
      artype = float
      M = max([N, D])
      Np = M
      Dp = M
      bprime = numpy.zeros(Np + 1, artype)
      aprime = numpy.zeros(Dp + 1, artype)
      for j in range(Np + 1):
          val = 0.0
          for i in range(N + 1):
              for k in range(i + 1):
                  for l in range(M - i + 1):
                      if k + l == j:
                          val += (comb(i, k) * comb(M - i, l) * b[N - i] *
                                  pow(2 * fs, i) * (-1) ** k)
          bprime[j] = real(val)
      for j in range(Dp + 1):
          val = 0.0
          for i in range(D + 1):
              for k in range(i + 1):
                  for l in range(M - i + 1):
                      if k + l == j:
                          val += (comb(i, k) * comb(M - i, l) * a[D - i] *
                                  pow(2 * fs, i) * (-1) ** k)
          aprime[j] = real(val)
  
      return normalize(bprime, aprime)
  
  
  def iirdesign(wp, ws, gpass, gstop, analog=False, ftype='ellip', output='ba'):
      """Complete IIR digital and analog filter design.
  
      Given passband and stopband frequencies and gains, construct an analog or
      digital IIR filter of minimum order for a given basic type.  Return the
      output in numerator, denominator ('ba'), pole-zero ('zpk') or second order
      sections ('sos') form.
  
      Parameters
      ----------
      wp, ws : float
          Passband and stopband edge frequencies.
          For digital filters, these are normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`wp` and `ws` are thus in
          half-cycles / sample.)  For example:
  
              - Lowpass:   wp = 0.2,          ws = 0.3
              - Highpass:  wp = 0.3,          ws = 0.2
              - Bandpass:  wp = [0.2, 0.5],   ws = [0.1, 0.6]
              - Bandstop:  wp = [0.1, 0.6],   ws = [0.2, 0.5]
  
          For analog filters, `wp` and `ws` are angular frequencies (e.g. rad/s).
  
      gpass : float
          The maximum loss in the passband (dB).
      gstop : float
          The minimum attenuation in the stopband (dB).
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.
      ftype : str, optional
          The type of IIR filter to design:
  
              - Butterworth   : 'butter'
              - Chebyshev I   : 'cheby1'
              - Chebyshev II  : 'cheby2'
              - Cauer/elliptic: 'ellip'
              - Bessel/Thomson: 'bessel'
  
      output : {'ba', 'zpk', 'sos'}, optional
          Type of output:  numerator/denominator ('ba'), pole-zero ('zpk'), or
          second-order sections ('sos'). Default is 'ba'.
  
      Returns
      -------
      b, a : ndarray, ndarray
          Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
          Only returned if ``output='ba'``.
      z, p, k : ndarray, ndarray, float
          Zeros, poles, and system gain of the IIR filter transfer
          function.  Only returned if ``output='zpk'``.
      sos : ndarray
          Second-order sections representation of the IIR filter.
          Only returned if ``output=='sos'``.
  
      See Also
      --------
      butter : Filter design using order and critical points
      cheby1, cheby2, ellip, bessel
      buttord : Find order and critical points from passband and stopband spec
      cheb1ord, cheb2ord, ellipord
      iirfilter : General filter design using order and critical frequencies
  
      Notes
      -----
      The ``'sos'`` output parameter was added in 0.16.0.
      """
      try:
          ordfunc = filter_dict[ftype][1]
      except KeyError:
          raise ValueError("Invalid IIR filter type: %s" % ftype)
      except IndexError:
          raise ValueError(("%s does not have order selection. Use "
                            "iirfilter function.") % ftype)
  
      wp = atleast_1d(wp)
      ws = atleast_1d(ws)
      band_type = 2 * (len(wp) - 1)
      band_type += 1
      if wp[0] >= ws[0]:
          band_type += 1
  
      btype = {1: 'lowpass', 2: 'highpass',
               3: 'bandstop', 4: 'bandpass'}[band_type]
  
      N, Wn = ordfunc(wp, ws, gpass, gstop, analog=analog)
      return iirfilter(N, Wn, rp=gpass, rs=gstop, analog=analog, btype=btype,
                       ftype=ftype, output=output)
  
  
  def iirfilter(N, Wn, rp=None, rs=None, btype='band', analog=False,
                ftype='butter', output='ba'):
      """
      IIR digital and analog filter design given order and critical points.
  
      Design an Nth-order digital or analog filter and return the filter
      coefficients.
  
      Parameters
      ----------
      N : int
          The order of the filter.
      Wn : array_like
          A scalar or length-2 sequence giving the critical frequencies.
          For digital filters, `Wn` is normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`Wn` is thus in
          half-cycles / sample.)
          For analog filters, `Wn` is an angular frequency (e.g. rad/s).
      rp : float, optional
          For Chebyshev and elliptic filters, provides the maximum ripple
          in the passband. (dB)
      rs : float, optional
          For Chebyshev and elliptic filters, provides the minimum attenuation
          in the stop band. (dB)
      btype : {'bandpass', 'lowpass', 'highpass', 'bandstop'}, optional
          The type of filter.  Default is 'bandpass'.
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.
      ftype : str, optional
          The type of IIR filter to design:
  
              - Butterworth   : 'butter'
              - Chebyshev I   : 'cheby1'
              - Chebyshev II  : 'cheby2'
              - Cauer/elliptic: 'ellip'
              - Bessel/Thomson: 'bessel'
  
      output : {'ba', 'zpk', 'sos'}, optional
          Type of output:  numerator/denominator ('ba'), pole-zero ('zpk'), or
          second-order sections ('sos'). Default is 'ba'.
  
      Returns
      -------
      b, a : ndarray, ndarray
          Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
          Only returned if ``output='ba'``.
      z, p, k : ndarray, ndarray, float
          Zeros, poles, and system gain of the IIR filter transfer
          function.  Only returned if ``output='zpk'``.
      sos : ndarray
          Second-order sections representation of the IIR filter.
          Only returned if ``output=='sos'``.
  
      See Also
      --------
      butter : Filter design using order and critical points
      cheby1, cheby2, ellip, bessel
      buttord : Find order and critical points from passband and stopband spec
      cheb1ord, cheb2ord, ellipord
      iirdesign : General filter design using passband and stopband spec
  
      Notes
      -----
      The ``'sos'`` output parameter was added in 0.16.0.
  
      Examples
      --------
      Generate a 17th-order Chebyshev II bandpass filter and plot the frequency
      response:
  
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      >>> b, a = signal.iirfilter(17, [50, 200], rs=60, btype='band',
      ...                         analog=True, ftype='cheby2')
      >>> w, h = signal.freqs(b, a, 1000)
      >>> fig = plt.figure()
      >>> ax = fig.add_subplot(111)
      >>> ax.semilogx(w, 20 * np.log10(abs(h)))
      >>> ax.set_title('Chebyshev Type II bandpass frequency response')
      >>> ax.set_xlabel('Frequency [radians / second]')
      >>> ax.set_ylabel('Amplitude [dB]')
      >>> ax.axis((10, 1000, -100, 10))
      >>> ax.grid(which='both', axis='both')
      >>> plt.show()
  
      """
      ftype, btype, output = [x.lower() for x in (ftype, btype, output)]
      Wn = asarray(Wn)
      try:
          btype = band_dict[btype]
      except KeyError:
          raise ValueError("'%s' is an invalid bandtype for filter." % btype)
  
      try:
          typefunc = filter_dict[ftype][0]
      except KeyError:
          raise ValueError("'%s' is not a valid basic IIR filter." % ftype)
  
      if output not in ['ba', 'zpk', 'sos']:
          raise ValueError("'%s' is not a valid output form." % output)
  
      if rp is not None and rp < 0:
          raise ValueError("passband ripple (rp) must be positive")
  
      if rs is not None and rs < 0:
          raise ValueError("stopband attenuation (rs) must be positive")
  
      # Get analog lowpass prototype
      if typefunc == buttap:
          z, p, k = typefunc(N)
      elif typefunc == besselap:
          z, p, k = typefunc(N, norm=bessel_norms[ftype])
      elif typefunc == cheb1ap:
          if rp is None:
              raise ValueError("passband ripple (rp) must be provided to "
                               "design a Chebyshev I filter.")
          z, p, k = typefunc(N, rp)
      elif typefunc == cheb2ap:
          if rs is None:
              raise ValueError("stopband attenuation (rs) must be provided to "
                               "design an Chebyshev II filter.")
          z, p, k = typefunc(N, rs)
      elif typefunc == ellipap:
          if rs is None or rp is None:
              raise ValueError("Both rp and rs must be provided to design an "
                               "elliptic filter.")
          z, p, k = typefunc(N, rp, rs)
      else:
          raise NotImplementedError("'%s' not implemented in iirfilter." % ftype)
  
      # Pre-warp frequencies for digital filter design
      if not analog:
          if numpy.any(Wn < 0) or numpy.any(Wn > 1):
              raise ValueError("Digital filter critical frequencies "
                               "must be 0 <= Wn <= 1")
          fs = 2.0
          warped = 2 * fs * tan(pi * Wn / fs)
      else:
          warped = Wn
  
      # transform to lowpass, bandpass, highpass, or bandstop
      if btype in ('lowpass', 'highpass'):
          if numpy.size(Wn) != 1:
              raise ValueError('Must specify a single critical frequency Wn')
  
          if btype == 'lowpass':
              z, p, k = _zpklp2lp(z, p, k, wo=warped)
          elif btype == 'highpass':
              z, p, k = _zpklp2hp(z, p, k, wo=warped)
      elif btype in ('bandpass', 'bandstop'):
          try:
              bw = warped[1] - warped[0]
              wo = sqrt(warped[0] * warped[1])
          except IndexError:
              raise ValueError('Wn must specify start and stop frequencies')
  
          if btype == 'bandpass':
              z, p, k = _zpklp2bp(z, p, k, wo=wo, bw=bw)
          elif btype == 'bandstop':
              z, p, k = _zpklp2bs(z, p, k, wo=wo, bw=bw)
      else:
          raise NotImplementedError("'%s' not implemented in iirfilter." % btype)
  
      # Find discrete equivalent if necessary
      if not analog:
          z, p, k = _zpkbilinear(z, p, k, fs=fs)
  
      # Transform to proper out type (pole-zero, state-space, numer-denom)
      if output == 'zpk':
          return z, p, k
      elif output == 'ba':
          return zpk2tf(z, p, k)
      elif output == 'sos':
          return zpk2sos(z, p, k)
  
  
  def _relative_degree(z, p):
      """
      Return relative degree of transfer function from zeros and poles
      """
      degree = len(p) - len(z)
      if degree < 0:
          raise ValueError("Improper transfer function. "
                           "Must have at least as many poles as zeros.")
      else:
          return degree
  
  
  # TODO: merge these into existing functions or make public versions
  
  def _zpkbilinear(z, p, k, fs):
      """
      Return a digital filter from an analog one using a bilinear transform.
  
