filter_design.py

"""Filter design.
"""
from __future__ import division, print_function, absolute_import

import warnings

import numpy
from numpy import atleast_1d, poly, polyval, roots, real, asarray, allclose, \
    resize, pi, absolute, logspace, r_, sqrt, tan, log10, arctan, arcsinh, \
    cos, exp, cosh, arccosh, ceil, conjugate, zeros, sinh
from numpy import mintypecode
from scipy import special, optimize
from scipy.misc import comb

__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',
           'filter_dict', 'band_dict', 'BadCoefficients']


class BadCoefficients(UserWarning):
    pass

abs = absolute


def findfreqs(num, den, N):
    """
    Find an array of frequencies for computing the response of a 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.  The coefficients are
        ordered from highest to lowest degree.
    N : int
        The length of the array to be computed.

    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)

    >>> 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])
    """
    ep = atleast_1d(roots(den)) + 0j
    tz = atleast_1d(roots(num)) + 0j

    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 numerator `b` and denominator `a` of a filter, compute its
    frequency response::

             b[0]*(jw)**(nb-1) + b[1]*(jw)**(nb-2) + ... + b[nb-1]
     H(w) = -------------------------------------------------------
             a[0]*(jw)**(na-1) + a[1]*(jw)**(na-2) + ... + a[na-1]

    Parameters
    ----------
    b : ndarray
        Numerator of a linear filter.
    a : ndarray
        Denominator of a linear filter.
    worN : {None, int}, 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
        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.

    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, abs(h))
    >>> plt.xlabel('Frequency')
    >>> plt.ylabel('Amplitude response')
    >>> 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 not plot is None:
        plot(w, h)
    return w, h


def freqz(b, a=1, worN=None, whole=0, plot=None):
    """
    Compute the frequency response of a digital filter.

    Given the numerator `b` and denominator `a` of a digital filter,
    compute its frequency response::

               jw               -jw            -jmw
        jw  B(e)    b[0] + b[1]e + .... + b[m]e
     H(e) = ---- = ------------------------------------
               jw               -jw            -jnw
            A(e)    a[0] + a[1]e + .... + a[n]e

    Parameters
    ----------
    b : ndarray
        numerator of a linear filter
    a : ndarray
        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.

    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.

    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.semilogy(w, np.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 not plot is None:
        plot(w, h)
    return w, h


def tf2zpk(b, a):
    """Return zero, pole, gain (z,p,k) representation from a numerator,
    denominator representation of a linear filter.

    Parameters
    ----------
    b : ndarray
        Numerator polynomial.
    a : ndarray
        Denominator polynomial.

    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.

    """
    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 : ndarray
        Zeros of the transfer function.
    p : ndarray
        Poles of the transfer function.
    k : float
        System gain.

    Returns
    -------
    b : ndarray
        Numerator polynomial.
    a : ndarray
        Denominator polynomial.

    """
    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))
    return b, a


def normalize(b, a):
    """Normalize polynomial representation of a transfer function.

    If values of `b` are too close to 0, they are removed. In that case, a
    BadCoefficients warning is emitted.

    """
    b, a = map(atleast_1d, (b, a))
    if len(a.shape) != 1:
        raise ValueError("Denominator polynomial must be rank-1 array.")
    if len(b.shape) > 2:
        raise ValueError("Numerator polynomial must be rank-1 or"
                         " rank-2 array.")
    if len(b.shape) == 1:
        b = asarray([b], b.dtype.char)
    while a[0] == 0.0 and len(a) > 1:
        a = a[1:]
    outb = b * (1.0) / a[0]
    outa = a * (1.0) / a[0]
    if allclose(0, outb[:, 0], atol=1e-14):
        warnings.warn("Badly conditioned filter coefficients (numerator): the "
                      "results may be meaningless", BadCoefficients)
        while allclose(0, outb[:, 0], atol=1e-14) and (outb.shape[-1] > 1):
            outb = outb[:, 1:]
    if outb.shape[0] == 1:
        outb = outb[0]
    return outb, outa


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 highpass 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') or pole-zero ('zpk') 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'}, optional
        Type of output:  numerator/denominator ('ba') or pole-zero ('zpk').
        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'``.

    """
    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 in (B,A) (numerator, denominator) or (Z,P,K) form.

    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'}, optional
        Type of output:  numerator/denominator ('ba') or pole-zero ('zpk').
        Default is 'ba'.

