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# -*- coding:utf-8 -*-
from __future__ import division
import numpy as np
__all__ = ['demo_signal']
_implemented_signals = [
'Blocks',
'Bumps',
'HeaviSine',
'Doppler',
'Ramp',
'HiSine',
'LoSine',
'LinChirp',
'TwoChirp',
'QuadChirp',
'MishMash',
'WernerSorrows',
'HypChirps',
'LinChirps',
'Chirps',
'Gabor',
'sineoneoverx',
'Piece-Regular',
'Piece-Polynomial',
'Riemann']
def demo_signal(name='Bumps', n=None):
"""Simple 1D wavelet test functions.
This function can generate a number of common 1D test signals used in
papers by David Donoho and colleagues (e.g. [1]_) as well as the wavelet
book by Stéphane Mallat [2]_.
Parameters
----------
name : {'Blocks', 'Bumps', 'HeaviSine', 'Doppler', ...}
The type of test signal to generate (`name` is case-insensitive). If
`name` is set to `'list'`, a list of the available test functions is
returned.
n : int or None
The length of the test signal. This should be provided for all test
signals except `'Gabor'` and `'sineoneoverx'` which have a fixed
length.
Returns
-------
f : np.ndarray
Array of length ``n`` corresponding to the specified test signal type.
References
----------
.. [1] D.L. Donoho and I.M. Johnstone. Ideal spatial adaptation by
wavelet shrinkage. Biometrika, vol. 81, pp. 425–455, 1994.
.. [2] S. Mallat. A Wavelet Tour of Signal Processing: The Sparse Way.
Academic Press. 2009.
Notes
-----
This function is a partial reimplementation of the `MakeSignal` function
from the [Wavelab](https://statweb.stanford.edu/~wavelab/) toolbox. These
test signals are provided with permission of Dr. Donoho to encourage
reproducible research.
"""
if name.lower() == 'list':
return _implemented_signals
if n is not None:
if n < 1 or (n % 1) != 0:
raise ValueError("n must be an integer >= 1")
t = np.arange(1/n, 1 + 1/n, 1/n)
# The following function types don't allow user-specified `n`.
n_hard_coded = ['gabor', 'sineoneoverx']
name = name.lower()
if name in n_hard_coded and n is not None:
raise ValueError(
"Parameter n must be set to None when name is {}".format(name))
elif n is None and name not in n_hard_coded:
raise ValueError(
"Parameter n must be provided when name is {}".format(name))
if name == 'blocks':
t0s = [.1, .13, .15, .23, .25, .4, .44, .65, .76, .78, .81]
hs = [4, -5, 3, -4, 5, -4.2, 2.1, 4.3, -3.1, 2.1, -4.2]
f = 0
for (t0, h) in zip(t0s, hs):
f += h * (1 + np.sign(t - t0)) / 2
elif name == 'bumps':
t0s = [.1, .13, .15, .23, .25, .4, .44, .65, .76, .78, .81]
hs = [4, 5, 3, 4, 5, 4.2, 2.1, 4.3, 3.1, 5.1, 4.2]
ws = [.005, .005, .006, .01, .01, .03, .01, .01, .005, .008, .005]