      Transform a set of poles and zeros from the analog s-plane to the digital
      z-plane using Tustin's method, which substitutes ``(z-1) / (z+1)`` for
      ``s``, maintaining the shape of the frequency response.
  
      Parameters
      ----------
      z : array_like
          Zeros of the analog IIR filter transfer function.
      p : array_like
          Poles of the analog IIR filter transfer function.
      k : float
          System gain of the analog IIR filter transfer function.
      fs : float
          Sample rate, as ordinary frequency (e.g. hertz). No prewarping is
          done in this function.
  
      Returns
      -------
      z : ndarray
          Zeros of the transformed digital filter transfer function.
      p : ndarray
          Poles of the transformed digital filter transfer function.
      k : float
          System gain of the transformed digital filter.
  
      """
      z = atleast_1d(z)
      p = atleast_1d(p)
  
      degree = _relative_degree(z, p)
  
      fs2 = 2*fs
  
      # Bilinear transform the poles and zeros
      z_z = (fs2 + z) / (fs2 - z)
      p_z = (fs2 + p) / (fs2 - p)
  
      # Any zeros that were at infinity get moved to the Nyquist frequency
      z_z = append(z_z, -ones(degree))
  
      # Compensate for gain change
      k_z = k * real(prod(fs2 - z) / prod(fs2 - p))
  
      return z_z, p_z, k_z
  
  
  def _zpklp2lp(z, p, k, wo=1.0):
      r"""
      Transform a lowpass filter prototype to a different frequency.
  
      Return an analog low-pass filter with cutoff frequency `wo`
      from an analog low-pass filter prototype with unity cutoff frequency,
      using zeros, poles, and gain ('zpk') representation.
  
      Parameters
      ----------
      z : array_like
          Zeros of the analog IIR filter transfer function.
      p : array_like
          Poles of the analog IIR filter transfer function.
      k : float
          System gain of the analog IIR filter transfer function.
      wo : float
          Desired cutoff, as angular frequency (e.g. rad/s).
          Defaults to no change.
  
      Returns
      -------
      z : ndarray
          Zeros of the transformed low-pass filter transfer function.
      p : ndarray
          Poles of the transformed low-pass filter transfer function.
      k : float
          System gain of the transformed low-pass filter.
  
      Notes
      -----
      This is derived from the s-plane substitution
  
      .. math:: s \rightarrow \frac{s}{\omega_0}
  
      """
      z = atleast_1d(z)
      p = atleast_1d(p)
      wo = float(wo)  # Avoid int wraparound
  
      degree = _relative_degree(z, p)
  
      # Scale all points radially from origin to shift cutoff frequency
      z_lp = wo * z
      p_lp = wo * p
  
      # Each shifted pole decreases gain by wo, each shifted zero increases it.
      # Cancel out the net change to keep overall gain the same
      k_lp = k * wo**degree
  
      return z_lp, p_lp, k_lp
  
  
  def _zpklp2hp(z, p, k, wo=1.0):
      r"""
      Transform a lowpass filter prototype to a highpass filter.
  
      Return an analog high-pass filter with cutoff frequency `wo`
      from an analog low-pass filter prototype with unity cutoff frequency,
      using zeros, poles, and gain ('zpk') representation.
  
      Parameters
      ----------
      z : array_like
          Zeros of the analog IIR filter transfer function.
      p : array_like
          Poles of the analog IIR filter transfer function.
      k : float
          System gain of the analog IIR filter transfer function.
      wo : float
          Desired cutoff, as angular frequency (e.g. rad/s).
          Defaults to no change.
  
      Returns
      -------
      z : ndarray
          Zeros of the transformed high-pass filter transfer function.
      p : ndarray
          Poles of the transformed high-pass filter transfer function.
      k : float
          System gain of the transformed high-pass filter.
  
      Notes
      -----
      This is derived from the s-plane substitution
  
      .. math:: s \rightarrow \frac{\omega_0}{s}
  
      This maintains symmetry of the lowpass and highpass responses on a
      logarithmic scale.
  
      """
      z = atleast_1d(z)
      p = atleast_1d(p)
      wo = float(wo)
  
      degree = _relative_degree(z, p)
  
      # Invert positions radially about unit circle to convert LPF to HPF
      # Scale all points radially from origin to shift cutoff frequency
      z_hp = wo / z
      p_hp = wo / p
  
      # If lowpass had zeros at infinity, inverting moves them to origin.
      z_hp = append(z_hp, zeros(degree))
  
      # Cancel out gain change caused by inversion
      k_hp = k * real(prod(-z) / prod(-p))
  
      return z_hp, p_hp, k_hp
  
  
  def _zpklp2bp(z, p, k, wo=1.0, bw=1.0):
      r"""
      Transform a lowpass filter prototype to a bandpass filter.
  
      Return an analog band-pass filter with center frequency `wo` and
      bandwidth `bw` from an analog low-pass filter prototype with unity
      cutoff frequency, using zeros, poles, and gain ('zpk') representation.
  
      Parameters
      ----------
      z : array_like
          Zeros of the analog IIR filter transfer function.
      p : array_like
          Poles of the analog IIR filter transfer function.
      k : float
          System gain of the analog IIR filter transfer function.
      wo : float
          Desired passband center, as angular frequency (e.g. rad/s).
          Defaults to no change.
      bw : float
          Desired passband width, as angular frequency (e.g. rad/s).
          Defaults to 1.
  
      Returns
      -------
      z : ndarray
          Zeros of the transformed band-pass filter transfer function.
      p : ndarray
          Poles of the transformed band-pass filter transfer function.
      k : float
          System gain of the transformed band-pass filter.
  
      Notes
      -----
      This is derived from the s-plane substitution
  
      .. math:: s \rightarrow \frac{s^2 + {\omega_0}^2}{s \cdot \mathrm{BW}}
  
      This is the "wideband" transformation, producing a passband with
      geometric (log frequency) symmetry about `wo`.
  
      """
      z = atleast_1d(z)
      p = atleast_1d(p)
      wo = float(wo)
      bw = float(bw)
  
      degree = _relative_degree(z, p)
  
      # Scale poles and zeros to desired bandwidth
      z_lp = z * bw/2
      p_lp = p * bw/2
  
      # Square root needs to produce complex result, not NaN
      z_lp = z_lp.astype(complex)
      p_lp = p_lp.astype(complex)
  
      # Duplicate poles and zeros and shift from baseband to +wo and -wo
      z_bp = concatenate((z_lp + sqrt(z_lp**2 - wo**2),
                          z_lp - sqrt(z_lp**2 - wo**2)))
      p_bp = concatenate((p_lp + sqrt(p_lp**2 - wo**2),
                          p_lp - sqrt(p_lp**2 - wo**2)))
  
      # Move degree zeros to origin, leaving degree zeros at infinity for BPF
      z_bp = append(z_bp, zeros(degree))
  
      # Cancel out gain change from frequency scaling
      k_bp = k * bw**degree
  
      return z_bp, p_bp, k_bp
  
  
  def _zpklp2bs(z, p, k, wo=1.0, bw=1.0):
      r"""
      Transform a lowpass filter prototype to a bandstop filter.
  
      Return an analog band-stop filter with center frequency `wo` and
      stopband width `bw` from an analog low-pass filter prototype with unity
      cutoff frequency, using zeros, poles, and gain ('zpk') representation.
  
      Parameters
      ----------
      z : array_like
          Zeros of the analog IIR filter transfer function.
      p : array_like
          Poles of the analog IIR filter transfer function.
      k : float
          System gain of the analog IIR filter transfer function.
      wo : float
          Desired stopband center, as angular frequency (e.g. rad/s).
          Defaults to no change.
      bw : float
          Desired stopband width, as angular frequency (e.g. rad/s).
          Defaults to 1.
  
      Returns
      -------
      z : ndarray
          Zeros of the transformed band-stop filter transfer function.
      p : ndarray
          Poles of the transformed band-stop filter transfer function.
      k : float
          System gain of the transformed band-stop filter.
  
      Notes
      -----
      This is derived from the s-plane substitution
  
      .. math:: s \rightarrow \frac{s \cdot \mathrm{BW}}{s^2 + {\omega_0}^2}
  
      This is the "wideband" transformation, producing a stopband with
      geometric (log frequency) symmetry about `wo`.
  
      """
      z = atleast_1d(z)
      p = atleast_1d(p)
      wo = float(wo)
      bw = float(bw)
  
      degree = _relative_degree(z, p)
  
      # Invert to a highpass filter with desired bandwidth
      z_hp = (bw/2) / z
      p_hp = (bw/2) / p
  
      # Square root needs to produce complex result, not NaN
      z_hp = z_hp.astype(complex)
      p_hp = p_hp.astype(complex)
  
      # Duplicate poles and zeros and shift from baseband to +wo and -wo
      z_bs = concatenate((z_hp + sqrt(z_hp**2 - wo**2),
                          z_hp - sqrt(z_hp**2 - wo**2)))
      p_bs = concatenate((p_hp + sqrt(p_hp**2 - wo**2),
                          p_hp - sqrt(p_hp**2 - wo**2)))
  
      # Move any zeros that were at infinity to the center of the stopband
      z_bs = append(z_bs, +1j*wo * ones(degree))
      z_bs = append(z_bs, -1j*wo * ones(degree))
  
      # Cancel out gain change caused by inversion
      k_bs = k * real(prod(-z) / prod(-p))
  
      return z_bs, p_bs, k_bs
  
  
  def butter(N, Wn, btype='low', analog=False, output='ba'):
      """
      Butterworth digital and analog filter design.
  
      Design an Nth-order digital or analog Butterworth filter and return
      the filter coefficients.
  
      Parameters
      ----------
      N : int
          The order of the filter.
      Wn : array_like
          A scalar or length-2 sequence giving the critical frequencies.
          For a Butterworth filter, this is the point at which the gain
          drops to 1/sqrt(2) that of the passband (the "-3 dB point").
          For digital filters, `Wn` is normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`Wn` is thus in
          half-cycles / sample.)
          For analog filters, `Wn` is an angular frequency (e.g. rad/s).
      btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
          The type of filter.  Default is 'lowpass'.
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.
      output : {'ba', 'zpk', 'sos'}, optional
          Type of output:  numerator/denominator ('ba'), pole-zero ('zpk'), or
          second-order sections ('sos'). Default is 'ba'.
  