    See Also
    --------
    buttord, cheb1ord, cheb2ord, ellipord

    """
    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']:
        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 in [buttap, besselap]:
        z, p, k = typefunc(N)
    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)

    b, a = zpk2tf(z, p, k)

    # Pre-warp frequencies for digital filter design
    if not analog:
        fs = 2.0
        warped = 2 * fs * tan(pi * Wn / fs)
    else:
        warped = Wn

    # transform to lowpass, bandpass, highpass, or bandstop
    if btype == 'lowpass':
        b, a = lp2lp(b, a, wo=warped)
    elif btype == 'highpass':
        b, a = lp2hp(b, a, wo=warped)
    elif btype == 'bandpass':
        bw = warped[1] - warped[0]
        wo = sqrt(warped[0] * warped[1])
        b, a = lp2bp(b, a, wo=wo, bw=bw)
    elif btype == 'bandstop':
        bw = warped[1] - warped[0]
        wo = sqrt(warped[0] * warped[1])
        b, a = lp2bs(b, a, wo=wo, bw=bw)
    else:
        raise NotImplementedError("%s not implemented in iirfilter." % btype)

    # Find discrete equivalent if necessary
    if not analog:
        b, a = bilinear(b, a, fs=fs)

    # Transform to proper out type (pole-zero, state-space, numer-denom)
    if output == 'zpk':
        return tf2zpk(b, a)
    else:
        return b, a


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 in (B,A) or (Z,P,K) form.

    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'}, optional
        Type of output:  numerator/denominator ('ba') or pole-zero ('zpk').
        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'``.

    See also
    --------
    buttord

    Notes
    -----
    The Butterworth filter has maximally flat frequency response in the
    passband.

    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.plot(w, 20 * np.log10(abs(h)))
    >>> plt.xscale('log')
    >>> 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 in (B,A) or (Z,P,K) form.

    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'}, optional
        Type of output:  numerator/denominator ('ba') or pole-zero ('zpk').
        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'``.

    See also
    --------
    cheb1ord

    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.

    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.plot(w, 20 * np.log10(abs(h)))
    >>> plt.xscale('log')
    >>> 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 in (B,A) or (Z,P,K) form.

    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'}, optional
        Type of output:  numerator/denominator ('ba') or pole-zero ('zpk').
        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'``.

    See also
    --------
    cheb2ord

    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`).

    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.plot(w, 20 * np.log10(abs(h)))
    >>> plt.xscale('log')
    >>> 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 in (B,A) or (Z,P,K) form.

    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'}, optional
        Type of output:  numerator/denominator ('ba') or pole-zero ('zpk').
        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'``.

    See also
    --------
    ellipord

    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`).

    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.plot(w, 20 * np.log10(abs(h)))
    >>> plt.xscale('log')
    >>> 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'):
    """Bessel/Thomson digital and analog filter design.

    Design an Nth order digital or analog Bessel filter and return the
    filter coefficients in (B,A) or (Z,P,K) form.

    Parameters
    ----------
    N : int
        The order of the filter.
    Wn : array_like
        A scalar or length-2 sequence giving the critical frequencies.
        For a Bessel filter, this is defined as the point at which the
        asymptotes of the response are the same as a Butterworth filter of
        the same order.
        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'}, optional
        Type of output:  numerator/denominator ('ba') or pole-zero ('zpk').
        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'``.

    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.

    As order increases, the Bessel filter approaches a Gaussian filter.

    The digital Bessel filter is generated 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.

    For a given `Wn`, the lowpass and highpass filter have the same phase vs
    frequency curves; they are "phase-matched".

    Examples
    --------
    Plot the filter's frequency response, showing the flat group delay and
    the relationship to the Butterworth's cutoff frequency:

    >>> 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.plot(w, 20 * np.log10(np.abs(h)), color='silver', ls='dashed')
    >>> b, a = signal.bessel(4, 100, 'low', analog=True)
    >>> w, h = signal.freqs(b, a)
    >>> plt.plot(w, 20 * np.log10(np.abs(h)))
    >>> plt.xscale('log')
    >>> plt.title('Bessel filter frequency 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()

    >>> plt.figure()
    >>> plt.plot(w[1:], -np.diff(np.unwrap(np.angle(h)))/np.diff(w))
    >>> plt.xscale('log')
    >>> 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()

    """
    return iirfilter(N, Wn, btype=btype, analog=analog,
                     output=output, ftype='bessel')


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.

