847 lines
40 KiB
Python
847 lines
40 KiB
Python
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import numpy as np
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from numpy import array
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from numpy.testing import (assert_array_almost_equal, assert_array_equal,
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assert_allclose,
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assert_equal, assert_, assert_array_less,
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suppress_warnings)
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from pytest import raises as assert_raises
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from scipy.fft import fft
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from scipy.signal import windows, get_window, resample
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window_funcs = [
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('boxcar', ()),
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('triang', ()),
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('parzen', ()),
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('bohman', ()),
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('blackman', ()),
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('nuttall', ()),
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('blackmanharris', ()),
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('flattop', ()),
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('bartlett', ()),
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('barthann', ()),
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('hamming', ()),
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('kaiser', (1,)),
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('dpss', (2,)),
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('gaussian', (0.5,)),
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('general_gaussian', (1.5, 2)),
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('chebwin', (1,)),
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('cosine', ()),
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('hann', ()),
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('exponential', ()),
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('taylor', ()),
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('tukey', (0.5,)),
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('lanczos', ()),
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]
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class TestBartHann:
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def test_basic(self):
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assert_allclose(windows.barthann(6, sym=True),
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[0, 0.35857354213752, 0.8794264578624801,
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0.8794264578624801, 0.3585735421375199, 0],
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rtol=1e-15, atol=1e-15)
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assert_allclose(windows.barthann(7),
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[0, 0.27, 0.73, 1.0, 0.73, 0.27, 0],
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rtol=1e-15, atol=1e-15)
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assert_allclose(windows.barthann(6, False),
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[0, 0.27, 0.73, 1.0, 0.73, 0.27],
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rtol=1e-15, atol=1e-15)
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class TestBartlett:
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def test_basic(self):
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assert_allclose(windows.bartlett(6), [0, 0.4, 0.8, 0.8, 0.4, 0])
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assert_allclose(windows.bartlett(7), [0, 1/3, 2/3, 1.0, 2/3, 1/3, 0])
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assert_allclose(windows.bartlett(6, False),
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[0, 1/3, 2/3, 1.0, 2/3, 1/3])
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class TestBlackman:
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def test_basic(self):
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assert_allclose(windows.blackman(6, sym=False),
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[0, 0.13, 0.63, 1.0, 0.63, 0.13], atol=1e-14)
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assert_allclose(windows.blackman(7, sym=False),
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[0, 0.09045342435412804, 0.4591829575459636,
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0.9203636180999081, 0.9203636180999081,
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0.4591829575459636, 0.09045342435412804], atol=1e-8)
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assert_allclose(windows.blackman(6),
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[0, 0.2007701432625305, 0.8492298567374694,
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0.8492298567374694, 0.2007701432625305, 0],
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atol=1e-14)
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assert_allclose(windows.blackman(7, True),
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[0, 0.13, 0.63, 1.0, 0.63, 0.13, 0], atol=1e-14)
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class TestBlackmanHarris:
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def test_basic(self):
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assert_allclose(windows.blackmanharris(6, False),
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[6.0e-05, 0.055645, 0.520575, 1.0, 0.520575, 0.055645])
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assert_allclose(windows.blackmanharris(7, sym=False),
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[6.0e-05, 0.03339172347815117, 0.332833504298565,
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0.8893697722232837, 0.8893697722232838,
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0.3328335042985652, 0.03339172347815122])
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assert_allclose(windows.blackmanharris(6),
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[6.0e-05, 0.1030114893456638, 0.7938335106543362,
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0.7938335106543364, 0.1030114893456638, 6.0e-05])
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assert_allclose(windows.blackmanharris(7, sym=True),
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[6.0e-05, 0.055645, 0.520575, 1.0, 0.520575, 0.055645,
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6.0e-05])
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class TestTaylor:
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def test_normalized(self):
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"""Tests windows of small length that are normalized to 1. See the
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documentation for the Taylor window for more information on
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normalization.
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"""
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assert_allclose(windows.taylor(1, 2, 15), 1.0)
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assert_allclose(
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windows.taylor(5, 2, 15),
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np.array([0.75803341, 0.90757699, 1.0, 0.90757699, 0.75803341])
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)
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assert_allclose(
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windows.taylor(6, 2, 15),
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np.array([
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0.7504082, 0.86624416, 0.98208011, 0.98208011, 0.86624416,
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0.7504082
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])
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)
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def test_non_normalized(self):
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"""Test windows of small length that are not normalized to 1. See
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the documentation for the Taylor window for more information on
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normalization.