      Returns
      -------
      b, a : ndarray, ndarray
          Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
          Only returned if ``output='ba'``.
      z, p, k : ndarray, ndarray, float
          Zeros, poles, and system gain of the IIR filter transfer
          function.  Only returned if ``output='zpk'``.
      sos : ndarray
          Second-order sections representation of the IIR filter.
          Only returned if ``output=='sos'``.
  
      See Also
      --------
      buttord, buttap
  
      Notes
      -----
      The Butterworth filter has maximally flat frequency response in the
      passband.
  
      The ``'sos'`` output parameter was added in 0.16.0.
  
      Examples
      --------
      Plot the filter's frequency response, showing the critical points:
  
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      >>> b, a = signal.butter(4, 100, 'low', analog=True)
      >>> w, h = signal.freqs(b, a)
      >>> plt.semilogx(w, 20 * np.log10(abs(h)))
      >>> plt.title('Butterworth filter frequency response')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.margins(0, 0.1)
      >>> plt.grid(which='both', axis='both')
      >>> plt.axvline(100, color='green') # cutoff frequency
      >>> plt.show()
  
      """
      return iirfilter(N, Wn, btype=btype, analog=analog,
                       output=output, ftype='butter')
  
  
  def cheby1(N, rp, Wn, btype='low', analog=False, output='ba'):
      """
      Chebyshev type I digital and analog filter design.
  
      Design an Nth-order digital or analog Chebyshev type I filter and
      return the filter coefficients.
  
      Parameters
      ----------
      N : int
          The order of the filter.
      rp : float
          The maximum ripple allowed below unity gain in the passband.
          Specified in decibels, as a positive number.
      Wn : array_like
          A scalar or length-2 sequence giving the critical frequencies.
          For Type I filters, this is the point in the transition band at which
          the gain first drops below -`rp`.
          For digital filters, `Wn` is normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`Wn` is thus in
          half-cycles / sample.)
          For analog filters, `Wn` is an angular frequency (e.g. rad/s).
      btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
          The type of filter.  Default is 'lowpass'.
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.
      output : {'ba', 'zpk', 'sos'}, optional
          Type of output:  numerator/denominator ('ba'), pole-zero ('zpk'), or
          second-order sections ('sos'). Default is 'ba'.
  
      Returns
      -------
      b, a : ndarray, ndarray
          Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
          Only returned if ``output='ba'``.
      z, p, k : ndarray, ndarray, float
          Zeros, poles, and system gain of the IIR filter transfer
          function.  Only returned if ``output='zpk'``.
      sos : ndarray
          Second-order sections representation of the IIR filter.
          Only returned if ``output=='sos'``.
  
      See Also
      --------
      cheb1ord, cheb1ap
  
      Notes
      -----
      The Chebyshev type I filter maximizes the rate of cutoff between the
      frequency response's passband and stopband, at the expense of ripple in
      the passband and increased ringing in the step response.
  
      Type I filters roll off faster than Type II (`cheby2`), but Type II
      filters do not have any ripple in the passband.
  
      The equiripple passband has N maxima or minima (for example, a
      5th-order filter has 3 maxima and 2 minima).  Consequently, the DC gain is
      unity for odd-order filters, or -rp dB for even-order filters.
  
      The ``'sos'`` output parameter was added in 0.16.0.
  
      Examples
      --------
      Plot the filter's frequency response, showing the critical points:
  
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      >>> b, a = signal.cheby1(4, 5, 100, 'low', analog=True)
      >>> w, h = signal.freqs(b, a)
      >>> plt.semilogx(w, 20 * np.log10(abs(h)))
      >>> plt.title('Chebyshev Type I frequency response (rp=5)')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.margins(0, 0.1)
      >>> plt.grid(which='both', axis='both')
      >>> plt.axvline(100, color='green') # cutoff frequency
      >>> plt.axhline(-5, color='green') # rp
      >>> plt.show()
  
      """
      return iirfilter(N, Wn, rp=rp, btype=btype, analog=analog,
                       output=output, ftype='cheby1')
  
  
  def cheby2(N, rs, Wn, btype='low', analog=False, output='ba'):
      """
      Chebyshev type II digital and analog filter design.
  
      Design an Nth-order digital or analog Chebyshev type II filter and
      return the filter coefficients.
  
      Parameters
      ----------
      N : int
          The order of the filter.
      rs : float
          The minimum attenuation required in the stop band.
          Specified in decibels, as a positive number.
      Wn : array_like
          A scalar or length-2 sequence giving the critical frequencies.
          For Type II filters, this is the point in the transition band at which
          the gain first reaches -`rs`.
          For digital filters, `Wn` is normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`Wn` is thus in
          half-cycles / sample.)
          For analog filters, `Wn` is an angular frequency (e.g. rad/s).
      btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
          The type of filter.  Default is 'lowpass'.
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.
      output : {'ba', 'zpk', 'sos'}, optional
          Type of output:  numerator/denominator ('ba'), pole-zero ('zpk'), or
          second-order sections ('sos'). Default is 'ba'.
  
      Returns
      -------
      b, a : ndarray, ndarray
          Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
          Only returned if ``output='ba'``.
      z, p, k : ndarray, ndarray, float
          Zeros, poles, and system gain of the IIR filter transfer
          function.  Only returned if ``output='zpk'``.
      sos : ndarray
          Second-order sections representation of the IIR filter.
          Only returned if ``output=='sos'``.
  
      See Also
      --------
      cheb2ord, cheb2ap
  
      Notes
      -----
      The Chebyshev type II filter maximizes the rate of cutoff between the
      frequency response's passband and stopband, at the expense of ripple in
      the stopband and increased ringing in the step response.
  
      Type II filters do not roll off as fast as Type I (`cheby1`).
  
      The ``'sos'`` output parameter was added in 0.16.0.
  
      Examples
      --------
      Plot the filter's frequency response, showing the critical points:
  
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      >>> b, a = signal.cheby2(4, 40, 100, 'low', analog=True)
      >>> w, h = signal.freqs(b, a)
      >>> plt.semilogx(w, 20 * np.log10(abs(h)))
      >>> plt.title('Chebyshev Type II frequency response (rs=40)')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.margins(0, 0.1)
      >>> plt.grid(which='both', axis='both')
      >>> plt.axvline(100, color='green') # cutoff frequency
      >>> plt.axhline(-40, color='green') # rs
      >>> plt.show()
  
      """
      return iirfilter(N, Wn, rs=rs, btype=btype, analog=analog,
                       output=output, ftype='cheby2')
  
  
  def ellip(N, rp, rs, Wn, btype='low', analog=False, output='ba'):
      """
      Elliptic (Cauer) digital and analog filter design.
  
      Design an Nth-order digital or analog elliptic filter and return
      the filter coefficients.
  
      Parameters
      ----------
      N : int
          The order of the filter.
      rp : float
          The maximum ripple allowed below unity gain in the passband.
          Specified in decibels, as a positive number.
      rs : float
          The minimum attenuation required in the stop band.
          Specified in decibels, as a positive number.
      Wn : array_like
          A scalar or length-2 sequence giving the critical frequencies.
          For elliptic filters, this is the point in the transition band at
          which the gain first drops below -`rp`.
          For digital filters, `Wn` is normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`Wn` is thus in
          half-cycles / sample.)
          For analog filters, `Wn` is an angular frequency (e.g. rad/s).
      btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
          The type of filter.  Default is 'lowpass'.
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.
      output : {'ba', 'zpk', 'sos'}, optional
          Type of output:  numerator/denominator ('ba'), pole-zero ('zpk'), or
          second-order sections ('sos'). Default is 'ba'.
  
      Returns
      -------
      b, a : ndarray, ndarray
          Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
          Only returned if ``output='ba'``.
      z, p, k : ndarray, ndarray, float
          Zeros, poles, and system gain of the IIR filter transfer
          function.  Only returned if ``output='zpk'``.
      sos : ndarray
          Second-order sections representation of the IIR filter.
          Only returned if ``output=='sos'``.
  
      See Also
      --------
      ellipord, ellipap
  
      Notes
      -----
      Also known as Cauer or Zolotarev filters, the elliptical filter maximizes
      the rate of transition between the frequency response's passband and
      stopband, at the expense of ripple in both, and increased ringing in the
      step response.
  
      As `rp` approaches 0, the elliptical filter becomes a Chebyshev
      type II filter (`cheby2`).  As `rs` approaches 0, it becomes a Chebyshev
      type I filter (`cheby1`).  As both approach 0, it becomes a Butterworth
      filter (`butter`).
  
      The equiripple passband has N maxima or minima (for example, a
      5th-order filter has 3 maxima and 2 minima).  Consequently, the DC gain is
      unity for odd-order filters, or -rp dB for even-order filters.
  
      The ``'sos'`` output parameter was added in 0.16.0.
  
      Examples
      --------
      Plot the filter's frequency response, showing the critical points:
  
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      >>> b, a = signal.ellip(4, 5, 40, 100, 'low', analog=True)
      >>> w, h = signal.freqs(b, a)
      >>> plt.semilogx(w, 20 * np.log10(abs(h)))
      >>> plt.title('Elliptic filter frequency response (rp=5, rs=40)')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.margins(0, 0.1)
      >>> plt.grid(which='both', axis='both')
      >>> plt.axvline(100, color='green') # cutoff frequency
      >>> plt.axhline(-40, color='green') # rs
      >>> plt.axhline(-5, color='green') # rp
      >>> plt.show()
  
      """
      return iirfilter(N, Wn, rs=rs, rp=rp, btype=btype, analog=analog,
                       output=output, ftype='elliptic')
  
  
  def bessel(N, Wn, btype='low', analog=False, output='ba', norm='phase'):
      """
      Bessel/Thomson digital and analog filter design.
  
      Design an Nth-order digital or analog Bessel filter and return the
      filter coefficients.
  
      Parameters
      ----------
      N : int
          The order of the filter.
      Wn : array_like
          A scalar or length-2 sequence giving the critical frequencies (defined
          by the `norm` parameter).
          For analog filters, `Wn` is an angular frequency (e.g. rad/s).
          For digital filters, `Wn` is normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`Wn` is thus in
          half-cycles / sample.)
      btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
          The type of filter.  Default is 'lowpass'.
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.  (See Notes.)
      output : {'ba', 'zpk', 'sos'}, optional
          Type of output:  numerator/denominator ('ba'), pole-zero ('zpk'), or
          second-order sections ('sos'). Default is 'ba'.
      norm : {'phase', 'delay', 'mag'}, optional
          Critical frequency normalization:
  
          ``phase``
              The filter is normalized such that the phase response reaches its
              midpoint at angular (e.g. rad/s) frequency `Wn`.  This happens for
              both low-pass and high-pass filters, so this is the
              "phase-matched" case.
  