    """
    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 W0 (or the "3dB" frequency")
    # to give exactly gstop at nat. W0 will be between 1 and nat
    try:
        W0 = nat / ((10 ** (0.1 * abs(gstop)) - 1) ** (1.0 / (2.0 * ord)))
    except ZeroDivisionError:
        W0 = nat
        print("Warning, order is zero...check input parametegstop.")

    # 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.

    """
    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.

    """
    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.

    """
    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) zero, pole, gain for analog prototype of an Nth
    order Butterworth filter with an angular (e.g. rad/s) cutoff frequency
    of 1.

    """
    z = numpy.array([])
    n = numpy.arange(1, N + 1)
    p = numpy.exp(1j * (2 * n - 1) / (2.0 * N) * pi) * 1j
    k = 1
    return z, p, k


def cheb1ap(N, rp):
    """Return (z,p,k) zero, pole, gain for Nth order Chebyshev type I lowpass
    analog filter prototype with `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`.

    """
    z = numpy.array([])
    eps = numpy.sqrt(10 ** (0.1 * rp) - 1.0)
    n = numpy.arange(1, N + 1)
    mu = 1.0 / N * numpy.log((1.0 + numpy.sqrt(1 + eps * eps)) / eps)
    theta = pi / 2.0 * (2 * n - 1.0) / N
    p = (-numpy.sinh(mu) * numpy.sin(theta) +
         1j * numpy.cosh(mu) * numpy.cos(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) zero, pole, gain for Nth order Chebyshev type II lowpass
    analog filter prototype with `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`.

    """
    de = 1.0 / sqrt(10 ** (0.1 * rs) - 1)
    mu = arcsinh(1.0 / de) / N

    if N % 2:
        n = numpy.concatenate((numpy.arange(1, N - 1, 2),
                               numpy.arange(N + 2, 2 * N, 2)))
    else:
        n = numpy.arange(1, 2 * N, 2)

    z = conjugate(1j / cos(n * pi / (2.0 * N)))
    p = exp(1j * (pi * numpy.arange(1, 2 * N, 2) / (2.0 * N) + pi / 2.0))
    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) zeros, poles, and gain of an Nth order normalized
    prototype elliptic analog lowpass filter with `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`.

    References
    ----------
    Lutova, Tosic, and Evans, "Filter Design for Signal Processing", Chapters 5
    and 12.

    """
    if 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


def besselap(N):
    """Return (z,p,k) zero, pole, gain for analog prototype of an Nth order
    Bessel filter.

    The filter is normalized such that the filter asymptotes are the same as
    a Butterworth filter of the same order with an angular (e.g. rad/s)
    cutoff frequency of 1.

