[SciPy-user] problem with signal.residue

Ryan Krauss ryanlists@gmail....
Wed Feb 14 14:36:17 CST 2007


I will add this to my ticket, but there is also a problem with
signal.residue when the first denominator coefficent isn't zero.  I
think I fixed it and I tested my fix against Maxima's partfrac command
and it looks right.  Bascially, I added the line

rscale = a[0]

near the beginning of the function and then change the last line to return

return r/rscale, p, k

This is (nearly) identical to the approach in residuez, so that and
the agreement with Maxima give me a good feeling about it.

The .wxm file I am attaching is a wxMaxima script for the test, but it
requires a fairly recent version of wxMaxima (I am using 0.7.1).

Ryan

On 2/13/07, Kumar Appaiah <akumar@iitm.ac.in> wrote:
> On Tue, Feb 13, 2007 at 01:57:46PM -0600, Ryan Krauss wrote:
> > I think I have found a small bug in signal.residue and may have found
> > a simple solution.  The problem seems to come from polydiv requiring
> > that the numerator polynomial be of degree at most 1 less than the
> > denominator.  If I have a denominator of s^2+3*s+2, the numerator must
> > have an s coefficient (even if that coefficient is 0) for
> > signal.residue to work:
> >
> > In [75]: a
> > Out[75]: array([1, 3, 2])
> >
> > In [76]: signal.residue([1],a)
> > ---------------------------------------------------------------------------
> > exceptions.ValueError                                Traceback (most recent cal
> >  last)
> [snip]
> > In [77]: signal.residue([0,1],a)
> > Out[77]:
> > (array([ 1.+0.j, -1.+0.j]),
> >  array([-1.+0.j, -2.+0.j]),
> >   array([], dtype=float64))
> >
> >
> > I think the simple solution is to replace line 1056 with these four lines:
> >     if len(b)<len(a):
> >         k=[]
> >     else:
> >         k,b = polydiv(b,a)
> >
> > where the last line above is the old line 1056.  Basically, specify
> > that there is no k term if the len of b is less than the len of a.
> >
> > Is this too simple?  What do I do to actually submit this if it is the
> > right solution?
>
> I think you are right. This seems to be a bug.
> Please register and open a ticket at
> http://projects.scipy.org/scipy/scipy and state the problem and the
> specified solution.
>
> Thanks.
>
> Kumar
> --
> Kumar Appaiah,
> 462, Jamuna Hostel,
> Indian Institute of Technology Madras,
> Chennai - 600 036
>
> _______________________________________________
> SciPy-user mailing list
> SciPy-user@scipy.org
> http://projects.scipy.org/mailman/listinfo/scipy-user
>
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-------------- next part --------------
from scipy import *
from pylab import figure, cla, clf, plot, subplot, show, ylabel, xlabel, xlim, ylim, semilogx, legend, title, savefig, yticks, grid, rcParams

from  IPython.Debugger import Pdb

import copy, os, sys

#Test 1
k=100.0
b=10.0
signal.residue([1.0],[b,k,0.0])

#Test2
num=array([1.0,3,-1,10,1,7])
den=array([15.0,3115.5,173311.5,1197300.0,2115000.0])
signal.residue(num,den)
-------------- next part --------------
# Author: Travis Oliphant
# signaltools.py -- 2002

import types
import sigtools
from scipy import special, linalg
from scipy.fftpack import fft, ifft, ifftshift, fft2, ifft2
from numpy import polyadd, polymul, polydiv, polysub, \
     roots, poly, polyval, polyder, cast, asarray, isscalar, atleast_1d, \
     ones, sin, linspace, real, extract, real_if_close, zeros, array, arange, \
     where, sqrt, rank, newaxis, argmax, product, cos, pi, exp, \
     ravel, size, less_equal, sum, r_, iscomplexobj, take, \
     argsort, allclose, expand_dims, unique, prod, sort, reshape, c_, \
     transpose, dot, any, minimum, maximum, mean
import numpy
from scipy.fftpack import fftn, ifftn
from scipy.misc import factorial

from  IPython.Debugger import Pdb

_modedict = {'valid':0, 'same':1, 'full':2}
_boundarydict = {'fill':0, 'pad':0, 'wrap':2, 'circular':2, 'symm':1, 'symmetric':1, 'reflect':4}

def _valfrommode(mode):
    try:
        val = _modedict[mode]
    except KeyError:
        if mode not in [0,1,2]:
            raise ValueError, "Acceptable mode flags are 'valid' (0), 'same' (1), or 'full' (2)."
        val = mode
    return val

def _bvalfromboundary(boundary):
    try:
        val = _boundarydict[boundary] << 2
    except KeyError:
        if val not in [0,1,2] :
            raise ValueError, "Acceptable boundary flags are 'fill', 'wrap' (or 'circular'), \n  and 'symm' (or 'symmetric')."
        val = boundary << 2
    return val


def correlate(in1, in2, mode='full'):
    """Cross-correlate two N-dimensional arrays.

  Description:

     Cross-correlate in1 and in2 with the output size determined by mode.

  Inputs:

    in1 -- an N-dimensional array.
    in2 -- an array with the same number of dimensions as in1.
    mode -- a flag indicating the size of the output
            'valid'  (0): The output consists only of those elements that
                            do not rely on the zero-padding.
            'same'   (1): The output is the same size as the largest input
                            centered with respect to the 'full' output.
            'full'   (2): The output is the full discrete linear
                            cross-correlation of the inputs. (Default)

  Outputs:  (out,)

    out -- an N-dimensional array containing a subset of the discrete linear
           cross-correlation of in1 with in2.

