endolith/readme.md

## readme.md

      
    Raw
  

              readme.md
            
          
    copied from Pyscellania: a selection of Python programs¶
Examples:
http://flic.kr/p/7oXfh6
http://flic.kr/p/7oXfbT
The complex Gabor/Morlet wavelet was the first continuous wavelet, very similar to the short-time Fourier transform, and is kind of the bridge between the Fourier transform world and the wavelet transform world.

  
## wavelets.py
import numpy as NP

"""
A module which implements the continuous wavelet transform

Wavelet classes:
Morlet
MorletReal
MexicanHat
Paul2      : Paul order 2
Paul4      : Paul order 4
DOG1       : 1st Derivative Of Gaussian
DOG4       : 4th Derivative Of Gaussian
Haar       : Unnormalised version of continuous Haar transform
HaarW      : Normalised Haar

Usage e.g.
wavelet=Morlet(data, largestscale=2, notes=0, order=2, scaling="log")
 data:  Numeric array of data (float), with length ndata.
        Optimum length is a power of 2 (for FFT)
        Worst-case length is a prime
 largestscale:
        largest scale as inverse fraction of length
        scale = len(data)/largestscale
        smallest scale should be >= 2 for meaningful data
 notes: number of scale intervals per octave
        if notes == 0, scales are on a linear increment
 order: order of wavelet for wavelets with variable order
        [Paul, DOG, ..]
 scaling: "linear" or "log" scaling of the wavelet scale.
        Note that feature width in the scale direction
        is constant on a log scale.

Attributes of instance:
wavelet.cwt:       2-d array of Wavelet coefficients, (nscales,ndata)
wavelet.nscale:    Number of scale intervals
wavelet.scales:    Array of scale values
                   Note that meaning of the scale will depend on the family
wavelet.fourierwl: Factor to multiply scale by to get scale
                   of equivalent FFT
                   Using this factor, different wavelet families will
                   have comparable scales

References:
A practical guide to wavelet analysis
C Torrance and GP Compo
Bull Amer Meteor Soc Vol 79 No 1 61-78 (1998)
naming below vaguely follows this.

updates:
(24/2/07):  Fix Morlet so can get MorletReal by cutting out H
(10/04/08): Numeric -> numpy
(25/07/08): log and lin scale increment in same direction!
            swap indices in 2-d coefficient matrix
            explicit scaling of scale axis
"""

class Cwt:
    """
    Base class for continuous wavelet transforms
    Implements cwt via the Fourier transform
    Used by subclass which provides the method wf(self,s_omega)
    wf is the Fourier transform of the wavelet function.
    Returns an instance.
    """

    fourierwl=1.00

    def _log2(self, x):
        # utility function to return (integer) log2
        return int( NP.log(float(x))/ NP.log(2.0)+0.0001 )

    def __init__(self, data, largestscale=1, notes=0, order=2, scaling='linear'):
        """
        Continuous wavelet transform of data

        data:    data in array to transform, length must be power of 2
        notes:   number of scale intervals per octave
        largestscale: largest scale as inverse fraction of length
                 of data array
                 scale = len(data)/largestscale
                 smallest scale should be >= 2 for meaningful data
        order:   Order of wavelet basis function for some families
        scaling: Linear or log
        """
        ndata = len(data)
        self.order=order
        self.scale=largestscale
        self._setscales(ndata,largestscale,notes,scaling)
        self.cwt= NP.zeros((self.nscale,ndata), NP.complex64)
        omega= NP.array(range(0,ndata/2)+range(-ndata/2,0))*(2.0*NP.pi/ndata)
        datahat=NP.fft.fft(data)
        self.fftdata=datahat
        #self.psihat0=self.wf(omega*self.scales[3*self.nscale/4])
        # loop over scales and compute wavelet coefficients at each scale
        # using the fft to do the convolution
        for scaleindex in range(self.nscale):
            currentscale=self.scales[scaleindex]
            self.currentscale=currentscale  # for internal use
            s_omega = omega*currentscale
            psihat=self.wf(s_omega)
            psihat = psihat *  NP.sqrt(2.0*NP.pi*currentscale)
            convhat = psihat * datahat
            W    = NP.fft.ifft(convhat)
            self.cwt[scaleindex,0:ndata] = W
        return

