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import numpy as np
from numpy.dual import eig
from scipy.misc import comb
from scipy import linspace, pi, exp
from scipy.signal import convolve
__all__ = ['daub', 'qmf', 'cascade', 'morlet', 'ricker', 'cwt']
def daub(p):
"""
The coefficients for the FIR low-pass filter producing Daubechies wavelets.
p>=1 gives the order of the zero at f=1/2.
There are 2p filter coefficients.
Parameters
----------
p : int
Order of the zero at f=1/2, can have values from 1 to 34.
Returns
-------
q : ndarray
"""
sqrt = np.sqrt
if p < 1:
raise ValueError("p must be at least 1.")
if p == 1:
c = 1 / sqrt(2)
return np.array([c, c])
elif p == 2:
f = sqrt(2) / 8
c = sqrt(3)
return f * np.array([1 + c, 3 + c, 3 - c, 1 - c])
elif p == 3:
tmp = 12 * sqrt(10)
z1 = 1.5 + sqrt(15 + tmp) / 6 - 1j * (sqrt(15) + sqrt(tmp - 15)) / 6
z1c = np.conj(z1)
f = sqrt(2) / 8
d0 = np.real((1 - z1) * (1 - z1c))
a0 = np.real(z1 * z1c)
a1 = 2 * np.real(z1)
return f / d0 * np.array([a0, 3 * a0 - a1, 3 * a0 - 3 * a1 + 1,
a0 - 3 * a1 + 3, 3 - a1, 1])
elif p < 35:
# construct polynomial and factor it
if p < 35:
P = [comb(p - 1 + k, k, exact=1) for k in range(p)][::-1]
yj = np.roots(P)
else: # try different polynomial --- needs work
P = [comb(p - 1 + k, k, exact=1) / 4.0**k
for k in range(p)][::-1]
yj = np.roots(P) / 4
# for each root, compute two z roots, select the one with |z|>1
# Build up final polynomial
c = np.poly1d([1, 1])**p
q = np.poly1d([1])
for k in range(p - 1):
yval = yj[k]
part = 2 * sqrt(yval * (yval - 1))
const = 1 - 2 * yval
z1 = const + part
if (abs(z1)) < 1:
z1 = const - part
q = q * [1, -z1]
q = c * np.real(q)
# Normalize result
q = q / np.sum(q) * sqrt(2)
return q.c[::-1]
else:
raise ValueError("Polynomial factorization does not work "
"well for p too large.")
def qmf(hk):
"""Return high-pass qmf filter from low-pass
"""
N = len(hk) - 1
asgn = [{0: 1, 1:-1}[k % 2] for k in range(N + 1)]
return hk[::-1] * np.array(asgn)
def wavedec(amn, hk):
gk = qmf(hk)
return NotImplemented
def cascade(hk, J=7):
"""
Return (x, phi, psi) at dyadic points ``K/2**J`` from filter coefficients.
Parameters
----------
hk : array_like
Coefficients of low-pass filter.
J : int, optional
Values will be computed at grid points ``K/2**J``. Default is 7.
Returns
-------
x : ndarray
The dyadic points ``K/2**J`` for ``K=0...N * (2**J)-1`` where
``len(hk) = len(gk) = N+1``.
phi : ndarray
The scaling function ``phi(x)`` at `x`::
N
phi(x) = sum hk * phi(2x-k)
k=0
psi : ndarray, optional
The wavelet function ``psi(x)`` at `x`::
N
phi(x) = sum gk * phi(2x-k)
k=0
`psi` is only returned if `gk` is not None.
Notes
-----
The algorithm uses the vector cascade algorithm described by Strang and
Nguyen in "Wavelets and Filter Banks". It builds a dictionary of values
and slices for quick reuse. Then inserts vectors into final vector at the
end.
"""
N = len(hk) - 1
if (J > 30 - np.log2(N + 1)):
raise ValueError("Too many levels.")
if (J < 1):
raise ValueError("Too few levels.")
# construct matrices needed
nn, kk = np.ogrid[:N, :N]
s2 = np.sqrt(2)
# append a zero so that take works
thk = np.r_[hk, 0]
gk = qmf(hk)
tgk = np.r_[gk, 0]
indx1 = np.clip(2 * nn - kk, -1, N + 1)
indx2 = np.clip(2 * nn - kk + 1, -1, N + 1)
m = np.zeros((2, 2, N, N), 'd')
m[0, 0] = np.take(thk, indx1, 0)
m[0, 1] = np.take(thk, indx2, 0)
m[1, 0] = np.take(tgk, indx1, 0)
m[1, 1] = np.take(tgk, indx2, 0)
m *= s2
# construct the grid of points
x = np.arange(0, N * (1 << J), dtype=np.float) / (1 << J)
phi = 0 * x
psi = 0 * x
# find phi0, and phi1
lam, v = eig(m[0, 0])
ind = np.argmin(np.absolute(lam - 1))
# a dictionary with a binary representation of the
# evaluation points x < 1 -- i.e. position is 0.xxxx
v = np.real(v[:, ind])
# need scaling function to integrate to 1 so find
# eigenvector normalized to sum(v,axis=0)=1
sm = np.sum(v)
if sm < 0: # need scaling function to integrate to 1
v = -v
sm = -sm
bitdic = {}
bitdic['0'] = v / sm
bitdic['1'] = np.dot(m[0, 1], bitdic['0'])
step = 1 << J
phi[::step] = bitdic['0']
phi[(1 << (J - 1))::step] = bitdic['1']
psi[::step] = np.dot(m[1, 0], bitdic['0'])
psi[(1 << (J - 1))::step] = np.dot(m[1, 1], bitdic['0'])