    """
    # Code is faster if kernel is smallest array.
    volume = asarray(in1)
    kernel = asarray(in2)
    if rank(volume) == rank(kernel) == 0:
        return volume*kernel
    if (product(kernel.shape,axis=0) > product(volume.shape,axis=0)):
        temp = kernel
        kernel = volume
        volume = temp
        del temp

    val = _valfrommode(mode)

    return sigtools._correlateND(volume, kernel, val)

def _centered(arr, newsize):
    # Return the center newsize portion of the array.
    newsize = asarray(newsize)
    currsize = array(arr.shape)
    startind = (currsize - newsize) / 2
    endind = startind + newsize
    myslice = [slice(startind[k], endind[k]) for k in range(len(endind))]
    return arr[tuple(myslice)]

def fftconvolve(in1, in2, mode="full"):
    """Convolve two N-dimensional arrays using FFT. See convolve.
    
    """
    s1 = array(in1.shape)
    s2 = array(in2.shape)
    if (s1.dtype.char in ['D','F']) or (s2.dtype.char in ['D', 'F']):
        cmplx=1
    else: cmplx=0
    size = s1+s2-1
    IN1 = fftn(in1,size)
    IN1 *= fftn(in2,size)
    ret = ifftn(IN1)
    del IN1
    if not cmplx:
        ret = real(ret)
    if mode == "full":
        return ret
    elif mode == "same":
        if product(s1,axis=0) > product(s2,axis=0):
            osize = s1
        else:
            osize = s2
        return _centered(ret,osize)
    elif mode == "valid":
        return _centered(ret,abs(s2-s1)+1)


def convolve(in1, in2, mode='full'):
    """Convolve two N-dimensional arrays.

  Description:

     Convolve in1 and in2 with output size determined by mode.

  Inputs:

    in1 -- an N-dimensional array.
    in2 -- an array with the same number of dimensions as in1.
    mode -- a flag indicating the size of the output
            'valid'  (0): The output consists only of those elements that
                            are computed by scaling the larger array with all
                            the values of the smaller array.
            'same'   (1): The output is the same size as the largest input
                            centered with respect to the 'full' output.
            'full'   (2): The output is the full discrete linear convolution
                            of the inputs. (Default)

  Outputs:  (out,)

    out -- an N-dimensional array containing a subset of the discrete linear
           convolution of in1 with in2.

    """
    volume = asarray(in1)
    kernel = asarray(in2)
    if rank(volume) == rank(kernel) == 0:
        return volume*kernel
    if (product(kernel.shape,axis=0) > product(volume.shape,axis=0)):
        temp = kernel
        kernel = volume
        volume = temp
        del temp

    slice_obj = [slice(None,None,-1)]*len(kernel.shape)
    val = _valfrommode(mode)

    return sigtools._correlateND(volume,kernel[slice_obj],val)

def order_filter(a, domain, order):
    """Perform an order filter on an N-dimensional array.

  Description:

    Perform an order filter on the array in.  The domain argument acts as a
    mask centered over each pixel.  The non-zero elements of domain are
    used to select elements surrounding each input pixel which are placed
    in a list.   The list is sorted, and the output for that pixel is the
    element corresponding to rank in the sorted list.

  Inputs:

    in -- an N-dimensional input array.
    domain -- a mask array with the same number of dimensions as in.  Each
              dimension should have an odd number of elements.
    rank -- an non-negative integer which selects the element from the
            sorted list (0 corresponds to the largest element, 1 is the
            next largest element, etc.)

  Output: (out,)

    out -- the results of the order filter in an array with the same
           shape as in.

    """
    domain = asarray(domain)
    size = domain.shape
    for k in range(len(size)):
        if (size[k] % 2) != 1:
            raise ValueError, "Each dimension of domain argument should have an odd number of elements."
    return sigtools._orderfilterND(a, domain, rank)


def medfilt(volume,kernel_size=None):
    """Perform a median filter on an N-dimensional array.

  Description:

    Apply a median filter to the input array using a local window-size
    given by kernel_size.

  Inputs:

    in -- An N-dimensional input array.
    kernel_size -- A scalar or an N-length list giving the size of the
                   median filter window in each dimension.  Elements of
                   kernel_size should be odd.  If kernel_size is a scalar,
                   then this scalar is used as the size in each dimension.

  Outputs: (out,)

    out -- An array the same size as input containing the median filtered
           result.

    """
    volume = asarray(volume)
    if kernel_size is None:
        kernel_size = [3] * len(volume.shape)
    kernel_size = asarray(kernel_size)
    if len(kernel_size.shape) == 0:
        kernel_size = [kernel_size.item()] * len(volume.shape)
    kernel_size = asarray(kernel_size)

    for k in range(len(volume.shape)):
        if (kernel_size[k] % 2) != 1:
            raise ValueError, "Each element of kernel_size should be odd."

    domain = ones(kernel_size)

    numels = product(kernel_size,axis=0)
    order = int(numels/2)
    return sigtools._order_filterND(volume,domain,order)


def wiener(im,mysize=None,noise=None):
    """Perform a Wiener filter on an N-dimensional array.

  Description:

    Apply a Wiener filter to the N-dimensional array in.

  Inputs:

    in -- an N-dimensional array.
    kernel_size -- A scalar or an N-length list giving the size of the
                   median filter window in each dimension.  Elements of
                   kernel_size should be odd.  If kernel_size is a scalar,
                   then this scalar is used as the size in each dimension.
    noise -- The noise-power to use.  If None, then noise is estimated as
             the average of the local variance of the input.

  Outputs: (out,)

    out -- Wiener filtered result with the same shape as in.