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"""
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assert_allclose(
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windows.taylor(5, 2, 15, norm=False),
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np.array([
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0.87508054, 1.04771499, 1.15440894, 1.04771499, 0.87508054
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])
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)
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assert_allclose(
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windows.taylor(6, 2, 15, norm=False),
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np.array([
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0.86627793, 1.0, 1.13372207, 1.13372207, 1.0, 0.86627793
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])
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)
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def test_correctness(self):
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"""This test ensures the correctness of the implemented Taylor
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Windowing function. A Taylor Window of 1024 points is created, its FFT
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is taken, and the Peak Sidelobe Level (PSLL) and 3dB and 18dB bandwidth
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are found and checked.
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A publication from Sandia National Laboratories was used as reference
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for the correctness values [1]_.
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References
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-----
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.. [1] Armin Doerry, "Catalog of Window Taper Functions for
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Sidelobe Control", 2017.
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https://www.researchgate.net/profile/Armin_Doerry/publication/316281181_Catalog_of_Window_Taper_Functions_for_Sidelobe_Control/links/58f92cb2a6fdccb121c9d54d/Catalog-of-Window-Taper-Functions-for-Sidelobe-Control.pdf
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"""
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M_win = 1024
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N_fft = 131072
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# Set norm=False for correctness as the values obtained from the
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# scientific publication do not normalize the values. Normalizing
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# changes the sidelobe level from the desired value.
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w = windows.taylor(M_win, nbar=4, sll=35, norm=False, sym=False)
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f = fft(w, N_fft)
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spec = 20 * np.log10(np.abs(f / np.amax(f)))
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first_zero = np.argmax(np.diff(spec) > 0)
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PSLL = np.amax(spec[first_zero:-first_zero])
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BW_3dB = 2*np.argmax(spec <= -3.0102999566398121) / N_fft * M_win
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BW_18dB = 2*np.argmax(spec <= -18.061799739838872) / N_fft * M_win
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assert_allclose(PSLL, -35.1672, atol=1)
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assert_allclose(BW_3dB, 1.1822, atol=0.1)
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assert_allclose(BW_18dB, 2.6112, atol=0.1)
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class TestBohman:
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def test_basic(self):
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assert_allclose(windows.bohman(6),
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[0, 0.1791238937062839, 0.8343114522576858,
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0.8343114522576858, 0.1791238937062838, 0])
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assert_allclose(windows.bohman(7, sym=True),
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[0, 0.1089977810442293, 0.6089977810442293, 1.0,
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0.6089977810442295, 0.1089977810442293, 0])
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assert_allclose(windows.bohman(6, False),
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[0, 0.1089977810442293, 0.6089977810442293, 1.0,
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0.6089977810442295, 0.1089977810442293])
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class TestBoxcar:
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def test_basic(self):
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assert_allclose(windows.boxcar(6), [1, 1, 1, 1, 1, 1])
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assert_allclose(windows.boxcar(7), [1, 1, 1, 1, 1, 1, 1])