    """
    z = []
    k = 1
    if N == 0:
        p = []
    elif N == 1:
        p = [-1]
    elif N == 2:
        p = [-.8660254037844386467637229 + .4999999999999999999999996j,
             -.8660254037844386467637229 - .4999999999999999999999996j]
    elif N == 3:
        p = [-.9416000265332067855971980,
             -.7456403858480766441810907 - .7113666249728352680992154j,
             -.7456403858480766441810907 + .7113666249728352680992154j]
    elif N == 4:
        p = [-.6572111716718829545787781 - .8301614350048733772399715j,
             -.6572111716718829545787788 + .8301614350048733772399715j,
             -.9047587967882449459642637 - .2709187330038746636700923j,
             -.9047587967882449459642624 + .2709187330038746636700926j]
    elif N == 5:
        p = [-.9264420773877602247196260,
             -.8515536193688395541722677 - .4427174639443327209850002j,
             -.8515536193688395541722677 + .4427174639443327209850002j,
             -.5905759446119191779319432 - .9072067564574549539291747j,
             -.5905759446119191779319432 + .9072067564574549539291747j]
    elif N == 6:
        p = [-.9093906830472271808050953 - .1856964396793046769246397j,
             -.9093906830472271808050953 + .1856964396793046769246397j,
             -.7996541858328288520243325 - .5621717346937317988594118j,
             -.7996541858328288520243325 + .5621717346937317988594118j,
             -.5385526816693109683073792 - .9616876881954277199245657j,
             -.5385526816693109683073792 + .9616876881954277199245657j]
    elif N == 7:
        p = [-.9194871556490290014311619,
             -.8800029341523374639772340 - .3216652762307739398381830j,
             -.8800029341523374639772340 + .3216652762307739398381830j,
             -.7527355434093214462291616 - .6504696305522550699212995j,
             -.7527355434093214462291616 + .6504696305522550699212995j,
             -.4966917256672316755024763 - 1.002508508454420401230220j,
             -.4966917256672316755024763 + 1.002508508454420401230220j]
    elif N == 8:
        p = [-.9096831546652910216327629 - .1412437976671422927888150j,
             -.9096831546652910216327629 + .1412437976671422927888150j,
             -.8473250802359334320103023 - .4259017538272934994996429j,
             -.8473250802359334320103023 + .4259017538272934994996429j,
             -.7111381808485399250796172 - .7186517314108401705762571j,
             -.7111381808485399250796172 + .7186517314108401705762571j,
             -.4621740412532122027072175 - 1.034388681126901058116589j,
             -.4621740412532122027072175 + 1.034388681126901058116589j]
    elif N == 9:
        p = [-.9154957797499037686769223,
             -.8911217017079759323183848 - .2526580934582164192308115j,
             -.8911217017079759323183848 + .2526580934582164192308115j,
             -.8148021112269012975514135 - .5085815689631499483745341j,
             -.8148021112269012975514135 + .5085815689631499483745341j,
             -.6743622686854761980403401 - .7730546212691183706919682j,
             -.6743622686854761980403401 + .7730546212691183706919682j,
             -.4331415561553618854685942 - 1.060073670135929666774323j,
             -.4331415561553618854685942 + 1.060073670135929666774323j]
    elif N == 10:
        p = [-.9091347320900502436826431 - .1139583137335511169927714j,
             -.9091347320900502436826431 + .1139583137335511169927714j,
             -.8688459641284764527921864 - .3430008233766309973110589j,
             -.8688459641284764527921864 + .3430008233766309973110589j,
             -.7837694413101441082655890 - .5759147538499947070009852j,
             -.7837694413101441082655890 + .5759147538499947070009852j,
             -.6417513866988316136190854 - .8175836167191017226233947j,
             -.6417513866988316136190854 + .8175836167191017226233947j,
             -.4083220732868861566219785 - 1.081274842819124562037210j,
             -.4083220732868861566219785 + 1.081274842819124562037210j]
    elif N == 11:
        p = [-.9129067244518981934637318,
             -.8963656705721166099815744 - .2080480375071031919692341j,
             -.8963656705721166099815744 + .2080480375071031919692341j,
             -.8453044014712962954184557 - .4178696917801248292797448j,
             -.8453044014712962954184557 + .4178696917801248292797448j,
             -.7546938934722303128102142 - .6319150050721846494520941j,
             -.7546938934722303128102142 + .6319150050721846494520941j,
             -.6126871554915194054182909 - .8547813893314764631518509j,
             -.6126871554915194054182909 + .8547813893314764631518509j,
             -.3868149510055090879155425 - 1.099117466763120928733632j,
             -.3868149510055090879155425 + 1.099117466763120928733632j]
    elif N == 12:
        p = [-.9084478234140682638817772 - 95506365213450398415258360.0e-27j,
             -.9084478234140682638817772 + 95506365213450398415258360.0e-27j,
             -.8802534342016826507901575 - .2871779503524226723615457j,
             -.8802534342016826507901575 + .2871779503524226723615457j,
             -.8217296939939077285792834 - .4810212115100676440620548j,
             -.8217296939939077285792834 + .4810212115100676440620548j,
             -.7276681615395159454547013 - .6792961178764694160048987j,
             -.7276681615395159454547013 + .6792961178764694160048987j,
             -.5866369321861477207528215 - .8863772751320727026622149j,
             -.5866369321861477207528215 + .8863772751320727026622149j,
             -.3679640085526312839425808 - 1.114373575641546257595657j,
             -.3679640085526312839425808 + 1.114373575641546257595657j]
    elif N == 13:
        p = [-.9110914665984182781070663,
             -.8991314665475196220910718 - .1768342956161043620980863j,
             -.8991314665475196220910718 + .1768342956161043620980863j,
             -.8625094198260548711573628 - .3547413731172988997754038j,
             -.8625094198260548711573628 + .3547413731172988997754038j,
             -.7987460692470972510394686 - .5350752120696801938272504j,
             -.7987460692470972510394686 + .5350752120696801938272504j,
             -.7026234675721275653944062 - .7199611890171304131266374j,
             -.7026234675721275653944062 + .7199611890171304131266374j,
             -.5631559842430199266325818 - .9135900338325109684927731j,
             -.5631559842430199266325818 + .9135900338325109684927731j,
             -.3512792323389821669401925 - 1.127591548317705678613239j,
             -.3512792323389821669401925 + 1.127591548317705678613239j]
    elif N == 14:
        p = [-.9077932138396487614720659 - 82196399419401501888968130.0e-27j,
             -.9077932138396487614720659 + 82196399419401501888968130.0e-27j,
             -.8869506674916445312089167 - .2470079178765333183201435j,
             -.8869506674916445312089167 + .2470079178765333183201435j,
             -.8441199160909851197897667 - .4131653825102692595237260j,
             -.8441199160909851197897667 + .4131653825102692595237260j,
             -.7766591387063623897344648 - .5819170677377608590492434j,
             -.7766591387063623897344648 + .5819170677377608590492434j,
             -.6794256425119233117869491 - .7552857305042033418417492j,
             -.6794256425119233117869491 + .7552857305042033418417492j,
             -.5418766775112297376541293 - .9373043683516919569183099j,
             -.5418766775112297376541293 + .9373043683516919569183099j,
             -.3363868224902037330610040 - 1.139172297839859991370924j,
             -.3363868224902037330610040 + 1.139172297839859991370924j]
    elif N == 15:
        p = [-.9097482363849064167228581,
             -.9006981694176978324932918 - .1537681197278439351298882j,
             -.9006981694176978324932918 + .1537681197278439351298882j,
             -.8731264620834984978337843 - .3082352470564267657715883j,
             -.8731264620834984978337843 + .3082352470564267657715883j,
             -.8256631452587146506294553 - .4642348752734325631275134j,
             -.8256631452587146506294553 + .4642348752734325631275134j,
             -.7556027168970728127850416 - .6229396358758267198938604j,
             -.7556027168970728127850416 + .6229396358758267198938604j,
             -.6579196593110998676999362 - .7862895503722515897065645j,
             -.6579196593110998676999362 + .7862895503722515897065645j,
             -.5224954069658330616875186 - .9581787261092526478889345j,
             -.5224954069658330616875186 + .9581787261092526478889345j,
             -.3229963059766444287113517 - 1.149416154583629539665297j,
             -.3229963059766444287113517 + 1.149416154583629539665297j]
    elif N == 16:
        p = [-.9072099595087001356491337 - 72142113041117326028823950.0e-27j,
             -.9072099595087001356491337 + 72142113041117326028823950.0e-27j,
             -.8911723070323647674780132 - .2167089659900576449410059j,
             -.8911723070323647674780132 + .2167089659900576449410059j,
             -.8584264231521330481755780 - .3621697271802065647661080j,
             -.8584264231521330481755780 + .3621697271802065647661080j,
             -.8074790293236003885306146 - .5092933751171800179676218j,
             -.8074790293236003885306146 + .5092933751171800179676218j,
             -.7356166304713115980927279 - .6591950877860393745845254j,
             -.7356166304713115980927279 + .6591950877860393745845254j,
             -.6379502514039066715773828 - .8137453537108761895522580j,
             -.6379502514039066715773828 + .8137453537108761895522580j,
             -.5047606444424766743309967 - .9767137477799090692947061j,