    """
    im = asarray(im)
    if mysize is None:
        mysize = [3] * len(im.shape)
    mysize = asarray(mysize);

    # Estimate the local mean
    lMean = correlate(im,ones(mysize),1) / product(mysize,axis=0)

    # Estimate the local variance
    lVar = correlate(im**2,ones(mysize),1) / product(mysize,axis=0) - lMean**2

    # Estimate the noise power if needed.
    if noise==None:
        noise = mean(ravel(lVar),axis=0)

    res = (im - lMean)
    res *= (1-noise / lVar)
    res += lMean
    out = where(lVar < noise, lMean, res)

    return out


def convolve2d(in1, in2, mode='full', boundary='fill', fillvalue=0):
    """Convolve two 2-dimensional arrays.

  Description:

     Convolve in1 and in2 with output size determined by mode and boundary
     conditions determined by boundary and fillvalue.

  Inputs:

    in1 -- a 2-dimensional array.
    in2 -- a 2-dimensional array.
    mode -- a flag indicating the size of the output
            'valid'  (0): The output consists only of those elements that
                            do not rely on the zero-padding.
            'same'   (1): The output is the same size as the input centered
                            with respect to the 'full' output.
            'full'   (2): The output is the full discrete linear convolution
                            of the inputs. (*Default*)
    boundary -- a flag indicating how to handle boundaries
                'fill' : pad input arrays with fillvalue. (*Default*)
                'wrap' : circular boundary conditions.
                'symm' : symmetrical boundary conditions.
    fillvalue -- value to fill pad input arrays with (*Default* = 0)

  Outputs:  (out,)

    out -- a 2-dimensional array containing a subset of the discrete linear
           convolution of in1 with in2.

    """
    val = _valfrommode(mode)
    bval = _bvalfromboundary(boundary)

    return sigtools._convolve2d(in1,in2,1,val,bval,fillvalue)

def correlate2d(in1, in2, mode='full', boundary='fill', fillvalue=0):
    """Cross-correlate two 2-dimensional arrays.

  Description:

     Cross correlate in1 and in2 with output size determined by mode
     and boundary conditions determined by boundary and fillvalue.

  Inputs:

    in1 -- a 2-dimensional array.
    in2 -- a 2-dimensional array.
    mode -- a flag indicating the size of the output
            'valid'  (0): The output consists only of those elements that
                            do not rely on the zero-padding.
            'same'   (1): The output is the same size as the input centered
                            with respect to the 'full' output.
            'full'   (2): The output is the full discrete linear convolution
                            of the inputs. (*Default*)
    boundary -- a flag indicating how to handle boundaries
                'fill' : pad input arrays with fillvalue. (*Default*)
                'wrap' : circular boundary conditions.
                'symm' : symmetrical boundary conditions.
    fillvalue -- value to fill pad input arrays with (*Default* = 0)

  Outputs:  (out,)

    out -- a 2-dimensional array containing a subset of the discrete linear
           cross-correlation of in1 with in2.

    """
    val = _valfrommode(mode)
    bval = _bvalfromboundary(boundary)

    return sigtools._convolve2d(in1, in2, 0,val,bval,fillvalue)

def medfilt2d(input, kernel_size=3):
    """Median filter two 2-dimensional arrays.

  Description:

    Apply a median filter to the input array using a local window-size
    given by kernel_size (must be odd).

  Inputs:

    in -- An 2 dimensional input array.
    kernel_size -- A scalar or an length-2 list giving the size of the
                   median filter window in each dimension.  Elements of
                   kernel_size should be odd.  If kernel_size is a scalar,
                   then this scalar is used as the size in each dimension.

  Outputs: (out,)

    out -- An array the same size as input containing the median filtered
           result.
           
    """
    image = asarray(input)
    if kernel_size is None:
        kernel_size = [3] * 2
    kernel_size = asarray(kernel_size)
    if len(kernel_size.shape) == 0:
        kernel_size = [kernel_size.item()] * 2
    kernel_size = asarray(kernel_size)

    for size in kernel_size:
        if (size % 2) != 1:
            raise ValueError, "Each element of kernel_size should be odd."

    return sigtools._medfilt2d(image, kernel_size)

def remez(numtaps, bands, desired, weight=None, Hz=1, type='bandpass',
          maxiter=25, grid_density=16):
    """Calculate the minimax optimal filter using Remez exchange algorithm.

  Description:

    Calculate the filter-coefficients for the finite impulse response
    (FIR) filter whose transfer function minimizes the maximum error
    between the desired gain and the realized gain in the specified bands
    using the remez exchange algorithm.

  Inputs:

    numtaps -- The desired number of taps in the filter.
    bands -- A montonic sequence containing the band edges.  All elements
             must be non-negative and less than 1/2 the sampling frequency
             as given by Hz.
    desired -- A sequency half the size of bands containing the desired gain
               in each of the specified bands
    weight -- A relative weighting to give to each band region.
    type --- The type of filter:
             'bandpass' : flat response in bands.
             'differentiator' : frequency proportional response in bands.

  Outputs: (out,)

    out -- A rank-1 array containing the coefficients of the optimal
           (in a minimax sense) filter.

    """
    # Convert type
    try:
        tnum = {'bandpass':1, 'differentiator':2}[type]
    except KeyError:
        raise ValueError, "Type must be 'bandpass', or 'differentiator'"

    # Convert weight
    if weight is None:
        weight = [1] * len(desired)

    bands = asarray(bands).copy()
    return sigtools._remez(numtaps, bands, desired, weight, tnum, Hz,
                           maxiter, grid_density)

def lfilter(b, a, x, axis=-1, zi=None):
    """Filter data along one-dimension with an IIR or FIR filter.

  Description

    Filter a data sequence, x, using a digital filter.  This works for many
    fundamental data types (including Object type).  The filter is a direct
    form II transposed implementation of the standard difference equation
    (see "Algorithm").