    def _setscales(self,ndata,largestscale,notes,scaling):
        """
        if notes non-zero, returns a log scale based on notes per octave
        else a linear scale
        (25/07/08): fix notes!=0 case so smallest scale at [0]
        """
        if scaling=="log":
            if notes<=0: notes=1
            # adjust nscale so smallest scale is 2
            noctave=self._log2( ndata/largestscale/2 )
            self.nscale=notes*noctave
            self.scales=NP.zeros(self.nscale,float)
            for j in range(self.nscale):
                self.scales[j] = ndata/(self.scale*(2.0**(float(self.nscale-1-j)/notes)))
        elif scaling=="linear":
            nmax=ndata/largestscale/2
            self.scales=NP.arange(float(2),float(nmax))
            self.nscale=len(self.scales)
        else: raise ValueError, "scaling must be linear or log"
        return

    def getdata(self):
        """
        returns wavelet coefficient array
        """
        return self.cwt

    def getcoefficients(self):
        return self.cwt

    def getpower(self):
        """
        returns square of wavelet coefficient array
        """
        return (self.cwt* NP.conjugate(self.cwt)).real

    def getscales(self):
        """
        returns array containing scales used in transform
        """
        return self.scales

    def getnscale(self):
        """
        return number of scales
        """
        return self.nscale

# wavelet classes
class Morlet(Cwt):
    """
    Complex Morlet wavelet
    """
    _omega0=5.0
    fourierwl=4* NP.pi/(_omega0+ NP.sqrt(2.0+_omega0**2))
    def wf(self, s_omega):
        H= NP.ones(len(s_omega))
        n=len(s_omega)
        for i in range(len(s_omega)):
            if s_omega[i] < 0.0: H[i]=0.0
        # !!!! note : was s_omega/8 before 17/6/03
        xhat=0.75112554*( NP.exp(-(s_omega-self._omega0)**2/2.0))*H
        return xhat

class MorletReal(Cwt):
    """
    Real Morlet wavelet
    """
    _omega0=5.0
    fourierwl=4* NP.pi/(_omega0+ NP.sqrt(2.0+_omega0**2))
    def wf(self, s_omega):
        H= NP.ones(len(s_omega))
        n=len(s_omega)
        for i in range(len(s_omega)):
            if s_omega[i] < 0.0: H[i]=0.0
        # !!!! note : was s_omega/8 before 17/6/03
        xhat=0.75112554*( NP.exp(-(s_omega-self._omega0)**2/2.0)+ NP.exp(-(s_omega+self._omega0)**2/2.0)- NP.exp(-(self._omega0)**2/2.0)+ NP.exp(-(self._omega0)**2/2.0))
        return xhat

class Paul4(Cwt):
    """
    Paul m=4 wavelet
    """
    fourierwl=4* NP.pi/(2.*4+1.)
    def wf(self, s_omega):
        n=len(s_omega)
        xhat= NP.zeros(n)
        xhat[0:n/2]=0.11268723*s_omega[0:n/2]**4* NP.exp(-s_omega[0:n/2])
        #return 0.11268723*s_omega**2*exp(-s_omega)*H
        return xhat

class Paul2(Cwt):
    """
    Paul m=2 wavelet
    """
    fourierwl=4* NP.pi/(2.*2+1.)
    def wf(self, s_omega):
        n=len(s_omega)
        xhat= NP.zeros(n)
        xhat[0:n/2]=1.1547005*s_omega[0:n/2]**2* NP.exp(-s_omega[0:n/2])
        #return 0.11268723*s_omega**2*exp(-s_omega)*H
        return xhat

class Paul(Cwt):
    """
    Paul order m wavelet
    """
    def wf(self, s_omega):
        Cwt.fourierwl=4* NP.pi/(2.*self.order+1.)
        m=self.order
        n=len(s_omega)
        normfactor=float(m)
        for i in range(1,2*m):
            normfactor=normfactor*i
        normfactor=2.0**m/ NP.sqrt(normfactor)
        xhat= NP.zeros(n)
        xhat[0:n/2]=normfactor*s_omega[0:n/2]**m* NP.exp(-s_omega[0:n/2])
        #return 0.11268723*s_omega**2*exp(-s_omega)*H
        return xhat