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assert_allclose(windows.boxcar(6, False), [1, 1, 1, 1, 1, 1])
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cheb_odd_true = array([0.200938, 0.107729, 0.134941, 0.165348,
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0.198891, 0.235450, 0.274846, 0.316836,
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0.361119, 0.407338, 0.455079, 0.503883,
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0.553248, 0.602637, 0.651489, 0.699227,
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0.745266, 0.789028, 0.829947, 0.867485,
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0.901138, 0.930448, 0.955010, 0.974482,
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0.988591, 0.997138, 1.000000, 0.997138,
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0.988591, 0.974482, 0.955010, 0.930448,
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0.901138, 0.867485, 0.829947, 0.789028,
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0.745266, 0.699227, 0.651489, 0.602637,
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0.553248, 0.503883, 0.455079, 0.407338,
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0.361119, 0.316836, 0.274846, 0.235450,
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0.198891, 0.165348, 0.134941, 0.107729,
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0.200938])
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cheb_even_true = array([0.203894, 0.107279, 0.133904,
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0.163608, 0.196338, 0.231986,
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0.270385, 0.311313, 0.354493,
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0.399594, 0.446233, 0.493983,
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0.542378, 0.590916, 0.639071,
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0.686302, 0.732055, 0.775783,
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0.816944, 0.855021, 0.889525,
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0.920006, 0.946060, 0.967339,
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0.983557, 0.994494, 1.000000,
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1.000000, 0.994494, 0.983557,
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0.967339, 0.946060, 0.920006,
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0.889525, 0.855021, 0.816944,
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0.775783, 0.732055, 0.686302,
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0.639071, 0.590916, 0.542378,
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0.493983, 0.446233, 0.399594,
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0.354493, 0.311313, 0.270385,
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0.231986, 0.196338, 0.163608,
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0.133904, 0.107279, 0.203894])
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class TestChebWin:
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def test_basic(self):
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with suppress_warnings() as sup:
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sup.filter(UserWarning, "This window is not suitable")
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assert_allclose(windows.chebwin(6, 100),
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[0.1046401879356917, 0.5075781475823447, 1.0, 1.0,
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0.5075781475823447, 0.1046401879356917])
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assert_allclose(windows.chebwin(7, 100),
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[0.05650405062850233, 0.316608530648474,
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0.7601208123539079, 1.0, 0.7601208123539079,
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0.316608530648474, 0.05650405062850233])
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assert_allclose(windows.chebwin(6, 10),
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[1.0, 0.6071201674458373, 0.6808391469897297,
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0.6808391469897297, 0.6071201674458373, 1.0])
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assert_allclose(windows.chebwin(7, 10),
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[1.0, 0.5190521247588651, 0.5864059018130382,
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0.6101519801307441, 0.5864059018130382,
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0.5190521247588651, 1.0])
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assert_allclose(windows.chebwin(6, 10, False),
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[1.0, 0.5190521247588651, 0.5864059018130382,
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0.6101519801307441, 0.5864059018130382,
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0.5190521247588651])
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def test_cheb_odd_high_attenuation(self):
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with suppress_warnings() as sup:
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sup.filter(UserWarning, "This window is not suitable")
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cheb_odd = windows.chebwin(53, at=-40)
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assert_array_almost_equal(cheb_odd, cheb_odd_true, decimal=4)
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def test_cheb_even_high_attenuation(self):
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with suppress_warnings() as sup:
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sup.filter(UserWarning, "This window is not suitable")
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cheb_even = windows.chebwin(54, at=40)
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assert_array_almost_equal(cheb_even, cheb_even_true, decimal=4)
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def test_cheb_odd_low_attenuation(self):
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cheb_odd_low_at_true = array([1.000000, 0.519052, 0.586405,
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0.610151, 0.586405, 0.519052,
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1.000000])
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with suppress_warnings() as sup:
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sup.filter(UserWarning, "This window is not suitable")
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cheb_odd = windows.chebwin(7, at=10)
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assert_array_almost_equal(cheb_odd, cheb_odd_low_at_true, decimal=4)
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def test_cheb_even_low_attenuation(self):
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cheb_even_low_at_true = array([1.000000, 0.451924, 0.51027,
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0.541338, 0.541338, 0.51027,
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0.451924, 1.000000])
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with suppress_warnings() as sup:
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sup.filter(UserWarning, "This window is not suitable")
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cheb_even = windows.chebwin(8, at=-10)
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assert_array_almost_equal(cheb_even, cheb_even_low_at_true, decimal=4)
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exponential_data = {
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(4, None, 0.2, False):
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array([4.53999297624848542e-05,
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6.73794699908546700e-03, 1.00000000000000000e+00,
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6.73794699908546700e-03]),
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(4, None, 0.2, True): array([0.00055308437014783, 0.0820849986238988,
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0.0820849986238988, 0.00055308437014783]),
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(4, None, 1.0, False): array([0.1353352832366127, 0.36787944117144233, 1.,
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0.36787944117144233]),
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(4, None, 1.0, True): array([0.22313016014842982, 0.60653065971263342,
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0.60653065971263342, 0.22313016014842982]),
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(4, 2, 0.2, False):
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array([4.53999297624848542e-05, 6.73794699908546700e-03,
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1.00000000000000000e+00, 6.73794699908546700e-03]),
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(4, 2, 0.2, True): None,
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(4, 2, 1.0, False): array([0.1353352832366127, 0.36787944117144233, 1.,
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0.36787944117144233]),
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(4, 2, 1.0, True): None,
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(5, None, 0.2, True):
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array([4.53999297624848542e-05,
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6.73794699908546700e-03, 1.00000000000000000e+00,
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6.73794699908546700e-03, 4.53999297624848542e-05]),
|
||
|
|
(5, None, 1.0, True): array([0.1353352832366127, 0.36787944117144233, 1.,
|
||
|
|
0.36787944117144233, 0.1353352832366127]),
|
||
|
|
(5, 2, 0.2, True): None,
|
||
|
|
(5, 2, 1.0, True): None
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
def test_exponential():
|
||
|
|
for k, v in exponential_data.items():
|
||
|
|
if v is None:
|
||
|
|
assert_raises(ValueError, windows.exponential, *k)
|
||
|
|
else:
|
||
|
|
win = windows.exponential(*k)
|
||
|
|
assert_allclose(win, v, rtol=1e-14)
|
||
|
|
|
||
|
|
|
||
|
|
class TestFlatTop:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
assert_allclose(windows.flattop(6, sym=False),
|
||
|
|
[-0.000421051, -0.051263156, 0.19821053, 1.0,
|
||
|
|
0.19821053, -0.051263156])
|
||
|
|
assert_allclose(windows.flattop(7, sym=False),