              The magnitude response asymptotes are the same as a Butterworth
              filter of the same order with a cutoff of `Wn`.
  
              This is the default, and matches MATLAB's implementation.
  
          ``delay``
              The filter is normalized such that the group delay in the passband
              is 1/`Wn` (e.g. seconds).  This is the "natural" type obtained by
              solving Bessel polynomials.
  
          ``mag``
              The filter is normalized such that the gain magnitude is -3 dB at
              angular frequency `Wn`.
  
          .. versionadded:: 0.18.0
  
      Returns
      -------
      b, a : ndarray, ndarray
          Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
          Only returned if ``output='ba'``.
      z, p, k : ndarray, ndarray, float
          Zeros, poles, and system gain of the IIR filter transfer
          function.  Only returned if ``output='zpk'``.
      sos : ndarray
          Second-order sections representation of the IIR filter.
          Only returned if ``output=='sos'``.
  
      Notes
      -----
      Also known as a Thomson filter, the analog Bessel filter has maximally
      flat group delay and maximally linear phase response, with very little
      ringing in the step response. [1]_
  
      The Bessel is inherently an analog filter.  This function generates digital
      Bessel filters using the bilinear transform, which does not preserve the
      phase response of the analog filter.  As such, it is only approximately
      correct at frequencies below about fs/4.  To get maximally-flat group
      delay at higher frequencies, the analog Bessel filter must be transformed
      using phase-preserving techniques.
  
      See `besselap` for implementation details and references.
  
      The ``'sos'`` output parameter was added in 0.16.0.
  
      Examples
      --------
      Plot the phase-normalized frequency response, showing the relationship
      to the Butterworth's cutoff frequency (green):
  
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      >>> b, a = signal.butter(4, 100, 'low', analog=True)
      >>> w, h = signal.freqs(b, a)
      >>> plt.semilogx(w, 20 * np.log10(np.abs(h)), color='silver', ls='dashed')
      >>> b, a = signal.bessel(4, 100, 'low', analog=True, norm='phase')
      >>> w, h = signal.freqs(b, a)
      >>> plt.semilogx(w, 20 * np.log10(np.abs(h)))
      >>> plt.title('Bessel filter magnitude response (with Butterworth)')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.margins(0, 0.1)
      >>> plt.grid(which='both', axis='both')
      >>> plt.axvline(100, color='green')  # cutoff frequency
      >>> plt.show()
  
      and the phase midpoint:
  
      >>> plt.figure()
      >>> plt.semilogx(w, np.unwrap(np.angle(h)))
      >>> plt.axvline(100, color='green')  # cutoff frequency
      >>> plt.axhline(-np.pi, color='red')  # phase midpoint
      >>> plt.title('Bessel filter phase response')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Phase [radians]')
      >>> plt.margins(0, 0.1)
      >>> plt.grid(which='both', axis='both')
      >>> plt.show()
  
      Plot the magnitude-normalized frequency response, showing the -3 dB cutoff:
  
      >>> b, a = signal.bessel(3, 10, 'low', analog=True, norm='mag')
      >>> w, h = signal.freqs(b, a)
      >>> plt.semilogx(w, 20 * np.log10(np.abs(h)))
      >>> plt.axhline(-3, color='red')  # -3 dB magnitude
      >>> plt.axvline(10, color='green')  # cutoff frequency
      >>> plt.title('Magnitude-normalized Bessel filter frequency response')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.margins(0, 0.1)
      >>> plt.grid(which='both', axis='both')
      >>> plt.show()
  
      Plot the delay-normalized filter, showing the maximally-flat group delay
      at 0.1 seconds:
  
      >>> b, a = signal.bessel(5, 1/0.1, 'low', analog=True, norm='delay')
      >>> w, h = signal.freqs(b, a)
      >>> plt.figure()
      >>> plt.semilogx(w[1:], -np.diff(np.unwrap(np.angle(h)))/np.diff(w))
      >>> plt.axhline(0.1, color='red')  # 0.1 seconds group delay
      >>> plt.title('Bessel filter group delay')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Group delay [seconds]')
      >>> plt.margins(0, 0.1)
      >>> plt.grid(which='both', axis='both')
      >>> plt.show()
  
      References
      ----------
      .. [1] Thomson, W.E., "Delay Networks having Maximally Flat Frequency
             Characteristics", Proceedings of the Institution of Electrical
             Engineers, Part III, November 1949, Vol. 96, No. 44, pp. 487-490.
  
      """
      return iirfilter(N, Wn, btype=btype, analog=analog,
                       output=output, ftype='bessel_'+norm)
  
  
  def maxflat():
      pass
  
  
  def yulewalk():
      pass
  
  
  def band_stop_obj(wp, ind, passb, stopb, gpass, gstop, type):
      """
      Band Stop Objective Function for order minimization.
  
      Returns the non-integer order for an analog band stop filter.
  
      Parameters
      ----------
      wp : scalar
          Edge of passband `passb`.
      ind : int, {0, 1}
          Index specifying which `passb` edge to vary (0 or 1).
      passb : ndarray
          Two element sequence of fixed passband edges.
      stopb : ndarray
          Two element sequence of fixed stopband edges.
      gstop : float
          Amount of attenuation in stopband in dB.
      gpass : float
          Amount of ripple in the passband in dB.
      type : {'butter', 'cheby', 'ellip'}
          Type of filter.
  
      Returns
      -------
      n : scalar
          Filter order (possibly non-integer).
  
      """
      passbC = passb.copy()
      passbC[ind] = wp
      nat = (stopb * (passbC[0] - passbC[1]) /
             (stopb ** 2 - passbC[0] * passbC[1]))
      nat = min(abs(nat))
  
      if type == 'butter':
          GSTOP = 10 ** (0.1 * abs(gstop))
          GPASS = 10 ** (0.1 * abs(gpass))
          n = (log10((GSTOP - 1.0) / (GPASS - 1.0)) / (2 * log10(nat)))
      elif type == 'cheby':
          GSTOP = 10 ** (0.1 * abs(gstop))
          GPASS = 10 ** (0.1 * abs(gpass))
          n = arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) / arccosh(nat)
      elif type == 'ellip':
          GSTOP = 10 ** (0.1 * gstop)
          GPASS = 10 ** (0.1 * gpass)
          arg1 = sqrt((GPASS - 1.0) / (GSTOP - 1.0))
          arg0 = 1.0 / nat
          d0 = special.ellipk([arg0 ** 2, 1 - arg0 ** 2])
          d1 = special.ellipk([arg1 ** 2, 1 - arg1 ** 2])
          n = (d0[0] * d1[1] / (d0[1] * d1[0]))
      else:
          raise ValueError("Incorrect type: %s" % type)
      return n
  
  
  def buttord(wp, ws, gpass, gstop, analog=False):
      """Butterworth filter order selection.
  
      Return the order of the lowest order digital or analog Butterworth filter
      that loses no more than `gpass` dB in the passband and has at least
      `gstop` dB attenuation in the stopband.
  
      Parameters
      ----------
      wp, ws : float
          Passband and stopband edge frequencies.
          For digital filters, these are normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`wp` and `ws` are thus in
          half-cycles / sample.)  For example:
  
              - Lowpass:   wp = 0.2,          ws = 0.3
              - Highpass:  wp = 0.3,          ws = 0.2
              - Bandpass:  wp = [0.2, 0.5],   ws = [0.1, 0.6]
              - Bandstop:  wp = [0.1, 0.6],   ws = [0.2, 0.5]
  
          For analog filters, `wp` and `ws` are angular frequencies (e.g. rad/s).
  
      gpass : float
          The maximum loss in the passband (dB).
      gstop : float
          The minimum attenuation in the stopband (dB).
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.
  
      Returns
      -------
      ord : int
          The lowest order for a Butterworth filter which meets specs.
      wn : ndarray or float
          The Butterworth natural frequency (i.e. the "3dB frequency").  Should
          be used with `butter` to give filter results.
  
      See Also
      --------
      butter : Filter design using order and critical points
      cheb1ord : Find order and critical points from passband and stopband spec
      cheb2ord, ellipord
      iirfilter : General filter design using order and critical frequencies
      iirdesign : General filter design using passband and stopband spec
  
      Examples
      --------
      Design an analog bandpass filter with passband within 3 dB from 20 to
      50 rad/s, while rejecting at least -40 dB below 14 and above 60 rad/s.
      Plot its frequency response, showing the passband and stopband
      constraints in gray.
  
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      >>> N, Wn = signal.buttord([20, 50], [14, 60], 3, 40, True)
      >>> b, a = signal.butter(N, Wn, 'band', True)
      >>> w, h = signal.freqs(b, a, np.logspace(1, 2, 500))
      >>> plt.semilogx(w, 20 * np.log10(abs(h)))
      >>> plt.title('Butterworth bandpass filter fit to constraints')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.grid(which='both', axis='both')
      >>> plt.fill([1,  14,  14,   1], [-40, -40, 99, 99], '0.9', lw=0) # stop
      >>> plt.fill([20, 20,  50,  50], [-99, -3, -3, -99], '0.9', lw=0) # pass
      >>> plt.fill([60, 60, 1e9, 1e9], [99, -40, -40, 99], '0.9', lw=0) # stop
      >>> plt.axis([10, 100, -60, 3])
      >>> plt.show()
  
      """
      wp = atleast_1d(wp)
      ws = atleast_1d(ws)
      filter_type = 2 * (len(wp) - 1)
      filter_type += 1
      if wp[0] >= ws[0]:
          filter_type += 1
  
      # Pre-warp frequencies for digital filter design
      if not analog:
          passb = tan(pi * wp / 2.0)
          stopb = tan(pi * ws / 2.0)
      else:
          passb = wp * 1.0
          stopb = ws * 1.0
  
      if filter_type == 1:            # low
          nat = stopb / passb
      elif filter_type == 2:          # high
          nat = passb / stopb
      elif filter_type == 3:          # stop
          wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12,
                                   args=(0, passb, stopb, gpass, gstop,
                                         'butter'),
                                   disp=0)
          passb[0] = wp0
          wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1],
                                   args=(1, passb, stopb, gpass, gstop,
                                         'butter'),
                                   disp=0)
          passb[1] = wp1
          nat = ((stopb * (passb[0] - passb[1])) /
                 (stopb ** 2 - passb[0] * passb[1]))
      elif filter_type == 4:          # pass
          nat = ((stopb ** 2 - passb[0] * passb[1]) /
                 (stopb * (passb[0] - passb[1])))
  
      nat = min(abs(nat))
  
      GSTOP = 10 ** (0.1 * abs(gstop))
      GPASS = 10 ** (0.1 * abs(gpass))
      ord = int(ceil(log10((GSTOP - 1.0) / (GPASS - 1.0)) / (2 * log10(nat))))
  
      # Find the Butterworth natural frequency WN (or the "3dB" frequency")
      # to give exactly gpass at passb.
      try:
          W0 = (GPASS - 1.0) ** (-1.0 / (2.0 * ord))
      except ZeroDivisionError:
          W0 = 1.0
          print("Warning, order is zero...check input parameters.")
  