  Inputs:

    b -- The numerator coefficient vector in a 1-D sequence.
    a -- The denominator coefficient vector in a 1-D sequence.  If a[0]
         is not 1, then both a and b are normalized by a[0].
    x -- An N-dimensional input array.
    axis -- The axis of the input data array along which to apply the
            linear filter. The filter is applied to each subarray along
            this axis (*Default* = -1)
    zi -- Initial conditions for the filter delays.  It is a vector
          (or array of vectors for an N-dimensional input) of length
          max(len(a),len(b)).  If zi=None or is not given then initial
          rest is assumed.  SEE signal.lfiltic for more information.

  Outputs: (y, {zf})

    y -- The output of the digital filter.
    zf -- If zi is None, this is not returned, otherwise, zf holds the
          final filter delay values.

  Algorithm:

    The filter function is implemented as a direct II transposed structure.
    This means that the filter implements

    y[n] = b[0]*x[n] + b[1]*x[n-1] + ... + b[nb]*x[n-nb]
                     - a[1]*y[n-1] + ... + a[na]*y[n-na]

    using the following difference equations:

    y[m] = b[0]*x[m] + z[0,m-1]
    z[0,m] = b[1]*x[m] + z[1,m-1] - a[1]*y[m]
    ...
    z[n-3,m] = b[n-2]*x[m] + z[n-2,m-1] - a[n-2]*y[m]
    z[n-2,m] = b[n-1]*x[m] - a[n-1]*y[m]

    where m is the output sample number and n=max(len(a),len(b)) is the
    model order.

    The rational transfer function describing this filter in the
    z-transform domain is
                                -1               -nb
                    b[0] + b[1]z  + ... + b[nb] z
            Y(z) = ---------------------------------- X(z)
                                -1               -na
                    a[0] + a[1]z  + ... + a[na] z

    """
    if isscalar(a):
        a = [a]
    if zi is None:
        return sigtools._linear_filter(b, a, x, axis)
    else:
        return sigtools._linear_filter(b, a, x, axis, zi)

def lfiltic(b,a,y,x=None):
    """Given a linear filter (b,a) and initial conditions on the output y
    and the input x, return the inital conditions on the state vector zi
    which is used by lfilter to generate the output given the input.

    If M=len(b)-1 and N=len(a)-1.  Then, the initial conditions are given
    in the vectors x and y as

    x = {x[-1],x[-2],...,x[-M]}
    y = {y[-1],y[-2],...,y[-N]}

    If x is not given, its inital conditions are assumed zero.
    If either vector is too short, then zeros are added
      to achieve the proper length.

    The output vector zi contains

    zi = {z_0[-1], z_1[-1], ..., z_K-1[-1]}  where K=max(M,N).
    
    """
    N = size(a)-1
    M = size(b)-1
    K = max(M,N)
    y = asarray(y)
    zi = zeros(K,y.dtype.char)
    if x is None:
        x = zeros(M,y.dtype.char)
    else:
        x = asarray(x)
        L = size(x)
        if L < M:
            x = r_[x,zeros(M-L)]
    L = size(y)
    if L < N:
        y = r_[y,zeros(N-L)]

    for m in range(M):
        zi[m] = sum(b[m+1:]*x[:M-m],axis=0)

    for m in range(N):
        zi[m] -= sum(a[m+1:]*y[:N-m],axis=0)

    return zi

def deconvolve(signal, divisor):
    """Deconvolves divisor out of signal.
    
    """
    num = atleast_1d(signal)
    den = atleast_1d(divisor)
    N = len(num)
    D = len(den)
    if D > N:
        quot = [];
        rem = num;
    else:
        input = ones(N-D+1, float)
        input[1:] = 0
        quot = lfilter(num, den, input)
        rem = num - convolve(den, quot, mode='full')
    return quot, rem


def boxcar(M,sym=1):
    """The M-point boxcar window.
    
    """
    return ones(M, float)

def triang(M,sym=1):
    """The M-point triangular window.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M + 1
    n = arange(1,int((M+1)/2)+1)
    if M % 2 == 0:
        w = (2*n-1.0)/M
        w = r_[w, w[::-1]]
    else:
        w = 2*n/(M+1.0)
        w = r_[w, w[-2::-1]]

    if not sym and not odd:
        w = w[:-1]
    return w

def parzen(M,sym=1):
    """The M-point Parzen window.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    n = arange(-(M-1)/2.0,(M-1)/2.0+0.5,1.0)
    na = extract(n < -(M-1)/4.0, n)
    nb = extract(abs(n) <= (M-1)/4.0, n)
    wa = 2*(1-abs(na)/(M/2.0))**3.0
    wb = 1-6*(abs(nb)/(M/2.0))**2.0 + 6*(abs(nb)/(M/2.0))**3.0
    w = r_[wa,wb,wa[::-1]]
    if not sym and not odd:
        w = w[:-1]
    return w

def bohman(M,sym=1):
    """The M-point Bohman window.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    fac = abs(linspace(-1,1,M)[1:-1])
    w = (1 - fac)* cos(pi*fac) + 1.0/pi*sin(pi*fac)
    w = r_[0,w,0]
    if not sym and not odd:
        w = w[:-1]
    return w

def blackman(M,sym=1):
    """The M-point Blackman window.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    n = arange(0,M)
    w = 0.42-0.5*cos(2.0*pi*n/(M-1)) + 0.08*cos(4.0*pi*n/(M-1))
    if not sym and not odd:
        w = w[:-1]
    return w

def nuttall(M,sym=1):
    """A minimum 4-term Blackman-Harris window according to Nuttall.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    a = [0.3635819, 0.4891775, 0.1365995, 0.0106411]
    n = arange(0,M)
    fac = n*2*pi/(M-1.0)
    w = a[0] - a[1]*cos(fac) + a[2]*cos(2*fac) - a[3]*cos(3*fac)
    if not sym and not odd:
        w = w[:-1]
    return w