f = 0
for (t0, h, w) in zip(t0s, hs, ws):
f += h / (1 + np.abs((t - t0) / w))**4
elif name == 'heavisine':
f = 4 * np.sin(4 * np.pi * t) - np.sign(t - 0.3) - np.sign(0.72 - t)
elif name == 'doppler':
f = np.sqrt(t * (1 - t)) * np.sin(2 * np.pi * 1.05 / (t + 0.05))
elif name == 'ramp':
f = t - (t >= .37)
elif name == 'hisine':
f = np.sin(np.pi * (n * .6902) * t)
elif name == 'losine':
f = np.sin(np.pi * (n * .3333) * t)
elif name == 'linchirp':
f = np.sin(np.pi * t * ((n * .500) * t))
elif name == 'twochirp':
f = np.sin(np.pi * t * (n * t)) + np.sin((np.pi / 3) * t * (n * t))
elif name == 'quadchirp':
f = np.sin((np.pi / 3) * t * (n * t**2))
elif name == 'mishmash': # QuadChirp + LinChirp + HiSine
f = np.sin((np.pi / 3) * t * (n * t**2))
f += np.sin(np.pi * (n * .6902) * t)
f += np.sin(np.pi * t * (n * .125 * t))
elif name == 'wernersorrows':
f = np.sin(np.pi * t * (n / 2 * t**2))
f = f + np.sin(np.pi * (n * .6902) * t)
f = f + np.sin(np.pi * t * (n * t))
pos = [.1, .13, .15, .23, .25, .40, .44, .65, .76, .78, .81]
hgt = [4, 5, 3, 4, 5, 4.2, 2.1, 4.3, 3.1, 5.1, 4.2]
wth = [.005, .005, .006, .01, .01, .03, .01, .01, .005, .008, .005]
for p, h, w in zip(pos, hgt, wth):
f += h / (1 + np.abs((t - p) / w))**4
elif name == 'hypchirps': # Hyperbolic Chirps of Mallat's book
alpha = 15 * n * np.pi / 1024
beta = 5 * n * np.pi / 1024
t = np.arange(1.001, n + .001 + 1) / n
f1 = np.zeros(n)
f2 = np.zeros(n)
f1 = np.sin(alpha / (.8 - t)) * (0.1 < t) * (t < 0.68)
f2 = np.sin(beta / (.8 - t)) * (0.1 < t) * (t < 0.75)
m = int(np.round(0.65 * n))
p = m // 4
envelope = np.ones(m) # the rinp.sing cutoff function
tmp = np.arange(1, p + 1)-np.ones(p)
envelope[:p] = (1 + np.sin(-np.pi / 2 + tmp / (p - 1) * np.pi)) / 2
envelope[m-p:m] = envelope[:p][::-1]
env = np.zeros(n)
env[int(np.ceil(n / 10)) - 1:m + int(np.ceil(n / 10)) - 1] = \
envelope[:m]
f = (f1 + f2) * env
elif name == 'linchirps': # Linear Chirps of Mallat's book
b = 100 * n * np.pi / 1024
a = 250 * n * np.pi / 1024
t = np.arange(1, n + 1) / n
A1 = np.sqrt((t - 1 / n) * (1 - t))
f = A1 * (np.cos(a * t**2) + np.cos(b * t + a * t**2))
elif name == 'chirps': # Mixture of Chirps of Mallat's book
t = np.arange(1, n + 1)/n * 10 * np.pi
f1 = np.cos(t**2 * n / 1024)
a = 30 * n / 1024
t = np.arange(1, n + 1)/n * np.pi
f2 = np.cos(a * (t**3))
f2 = f2[::-1]
ix = np.arange(-n, n + 1) / n * 20
g = np.exp(-ix**2 * 4 * n / 1024)
i1 = slice(n // 2, n // 2 + n)
i2 = slice(n // 8, n // 8 + n)
j = np.arange(1, n + 1) / n
f3 = g[i1] * np.cos(50 * np.pi * j * n / 1024)
f4 = g[i2] * np.cos(350 * np.pi * j * n / 1024)
f = f1 + f2 + f3 + f4