             -.5047606444424766743309967 + .9767137477799090692947061j,
             -.3108782755645387813283867 - 1.158552841199330479412225j,
             -.3108782755645387813283867 + 1.158552841199330479412225j]
    elif N == 17:
        p = [-.9087141161336397432860029,
             -.9016273850787285964692844 - .1360267995173024591237303j,
             -.9016273850787285964692844 + .1360267995173024591237303j,
             -.8801100704438627158492165 - .2725347156478803885651973j,
             -.8801100704438627158492165 + .2725347156478803885651973j,
             -.8433414495836129204455491 - .4100759282910021624185986j,
             -.8433414495836129204455491 + .4100759282910021624185986j,
             -.7897644147799708220288138 - .5493724405281088674296232j,
             -.7897644147799708220288138 + .5493724405281088674296232j,
             -.7166893842372349049842743 - .6914936286393609433305754j,
             -.7166893842372349049842743 + .6914936286393609433305754j,
             -.6193710717342144521602448 - .8382497252826992979368621j,
             -.6193710717342144521602448 + .8382497252826992979368621j,
             -.4884629337672704194973683 - .9932971956316781632345466j,
             -.4884629337672704194973683 + .9932971956316781632345466j,
             -.2998489459990082015466971 - 1.166761272925668786676672j,
             -.2998489459990082015466971 + 1.166761272925668786676672j]
    elif N == 18:
        p = [-.9067004324162775554189031 - 64279241063930693839360680.0e-27j,
             -.9067004324162775554189031 + 64279241063930693839360680.0e-27j,
             -.8939764278132455733032155 - .1930374640894758606940586j,
             -.8939764278132455733032155 + .1930374640894758606940586j,
             -.8681095503628830078317207 - .3224204925163257604931634j,
             -.8681095503628830078317207 + .3224204925163257604931634j,
             -.8281885016242836608829018 - .4529385697815916950149364j,
             -.8281885016242836608829018 + .4529385697815916950149364j,
             -.7726285030739558780127746 - .5852778162086640620016316j,
             -.7726285030739558780127746 + .5852778162086640620016316j,
             -.6987821445005273020051878 - .7204696509726630531663123j,
             -.6987821445005273020051878 + .7204696509726630531663123j,
             -.6020482668090644386627299 - .8602708961893664447167418j,
             -.6020482668090644386627299 + .8602708961893664447167418j,
             -.4734268069916151511140032 - 1.008234300314801077034158j,
             -.4734268069916151511140032 + 1.008234300314801077034158j,
             -.2897592029880489845789953 - 1.174183010600059128532230j,
             -.2897592029880489845789953 + 1.174183010600059128532230j]
    elif N == 19:
        p = [-.9078934217899404528985092,
             -.9021937639390660668922536 - .1219568381872026517578164j,
             -.9021937639390660668922536 + .1219568381872026517578164j,
             -.8849290585034385274001112 - .2442590757549818229026280j,
             -.8849290585034385274001112 + .2442590757549818229026280j,
             -.8555768765618421591093993 - .3672925896399872304734923j,
             -.8555768765618421591093993 + .3672925896399872304734923j,
             -.8131725551578197705476160 - .4915365035562459055630005j,
             -.8131725551578197705476160 + .4915365035562459055630005j,
             -.7561260971541629355231897 - .6176483917970178919174173j,
             -.7561260971541629355231897 + .6176483917970178919174173j,
             -.6818424412912442033411634 - .7466272357947761283262338j,
             -.6818424412912442033411634 + .7466272357947761283262338j,
             -.5858613321217832644813602 - .8801817131014566284786759j,
             -.5858613321217832644813602 + .8801817131014566284786759j,
             -.4595043449730988600785456 - 1.021768776912671221830298j,
             -.4595043449730988600785456 + 1.021768776912671221830298j,
             -.2804866851439370027628724 - 1.180931628453291873626003j,
             -.2804866851439370027628724 + 1.180931628453291873626003j]
    elif N == 20:
        p = [-.9062570115576771146523497 - 57961780277849516990208850.0e-27j,
             -.9062570115576771146523497 + 57961780277849516990208850.0e-27j,
             -.8959150941925768608568248 - .1740317175918705058595844j,
             -.8959150941925768608568248 + .1740317175918705058595844j,
             -.8749560316673332850673214 - .2905559296567908031706902j,
             -.8749560316673332850673214 + .2905559296567908031706902j,
             -.8427907479956670633544106 - .4078917326291934082132821j,
             -.8427907479956670633544106 + .4078917326291934082132821j,
             -.7984251191290606875799876 - .5264942388817132427317659j,