def blackmanharris(M,sym=1):
    """The M-point minimum 4-term Blackman-Harris window.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    a = [0.35875, 0.48829, 0.14128, 0.01168];
    n = arange(0,M)
    fac = n*2*pi/(M-1.0)
    w = a[0] - a[1]*cos(fac) + a[2]*cos(2*fac) - a[3]*cos(3*fac)
    if not sym and not odd:
        w = w[:-1]
    return w

def flattop(M,sym=1):
    """The M-point Flat top window.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    a = [0.2156, 0.4160, 0.2781, 0.0836, 0.0069]
    n = arange(0,M)
    fac = n*2*pi/(M-1.0)
    w = a[0] - a[1]*cos(fac) + a[2]*cos(2*fac) - a[3]*cos(3*fac) + a[4]*cos(4*fac)
    if not sym and not odd:
        w = w[:-1]
    return w


def bartlett(M,sym=1):
    """The M-point Bartlett window.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    n = arange(0,M)
    w = where(less_equal(n,(M-1)/2.0),2.0*n/(M-1),2.0-2.0*n/(M-1))
    if not sym and not odd:
        w = w[:-1]
    return w

def hanning(M,sym=1):
    """The M-point Hanning window.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    n = arange(0,M)
    w = 0.5-0.5*cos(2.0*pi*n/(M-1))
    if not sym and not odd:
        w = w[:-1]
    return w

hann = hanning

def barthann(M,sym=1):
    """Return the M-point modified Bartlett-Hann window.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    n = arange(0,M)
    fac = abs(n/(M-1.0)-0.5)
    w = 0.62 - 0.48*fac + 0.38*cos(2*pi*fac)
    if not sym and not odd:
        w = w[:-1]
    return w

def hamming(M,sym=1):
    """The M-point Hamming window.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    n = arange(0,M)
    w = 0.54-0.46*cos(2.0*pi*n/(M-1))
    if not sym and not odd:
        w = w[:-1]
    return w

    

def kaiser(M,beta,sym=1):
    """Return a Kaiser window of length M with shape parameter beta.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    n = arange(0,M)
    alpha = (M-1)/2.0
    w = special.i0(beta * sqrt(1-((n-alpha)/alpha)**2.0))/special.i0(beta)
    if not sym and not odd:
        w = w[:-1]
    return w

def gaussian(M,std,sym=1):
    """Return a Gaussian window of length M with standard-deviation std.
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M + 1
    n = arange(0,M)-(M-1.0)/2.0
    sig2 = 2*std*std
    w = exp(-n**2 / sig2)
    if not sym and not odd:
        w = w[:-1]
    return w

def general_gaussian(M,p,sig,sym=1):
    """Return a window with a generalized Gaussian shape.

    exp(-0.5*(x/sig)**(2*p))

    half power point is at (2*log(2)))**(1/(2*p))*sig
    
    """
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1
    n = arange(0,M)-(M-1.0)/2.0
    w = exp(-0.5*(n/sig)**(2*p))
    if not sym and not odd:
        w = w[:-1]
    return w


def slepian(M,width,sym=1):
    """Return the M-point slepian window.

    """
    if (M*width > 27.38):
        raise ValueError, "Cannot reliably obtain slepian sequences for"\
              " M*width > 27.38."
    if M < 1:
        return array([])
    if M == 1:
        return ones(1,'d')
    odd = M % 2
    if not sym and not odd:
        M = M+1

    twoF = width/2.0
    alpha = (M-1)/2.0
    m = arange(0,M)-alpha
    n = m[:,newaxis]
    k = m[newaxis,:]
    AF = twoF*special.sinc(twoF*(n-k))
    [lam,vec] = linalg.eig(AF)
    ind = argmax(abs(lam),axis=-1)
    w = abs(vec[:,ind])
    w = w / max(w)

    if not sym and not odd:
        w = w[:-1]
    return w


def hilbert(x, N=None):
    """Return the hilbert transform of x of length N.
    """
    x = asarray(x)
    if N is None:
        N = len(x)
    if N <=0:
        raise ValueError, "N must be positive."
    if iscomplexobj(x):
        print "Warning: imaginary part of x ignored."
        x = real(x)
    Xf = fft(x,N,axis=0)
    h = zeros(N)
    if N % 2 == 0:
        h[0] = h[N/2] = 1
        h[1:N/2] = 2
    else:
        h[0] = 1
        h[1:(N+1)/2] = 2

    if len(x.shape) > 1:
        h = h[:, newaxis]
    x = ifft(Xf*h)
    return x

def hilbert2(x,N=None):
    """Return the '2-D' hilbert transform of x of length N.
    """
    x = asarray(x)
    x = asarray(x)
    if N is None:
        N = x.shape
    if len(N) < 2:
        if N <=0:
            raise ValueError, "N must be positive."
        N = (N,N)
    if iscomplexobj(x):
        print "Warning: imaginary part of x ignored."
        x = real(x)
    print N
    Xf = fft2(x,N,axes=(0,1))
    h1 = zeros(N[0],'d')
    h2 = zeros(N[1],'d')
    for p in range(2):
        h = eval("h%d"%(p+1))
        N1 = N[p]
        if N1 % 2 == 0:
            h[0] = h[N1/2] = 1
            h[1:N1/2] = 2
        else:
            h[0] = 1
            h[1:(N1+1)/2] = 2
        exec("h%d = h" % (p+1), globals(), locals())

    h = h1[:,newaxis] * h2[newaxis,:]
    k = len(x.shape)
    while k > 2:
        h = h[:, newaxis]
        k -= 1
    x = ifft2(Xf*h,axes=(0,1))
    return x


def cmplx_sort(p):
    "sort roots based on magnitude."
    p = asarray(p)
    if iscomplexobj(p):
        indx = argsort(abs(p))
    else:
        indx = argsort(p)
    return take(p,indx,0), indx

def unique_roots(p,tol=1e-3,rtype='min'):
    """Determine the unique roots and their multiplicities in two lists