class MexicanHat(Cwt):
    """
    2nd Derivative Gaussian (Mexican hat) wavelet
    """
    fourierwl=2.0* NP.pi/ NP.sqrt(2.5)
    def wf(self, s_omega):
        # should this number be 1/sqrt(3/4) (no pi)?
        #s_omega = s_omega/self.fourierwl
        #print max(s_omega)
        a=s_omega**2
        b=s_omega**2/2
        return a* NP.exp(-b)/1.1529702
        #return s_omega**2*exp(-s_omega**2/2.0)/1.1529702

class DOG4(Cwt):
    """
    4th Derivative Gaussian wavelet
    see also T&C errata for - sign
    but reconstruction seems to work best with +!
    """
    fourierwl=2.0* NP.pi/ NP.sqrt(4.5)
    def wf(self, s_omega):
        return s_omega**4* NP.exp(-s_omega**2/2.0)/3.4105319

class DOG1(Cwt):
    """
    1st Derivative Gaussian wavelet
    but reconstruction seems to work best with +!
    """
    fourierwl=2.0* NP.pi/ NP.sqrt(1.5)
    def wf(self, s_omega):
        dog1= NP.zeros(len(s_omega),complex64)
        dog1.imag=s_omega* NP.exp(-s_omega**2/2.0)/sqrt(pi)
        return dog1

class DOG(Cwt):
    """
    Derivative Gaussian wavelet of order m
    but reconstruction seems to work best with +!
    """
    def wf(self, s_omega):
        try:
            from scipy.special import gamma
        except ImportError:
            print "Requires scipy gamma function"
            raise ImportError
        Cwt.fourierwl=2* NP.pi/ NP.sqrt(self.order+0.5)
        m=self.order
        dog=1.0J**m*s_omega**m* NP.exp(-s_omega**2/2)/ NP.sqrt(gamma(self.order+0.5))
        return dog

class Haar(Cwt):
    """
    Continuous version of Haar wavelet
    """
    #    note: not orthogonal!
    #    note: s_omega/4 matches Lecroix scale defn.
    #          s_omega/2 matches orthogonal Haar
    # 2/8/05 constants adjusted to match artem eim

    fourierwl=1.0#1.83129  #2.0
    def wf(self, s_omega):
        haar= NP.zeros(len(s_omega),complex64)
        om = s_omega[:]/self.currentscale
        om[0]=1.0  #prevent divide error
        #haar.imag=4.0*sin(s_omega/2)**2/om
        haar.imag=4.0* NP.sin(s_omega/4)**2/om
        return haar

class HaarW(Cwt):
    """
    Continuous version of Haar wavelet (norm)
    """
    #    note: not orthogonal!
    #    note: s_omega/4 matches Lecroix scale defn.
    #          s_omega/2 matches orthogonal Haar
    # normalised to unit power

    fourierwl=1.83129*1.2  #2.0
    def wf(self, s_omega):
        haar= NP.zeros(len(s_omega),complex64)
        om = s_omega[:]#/self.currentscale
        om[0]=1.0  #prevent divide error
        #haar.imag=4.0*sin(s_omega/2)**2/om
        haar.imag=4.0* NP.sin(s_omega/2)**2/om
        return haar


if __name__=="__main__":
    import numpy as np
    import pylab as mpl

    wavelet=Morlet
    maxscale=4
    notes=16
    scaling="log" #or "linear"
    #scaling="linear"
    plotpower2d=True #False

    # set up some data
    Ns=1024
    #limits of analysis
    Nlo=0
    Nhi=Ns
    # sinusoids of two periods, 128 and 32.
    x=np.arange(0.0,1.0*Ns,1.0)
    A=np.sin(2.0*np.pi*x/128.0)
    B=np.sin(2.0*np.pi*x/32.0)
    A[512:768]+=B[0:256]

