493 lines
14 KiB
Python
493 lines
14 KiB
Python
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import numpy as np
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from scipy.linalg import eig
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from scipy.special import comb
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from scipy.signal import convolve
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__all__ = ['daub', 'qmf', 'cascade', 'morlet', 'ricker', 'morlet2', 'cwt']
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def daub(p):
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"""
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The coefficients for the FIR low-pass filter producing Daubechies wavelets.
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p>=1 gives the order of the zero at f=1/2.
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There are 2p filter coefficients.
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Parameters
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----------
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p : int
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Order of the zero at f=1/2, can have values from 1 to 34.
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Returns
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-------
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daub : ndarray
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Return
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"""
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sqrt = np.sqrt
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if p < 1:
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raise ValueError("p must be at least 1.")
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if p == 1:
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c = 1 / sqrt(2)
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return np.array([c, c])
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elif p == 2:
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f = sqrt(2) / 8
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c = sqrt(3)
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return f * np.array([1 + c, 3 + c, 3 - c, 1 - c])
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elif p == 3:
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tmp = 12 * sqrt(10)
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z1 = 1.5 + sqrt(15 + tmp) / 6 - 1j * (sqrt(15) + sqrt(tmp - 15)) / 6
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z1c = np.conj(z1)
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f = sqrt(2) / 8
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d0 = np.real((1 - z1) * (1 - z1c))
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a0 = np.real(z1 * z1c)
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a1 = 2 * np.real(z1)
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return f / d0 * np.array([a0, 3 * a0 - a1, 3 * a0 - 3 * a1 + 1,
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a0 - 3 * a1 + 3, 3 - a1, 1])
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elif p < 35:
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# construct polynomial and factor it
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if p < 35:
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P = [comb(p - 1 + k, k, exact=1) for k in range(p)][::-1]
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yj = np.roots(P)
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else: # try different polynomial --- needs work
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P = [comb(p - 1 + k, k, exact=1) / 4.0**k
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for k in range(p)][::-1]
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yj = np.roots(P) / 4
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# for each root, compute two z roots, select the one with |z|>1
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# Build up final polynomial
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c = np.poly1d([1, 1])**p
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q = np.poly1d([1])
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for k in range(p - 1):
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yval = yj[k]
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part = 2 * sqrt(yval * (yval - 1))
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const = 1 - 2 * yval
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z1 = const + part
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if (abs(z1)) < 1:
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z1 = const - part
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q = q * [1, -z1]
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q = c * np.real(q)
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# Normalize result
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q = q / np.sum(q) * sqrt(2)
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return q.c[::-1]
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else:
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raise ValueError("Polynomial factorization does not work "
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"well for p too large.")
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def qmf(hk):
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"""
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Return high-pass qmf filter from low-pass
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Parameters
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----------
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hk : array_like
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Coefficients of high-pass filter.
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Returns
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-------
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array_like
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High-pass filter coefficients.
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"""
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N = len(hk) - 1
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asgn = [{0: 1, 1: -1}[k % 2] for k in range(N + 1)]
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return hk[::-1] * np.array(asgn)
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def cascade(hk, J=7):
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"""
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Return (x, phi, psi) at dyadic points ``K/2**J`` from filter coefficients.
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Parameters
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----------
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hk : array_like
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Coefficients of low-pass filter.
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J : int, optional
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Values will be computed at grid points ``K/2**J``. Default is 7.
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Returns
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-------
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x : ndarray
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The dyadic points ``K/2**J`` for ``K=0...N * (2**J)-1`` where
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``len(hk) = len(gk) = N+1``.
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phi : ndarray
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The scaling function ``phi(x)`` at `x`:
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``phi(x) = sum(hk * phi(2x-k))``, where k is from 0 to N.
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psi : ndarray, optional
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The wavelet function ``psi(x)`` at `x`:
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``phi(x) = sum(gk * phi(2x-k))``, where k is from 0 to N.
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`psi` is only returned if `gk` is not None.
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Notes
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-----
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The algorithm uses the vector cascade algorithm described by Strang and
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Nguyen in "Wavelets and Filter Banks". It builds a dictionary of values
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and slices for quick reuse. Then inserts vectors into final vector at the
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end.