# descend down the levels inserting more and more values
# into bitdic -- store the values in the correct location once we
# have computed them -- stored in the dictionary
# for quicker use later.
prevkeys = ['1']
for level in range(2, J + 1):
newkeys = ['%d%s' % (xx, yy) for xx in [0, 1] for yy in prevkeys]
fac = 1 << (J - level)
for key in newkeys:
# convert key to number
num = 0
for pos in range(level):
if key[pos] == '1':
num += (1 << (level - 1 - pos))
pastphi = bitdic[key[1:]]
ii = int(key[0])
temp = np.dot(m[0, ii], pastphi)
bitdic[key] = temp
phi[num * fac::step] = temp
psi[num * fac::step] = np.dot(m[1, ii], pastphi)
prevkeys = newkeys
return x, phi, psi
def morlet(M, w=5.0, s=1.0, complete=True):
"""
Complex Morlet wavelet.
Parameters
----------
M : int
Length of the wavelet.
w : float
Omega0. Default is 5
s : float
Scaling factor, windowed from ``-s*2*pi`` to ``+s*2*pi``. Default is 1.
complete : bool
Whether to use the complete or the standard version.
Returns
-------
out : 1-D ndarray
Notes
-----
The standard version::
pi**-0.25 * exp(1j*w*x) * exp(-0.5*(x**2))
This commonly used wavelet is often referred to simply as the
Morlet wavelet. Note that this simplified version can cause
admissibility problems at low values of w.
The complete version::
pi**-0.25 * (exp(1j*w*x) - exp(-0.5*(w**2))) * exp(-0.5*(x**2))
The complete version of the Morlet wavelet, with a correction
term to improve admissibility. For w greater than 5, the
correction term is negligible.
Note that the energy of the return wavelet is not normalised
according to s.
The fundamental frequency of this wavelet in Hz is given
by ``f = 2*s*w*r / M`` where r is the sampling rate.
See Also
--------
scipy.signal.gausspulse
"""
x = linspace(-s * 2 * pi, s * 2 * pi, M)
output = exp(1j * w * x)
if complete:
output -= exp(-0.5 * (w**2))
output *= exp(-0.5 * (x**2)) * pi**(-0.25)
return output
def ricker(points, a):
"""
Return a Ricker wavelet, also known as the "Mexican hat wavelet".
It models the function:
``A (1 - x^2/a^2) exp(-t^2/a^2)``,
where ``A = 2/sqrt(3a)pi^1/3``.
Parameters
----------
points : int
Number of points in `vector`. Default is ``10 * a``.
Will be centered around 0.
a : scalar
Width parameter of the wavelet.
Returns
-------
vector : 1-D ndarray
Array of length `points` in shape of Ricker curve.
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> points = 100
>>> a = 4.0
>>> vec2 = signal.ricker(points, a)
>>> print len(vec2)
100
>>> plt.plot(vec2)
>>> plt.show()
"""
A = 2 / (np.sqrt(3 * a) * (np.pi**0.25))
wsq = a**2
vec = np.arange(0, points) - (points - 1.0) / 2
tsq = vec**2
mod = (1 - tsq / wsq)
gauss = np.exp(-tsq / (2 * wsq))
total = A * mod * gauss
return total
def cwt(data, wavelet, widths):
"""
Performs a continuous wavelet transform on `data`,
using the `wavelet` function. A CWT performs a convolution
with `data` using the `wavelet` function, which is characterized
by a width parameter and length parameter.
Parameters
----------
data : 1-D ndarray
data on which to perform the transform.
wavelet : function
Wavelet function, which should take 2 arguments.
The first argument is the number of points that the returned vector
will have (len(wavelet(width,length)) == length).
The second is a width parameter, defining the size of the wavelet
(e.g. standard deviation of a gaussian). See `ricker`, which
satisfies these requirements.
widths : sequence
Widths to use for transform.
Returns
-------
cwt : 2-D ndarray
Will be ``len(widths) * len(data)``.
Notes
------
``cwt[ii,:] = scipy.signal.convolve(data,wavelet(width[ii], length), mode='same')``,
where length = ``min(10 * width[ii], len(data))``.
Examples
--------
>>> from scipy import signal
>>> sig = np.random.rand(20) - 0.5
>>> wavelet = signal.ricker
>>> widths = np.arange(1, 11)
>>> cwtmatr = signal.cwt(sig, wavelet, widths)
"""
output = np.zeros([len(widths), len(data)])
for ind, width in enumerate(widths):
wavelet_data = wavelet(min(10 * width, len(data)), width)
output[ind, :] = convolve(data, wavelet_data,
mode='same')
return output
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