      # now convert this frequency back from lowpass prototype
      # to the original analog filter
  
      if filter_type == 1:  # low
          WN = W0 * passb
      elif filter_type == 2:  # high
          WN = passb / W0
      elif filter_type == 3:  # stop
          WN = numpy.zeros(2, float)
          discr = sqrt((passb[1] - passb[0]) ** 2 +
                       4 * W0 ** 2 * passb[0] * passb[1])
          WN[0] = ((passb[1] - passb[0]) + discr) / (2 * W0)
          WN[1] = ((passb[1] - passb[0]) - discr) / (2 * W0)
          WN = numpy.sort(abs(WN))
      elif filter_type == 4:  # pass
          W0 = numpy.array([-W0, W0], float)
          WN = (-W0 * (passb[1] - passb[0]) / 2.0 +
                sqrt(W0 ** 2 / 4.0 * (passb[1] - passb[0]) ** 2 +
                     passb[0] * passb[1]))
          WN = numpy.sort(abs(WN))
      else:
          raise ValueError("Bad type: %s" % filter_type)
  
      if not analog:
          wn = (2.0 / pi) * arctan(WN)
      else:
          wn = WN
  
      if len(wn) == 1:
          wn = wn[0]
      return ord, wn
  
  
  def cheb1ord(wp, ws, gpass, gstop, analog=False):
      """Chebyshev type I filter order selection.
  
      Return the order of the lowest order digital or analog Chebyshev Type I
      filter that loses no more than `gpass` dB in the passband and has at
      least `gstop` dB attenuation in the stopband.
  
      Parameters
      ----------
      wp, ws : float
          Passband and stopband edge frequencies.
          For digital filters, these are normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`wp` and `ws` are thus in
          half-cycles / sample.)  For example:
  
              - Lowpass:   wp = 0.2,          ws = 0.3
              - Highpass:  wp = 0.3,          ws = 0.2
              - Bandpass:  wp = [0.2, 0.5],   ws = [0.1, 0.6]
              - Bandstop:  wp = [0.1, 0.6],   ws = [0.2, 0.5]
  
          For analog filters, `wp` and `ws` are angular frequencies (e.g. rad/s).
  
      gpass : float
          The maximum loss in the passband (dB).
      gstop : float
          The minimum attenuation in the stopband (dB).
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.
  
      Returns
      -------
      ord : int
          The lowest order for a Chebyshev type I filter that meets specs.
      wn : ndarray or float
          The Chebyshev natural frequency (the "3dB frequency") for use with
          `cheby1` to give filter results.
  
      See Also
      --------
      cheby1 : Filter design using order and critical points
      buttord : Find order and critical points from passband and stopband spec
      cheb2ord, ellipord
      iirfilter : General filter design using order and critical frequencies
      iirdesign : General filter design using passband and stopband spec
  
      Examples
      --------
      Design a digital lowpass filter such that the passband is within 3 dB up
      to 0.2*(fs/2), while rejecting at least -40 dB above 0.3*(fs/2).  Plot its
      frequency response, showing the passband and stopband constraints in gray.
  
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      >>> N, Wn = signal.cheb1ord(0.2, 0.3, 3, 40)
      >>> b, a = signal.cheby1(N, 3, Wn, 'low')
      >>> w, h = signal.freqz(b, a)
      >>> plt.semilogx(w / np.pi, 20 * np.log10(abs(h)))
      >>> plt.title('Chebyshev I lowpass filter fit to constraints')
      >>> plt.xlabel('Normalized frequency')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.grid(which='both', axis='both')
      >>> plt.fill([.01, 0.2, 0.2, .01], [-3, -3, -99, -99], '0.9', lw=0) # stop
      >>> plt.fill([0.3, 0.3,   2,   2], [ 9, -40, -40,  9], '0.9', lw=0) # pass
      >>> plt.axis([0.08, 1, -60, 3])
      >>> plt.show()
  
      """
      wp = atleast_1d(wp)
      ws = atleast_1d(ws)
      filter_type = 2 * (len(wp) - 1)
      if wp[0] < ws[0]:
          filter_type += 1
      else:
          filter_type += 2
  
      # Pre-warp frequencies for digital filter design
      if not analog:
          passb = tan(pi * wp / 2.0)
          stopb = tan(pi * ws / 2.0)
      else:
          passb = wp * 1.0
          stopb = ws * 1.0
  
      if filter_type == 1:           # low
          nat = stopb / passb
      elif filter_type == 2:          # high
          nat = passb / stopb
      elif filter_type == 3:     # stop
          wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12,
                                   args=(0, passb, stopb, gpass, gstop, 'cheby'),
                                   disp=0)
          passb[0] = wp0
          wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1],
                                   args=(1, passb, stopb, gpass, gstop, 'cheby'),
                                   disp=0)
          passb[1] = wp1
          nat = ((stopb * (passb[0] - passb[1])) /
                 (stopb ** 2 - passb[0] * passb[1]))
      elif filter_type == 4:  # pass
          nat = ((stopb ** 2 - passb[0] * passb[1]) /
                 (stopb * (passb[0] - passb[1])))
  
      nat = min(abs(nat))
  
      GSTOP = 10 ** (0.1 * abs(gstop))
      GPASS = 10 ** (0.1 * abs(gpass))
      ord = int(ceil(arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) /
                     arccosh(nat)))
  
      # Natural frequencies are just the passband edges
      if not analog:
          wn = (2.0 / pi) * arctan(passb)
      else:
          wn = passb
  
      if len(wn) == 1:
          wn = wn[0]
      return ord, wn
  
  
  def cheb2ord(wp, ws, gpass, gstop, analog=False):
      """Chebyshev type II filter order selection.
  
      Return the order of the lowest order digital or analog Chebyshev Type II
      filter that loses no more than `gpass` dB in the passband and has at least
      `gstop` dB attenuation in the stopband.
  
      Parameters
      ----------
      wp, ws : float
          Passband and stopband edge frequencies.
          For digital filters, these are normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`wp` and `ws` are thus in
          half-cycles / sample.)  For example:
  
              - Lowpass:   wp = 0.2,          ws = 0.3
              - Highpass:  wp = 0.3,          ws = 0.2
              - Bandpass:  wp = [0.2, 0.5],   ws = [0.1, 0.6]
              - Bandstop:  wp = [0.1, 0.6],   ws = [0.2, 0.5]
  
          For analog filters, `wp` and `ws` are angular frequencies (e.g. rad/s).
  
      gpass : float
          The maximum loss in the passband (dB).
      gstop : float
          The minimum attenuation in the stopband (dB).
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.
  
      Returns
      -------
      ord : int
          The lowest order for a Chebyshev type II filter that meets specs.
      wn : ndarray or float
          The Chebyshev natural frequency (the "3dB frequency") for use with
          `cheby2` to give filter results.
  
      See Also
      --------
      cheby2 : Filter design using order and critical points
      buttord : Find order and critical points from passband and stopband spec
      cheb1ord, ellipord
      iirfilter : General filter design using order and critical frequencies
      iirdesign : General filter design using passband and stopband spec
  
      Examples
      --------
      Design a digital bandstop filter which rejects -60 dB from 0.2*(fs/2) to
      0.5*(fs/2), while staying within 3 dB below 0.1*(fs/2) or above
      0.6*(fs/2).  Plot its frequency response, showing the passband and
      stopband constraints in gray.
  
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      >>> N, Wn = signal.cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 60)
      >>> b, a = signal.cheby2(N, 60, Wn, 'stop')
      >>> w, h = signal.freqz(b, a)
      >>> plt.semilogx(w / np.pi, 20 * np.log10(abs(h)))
      >>> plt.title('Chebyshev II bandstop filter fit to constraints')
      >>> plt.xlabel('Normalized frequency')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.grid(which='both', axis='both')
      >>> plt.fill([.01, .1, .1, .01], [-3,  -3, -99, -99], '0.9', lw=0) # stop
      >>> plt.fill([.2,  .2, .5,  .5], [ 9, -60, -60,   9], '0.9', lw=0) # pass
      >>> plt.fill([.6,  .6,  2,   2], [-99, -3,  -3, -99], '0.9', lw=0) # stop
      >>> plt.axis([0.06, 1, -80, 3])
      >>> plt.show()
  
      """
      wp = atleast_1d(wp)
      ws = atleast_1d(ws)
      filter_type = 2 * (len(wp) - 1)
      if wp[0] < ws[0]:
          filter_type += 1
      else:
          filter_type += 2
  
      # Pre-warp frequencies for digital filter design
      if not analog:
          passb = tan(pi * wp / 2.0)
          stopb = tan(pi * ws / 2.0)
      else:
          passb = wp * 1.0
          stopb = ws * 1.0
  
      if filter_type == 1:           # low
          nat = stopb / passb
      elif filter_type == 2:          # high
          nat = passb / stopb
      elif filter_type == 3:     # stop
          wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12,
                                   args=(0, passb, stopb, gpass, gstop, 'cheby'),
                                   disp=0)
          passb[0] = wp0
          wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1],
                                   args=(1, passb, stopb, gpass, gstop, 'cheby'),
                                   disp=0)
          passb[1] = wp1
          nat = ((stopb * (passb[0] - passb[1])) /
                 (stopb ** 2 - passb[0] * passb[1]))
      elif filter_type == 4:  # pass
          nat = ((stopb ** 2 - passb[0] * passb[1]) /
                 (stopb * (passb[0] - passb[1])))
  
      nat = min(abs(nat))
  
      GSTOP = 10 ** (0.1 * abs(gstop))
      GPASS = 10 ** (0.1 * abs(gpass))
      ord = int(ceil(arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) /
                     arccosh(nat)))
  
      # Find frequency where analog response is -gpass dB.
      # Then convert back from low-pass prototype to the original filter.
  
      new_freq = cosh(1.0 / ord * arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))))
      new_freq = 1.0 / new_freq
  
      if filter_type == 1:
          nat = passb / new_freq
      elif filter_type == 2:
          nat = passb * new_freq
      elif filter_type == 3:
          nat = numpy.zeros(2, float)
          nat[0] = (new_freq / 2.0 * (passb[0] - passb[1]) +
                    sqrt(new_freq ** 2 * (passb[1] - passb[0]) ** 2 / 4.0 +
                         passb[1] * passb[0]))
          nat[1] = passb[1] * passb[0] / nat[0]
      elif filter_type == 4:
          nat = numpy.zeros(2, float)
          nat[0] = (1.0 / (2.0 * new_freq) * (passb[0] - passb[1]) +
                    sqrt((passb[1] - passb[0]) ** 2 / (4.0 * new_freq ** 2) +
                         passb[1] * passb[0]))
          nat[1] = passb[0] * passb[1] / nat[0]
  
      if not analog:
          wn = (2.0 / pi) * arctan(nat)
      else:
          wn = nat
  
      if len(wn) == 1:
          wn = wn[0]
      return ord, wn
  
  
  def ellipord(wp, ws, gpass, gstop, analog=False):
      """Elliptic (Cauer) filter order selection.
  