    Inputs:

      p -- The list of roots
      tol --- The tolerance for two roots to be considered equal.
      rtype --- How to determine the returned root from the close
                  ones:  'max': pick the maximum
                         'min': pick the minimum
                         'avg': average roots
    Outputs: (pout, mult)

      pout -- The list of sorted roots
      mult -- The multiplicity of each root
    """
    if rtype in ['max','maximum']:
        comproot = numpy.maximum
    elif rtype in ['min','minimum']:
        comproot = numpy.minimum
    elif rtype in ['avg','mean']:
        comproot = numpy.mean
    p = asarray(p)*1.0
    tol = abs(tol)
    p, indx = cmplx_sort(p)
    pout = []
    mult = []
    indx = -1
    curp = p[0] + 5*tol
    sameroots = []
    for k in range(len(p)):
        tr = p[k]
        if abs(tr-curp) < tol:
            sameroots.append(tr)
            curp = comproot(sameroots)
            pout[indx] = curp
            mult[indx] += 1
        else:
            pout.append(tr)
            curp = tr
            sameroots = [tr]
            indx += 1
            mult.append(1)
    return array(pout), array(mult)


def invres(r,p,k,tol=1e-3,rtype='avg'):
    """Compute b(s) and a(s) from partial fraction expansion: r,p,k

    If M = len(b) and N = len(a)

            b(s)     b[0] x**(M-1) + b[1] x**(M-2) + ... + b[M-1]
    H(s) = ------ = ----------------------------------------------
            a(s)     a[0] x**(N-1) + a[1] x**(N-2) + ... + a[N-1]

             r[0]       r[1]             r[-1]
         = -------- + -------- + ... + --------- + k(s)
           (s-p[0])   (s-p[1])         (s-p[-1])

    If there are any repeated roots (closer than tol), then the partial
    fraction expansion has terms like

            r[i]      r[i+1]              r[i+n-1]
          -------- + ----------- + ... + -----------
          (s-p[i])  (s-p[i])**2          (s-p[i])**n

    See also:  residue, poly, polyval, unique_roots
    """
    extra = k
    p, indx = cmplx_sort(p)
    r = take(r,indx,0)
    pout, mult = unique_roots(p,tol=tol,rtype=rtype)
    p = []
    for k in range(len(pout)):
        p.extend([pout[k]]*mult[k])
    a = atleast_1d(poly(p))
    if len(extra) > 0:
        b = polymul(extra,a)
    else:
        b = [0]
    indx = 0
    for k in range(len(pout)):
        temp = []
        for l in range(len(pout)):
            if l != k:
                temp.extend([pout[l]]*mult[l])
        for m in range(mult[k]):
            t2 = temp[:]
            t2.extend([pout[k]]*(mult[k]-m-1))
            b = polyadd(b,r[indx]*poly(t2))
            indx += 1
    b = real_if_close(b)
    while allclose(b[0], 0, rtol=1e-14) and (b.shape[-1] > 1):
        b = b[1:]
    return b, a

def residue(b,a,tol=1e-3,rtype='avg'):
    """Compute partial-fraction expansion of b(s) / a(s).

    If M = len(b) and N = len(a)

            b(s)     b[0] s**(M-1) + b[1] s**(M-2) + ... + b[M-1]
    H(s) = ------ = ----------------------------------------------
            a(s)     a[0] s**(N-1) + a[1] s**(N-2) + ... + a[N-1]

             r[0]       r[1]             r[-1]
         = -------- + -------- + ... + --------- + k(s)
           (s-p[0])   (s-p[1])         (s-p[-1])

    If there are any repeated roots (closer than tol), then the partial
    fraction expansion has terms like

            r[i]      r[i+1]              r[i+n-1]
          -------- + ----------- + ... + -----------
          (s-p[i])  (s-p[i])**2          (s-p[i])**n

    See also:  invres, poly, polyval, unique_roots
    """

    b,a = map(asarray,(b,a))
    rscale = a[0]
    if len(b)<len(a):
        k=[]
    else:
        k,b = polydiv(b,a)
#    k,b = polydiv(b,a)
#    Pdb().set_trace()
    p = roots(a)
    r = p*0.0
    pout, mult = unique_roots(p,tol=tol,rtype=rtype)
    p = []
    for n in range(len(pout)):
        p.extend([pout[n]]*mult[n])
    p = asarray(p)
    # Compute the residue from the general formula
    indx = 0
    for n in range(len(pout)):
        bn = b.copy()
        pn = []
        for l in range(len(pout)):
            if l != n:
                pn.extend([pout[l]]*mult[l])
        an = atleast_1d(poly(pn))
        # bn(s) / an(s) is (s-po[n])**Nn * b(s) / a(s) where Nn is
        # multiplicity of pole at po[n]
        sig = mult[n]
        for m in range(sig,0,-1):
            if sig > m:
                # compute next derivative of bn(s) / an(s)
                term1 = polymul(polyder(bn,1),an)
                term2 = polymul(bn,polyder(an,1))
                bn = polysub(term1,term2)
                an = polymul(an,an)
            r[indx+m-1] = polyval(bn,pout[n]) / polyval(an,pout[n]) \
                          / factorial(sig-m)
        indx += sig
    return r/rscale, p, k

def residuez(b,a,tol=1e-3,rtype='avg'):
    """Compute partial-fraction expansion of b(z) / a(z).