    # Wavelet transform the data
    cw=wavelet(A,maxscale,notes,scaling=scaling)
    scales=cw.getscales()
    cwt=cw.getdata()
    # power spectrum
    pwr=cw.getpower()
    scalespec=np.sum(pwr,axis=1)/scales # calculate scale spectrum
    # scales
    y=cw.fourierwl*scales
    x=np.arange(Nlo*1.0,Nhi*1.0,1.0)

    fig=mpl.figure(1)

    # 2-d coefficient plot
    ax=mpl.axes([0.4,0.1,0.55,0.4])
    mpl.xlabel('Time [s]')
    plotcwt=np.clip(np.fabs(cwt.real), 0., 1000.)
    if plotpower2d: plotcwt=pwr
    im=mpl.imshow(plotcwt,cmap=mpl.cm.hot,extent=[x[0],x[-1],y[-1],y[0]],aspect='auto')
    #colorbar()
    if scaling=="log": ax.set_yscale('log')
    mpl.ylim(y[0],y[-1])
    ax.xaxis.set_ticks(np.arange(Nlo*1.0,(Nhi+1)*1.0,100.0))
    ax.yaxis.set_ticklabels(["",""])
    theposition=mpl.gca().get_position()

    # data plot
    ax2=mpl.axes([0.4,0.54,0.55,0.3])
    mpl.ylabel('Data')
    pos=ax.get_position()
    mpl.plot(x,A,'b-')
    mpl.xlim(Nlo*1.0,Nhi*1.0)
    ax2.xaxis.set_ticklabels(["",""])
    mpl.text(0.5,0.9,"Wavelet example with extra panes",
         fontsize=14,bbox=dict(facecolor='green',alpha=0.2),
         transform = fig.transFigure,horizontalalignment='center')

    # projected power spectrum
    ax3=mpl.axes([0.08,0.1,0.29,0.4])
    mpl.xlabel('Power')
    mpl.ylabel('Period [s]')
    vara=1.0
    if scaling=="log":
        mpl.loglog(scalespec/vara+0.01,y,'b-')
    else:
        mpl.semilogx(scalespec/vara+0.01,y,'b-')
    mpl.ylim(y[0],y[-1])
    mpl.xlim(1000.0,0.01)

    mpl.show()
	import numpy as NP

	"""
	A module which implements the continuous wavelet transform

	Wavelet classes:
	Morlet
	MorletReal
	MexicanHat
	Paul2 : Paul order 2
	Paul4 : Paul order 4
	DOG1 : 1st Derivative Of Gaussian
	DOG4 : 4th Derivative Of Gaussian
	Haar : Unnormalised version of continuous Haar transform
	HaarW : Normalised Haar

	Usage e.g.
	wavelet=Morlet(data, largestscale=2, notes=0, order=2, scaling="log")
	data: Numeric array of data (float), with length ndata.
	Optimum length is a power of 2 (for FFT)
	Worst-case length is a prime
	largestscale:
	largest scale as inverse fraction of length
	scale = len(data)/largestscale
	smallest scale should be >= 2 for meaningful data
	notes: number of scale intervals per octave
	if notes == 0, scales are on a linear increment
	order: order of wavelet for wavelets with variable order
	[Paul, DOG, ..]
	scaling: "linear" or "log" scaling of the wavelet scale.
	Note that feature width in the scale direction
	is constant on a log scale.

	Attributes of instance:
	wavelet.cwt: 2-d array of Wavelet coefficients, (nscales,ndata)
	wavelet.nscale: Number of scale intervals
	wavelet.scales: Array of scale values
	Note that meaning of the scale will depend on the family
	wavelet.fourierwl: Factor to multiply scale by to get scale
	of equivalent FFT
	Using this factor, different wavelet families will
	have comparable scales

	References:
	A practical guide to wavelet analysis
	C Torrance and GP Compo
	Bull Amer Meteor Soc Vol 79 No 1 61-78 (1998)
	naming below vaguely follows this.