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"""
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N = len(hk) - 1
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if (J > 30 - np.log2(N + 1)):
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raise ValueError("Too many levels.")
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if (J < 1):
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raise ValueError("Too few levels.")
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# construct matrices needed
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nn, kk = np.ogrid[:N, :N]
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s2 = np.sqrt(2)
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# append a zero so that take works
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thk = np.r_[hk, 0]
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gk = qmf(hk)
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tgk = np.r_[gk, 0]
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indx1 = np.clip(2 * nn - kk, -1, N + 1)
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indx2 = np.clip(2 * nn - kk + 1, -1, N + 1)
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m = np.empty((2, 2, N, N), 'd')
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m[0, 0] = np.take(thk, indx1, 0)
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m[0, 1] = np.take(thk, indx2, 0)
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m[1, 0] = np.take(tgk, indx1, 0)
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m[1, 1] = np.take(tgk, indx2, 0)
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m *= s2
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# construct the grid of points
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x = np.arange(0, N * (1 << J), dtype=float) / (1 << J)
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phi = 0 * x
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psi = 0 * x
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# find phi0, and phi1
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lam, v = eig(m[0, 0])
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ind = np.argmin(np.absolute(lam - 1))
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# a dictionary with a binary representation of the
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# evaluation points x < 1 -- i.e. position is 0.xxxx
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v = np.real(v[:, ind])
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# need scaling function to integrate to 1 so find
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# eigenvector normalized to sum(v,axis=0)=1
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sm = np.sum(v)
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if sm < 0: # need scaling function to integrate to 1
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v = -v
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sm = -sm
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bitdic = {'0': v / sm}
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bitdic['1'] = np.dot(m[0, 1], bitdic['0'])
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step = 1 << J
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phi[::step] = bitdic['0']
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phi[(1 << (J - 1))::step] = bitdic['1']
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psi[::step] = np.dot(m[1, 0], bitdic['0'])
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psi[(1 << (J - 1))::step] = np.dot(m[1, 1], bitdic['0'])
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# descend down the levels inserting more and more values
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# into bitdic -- store the values in the correct location once we
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# have computed them -- stored in the dictionary
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# for quicker use later.
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prevkeys = ['1']
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for level in range(2, J + 1):
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newkeys = ['%d%s' % (xx, yy) for xx in [0, 1] for yy in prevkeys]
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fac = 1 << (J - level)
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for key in newkeys:
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# convert key to number
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num = 0
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for pos in range(level):
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if key[pos] == '1':
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num += (1 << (level - 1 - pos))
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pastphi = bitdic[key[1:]]
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ii = int(key[0])
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temp = np.dot(m[0, ii], pastphi)
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bitdic[key] = temp
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phi[num * fac::step] = temp
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psi[num * fac::step] = np.dot(m[1, ii], pastphi)
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prevkeys = newkeys
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return x, phi, psi
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def morlet(M, w=5.0, s=1.0, complete=True):
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"""
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Complex Morlet wavelet.
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Parameters
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----------
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M : int
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Length of the wavelet.
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w : float, optional
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Omega0. Default is 5
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s : float, optional
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Scaling factor, windowed from ``-s*2*pi`` to ``+s*2*pi``. Default is 1.
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complete : bool, optional
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Whether to use the complete or the standard version.
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Returns
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-------
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morlet : (M,) ndarray
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See Also
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--------
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morlet2 : Implementation of Morlet wavelet, compatible with `cwt`.
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scipy.signal.gausspulse
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Notes
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-----
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The standard version::
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pi**-0.25 * exp(1j*w*x) * exp(-0.5*(x**2))
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This commonly used wavelet is often referred to simply as the
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Morlet wavelet. Note that this simplified version can cause
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admissibility problems at low values of `w`.
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The complete version::
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pi**-0.25 * (exp(1j*w*x) - exp(-0.5*(w**2))) * exp(-0.5*(x**2))
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This version has a correction
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term to improve admissibility. For `w` greater than 5, the
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correction term is negligible.
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Note that the energy of the return wavelet is not normalised
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according to `s`.