             -.7984251191290606875799876 + .5264942388817132427317659j,
             -.7402780309646768991232610 - .6469975237605228320268752j,
             -.7402780309646768991232610 + .6469975237605228320268752j,
             -.6658120544829934193890626 - .7703721701100763015154510j,
             -.6658120544829934193890626 + .7703721701100763015154510j,
             -.5707026806915714094398061 - .8982829066468255593407161j,
             -.5707026806915714094398061 + .8982829066468255593407161j,
             -.4465700698205149555701841 - 1.034097702560842962315411j,
             -.4465700698205149555701841 + 1.034097702560842962315411j,
             -.2719299580251652601727704 - 1.187099379810885886139638j,
             -.2719299580251652601727704 + 1.187099379810885886139638j]
    elif N == 21:
        p = [-.9072262653142957028884077,
             -.9025428073192696303995083 - .1105252572789856480992275j,
             -.9025428073192696303995083 + .1105252572789856480992275j,
             -.8883808106664449854431605 - .2213069215084350419975358j,
             -.8883808106664449854431605 + .2213069215084350419975358j,
             -.8643915813643204553970169 - .3326258512522187083009453j,
             -.8643915813643204553970169 + .3326258512522187083009453j,
             -.8299435470674444100273463 - .4448177739407956609694059j,
             -.8299435470674444100273463 + .4448177739407956609694059j,
             -.7840287980408341576100581 - .5583186348022854707564856j,
             -.7840287980408341576100581 + .5583186348022854707564856j,
             -.7250839687106612822281339 - .6737426063024382240549898j,
             -.7250839687106612822281339 + .6737426063024382240549898j,
             -.6506315378609463397807996 - .7920349342629491368548074j,
             -.6506315378609463397807996 + .7920349342629491368548074j,
             -.5564766488918562465935297 - .9148198405846724121600860j,
             -.5564766488918562465935297 + .9148198405846724121600860j,
             -.4345168906815271799687308 - 1.045382255856986531461592j,
             -.4345168906815271799687308 + 1.045382255856986531461592j,
             -.2640041595834031147954813 - 1.192762031948052470183960j,
             -.2640041595834031147954813 + 1.192762031948052470183960j]
    elif N == 22:
        p = [-.9058702269930872551848625 - 52774908289999045189007100.0e-27j,
             -.9058702269930872551848625 + 52774908289999045189007100.0e-27j,
             -.8972983138153530955952835 - .1584351912289865608659759j,
             -.8972983138153530955952835 + .1584351912289865608659759j,
             -.8799661455640176154025352 - .2644363039201535049656450j,
             -.8799661455640176154025352 + .2644363039201535049656450j,
             -.8534754036851687233084587 - .3710389319482319823405321j,
             -.8534754036851687233084587 + .3710389319482319823405321j,
             -.8171682088462720394344996 - .4785619492202780899653575j,
             -.8171682088462720394344996 + .4785619492202780899653575j,
             -.7700332930556816872932937 - .5874255426351153211965601j,
             -.7700332930556816872932937 + .5874255426351153211965601j,
             -.7105305456418785989070935 - .6982266265924524000098548j,
             -.7105305456418785989070935 + .6982266265924524000098548j,
             -.6362427683267827226840153 - .8118875040246347267248508j,
             -.6362427683267827226840153 + .8118875040246347267248508j,
             -.5430983056306302779658129 - .9299947824439872998916657j,
             -.5430983056306302779658129 + .9299947824439872998916657j,
             -.4232528745642628461715044 - 1.055755605227545931204656j,
             -.4232528745642628461715044 + 1.055755605227545931204656j,
             -.2566376987939318038016012 - 1.197982433555213008346532j,
             -.2566376987939318038016012 + 1.197982433555213008346532j]
    elif N == 23:
        p = [-.9066732476324988168207439,
             -.9027564979912504609412993 - .1010534335314045013252480j,
             -.9027564979912504609412993 + .1010534335314045013252480j,
             -.8909283242471251458653994 - .2023024699381223418195228j,
             -.8909283242471251458653994 + .2023024699381223418195228j,
             -.8709469395587416239596874 - .3039581993950041588888925j,
             -.8709469395587416239596874 + .3039581993950041588888925j,
             -.8423805948021127057054288 - .4062657948237602726779246j,
             -.8423805948021127057054288 + .4062657948237602726779246j,
             -.8045561642053176205623187 - .5095305912227258268309528j,
             -.8045561642053176205623187 + .5095305912227258268309528j,
             -.7564660146829880581478138 - .6141594859476032127216463j,
             -.7564660146829880581478138 + .6141594859476032127216463j,