      Return the order of the lowest order digital or analog elliptic filter
      that loses no more than `gpass` dB in the passband and has at least
      `gstop` dB attenuation in the stopband.
  
      Parameters
      ----------
      wp, ws : float
          Passband and stopband edge frequencies.
          For digital filters, these are normalized from 0 to 1, where 1 is the
          Nyquist frequency, pi radians/sample.  (`wp` and `ws` are thus in
          half-cycles / sample.)  For example:
  
              - Lowpass:   wp = 0.2,          ws = 0.3
              - Highpass:  wp = 0.3,          ws = 0.2
              - Bandpass:  wp = [0.2, 0.5],   ws = [0.1, 0.6]
              - Bandstop:  wp = [0.1, 0.6],   ws = [0.2, 0.5]
  
          For analog filters, `wp` and `ws` are angular frequencies (e.g. rad/s).
  
      gpass : float
          The maximum loss in the passband (dB).
      gstop : float
          The minimum attenuation in the stopband (dB).
      analog : bool, optional
          When True, return an analog filter, otherwise a digital filter is
          returned.
  
      Returns
      -------
      ord : int
          The lowest order for an Elliptic (Cauer) filter that meets specs.
      wn : ndarray or float
          The Chebyshev natural frequency (the "3dB frequency") for use with
          `ellip` to give filter results.
  
      See Also
      --------
      ellip : Filter design using order and critical points
      buttord : Find order and critical points from passband and stopband spec
      cheb1ord, cheb2ord
      iirfilter : General filter design using order and critical frequencies
      iirdesign : General filter design using passband and stopband spec
  
      Examples
      --------
      Design an analog highpass filter such that the passband is within 3 dB
      above 30 rad/s, while rejecting -60 dB at 10 rad/s.  Plot its
      frequency response, showing the passband and stopband constraints in gray.
  
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      >>> N, Wn = signal.ellipord(30, 10, 3, 60, True)
      >>> b, a = signal.ellip(N, 3, 60, Wn, 'high', True)
      >>> w, h = signal.freqs(b, a, np.logspace(0, 3, 500))
      >>> plt.semilogx(w, 20 * np.log10(abs(h)))
      >>> plt.title('Elliptical highpass filter fit to constraints')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.grid(which='both', axis='both')
      >>> plt.fill([.1, 10,  10,  .1], [1e4, 1e4, -60, -60], '0.9', lw=0) # stop
      >>> plt.fill([30, 30, 1e9, 1e9], [-99,  -3,  -3, -99], '0.9', lw=0) # pass
      >>> plt.axis([1, 300, -80, 3])
      >>> plt.show()
  
      """
      wp = atleast_1d(wp)
      ws = atleast_1d(ws)
      filter_type = 2 * (len(wp) - 1)
      filter_type += 1
      if wp[0] >= ws[0]:
          filter_type += 1
  
      # Pre-warp frequencies for digital filter design
      if not analog:
          passb = tan(pi * wp / 2.0)
          stopb = tan(pi * ws / 2.0)
      else:
          passb = wp * 1.0
          stopb = ws * 1.0
  
      if filter_type == 1:           # low
          nat = stopb / passb
      elif filter_type == 2:          # high
          nat = passb / stopb
      elif filter_type == 3:     # stop
          wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12,
                                   args=(0, passb, stopb, gpass, gstop, 'ellip'),
                                   disp=0)
          passb[0] = wp0
          wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1],
                                   args=(1, passb, stopb, gpass, gstop, 'ellip'),
                                   disp=0)
          passb[1] = wp1
          nat = ((stopb * (passb[0] - passb[1])) /
                 (stopb ** 2 - passb[0] * passb[1]))
      elif filter_type == 4:  # pass
          nat = ((stopb ** 2 - passb[0] * passb[1]) /
                 (stopb * (passb[0] - passb[1])))
  
      nat = min(abs(nat))
  
      GSTOP = 10 ** (0.1 * gstop)
      GPASS = 10 ** (0.1 * gpass)
      arg1 = sqrt((GPASS - 1.0) / (GSTOP - 1.0))
      arg0 = 1.0 / nat
      d0 = special.ellipk([arg0 ** 2, 1 - arg0 ** 2])
      d1 = special.ellipk([arg1 ** 2, 1 - arg1 ** 2])
      ord = int(ceil(d0[0] * d1[1] / (d0[1] * d1[0])))
  
      if not analog:
          wn = arctan(passb) * 2.0 / pi
      else:
          wn = passb
  
      if len(wn) == 1:
          wn = wn[0]
      return ord, wn
  
  
  def buttap(N):
      """Return (z,p,k) for analog prototype of Nth-order Butterworth filter.
  
      The filter will have an angular (e.g. rad/s) cutoff frequency of 1.
  
      See Also
      --------
      butter : Filter design function using this prototype
  
      """
      if abs(int(N)) != N:
          raise ValueError("Filter order must be a nonnegative integer")
      z = numpy.array([])
      m = numpy.arange(-N+1, N, 2)
      # Middle value is 0 to ensure an exactly real pole
      p = -numpy.exp(1j * pi * m / (2 * N))
      k = 1
      return z, p, k
  
  
  def cheb1ap(N, rp):
      """
      Return (z,p,k) for Nth-order Chebyshev type I analog lowpass filter.
  
      The returned filter prototype has `rp` decibels of ripple in the passband.
  
      The filter's angular (e.g. rad/s) cutoff frequency is normalized to 1,
      defined as the point at which the gain first drops below ``-rp``.
  
      See Also
      --------
      cheby1 : Filter design function using this prototype
  
      """
      if abs(int(N)) != N:
          raise ValueError("Filter order must be a nonnegative integer")
      elif N == 0:
          # Avoid divide-by-zero error
          # Even order filters have DC gain of -rp dB
          return numpy.array([]), numpy.array([]), 10**(-rp/20)
      z = numpy.array([])
  
      # Ripple factor (epsilon)
      eps = numpy.sqrt(10 ** (0.1 * rp) - 1.0)
      mu = 1.0 / N * arcsinh(1 / eps)
  
      # Arrange poles in an ellipse on the left half of the S-plane
      m = numpy.arange(-N+1, N, 2)
      theta = pi * m / (2*N)
      p = -sinh(mu + 1j*theta)
  
      k = numpy.prod(-p, axis=0).real
      if N % 2 == 0:
          k = k / sqrt((1 + eps * eps))
  
      return z, p, k
  
  
  def cheb2ap(N, rs):
      """
      Return (z,p,k) for Nth-order Chebyshev type I analog lowpass filter.
  
      The returned filter prototype has `rs` decibels of ripple in the stopband.
  
      The filter's angular (e.g. rad/s) cutoff frequency is normalized to 1,
      defined as the point at which the gain first reaches ``-rs``.
  
      See Also
      --------
      cheby2 : Filter design function using this prototype
  
      """
      if abs(int(N)) != N:
          raise ValueError("Filter order must be a nonnegative integer")
      elif N == 0:
          # Avoid divide-by-zero warning
          return numpy.array([]), numpy.array([]), 1
  
      # Ripple factor (epsilon)
      de = 1.0 / sqrt(10 ** (0.1 * rs) - 1)
      mu = arcsinh(1.0 / de) / N
  
      if N % 2:
          m = numpy.concatenate((numpy.arange(-N+1, 0, 2),
                                 numpy.arange(2, N, 2)))
      else:
          m = numpy.arange(-N+1, N, 2)
  
      z = -conjugate(1j / sin(m * pi / (2.0 * N)))
  
      # Poles around the unit circle like Butterworth
      p = -exp(1j * pi * numpy.arange(-N+1, N, 2) / (2 * N))
      # Warp into Chebyshev II
      p = sinh(mu) * p.real + 1j * cosh(mu) * p.imag
      p = 1.0 / p
  
      k = (numpy.prod(-p, axis=0) / numpy.prod(-z, axis=0)).real
      return z, p, k
  
  
  EPSILON = 2e-16
  
  
  def _vratio(u, ineps, mp):
      [s, c, d, phi] = special.ellipj(u, mp)
      ret = abs(ineps - s / c)
      return ret
  
  
  def _kratio(m, k_ratio):
      m = float(m)
      if m < 0:
          m = 0.0
      if m > 1:
          m = 1.0
      if abs(m) > EPSILON and (abs(m) + EPSILON) < 1:
          k = special.ellipk([m, 1 - m])
          r = k[0] / k[1] - k_ratio
      elif abs(m) > EPSILON:
          r = -k_ratio
      else:
          r = 1e20
      return abs(r)
  
  
  def ellipap(N, rp, rs):
      """Return (z,p,k) of Nth-order elliptic analog lowpass filter.
  
      The filter is a normalized prototype that has `rp` decibels of ripple
      in the passband and a stopband `rs` decibels down.
  
      The filter's angular (e.g. rad/s) cutoff frequency is normalized to 1,
      defined as the point at which the gain first drops below ``-rp``.
  
      See Also
      --------
      ellip : Filter design function using this prototype
  
      References
      ----------
      .. [1] Lutova, Tosic, and Evans, "Filter Design for Signal Processing",
             Chapters 5 and 12.
  