	updates:
	(24/2/07): Fix Morlet so can get MorletReal by cutting out H
	(10/04/08): Numeric -> numpy
	(25/07/08): log and lin scale increment in same direction!
	swap indices in 2-d coefficient matrix
	explicit scaling of scale axis
	"""

	class Cwt:
	"""
	Base class for continuous wavelet transforms
	Implements cwt via the Fourier transform
	Used by subclass which provides the method wf(self,s_omega)
	wf is the Fourier transform of the wavelet function.
	Returns an instance.
	"""

	fourierwl=1.00

	def _log2(self, x):
	# utility function to return (integer) log2
	return int( NP.log(float(x))/ NP.log(2.0)+0.0001 )

	def __init__(self, data, largestscale=1, notes=0, order=2, scaling='linear'):
	"""
	Continuous wavelet transform of data

	data: data in array to transform, length must be power of 2
	notes: number of scale intervals per octave
	largestscale: largest scale as inverse fraction of length
	of data array
	scale = len(data)/largestscale
	smallest scale should be >= 2 for meaningful data
	order: Order of wavelet basis function for some families
	scaling: Linear or log
	"""
	ndata = len(data)
	self.order=order
	self.scale=largestscale
	self._setscales(ndata,largestscale,notes,scaling)
	self.cwt= NP.zeros((self.nscale,ndata), NP.complex64)
	omega= NP.array(range(0,ndata/2)+range(-ndata/2,0))(2.0NP.pi/ndata)
	datahat=NP.fft.fft(data)
	self.fftdata=datahat
	#self.psihat0=self.wf(omegaself.scales[3self.nscale/4])
	# loop over scales and compute wavelet coefficients at each scale
	# using the fft to do the convolution
	for scaleindex in range(self.nscale):
	currentscale=self.scales[scaleindex]
	self.currentscale=currentscale # for internal use
	s_omega = omega*currentscale
	psihat=self.wf(s_omega)
	psihat = psihat * NP.sqrt(2.0NP.picurrentscale)
	convhat = psihat * datahat
	W = NP.fft.ifft(convhat)
	self.cwt[scaleindex,0:ndata] = W
	return

	def _setscales(self,ndata,largestscale,notes,scaling):
	"""
	if notes non-zero, returns a log scale based on notes per octave
	else a linear scale
	(25/07/08): fix notes!=0 case so smallest scale at [0]
	"""
	if scaling=="log":
	if notes<=0: notes=1
	# adjust nscale so smallest scale is 2
	noctave=self._log2( ndata/largestscale/2 )
	self.nscale=notes*noctave
	self.scales=NP.zeros(self.nscale,float)
	for j in range(self.nscale):
	self.scales[j] = ndata/(self.scale(2.0*(float(self.nscale-1-j)/notes)))
	elif scaling=="linear":
	nmax=ndata/largestscale/2
	self.scales=NP.arange(float(2),float(nmax))
	self.nscale=len(self.scales)
	else: raise ValueError, "scaling must be linear or log"
	return

	def getdata(self):
	"""
	returns wavelet coefficient array
	"""
	return self.cwt

	def getcoefficients(self):
	return self.cwt

	def getpower(self):
	"""
	returns square of wavelet coefficient array
	"""
	return (self.cwt* NP.conjugate(self.cwt)).real

	def getscales(self):
	"""
	returns array containing scales used in transform
	"""
	return self.scales

	def getnscale(self):
	"""
	return number of scales
	"""
	return self.nscale

	# wavelet classes
	class Morlet(Cwt):
	"""
	Complex Morlet wavelet
	"""
	_omega0=5.0
	fourierwl=4* NP.pi/(_omega0+ NP.sqrt(2.0+_omega0**2))
	def wf(self, s_omega):
	H= NP.ones(len(s_omega))
	n=len(s_omega)
	for i in range(len(s_omega)):
	if s_omega[i] < 0.0: H[i]=0.0
	# !!!! note : was s_omega/8 before 17/6/03
	xhat=0.75112554( NP.exp(-(s_omega-self._omega0)2/2.0))H
	return xhat