    If M = len(b) and N = len(a)

            b(z)     b[0] + b[1] z**(-1) + ... + b[M-1] z**(-M+1)
    H(z) = ------ = ----------------------------------------------
            a(z)     a[0] + a[1] z**(-1) + ... + a[N-1] z**(-N+1)

                 r[0]                   r[-1]
         = --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ...
           (1-p[0]z**(-1))         (1-p[-1]z**(-1))

    If there are any repeated roots (closer than tol), then the partial
    fraction expansion has terms like

               r[i]              r[i+1]                    r[i+n-1]
          -------------- + ------------------ + ... + ------------------
          (1-p[i]z**(-1))  (1-p[i]z**(-1))**2         (1-p[i]z**(-1))**n

    See also:  invresz, poly, polyval, unique_roots
    """
    b,a = map(asarray,(b,a))
    gain = a[0]
    brev, arev = b[::-1],a[::-1]
    krev,brev = polydiv(brev,arev)
    if krev == []:
        k = []
    else:
        k = krev[::-1]
    b = brev[::-1]
    p = roots(a)
    r = p*0.0
    pout, mult = unique_roots(p,tol=tol,rtype=rtype)
    p = []
    for n in range(len(pout)):
        p.extend([pout[n]]*mult[n])
    p = asarray(p)
    # Compute the residue from the general formula (for discrete-time)
    #  the polynomial is in z**(-1) and the multiplication is by terms
    #  like this (1-p[i] z**(-1))**mult[i].  After differentiation,
    #  we must divide by (-p[i])**(m-k) as well as (m-k)!
    indx = 0
    for n in range(len(pout)):
        bn = brev.copy()
        pn = []
        for l in range(len(pout)):
            if l != n:
                pn.extend([pout[l]]*mult[l])
        an = atleast_1d(poly(pn))[::-1]
        # bn(z) / an(z) is (1-po[n] z**(-1))**Nn * b(z) / a(z) where Nn is
        # multiplicity of pole at po[n] and b(z) and a(z) are polynomials.
        sig = mult[n]
        for m in range(sig,0,-1):
            if sig > m:
                # compute next derivative of bn(s) / an(s)
                term1 = polymul(polyder(bn,1),an)
                term2 = polymul(bn,polyder(an,1))
                bn = polysub(term1,term2)
                an = polymul(an,an)
            r[indx+m-1] = polyval(bn,1.0/pout[n]) / polyval(an,1.0/pout[n]) \
                          / factorial(sig-m) / (-pout[n])**(sig-m)
        indx += sig
    return r/gain, p, k

def invresz(r,p,k,tol=1e-3,rtype='avg'):
    """Compute b(z) and a(z) from partial fraction expansion: r,p,k

    If M = len(b) and N = len(a)

            b(z)     b[0] + b[1] z**(-1) + ... + b[M-1] z**(-M+1)
    H(z) = ------ = ----------------------------------------------
            a(z)     a[0] + a[1] z**(-1) + ... + a[N-1] z**(-N+1)

                 r[0]                   r[-1]
         = --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ...
           (1-p[0]z**(-1))         (1-p[-1]z**(-1))

    If there are any repeated roots (closer than tol), then the partial
    fraction expansion has terms like

               r[i]              r[i+1]                    r[i+n-1]
          -------------- + ------------------ + ... + ------------------
          (1-p[i]z**(-1))  (1-p[i]z**(-1))**2         (1-p[i]z**(-1))**n

    See also:  residuez, poly, polyval, unique_roots
    """
    extra = asarray(k)
    p, indx = cmplx_sort(p)
    r = take(r,indx,0)
    pout, mult = unique_roots(p,tol=tol,rtype=rtype)
    p = []
    for k in range(len(pout)):
        p.extend([pout[k]]*mult[k])
    a = atleast_1d(poly(p))
    if len(extra) > 0:
        b = polymul(extra,a)
    else:
        b = [0]
    indx = 0
    brev = asarray(b)[::-1]
    for k in range(len(pout)):
        temp = []
        # Construct polynomial which does not include any of this root
        for l in range(len(pout)):
            if l != k:
                temp.extend([pout[l]]*mult[l])
        for m in range(mult[k]):
            t2 = temp[:]
            t2.extend([pout[k]]*(mult[k]-m-1))
            brev = polyadd(brev,(r[indx]*poly(t2))[::-1])
            indx += 1
    b = real_if_close(brev[::-1])
    return b, a


def get_window(window,Nx,fftbins=1):
    """Return a window of length Nx and type window.

    If fftbins is 1, create a "periodic" window ready to use with ifftshift
    and be multiplied by the result of an fft (SEE ALSO fftfreq).

    Window types:  boxcar, triang, blackman, hamming, hanning, bartlett,
                   parzen, bohman, blackmanharris, nuttall, barthann,
                   kaiser (needs beta), gaussian (needs std),
                   general_gaussian (needs power, width),
                   slepian (needs width)

    If the window requires no parameters, then it can be a string.
    If the window requires parameters, the window argument should be a tuple
        with the first argument the string name of the window, and the next
        arguments the needed parameters.
    If window is a floating point number, it is interpreted as the beta
        parameter of the kaiser window.
    """

    sym = not fftbins
    try:
        beta = float(window)
    except (TypeError, ValueError):
        args = ()
        if isinstance(window, types.TupleType):
            winstr = window[0]
            if len(window) > 1:
                args = window[1:]
        elif isinstance(window, types.StringType):
            if window in ['kaiser', 'ksr', 'gaussian', 'gauss', 'gss',
                        'general gaussian', 'general_gaussian',
                        'general gauss', 'general_gauss', 'ggs']:
                raise ValueError, "That window needs a parameter -- pass a tuple"
            else:
                winstr = window