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The fundamental frequency of this wavelet in Hz is given
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by ``f = 2*s*w*r / M`` where `r` is the sampling rate.
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Note: This function was created before `cwt` and is not compatible
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with it.
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Examples
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--------
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>>> from scipy import signal
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>>> import matplotlib.pyplot as plt
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>>> M = 100
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>>> s = 4.0
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>>> w = 2.0
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>>> wavelet = signal.morlet(M, s, w)
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>>> plt.plot(wavelet)
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>>> plt.show()
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"""
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x = np.linspace(-s * 2 * np.pi, s * 2 * np.pi, M)
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output = np.exp(1j * w * x)
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if complete:
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output -= np.exp(-0.5 * (w**2))
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output *= np.exp(-0.5 * (x**2)) * np.pi**(-0.25)
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return output
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def ricker(points, a):
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"""
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Return a Ricker wavelet, also known as the "Mexican hat wavelet".
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It models the function:
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``A * (1 - (x/a)**2) * exp(-0.5*(x/a)**2)``,
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where ``A = 2/(sqrt(3*a)*(pi**0.25))``.
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Parameters
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----------
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points : int
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Number of points in `vector`.
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Will be centered around 0.
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a : scalar
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Width parameter of the wavelet.
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Returns
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-------
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vector : (N,) ndarray
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Array of length `points` in shape of ricker curve.
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Examples
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--------
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>>> from scipy import signal
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>>> import matplotlib.pyplot as plt
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>>> points = 100
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>>> a = 4.0
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>>> vec2 = signal.ricker(points, a)
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>>> print(len(vec2))
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100
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>>> plt.plot(vec2)
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>>> plt.show()
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"""
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A = 2 / (np.sqrt(3 * a) * (np.pi**0.25))
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wsq = a**2
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vec = np.arange(0, points) - (points - 1.0) / 2
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xsq = vec**2
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mod = (1 - xsq / wsq)
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gauss = np.exp(-xsq / (2 * wsq))
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total = A * mod * gauss
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return total
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def morlet2(M, s, w=5):
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"""
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Complex Morlet wavelet, designed to work with `cwt`.
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Returns the complete version of morlet wavelet, normalised
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according to `s`::
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exp(1j*w*x/s) * exp(-0.5*(x/s)**2) * pi**(-0.25) * sqrt(1/s)
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Parameters
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----------
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M : int
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Length of the wavelet.
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s : float
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Width parameter of the wavelet.
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w : float, optional
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Omega0. Default is 5
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Returns
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-------
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morlet : (M,) ndarray
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See Also
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--------
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morlet : Implementation of Morlet wavelet, incompatible with `cwt`
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Notes
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-----
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.. versionadded:: 1.4.0
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This function was designed to work with `cwt`. Because `morlet2`
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returns an array of complex numbers, the `dtype` argument of `cwt`
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should be set to `complex128` for best results.
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Note the difference in implementation with `morlet`.
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The fundamental frequency of this wavelet in Hz is given by::
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f = w*fs / (2*s*np.pi)
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where ``fs`` is the sampling rate and `s` is the wavelet width parameter.
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Similarly we can get the wavelet width parameter at ``f``::
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s = w*fs / (2*f*np.pi)
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Examples
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--------
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>>> import numpy as np
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>>> from scipy import signal
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>>> import matplotlib.pyplot as plt
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>>> M = 100
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>>> s = 4.0
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>>> w = 2.0
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>>> wavelet = signal.morlet2(M, s, w)
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>>> plt.plot(abs(wavelet))
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|
>>> plt.show()
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|
|
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||
|
|
This example shows basic use of `morlet2` with `cwt` in time-frequency
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||
|
|
analysis:
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||
|
|
|
||
|
|
>>> t, dt = np.linspace(0, 1, 200, retstep=True)
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|
|
>>> fs = 1/dt
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|
|
>>> w = 6.