      """
      if abs(int(N)) != N:
          raise ValueError("Filter order must be a nonnegative integer")
      elif N == 0:
          # Avoid divide-by-zero warning
          # Even order filters have DC gain of -rp dB
          return numpy.array([]), numpy.array([]), 10**(-rp/20)
      elif N == 1:
          p = -sqrt(1.0 / (10 ** (0.1 * rp) - 1.0))
          k = -p
          z = []
          return asarray(z), asarray(p), k
  
      eps = numpy.sqrt(10 ** (0.1 * rp) - 1)
      ck1 = eps / numpy.sqrt(10 ** (0.1 * rs) - 1)
      ck1p = numpy.sqrt(1 - ck1 * ck1)
      if ck1p == 1:
          raise ValueError("Cannot design a filter with given rp and rs"
                           " specifications.")
  
      val = special.ellipk([ck1 * ck1, ck1p * ck1p])
      if abs(1 - ck1p * ck1p) < EPSILON:
          krat = 0
      else:
          krat = N * val[0] / val[1]
  
      m = optimize.fmin(_kratio, [0.5], args=(krat,), maxfun=250, maxiter=250,
                        disp=0)
      if m < 0 or m > 1:
          m = optimize.fminbound(_kratio, 0, 1, args=(krat,), maxfun=250,
                                 maxiter=250, disp=0)
  
      capk = special.ellipk(m)
  
      j = numpy.arange(1 - N % 2, N, 2)
      jj = len(j)
  
      [s, c, d, phi] = special.ellipj(j * capk / N, m * numpy.ones(jj))
      snew = numpy.compress(abs(s) > EPSILON, s, axis=-1)
      z = 1.0 / (sqrt(m) * snew)
      z = 1j * z
      z = numpy.concatenate((z, conjugate(z)))
  
      r = optimize.fmin(_vratio, special.ellipk(m), args=(1. / eps, ck1p * ck1p),
                        maxfun=250, maxiter=250, disp=0)
      v0 = capk * r / (N * val[0])
  
      [sv, cv, dv, phi] = special.ellipj(v0, 1 - m)
      p = -(c * d * sv * cv + 1j * s * dv) / (1 - (d * sv) ** 2.0)
  
      if N % 2:
          newp = numpy.compress(abs(p.imag) > EPSILON *
                                numpy.sqrt(numpy.sum(p * numpy.conjugate(p),
                                                     axis=0).real),
                                p, axis=-1)
          p = numpy.concatenate((p, conjugate(newp)))
      else:
          p = numpy.concatenate((p, conjugate(p)))
  
      k = (numpy.prod(-p, axis=0) / numpy.prod(-z, axis=0)).real
      if N % 2 == 0:
          k = k / numpy.sqrt((1 + eps * eps))
  
      return z, p, k
  
  
  # TODO: Make this a real public function scipy.misc.ff
  def _falling_factorial(x, n):
      r"""
      Return the factorial of `x` to the `n` falling.
  
      This is defined as:
  
      .. math::   x^\underline n = (x)_n = x (x-1) \cdots (x-n+1)
  
      This can more efficiently calculate ratios of factorials, since:
  
      n!/m! == falling_factorial(n, n-m)
  
      where n >= m
  
      skipping the factors that cancel out
  
      the usual factorial n! == ff(n, n)
      """
      val = 1
      for k in range(x - n + 1, x + 1):
          val *= k
      return val
  
  
  def _bessel_poly(n, reverse=False):
      """
      Return the coefficients of Bessel polynomial of degree `n`
  
      If `reverse` is true, a reverse Bessel polynomial is output.
  
      Output is a list of coefficients:
      [1]                   = 1
      [1,  1]               = 1*s   +  1
      [1,  3,  3]           = 1*s^2 +  3*s   +  3
      [1,  6, 15, 15]       = 1*s^3 +  6*s^2 + 15*s   +  15
      [1, 10, 45, 105, 105] = 1*s^4 + 10*s^3 + 45*s^2 + 105*s + 105
      etc.
  
      Output is a Python list of arbitrary precision long ints, so n is only
      limited by your hardware's memory.
  
      Sequence is http://oeis.org/A001498 , and output can be confirmed to
      match http://oeis.org/A001498/b001498.txt :
  
      >>> i = 0
      >>> for n in range(51):
      ...     for x in _bessel_poly(n, reverse=True):
      ...         print(i, x)
      ...         i += 1
  
      """
      if abs(int(n)) != n:
          raise ValueError("Polynomial order must be a nonnegative integer")
      else:
          n = int(n)  # np.int32 doesn't work, for instance
  
      out = []
      for k in range(n + 1):
          num = _falling_factorial(2*n - k, n)
          den = 2**(n - k) * factorial(k, exact=True)
          out.append(num // den)
  
      if reverse:
          return out[::-1]
      else:
          return out
  
  
  def _campos_zeros(n):
      """
      Return approximate zero locations of Bessel polynomials y_n(x) for order
      `n` using polynomial fit (Campos-Calderon 2011)
      """
      if n == 1:
          return asarray([-1+0j])
  
      s = npp_polyval(n, [0, 0, 2, 0, -3, 1])
      b3 = npp_polyval(n, [16, -8]) / s
      b2 = npp_polyval(n, [-24, -12, 12]) / s
      b1 = npp_polyval(n, [8, 24, -12, -2]) / s
      b0 = npp_polyval(n, [0, -6, 0, 5, -1]) / s
  
      r = npp_polyval(n, [0, 0, 2, 1])
      a1 = npp_polyval(n, [-6, -6]) / r
      a2 = 6 / r
  
      k = np.arange(1, n+1)
      x = npp_polyval(k, [0, a1, a2])
      y = npp_polyval(k, [b0, b1, b2, b3])
  
      return x + 1j*y
  
  
  def _aberth(f, fp, x0, tol=1e-15, maxiter=50):
      """
      Given a function `f`, its first derivative `fp`, and a set of initial
      guesses `x0`, simultaneously find the roots of the polynomial using the
      Aberth-Ehrlich method.
  
      ``len(x0)`` should equal the number of roots of `f`.
  
      (This is not a complete implementation of Bini's algorithm.)
      """
  
      N = len(x0)
  
      x = array(x0, complex)
      beta = np.empty_like(x0)
  
      for iteration in range(maxiter):
          alpha = -f(x) / fp(x)  # Newton's method
  
          # Model "repulsion" between zeros
          for k in range(N):
              beta[k] = np.sum(1/(x[k] - x[k+1:]))
              beta[k] += np.sum(1/(x[k] - x[:k]))
  
          x += alpha / (1 + alpha * beta)
  
          if not all(np.isfinite(x)):
              raise RuntimeError('Root-finding calculation failed')
  
          # Mekwi: The iterative process can be stopped when |hn| has become
          # less than the largest error one is willing to permit in the root.
          if all(abs(alpha) <= tol):
              break
      else:
          raise Exception('Zeros failed to converge')
  
      return x
  
  
  def _bessel_zeros(N):
      """
      Find zeros of ordinary Bessel polynomial of order `N`, by root-finding of
      modified Bessel function of the second kind
      """
      if N == 0:
          return asarray([])
  
      # Generate starting points
      x0 = _campos_zeros(N)
  
      # Zeros are the same for exp(1/x)*K_{N+0.5}(1/x) and Nth-order ordinary
      # Bessel polynomial y_N(x)
      def f(x):
          return special.kve(N+0.5, 1/x)
  
      # First derivative of above
      def fp(x):
          return (special.kve(N-0.5, 1/x)/(2*x**2) -
                  special.kve(N+0.5, 1/x)/(x**2) +
                  special.kve(N+1.5, 1/x)/(2*x**2))
  
      # Starting points converge to true zeros
      x = _aberth(f, fp, x0)
  
      # Improve precision using Newton's method on each
      for i in range(len(x)):
          x[i] = optimize.newton(f, x[i], fp, tol=1e-15)
  
      # Average complex conjugates to make them exactly symmetrical
      x = np.mean((x, x[::-1].conj()), 0)
  
      # Zeros should sum to -1
      if abs(np.sum(x) + 1) > 1e-15:
          raise RuntimeError('Generated zeros are inaccurate')
  
      return x
  
  
  def _norm_factor(p, k):
      """
      Numerically find frequency shift to apply to delay-normalized filter such
      that -3 dB point is at 1 rad/sec.
  
      `p` is an array_like of polynomial poles
      `k` is a float gain
  
      First 10 values are listed in "Bessel Scale Factors" table,
      "Bessel Filters Polynomials, Poles and Circuit Elements 2003, C. Bond."
      """
      p = asarray(p, dtype=complex)
  
      def G(w):
          """
          Gain of filter
          """
          return abs(k / prod(1j*w - p))
  
      def cutoff(w):
          """
          When gain = -3 dB, return 0
          """
          return G(w) - 1/np.sqrt(2)
  
      return optimize.newton(cutoff, 1.5)
  
  
  def besselap(N, norm='phase'):
      """
      Return (z,p,k) for analog prototype of an Nth-order Bessel filter.
  
      Parameters
      ----------
      N : int
          The order of the filter.
      norm : {'phase', 'delay', 'mag'}, optional
          Frequency normalization:
  
          ``phase``
              The filter is normalized such that the phase response reaches its
              midpoint at an angular (e.g. rad/s) cutoff frequency of 1.  This
              happens for both low-pass and high-pass filters, so this is the
              "phase-matched" case. [6]_
  
              The magnitude response asymptotes are the same as a Butterworth
              filter of the same order with a cutoff of `Wn`.
  
              This is the default, and matches MATLAB's implementation.
  
          ``delay``
              The filter is normalized such that the group delay in the passband
              is 1 (e.g. 1 second).  This is the "natural" type obtained by
              solving Bessel polynomials
  
          ``mag``
              The filter is normalized such that the gain magnitude is -3 dB at
              angular frequency 1.  This is called "frequency normalization" by
              Bond. [1]_
  
          .. versionadded:: 0.18.0
  
      Returns
      -------
      z : ndarray
          Zeros of the transfer function. Is always an empty array.
      p : ndarray
          Poles of the transfer function.
      k : scalar
          Gain of the transfer function.  For phase-normalized, this is always 1.
  
      See Also
      --------
      bessel : Filter design function using this prototype
  
      Notes
      -----
      To find the pole locations, approximate starting points are generated [2]_
      for the zeros of the ordinary Bessel polynomial [3]_, then the
      Aberth-Ehrlich method [4]_ [5]_ is used on the Kv(x) Bessel function to
      calculate more accurate zeros, and these locations are then inverted about
      the unit circle.
  