             -.6965966033912705387505040 - .7207341374753046970247055j,
             -.6965966033912705387505040 + .7207341374753046970247055j,
             -.6225903228771341778273152 - .8301558302812980678845563j,
             -.6225903228771341778273152 + .8301558302812980678845563j,
             -.5304922463810191698502226 - .9439760364018300083750242j,
             -.5304922463810191698502226 + .9439760364018300083750242j,
             -.4126986617510148836149955 - 1.065328794475513585531053j,
             -.4126986617510148836149955 + 1.065328794475513585531053j,
             -.2497697202208956030229911 - 1.202813187870697831365338j,
             -.2497697202208956030229911 + 1.202813187870697831365338j]
    elif N == 24:
        p = [-.9055312363372773709269407 - 48440066540478700874836350.0e-27j,
             -.9055312363372773709269407 + 48440066540478700874836350.0e-27j,
             -.8983105104397872954053307 - .1454056133873610120105857j,
             -.8983105104397872954053307 + .1454056133873610120105857j,
             -.8837358034555706623131950 - .2426335234401383076544239j,
             -.8837358034555706623131950 + .2426335234401383076544239j,
             -.8615278304016353651120610 - .3403202112618624773397257j,
             -.8615278304016353651120610 + .3403202112618624773397257j,
             -.8312326466813240652679563 - .4386985933597305434577492j,
             -.8312326466813240652679563 + .4386985933597305434577492j,
             -.7921695462343492518845446 - .5380628490968016700338001j,
             -.7921695462343492518845446 + .5380628490968016700338001j,
             -.7433392285088529449175873 - .6388084216222567930378296j,
             -.7433392285088529449175873 + .6388084216222567930378296j,
             -.6832565803536521302816011 - .7415032695091650806797753j,
             -.6832565803536521302816011 + .7415032695091650806797753j,
             -.6096221567378335562589532 - .8470292433077202380020454j,
             -.6096221567378335562589532 + .8470292433077202380020454j,
             -.5185914574820317343536707 - .9569048385259054576937721j,
             -.5185914574820317343536707 + .9569048385259054576937721j,
             -.4027853855197518014786978 - 1.074195196518674765143729j,
             -.4027853855197518014786978 + 1.074195196518674765143729j,
             -.2433481337524869675825448 - 1.207298683731972524975429j,
             -.2433481337524869675825448 + 1.207298683731972524975429j]
    elif N == 25:
        p = [-.9062073871811708652496104,
             -.9028833390228020537142561 - 93077131185102967450643820.0e-27j,
             -.9028833390228020537142561 + 93077131185102967450643820.0e-27j,
             -.8928551459883548836774529 - .1863068969804300712287138j,
             -.8928551459883548836774529 + .1863068969804300712287138j,
             -.8759497989677857803656239 - .2798521321771408719327250j,
             -.8759497989677857803656239 + .2798521321771408719327250j,
             -.8518616886554019782346493 - .3738977875907595009446142j,
             -.8518616886554019782346493 + .3738977875907595009446142j,
             -.8201226043936880253962552 - .4686668574656966589020580j,
             -.8201226043936880253962552 + .4686668574656966589020580j,
             -.7800496278186497225905443 - .5644441210349710332887354j,
             -.7800496278186497225905443 + .5644441210349710332887354j,
             -.7306549271849967721596735 - .6616149647357748681460822j,
             -.7306549271849967721596735 + .6616149647357748681460822j,
             -.6704827128029559528610523 - .7607348858167839877987008j,
             -.6704827128029559528610523 + .7607348858167839877987008j,
             -.5972898661335557242320528 - .8626676330388028512598538j,
             -.5972898661335557242320528 + .8626676330388028512598538j,
             -.5073362861078468845461362 - .9689006305344868494672405j,
             -.5073362861078468845461362 + .9689006305344868494672405j,
             -.3934529878191079606023847 - 1.082433927173831581956863j,
             -.3934529878191079606023847 + 1.082433927173831581956863j,
             -.2373280669322028974199184 - 1.211476658382565356579418j,
             -.2373280669322028974199184 + 1.211476658382565356579418j]
    else:
        raise ValueError("Bessel Filter not supported for order %d" % N)

    return asarray(z), asarray(p), k

filter_dict = {'butter': [buttap, buttord],
               'butterworth': [buttap, buttord],
               'cauer': [ellipap, ellipord],
               'elliptic': [ellipap, ellipord],
               'ellip': [ellipap, ellipord],
               'bessel': [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',
             'high': 'highpass',
             'highpass': 'highpass',
             'h': 'highpass',
             }