envelope = np.ones(n) # the rinp.sing cutoff function
tmp = np.arange(1, n // 8 + 1) - np.ones(n // 8)
envelope[:n // 8] = (
1 + np.sin(-np.pi / 2 + tmp / (n / 8 - 1) * np.pi)) / 2
envelope[7 * n // 8:n] = envelope[:n // 8][::-1]
f = f*envelope
elif name == 'gabor': # two modulated Gabor functions in Mallat's book
n = 512
t = np.arange(-n, n + 1)*5 / n
j = np.arange(1, n + 1) / n
g = np.exp(-t**2 * 20)
i1 = slice(2*n // 4, 2 * n // 4 + n)
i2 = slice(n // 4, n // 4 + n)
f1 = 3 * g[i1] * np.exp(1j * (n // 16) * np.pi * j)
f2 = 3 * g[i2] * np.exp(1j * (n // 4) * np.pi * j)
f = f1 + f2
elif name == 'sineoneoverx': # np.sin(1/x) in Mallat's book
n = 1024
i1 = np.arange(-n + 1, n + 1, dtype=float)
i1[i1 == 0] = 1 / 100
i1 = i1 / (n - 1)
f = np.sin(1.5 / i1)
f = f[512:1536]
elif name == 'piece-regular':
f = np.zeros(n)
n_12 = int(np.fix(n / 12))
n_7 = int(np.fix(n / 7))
n_5 = int(np.fix(n / 5))
n_3 = int(np.fix(n / 3))
n_2 = int(np.fix(n / 2))
n_20 = int(np.fix(n / 20))
f1 = -15 * demo_signal('bumps', n)
t = np.arange(1, n_12 + 1) / n_12
f2 = -np.exp(4 * t)
t = np.arange(1, n_7 + 1) / n_7
f5 = np.exp(4 * t)-np.exp(4)
t = np.arange(1, n_3 + 1) / n_3
fma = 6 / 40
f6 = -70 * np.exp(-((t - 0.5) * (t - 0.5)) / (2 * fma**2))
f[:n_7] = f6[:n_7]
f[n_7:n_5] = 0.5 * f6[n_7:n_5]
f[n_5:n_3] = f6[n_5:n_3]
f[n_3:n_2] = f1[n_3:n_2]
f[n_2:n_2 + n_12] = f2
f[n_2 + 2 * n_12 - 1:n_2 + n_12 - 1:-1] = f2
f[n_2 + 2 * n_12 + n_20:n_2 + 2 * n_12 + 3 * n_20] = -np.ones(
n_2 + 2*n_12 + 3*n_20 - n_2 - 2*n_12 - n_20) * 25
k = n_2 + 2 * n_12 + 3 * n_20
f[k:k + n_7] = f5
diff = n - 5 * n_5
f[5 * n_5:n] = f[diff - 1::-1]
# zero-mean
bias = np.sum(f) / n
f = bias - f
elif name == 'piece-polynomial':
f = np.zeros(n)
n_5 = int(np.fix(n / 5))
n_10 = int(np.fix(n / 10))
n_20 = int(np.fix(n / 20))
t = np.arange(1, n_5 + 1) / n_5
f1 = 20 * (t**3 + t**2 + 4)
f3 = 40 * (2 * t**3 + t) + 100
f2 = 10 * t**3 + 45
f4 = 16 * t**2 + 8 * t + 16
f5 = 20 * (t + 4)
f6 = np.ones(n_10) * 20
f[:n_5] = f1
f[2 * n_5 - 1:n_5 - 1:-1] = f2
f[2 * n_5:3 * n_5] = f3
f[3 * n_5:4 * n_5] = f4
f[4 * n_5:5 * n_5] = f5[n_5::-1]
diff = n - 5*n_5
f[5 * n_5:n] = f[diff - 1::-1]
f[n_20:n_20 + n_10] = np.ones(n_10) * 10
f[n - n_10:n + n_20 - n_10] = np.ones(n_20) * 150
# zero-mean
bias = np.sum(f) / n
f = f - bias
elif name == 'riemann':
# Riemann's Non-differentiable Function
sqn = int(np.round(np.sqrt(n)))
idx = np.arange(1, sqn + 1)
idx *= idx
f = np.zeros_like(t)
f[idx - 1] = 1. / np.arange(1, sqn + 1)
f = np.real(np.fft.ifft(f))
else:
raise ValueError(
"unknown name: {}. name must be one of: {}".format(
name, _implemented_signals))
return f
|