	class MorletReal(Cwt):
	"""
	Real Morlet wavelet
	"""
	_omega0=5.0
	fourierwl=4* NP.pi/(_omega0+ NP.sqrt(2.0+_omega0**2))
	def wf(self, s_omega):
	H= NP.ones(len(s_omega))
	n=len(s_omega)
	for i in range(len(s_omega)):
	if s_omega[i] < 0.0: H[i]=0.0
	# !!!! note : was s_omega/8 before 17/6/03
	xhat=0.75112554( NP.exp(-(s_omega-self._omega0)2/2.0)+ NP.exp(-(s_omega+self._omega0)2/2.0)- NP.exp(-(self._omega0)2/2.0)+ NP.exp(-(self._omega0)*2/2.0))
	return xhat

	class Paul4(Cwt):
	"""
	Paul m=4 wavelet
	"""
	fourierwl=4* NP.pi/(2.*4+1.)
	def wf(self, s_omega):
	n=len(s_omega)
	xhat= NP.zeros(n)
	xhat[0:n/2]=0.11268723s_omega[0:n/2]4 NP.exp(-s_omega[0:n/2])
	#return 0.11268723s_omega2exp(-s_omega)*H
	return xhat

	class Paul2(Cwt):
	"""
	Paul m=2 wavelet
	"""
	fourierwl=4* NP.pi/(2.*2+1.)
	def wf(self, s_omega):
	n=len(s_omega)
	xhat= NP.zeros(n)
	xhat[0:n/2]=1.1547005s_omega[0:n/2]2 NP.exp(-s_omega[0:n/2])
	#return 0.11268723s_omega2exp(-s_omega)*H
	return xhat

	class Paul(Cwt):
	"""
	Paul order m wavelet
	"""
	def wf(self, s_omega):
	Cwt.fourierwl=4* NP.pi/(2.*self.order+1.)
	m=self.order
	n=len(s_omega)
	normfactor=float(m)
	for i in range(1,2*m):
	normfactor=normfactor*i
	normfactor=2.0**m/ NP.sqrt(normfactor)
	xhat= NP.zeros(n)
	xhat[0:n/2]=normfactors_omega[0:n/2]m NP.exp(-s_omega[0:n/2])
	#return 0.11268723s_omega2exp(-s_omega)*H
	return xhat

	class MexicanHat(Cwt):
	"""
	2nd Derivative Gaussian (Mexican hat) wavelet
	"""
	fourierwl=2.0* NP.pi/ NP.sqrt(2.5)
	def wf(self, s_omega):
	# should this number be 1/sqrt(3/4) (no pi)?
	#s_omega = s_omega/self.fourierwl
	#print max(s_omega)
	a=s_omega**2
	b=s_omega**2/2
	return a* NP.exp(-b)/1.1529702
	#return s_omega*2exp(-s_omega**2/2.0)/1.1529702

	class DOG4(Cwt):
	"""
	4th Derivative Gaussian wavelet
	see also T&C errata for - sign
	but reconstruction seems to work best with +!
	"""
	fourierwl=2.0* NP.pi/ NP.sqrt(4.5)
	def wf(self, s_omega):
	return s_omega*4 NP.exp(-s_omega**2/2.0)/3.4105319

	class DOG1(Cwt):
	"""
	1st Derivative Gaussian wavelet
	but reconstruction seems to work best with +!
	"""
	fourierwl=2.0* NP.pi/ NP.sqrt(1.5)
	def wf(self, s_omega):
	dog1= NP.zeros(len(s_omega),complex64)
	dog1.imag=s_omega* NP.exp(-s_omega**2/2.0)/sqrt(pi)
	return dog1

	class DOG(Cwt):
	"""
	Derivative Gaussian wavelet of order m
	but reconstruction seems to work best with +!
	"""
	def wf(self, s_omega):
	try:
	from scipy.special import gamma
	except ImportError:
	print "Requires scipy gamma function"
	raise ImportError
	Cwt.fourierwl=2* NP.pi/ NP.sqrt(self.order+0.5)
	m=self.order
	dog=1.0J*ms_omega*m NP.exp(-s_omega**2/2)/ NP.sqrt(gamma(self.order+0.5))
	return dog