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||
|
|
>>> sig = np.cos(2*np.pi*(50 + 10*t)*t) + np.sin(40*np.pi*t)
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|
|
>>> freq = np.linspace(1, fs/2, 100)
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|
|
>>> widths = w*fs / (2*freq*np.pi)
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||
|
|
>>> cwtm = signal.cwt(sig, signal.morlet2, widths, w=w)
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|
|
>>> plt.pcolormesh(t, freq, np.abs(cwtm), cmap='viridis', shading='gouraud')
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||
|
|
>>> plt.show()
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||
|
|
|
||
|
|
"""
|
||
|
|
x = np.arange(0, M) - (M - 1.0) / 2
|
||
|
|
x = x / s
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||
|
|
wavelet = np.exp(1j * w * x) * np.exp(-0.5 * x**2) * np.pi**(-0.25)
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||
|
|
output = np.sqrt(1/s) * wavelet
|
||
|
|
return output
|
||
|
|
|
||
|
|
|
||
|
|
def cwt(data, wavelet, widths, dtype=None, **kwargs):
|
||
|
|
"""
|
||
|
|
Continuous wavelet transform.
|
||
|
|
|
||
|
|
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. The `wavelet` function
|
||
|
|
is allowed to be complex.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : (N,) 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(length,width)) == 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 : (M,) sequence
|
||
|
|
Widths to use for transform.
|
||
|
|
dtype : data-type, optional
|
||
|
|
The desired data type of output. Defaults to ``float64`` if the
|
||
|
|
output of `wavelet` is real and ``complex128`` if it is complex.
|
||
|
|
|
||
|
|
.. versionadded:: 1.4.0
|
||
|
|
|
||
|
|
kwargs
|
||
|
|
Keyword arguments passed to wavelet function.
|
||
|
|
|
||
|
|
.. versionadded:: 1.4.0
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
cwt: (M, N) ndarray
|
||
|
|
Will have shape of (len(widths), len(data)).
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
|
||
|
|
.. versionadded:: 1.4.0
|
||
|
|
|
||
|
|
For non-symmetric, complex-valued wavelets, the input signal is convolved
|
||
|
|
with the time-reversed complex-conjugate of the wavelet data [1].
|
||
|
|
|
||
|
|
::
|
||
|
|
|
||
|
|
length = min(10 * width[ii], len(data))
|
||
|
|
cwt[ii,:] = signal.convolve(data, np.conj(wavelet(length, width[ii],
|
||
|
|
**kwargs))[::-1], mode='same')
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] S. Mallat, "A Wavelet Tour of Signal Processing (3rd Edition)",
|
||
|
|
Academic Press, 2009.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from scipy import signal
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> t = np.linspace(-1, 1, 200, endpoint=False)
|
||
|
|
>>> sig = np.cos(2 * np.pi * 7 * t) + signal.gausspulse(t - 0.4, fc=2)
|
||
|
|
>>> widths = np.arange(1, 31)
|
||
|
|
>>> cwtmatr = signal.cwt(sig, signal.ricker, widths)
|
||
|
|
|
||
|
|
.. note:: For cwt matrix plotting it is advisable to flip the y-axis
|
||
|
|
|
||
|
|
>>> cwtmatr_yflip = np.flipud(cwtmatr)
|
||
|
|
>>> plt.imshow(cwtmatr_yflip, extent=[-1, 1, 1, 31], cmap='PRGn', aspect='auto',
|
||
|
|
... vmax=abs(cwtmatr).max(), vmin=-abs(cwtmatr).max())
|
||
|
|
>>> plt.show()
|
||
|
|
"""
|
||
|
|
# Determine output type
|
||
|
|
if dtype is None:
|
||
|
|
if np.asarray(wavelet(1, widths[0], **kwargs)).dtype.char in 'FDG':
|
||
|
|
dtype = np.complex128
|
||
|
|
else:
|
||
|
|
dtype = np.float64
|
||
|
|
|
||
|
|
output = np.empty((len(widths), len(data)), dtype=dtype)
|
||
|
|
for ind, width in enumerate(widths):
|
||
|
|
N = np.min([10 * width, len(data)])
|
||
|
|
wavelet_data = np.conj(wavelet(N, width, **kwargs)[::-1])
|
||
|
|
output[ind] = convolve(data, wavelet_data, mode='same')
|
||
|
|
return output
|