        if winstr in ['blackman', 'black', 'blk']:
            winfunc = blackman
        elif winstr in ['triangle', 'triang', 'tri']:
            winfunc = triang
        elif winstr in ['hamming', 'hamm', 'ham']:
            winfunc = hamming
        elif winstr in ['bartlett', 'bart', 'brt']:
            winfunc = bartlett
        elif winstr in ['hanning', 'hann', 'han']:
            winfunc = hanning
        elif winstr in ['blackmanharris', 'blackharr','bkh']:
            winfunc = blackmanharris
        elif winstr in ['parzen', 'parz', 'par']:
            winfunc = parzen
        elif winstr in ['bohman', 'bman', 'bmn']:
            winfunc = bohman
        elif winstr in ['nuttall', 'nutl', 'nut']:
            winfunc = nuttall
        elif winstr in ['barthann', 'brthan', 'bth']:
            winfunc = barthann
        elif winstr in ['flattop', 'flat', 'flt']:
            winfunc = flattop
        elif winstr in ['kaiser', 'ksr']:
            winfunc = kaiser
        elif winstr in ['gaussian', 'gauss', 'gss']:
            winfunc = gaussian
        elif winstr in ['general gaussian', 'general_gaussian',
                        'general gauss', 'general_gauss', 'ggs']:
            winfunc = general_gaussian
        elif winstr in ['boxcar', 'box', 'ones']:
            winfunc = boxcar
        elif winstr in ['slepian', 'slep', 'optimal', 'dss']:
            winfunc = slepian
        else:
            raise ValueError, "Unknown window type."

        params = (Nx,)+args + (sym,)
    else:
        winfunc = kaiser
        params = (Nx,beta,sym)

    return winfunc(*params)


def resample(x,num,t=None,axis=0,window=None):
    """Resample to num samples using Fourier method along the given axis.

    The resampled signal starts at the same value of x but is sampled
    with a spacing of len(x) / num * (spacing of x).  Because a Fourier method
    is used, the signal is assumed periodic.

    Window controls a Fourier-domain window that tapers the Fourier spectrum
    before zero-padding to aleviate ringing in the resampled values for
    sampled signals you didn't intend to be interpreted as band-limited.

    If window is a string then use the named window.  If window is a float, then
    it represents a value of beta for a kaiser window.  If window is a tuple,
    then the first component is a string representing the window, and the next
    arguments are parameters for that window.

    Possible windows are:
           'blackman'       ('black',   'blk')
           'hamming'        ('hamm',    'ham')
           'bartlett'       ('bart',    'brt')
           'hanning'        ('hann',    'han')
           'kaiser'         ('ksr')             # requires parameter (beta)
           'gaussian'       ('gauss',   'gss')  # requires parameter (std.)
           'general gauss'  ('general', 'ggs')  # requires two parameters
                                                      (power, width)

    The first sample of the returned vector is the same as the first sample of the
        input vector, the spacing between samples is changed from dx to
        dx * len(x) / num

    If t is not None, then it represents the old sample positions, and the new
       sample positions will be returned as well as the new samples.
    """
    x = asarray(x)
    X = fft(x,axis=axis)
    Nx = x.shape[axis]
    if window is not None:
        W = ifftshift(get_window(window,Nx))
        newshape = ones(len(x.shape))
        newshape[axis] = len(W)
        W=W.reshape(newshape)
        X = X*W
    sl = [slice(None)]*len(x.shape)
    newshape = list(x.shape)
    newshape[axis] = num
    N = int(numpy.minimum(num,Nx))
    Y = zeros(newshape,'D')
    sl[axis] = slice(0,(N+1)/2)
    Y[sl] = X[sl]
    sl[axis] = slice(-(N-1)/2,None)
    Y[sl] = X[sl]
    y = ifft(Y,axis=axis)*(float(num)/float(Nx))

    if x.dtype.char not in ['F','D']:
        y = y.real

    if t is None:
        return y
    else:
        new_t = arange(0,num)*(t[1]-t[0])* Nx / float(num) + t[0]
        return y, new_t

def detrend(data, axis=-1, type='linear', bp=0):
    """Remove linear trend along axis from data.

    If type is 'constant' then remove mean only.

    If bp is given, then it is a sequence of points at which to
       break a piecewise-linear fit to the data.
       
    """
    if type not in ['linear','l','constant','c']:
        raise ValueError, "Trend type must be linear or constant"
    data = asarray(data)
    dtype = data.dtype.char
    if dtype not in 'dfDF':
        dtype = 'd'
    if type in ['constant','c']:
        ret = data - expand_dims(mean(data,axis),axis)
        return ret
    else:
        dshape = data.shape
        N = dshape[axis]
        bp = sort(unique(r_[0,bp,N]))
        if any(bp > N):
            raise ValueError, "Breakpoints must be less than length of data along given axis."
        Nreg = len(bp) - 1
        # Restructure data so that axis is along first dimension and
        #  all other dimensions are collapsed into second dimension
        rnk = len(dshape)
        if axis < 0: axis = axis + rnk
        newdims = r_[axis,0:axis,axis+1:rnk]
        newdata = reshape(transpose(data,tuple(newdims)),(N,prod(dshape,axis=0)/N))
        newdata = newdata.copy()  # make sure we have a copy
        if newdata.dtype.char not in 'dfDF':
            newdata = newdata.astype(dtype)
        # Find leastsq fit and remove it for each piece
        for m in range(Nreg):
            Npts = bp[m+1] - bp[m]
            A = ones((Npts,2),dtype)
            A[:,0] = cast[dtype](arange(1,Npts+1)*1.0/Npts)
            sl = slice(bp[m],bp[m+1])
            coef,resids,rank,s = linalg.lstsq(A,newdata[sl])
            newdata[sl] = newdata[sl] - dot(A,coef)
        # Put data back in original shape.
        tdshape = take(dshape,newdims,0)
        ret = reshape(newdata,tuple(tdshape))
        vals = range(1,rnk)
        olddims = vals[:axis] + [0] + vals[axis:]
        ret = transpose(ret,tuple(olddims))
        return ret


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