      References
      ----------
      .. [1] C.R. Bond, "Bessel Filter Constants",
             http://www.crbond.com/papers/bsf.pdf
      .. [2] Campos and Calderon, "Approximate closed-form formulas for the
             zeros of the Bessel Polynomials", :arXiv:`1105.0957`.
      .. [3] Thomson, W.E., "Delay Networks having Maximally Flat Frequency
             Characteristics", Proceedings of the Institution of Electrical
             Engineers, Part III, November 1949, Vol. 96, No. 44, pp. 487-490.
      .. [4] Aberth, "Iteration Methods for Finding all Zeros of a Polynomial
             Simultaneously", Mathematics of Computation, Vol. 27, No. 122,
             April 1973
      .. [5] Ehrlich, "A modified Newton method for polynomials", Communications
             of the ACM, Vol. 10, Issue 2, pp. 107-108, Feb. 1967,
             :DOI:`10.1145/363067.363115`
      .. [6] Miller and Bohn, "A Bessel Filter Crossover, and Its Relation to
             Others", RaneNote 147, 1998, http://www.rane.com/note147.html
  
      """
      if abs(int(N)) != N:
          raise ValueError("Filter order must be a nonnegative integer")
      if N == 0:
          p = []
          k = 1
      else:
          # Find roots of reverse Bessel polynomial
          p = 1/_bessel_zeros(N)
  
          a_last = _falling_factorial(2*N, N) // 2**N
  
          # Shift them to a different normalization if required
          if norm in ('delay', 'mag'):
              # Normalized for group delay of 1
              k = a_last
              if norm == 'mag':
                  # -3 dB magnitude point is at 1 rad/sec
                  norm_factor = _norm_factor(p, k)
                  p /= norm_factor
                  k = norm_factor**-N * a_last
          elif norm == 'phase':
              # Phase-matched (1/2 max phase shift at 1 rad/sec)
              # Asymptotes are same as Butterworth filter
              p *= 10**(-math.log10(a_last)/N)
              k = 1
          else:
              raise ValueError('normalization not understood')
  
      return asarray([]), asarray(p, dtype=complex), float(k)
  
  
  def iirnotch(w0, Q):
      """
      Design second-order IIR notch digital filter.
  
      A notch filter is a band-stop filter with a narrow bandwidth
      (high quality factor). It rejects a narrow frequency band and
      leaves the rest of the spectrum little changed.
  
      Parameters
      ----------
      w0 : float
          Normalized frequency to remove from a signal. It is a
          scalar that must satisfy  ``0 < w0 < 1``, with ``w0 = 1``
          corresponding to half of the sampling frequency.
      Q : float
          Quality factor. Dimensionless parameter that characterizes
          notch filter -3 dB bandwidth ``bw`` relative to its center
          frequency, ``Q = w0/bw``.
  
      Returns
      -------
      b, a : ndarray, ndarray
          Numerator (``b``) and denominator (``a``) polynomials
          of the IIR filter.
  
      See Also
      --------
      iirpeak
  
      Notes
      -----
      .. versionadded: 0.19.0
  
      References
      ----------
      .. [1] Sophocles J. Orfanidis, "Introduction To Signal Processing",
             Prentice-Hall, 1996
  
      Examples
      --------
      Design and plot filter to remove the 60Hz component from a
      signal sampled at 200Hz, using a quality factor Q = 30
  
      >>> from scipy import signal
      >>> import numpy as np
      >>> import matplotlib.pyplot as plt
  
      >>> fs = 200.0  # Sample frequency (Hz)
      >>> f0 = 60.0  # Frequency to be removed from signal (Hz)
      >>> Q = 30.0  # Quality factor
      >>> w0 = f0/(fs/2)  # Normalized Frequency
      >>> # Design notch filter
      >>> b, a = signal.iirnotch(w0, Q)
  
      >>> # Frequency response
      >>> w, h = signal.freqz(b, a)
      >>> # Generate frequency axis
      >>> freq = w*fs/(2*np.pi)
      >>> # Plot
      >>> fig, ax = plt.subplots(2, 1, figsize=(8, 6))
      >>> ax[0].plot(freq, 20*np.log10(abs(h)), color='blue')
      >>> ax[0].set_title("Frequency Response")
      >>> ax[0].set_ylabel("Amplitude (dB)", color='blue')
      >>> ax[0].set_xlim([0, 100])
      >>> ax[0].set_ylim([-25, 10])
      >>> ax[0].grid()
      >>> ax[1].plot(freq, np.unwrap(np.angle(h))*180/np.pi, color='green')
      >>> ax[1].set_ylabel("Angle (degrees)", color='green')
      >>> ax[1].set_xlabel("Frequency (Hz)")
      >>> ax[1].set_xlim([0, 100])
      >>> ax[1].set_yticks([-90, -60, -30, 0, 30, 60, 90])
      >>> ax[1].set_ylim([-90, 90])
      >>> ax[1].grid()
      >>> plt.show()
      """
  
      return _design_notch_peak_filter(w0, Q, "notch")
  
  
  def iirpeak(w0, Q):
      """
      Design second-order IIR peak (resonant) digital filter.
  
      A peak filter is a band-pass filter with a narrow bandwidth
      (high quality factor). It rejects components outside a narrow
      frequency band.
  
      Parameters
      ----------
      w0 : float
          Normalized frequency to be retained in a signal. It is a
          scalar that must satisfy  ``0 < w0 < 1``, with ``w0 = 1`` corresponding
          to half of the sampling frequency.
      Q : float
          Quality factor. Dimensionless parameter that characterizes
          peak filter -3 dB bandwidth ``bw`` relative to its center
          frequency, ``Q = w0/bw``.
  
      Returns
      -------
      b, a : ndarray, ndarray
          Numerator (``b``) and denominator (``a``) polynomials
          of the IIR filter.
  
      See Also
      --------
      iirnotch
  
      Notes
      -----
      .. versionadded: 0.19.0
  
      References
      ----------
      .. [1] Sophocles J. Orfanidis, "Introduction To Signal Processing",
             Prentice-Hall, 1996
  
      Examples
      --------
      Design and plot filter to remove the frequencies other than the 300Hz
      component from a signal sampled at 1000Hz, using a quality factor Q = 30
  
      >>> from scipy import signal
      >>> import numpy as np
      >>> import matplotlib.pyplot as plt
  
      >>> fs = 1000.0  # Sample frequency (Hz)
      >>> f0 = 300.0  # Frequency to be retained (Hz)
      >>> Q = 30.0  # Quality factor
      >>> w0 = f0/(fs/2)  # Normalized Frequency
      >>> # Design peak filter
      >>> b, a = signal.iirpeak(w0, Q)
  
      >>> # Frequency response
      >>> w, h = signal.freqz(b, a)
      >>> # Generate frequency axis
      >>> freq = w*fs/(2*np.pi)
      >>> # Plot
      >>> fig, ax = plt.subplots(2, 1, figsize=(8, 6))
      >>> ax[0].plot(freq, 20*np.log10(abs(h)), color='blue')
      >>> ax[0].set_title("Frequency Response")
      >>> ax[0].set_ylabel("Amplitude (dB)", color='blue')
      >>> ax[0].set_xlim([0, 500])
      >>> ax[0].set_ylim([-50, 10])
      >>> ax[0].grid()
      >>> ax[1].plot(freq, np.unwrap(np.angle(h))*180/np.pi, color='green')
      >>> ax[1].set_ylabel("Angle (degrees)", color='green')
      >>> ax[1].set_xlabel("Frequency (Hz)")
      >>> ax[1].set_xlim([0, 500])
      >>> ax[1].set_yticks([-90, -60, -30, 0, 30, 60, 90])
      >>> ax[1].set_ylim([-90, 90])
      >>> ax[1].grid()
      >>> plt.show()
      """
  
      return _design_notch_peak_filter(w0, Q, "peak")
  
  
  def _design_notch_peak_filter(w0, Q, ftype):
      """
      Design notch or peak digital filter.
  
      Parameters
      ----------
      w0 : float
          Normalized frequency to remove from a signal. It is a
          scalar that must satisfy  ``0 < w0 < 1``, with ``w0 = 1``
          corresponding to half of the sampling frequency.
      Q : float
          Quality factor. Dimensionless parameter that characterizes
          notch filter -3 dB bandwidth ``bw`` relative to its center
          frequency, ``Q = w0/bw``.
      ftype : str
          The type of IIR filter to design:
  
              - notch filter : ``notch``
              - peak filter  : ``peak``
  
      Returns
      -------
      b, a : ndarray, ndarray
          Numerator (``b``) and denominator (``a``) polynomials
          of the IIR filter.
      """
  
      # Guarantee that the inputs are floats
      w0 = float(w0)
      Q = float(Q)
  
      # Checks if w0 is within the range
      if w0 > 1.0 or w0 < 0.0:
          raise ValueError("w0 should be such that 0 < w0 < 1")
  
      # Get bandwidth
      bw = w0/Q
  
      # Normalize inputs
      bw = bw*np.pi
      w0 = w0*np.pi
  
      # Compute -3dB atenuation
      gb = 1/np.sqrt(2)
  
      if ftype == "notch":
          # Compute beta: formula 11.3.4 (p.575) from reference [1]
          beta = (np.sqrt(1.0-gb**2.0)/gb)*np.tan(bw/2.0)
      elif ftype == "peak":
          # Compute beta: formula 11.3.19 (p.579) from reference [1]
          beta = (gb/np.sqrt(1.0-gb**2.0))*np.tan(bw/2.0)
      else:
          raise ValueError("Unknown ftype.")
  
      # Compute gain: formula 11.3.6 (p.575) from reference [1]
      gain = 1.0/(1.0+beta)
  
      # Compute numerator b and denominator a
      # formulas 11.3.7 (p.575) and 11.3.21 (p.579)
      # from reference [1]
      if ftype == "notch":
          b = gain*np.array([1.0, -2.0*np.cos(w0), 1.0])
      else:
          b = (1.0-gain)*np.array([1.0, 0.0, -1.0])
      a = np.array([1.0, -2.0*gain*np.cos(w0), (2.0*gain-1.0)])
  
      return b, a
  
  
  filter_dict = {'butter': [buttap, buttord],
                 'butterworth': [buttap, buttord],
  
                 'cauer': [ellipap, ellipord],
                 'elliptic': [ellipap, ellipord],
                 'ellip': [ellipap, ellipord],
  
                 'bessel': [besselap],
                 'bessel_phase': [besselap],
                 'bessel_delay': [besselap],
                 'bessel_mag': [besselap],
  
                 'cheby1': [cheb1ap, cheb1ord],
                 'chebyshev1': [cheb1ap, cheb1ord],
                 'chebyshevi': [cheb1ap, cheb1ord],
  
                 'cheby2': [cheb2ap, cheb2ord],
                 'chebyshev2': [cheb2ap, cheb2ord],
                 'chebyshevii': [cheb2ap, cheb2ord],
                 }
  
  band_dict = {'band': 'bandpass',
               'bandpass': 'bandpass',
               'pass': 'bandpass',
               'bp': 'bandpass',
  
               'bs': 'bandstop',
               'bandstop': 'bandstop',
               'bands': 'bandstop',
               'stop': 'bandstop',
  
               'l': 'lowpass',
               'low': 'lowpass',
               'lowpass': 'lowpass',
               'lp': 'lowpass',
  
               'high': 'highpass',
               'highpass': 'highpass',
               'h': 'highpass',
               'hp': 'highpass',
               }
  
  bessel_norms = {'bessel': 'phase',
                  'bessel_phase': 'phase',
                  'bessel_delay': 'delay',
                  'bessel_mag': 'mag'}
  
  