	class Haar(Cwt):
	"""
	Continuous version of Haar wavelet
	"""
	# note: not orthogonal!
	# note: s_omega/4 matches Lecroix scale defn.
	# s_omega/2 matches orthogonal Haar
	# 2/8/05 constants adjusted to match artem eim

	fourierwl=1.0#1.83129 #2.0
	def wf(self, s_omega):
	haar= NP.zeros(len(s_omega),complex64)
	om = s_omega[:]/self.currentscale
	om[0]=1.0 #prevent divide error
	#haar.imag=4.0sin(s_omega/2)*2/om
	haar.imag=4.0* NP.sin(s_omega/4)**2/om
	return haar

	class HaarW(Cwt):
	"""
	Continuous version of Haar wavelet (norm)
	"""
	# note: not orthogonal!
	# note: s_omega/4 matches Lecroix scale defn.
	# s_omega/2 matches orthogonal Haar
	# normalised to unit power

	fourierwl=1.83129*1.2 #2.0
	def wf(self, s_omega):
	haar= NP.zeros(len(s_omega),complex64)
	om = s_omega[:]#/self.currentscale
	om[0]=1.0 #prevent divide error
	#haar.imag=4.0sin(s_omega/2)*2/om
	haar.imag=4.0* NP.sin(s_omega/2)**2/om
	return haar


	if __name__=="__main__":
	import numpy as np
	import pylab as mpl

	wavelet=Morlet
	maxscale=4
	notes=16
	scaling="log" #or "linear"
	#scaling="linear"
	plotpower2d=True #False

	# set up some data
	Ns=1024
	#limits of analysis
	Nlo=0
	Nhi=Ns
	# sinusoids of two periods, 128 and 32.
	x=np.arange(0.0,1.0*Ns,1.0)
	A=np.sin(2.0np.pix/128.0)
	B=np.sin(2.0np.pix/32.0)
	A[512:768]+=B[0:256]

	# Wavelet transform the data
	cw=wavelet(A,maxscale,notes,scaling=scaling)
	scales=cw.getscales()
	cwt=cw.getdata()
	# power spectrum
	pwr=cw.getpower()
	scalespec=np.sum(pwr,axis=1)/scales # calculate scale spectrum
	# scales
	y=cw.fourierwl*scales
	x=np.arange(Nlo1.0,Nhi1.0,1.0)

	fig=mpl.figure(1)

	# 2-d coefficient plot
	ax=mpl.axes([0.4,0.1,0.55,0.4])
	mpl.xlabel('Time [s]')
	plotcwt=np.clip(np.fabs(cwt.real), 0., 1000.)
	if plotpower2d: plotcwt=pwr
	im=mpl.imshow(plotcwt,cmap=mpl.cm.hot,extent=[x[0],x[-1],y[-1],y[0]],aspect='auto')
	#colorbar()
	if scaling=="log": ax.set_yscale('log')
	mpl.ylim(y[0],y[-1])
	ax.xaxis.set_ticks(np.arange(Nlo1.0,(Nhi+1)1.0,100.0))
	ax.yaxis.set_ticklabels(["",""])
	theposition=mpl.gca().get_position()

	# data plot
	ax2=mpl.axes([0.4,0.54,0.55,0.3])
	mpl.ylabel('Data')
	pos=ax.get_position()
	mpl.plot(x,A,'b-')
	mpl.xlim(Nlo1.0,Nhi1.0)
	ax2.xaxis.set_ticklabels(["",""])
	mpl.text(0.5,0.9,"Wavelet example with extra panes",
	fontsize=14,bbox=dict(facecolor='green',alpha=0.2),
	transform = fig.transFigure,horizontalalignment='center')

	# projected power spectrum
	ax3=mpl.axes([0.08,0.1,0.29,0.4])
	mpl.xlabel('Power')
	mpl.ylabel('Period [s]')
	vara=1.0
	if scaling=="log":
	mpl.loglog(scalespec/vara+0.01,y,'b-')
	else:
	mpl.semilogx(scalespec/vara+0.01,y,'b-')
	mpl.ylim(y[0],y[-1])
	mpl.xlim(1000.0,0.01)

	mpl.show()