|
||
|
|
[-0.000421051, -0.03684078115492348,
|
||
|
|
0.01070371671615342, 0.7808739149387698,
|
||
|
|
0.7808739149387698, 0.01070371671615342,
|
||
|
|
-0.03684078115492348])
|
||
|
|
assert_allclose(windows.flattop(6),
|
||
|
|
[-0.000421051, -0.0677142520762119, 0.6068721525762117,
|
||
|
|
0.6068721525762117, -0.0677142520762119,
|
||
|
|
-0.000421051])
|
||
|
|
assert_allclose(windows.flattop(7, True),
|
||
|
|
[-0.000421051, -0.051263156, 0.19821053, 1.0,
|
||
|
|
0.19821053, -0.051263156, -0.000421051])
|
||
|
|
|
||
|
|
|
||
|
|
class TestGaussian:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
assert_allclose(windows.gaussian(6, 1.0),
|
||
|
|
[0.04393693362340742, 0.3246524673583497,
|
||
|
|
0.8824969025845955, 0.8824969025845955,
|
||
|
|
0.3246524673583497, 0.04393693362340742])
|
||
|
|
assert_allclose(windows.gaussian(7, 1.2),
|
||
|
|
[0.04393693362340742, 0.2493522087772962,
|
||
|
|
0.7066482778577162, 1.0, 0.7066482778577162,
|
||
|
|
0.2493522087772962, 0.04393693362340742])
|
||
|
|
assert_allclose(windows.gaussian(7, 3),
|
||
|
|
[0.6065306597126334, 0.8007374029168081,
|
||
|
|
0.9459594689067654, 1.0, 0.9459594689067654,
|
||
|
|
0.8007374029168081, 0.6065306597126334])
|
||
|
|
assert_allclose(windows.gaussian(6, 3, False),
|
||
|
|
[0.6065306597126334, 0.8007374029168081,
|
||
|
|
0.9459594689067654, 1.0, 0.9459594689067654,
|
||
|
|
0.8007374029168081])
|
||
|
|
|
||
|
|
|
||
|
|
class TestGeneralCosine:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
assert_allclose(windows.general_cosine(5, [0.5, 0.3, 0.2]),
|
||
|
|
[0.4, 0.3, 1, 0.3, 0.4])
|
||
|
|
assert_allclose(windows.general_cosine(4, [0.5, 0.3, 0.2], sym=False),
|
||
|
|
[0.4, 0.3, 1, 0.3])
|
||
|
|
|
||
|
|
|
||
|
|
class TestGeneralHamming:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
assert_allclose(windows.general_hamming(5, 0.7),
|
||
|
|
[0.4, 0.7, 1.0, 0.7, 0.4])
|
||
|
|
assert_allclose(windows.general_hamming(5, 0.75, sym=False),
|
||
|
|
[0.5, 0.6727457514, 0.9522542486,
|
||
|
|
0.9522542486, 0.6727457514])
|
||
|
|
assert_allclose(windows.general_hamming(6, 0.75, sym=True),
|
||
|
|
[0.5, 0.6727457514, 0.9522542486,
|
||
|
|
0.9522542486, 0.6727457514, 0.5])
|
||
|
|
|
||
|
|
|
||
|
|
class TestHamming:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
assert_allclose(windows.hamming(6, False),
|
||
|
|
[0.08, 0.31, 0.77, 1.0, 0.77, 0.31])
|
||
|
|
assert_allclose(windows.hamming(7, sym=False),
|
||
|
|
[0.08, 0.2531946911449826, 0.6423596296199047,
|
||
|
|
0.9544456792351128, 0.9544456792351128,
|
||
|
|
0.6423596296199047, 0.2531946911449826])
|
||
|
|
assert_allclose(windows.hamming(6),
|
||
|
|
[0.08, 0.3978521825875242, 0.9121478174124757,
|
||
|
|
0.9121478174124757, 0.3978521825875242, 0.08])
|
||
|
|
assert_allclose(windows.hamming(7, sym=True),
|
||
|
|
[0.08, 0.31, 0.77, 1.0, 0.77, 0.31, 0.08])
|
||
|
|
|
||
|
|
|
||
|
|
class TestHann:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
assert_allclose(windows.hann(6, sym=False),
|
||
|
|
[0, 0.25, 0.75, 1.0, 0.75, 0.25],
|
||
|
|
rtol=1e-15, atol=1e-15)
|
||
|
|
assert_allclose(windows.hann(7, sym=False),
|
||
|
|
[0, 0.1882550990706332, 0.6112604669781572,
|
||
|
|
0.9504844339512095, 0.9504844339512095,
|
||
|
|
0.6112604669781572, 0.1882550990706332],
|
||
|
|
rtol=1e-15, atol=1e-15)
|
||
|
|
assert_allclose(windows.hann(6, True),
|
||
|
|
[0, 0.3454915028125263, 0.9045084971874737,
|
||
|
|
0.9045084971874737, 0.3454915028125263, 0],
|
||
|
|
rtol=1e-15, atol=1e-15)
|
||
|
|
assert_allclose(windows.hann(7),
|
||
|
|
[0, 0.25, 0.75, 1.0, 0.75, 0.25, 0],
|
||
|
|
rtol=1e-15, atol=1e-15)
|
||
|
|
|
||
|
|
|
||
|
|
class TestKaiser:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
assert_allclose(windows.kaiser(6, 0.5),
|
||
|
|
[0.9403061933191572, 0.9782962393705389,
|
||
|
|
0.9975765035372042, 0.9975765035372042,
|
||
|
|
0.9782962393705389, 0.9403061933191572])
|
||
|
|
assert_allclose(windows.kaiser(7, 0.5),
|
||
|
|
[0.9403061933191572, 0.9732402256999829,
|
||
|
|
0.9932754654413773, 1.0, 0.9932754654413773,
|
||
|
|
0.9732402256999829, 0.9403061933191572])
|
||
|
|
assert_allclose(windows.kaiser(6, 2.7),
|
||
|
|
[0.2603047507678832, 0.6648106293528054,
|
||
|
|
0.9582099802511439, 0.9582099802511439,
|
||
|
|
0.6648106293528054, 0.2603047507678832])
|
||
|
|
assert_allclose(windows.kaiser(7, 2.7),
|
||
|
|
[0.2603047507678832, 0.5985765418119844,
|
||
|
|
0.8868495172060835, 1.0, 0.8868495172060835,
|
||
|
|
0.5985765418119844, 0.2603047507678832])
|
||
|
|
assert_allclose(windows.kaiser(6, 2.7, False),
|
||
|
|
[0.2603047507678832, 0.5985765418119844,
|
||
|
|
0.8868495172060835, 1.0, 0.8868495172060835,
|
||
|
|
0.5985765418119844])
|
||
|
|
|
||
|
|
|
||
|
|
class TestKaiserBesselDerived:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
M = 100
|
||
|
|
w = windows.kaiser_bessel_derived(M, beta=4.0)
|
||
|
|
w2 = windows.get_window(('kaiser bessel derived', 4.0),
|
||
|
|
M, fftbins=False)
|
||
|
|
assert_allclose(w, w2)
|
||
|
|
|
||
|
|
# Test for Princen-Bradley condition
|
||
|
|
assert_allclose(w[:M // 2] ** 2 + w[-M // 2:] ** 2, 1.)
|
||
|
|
|
||
|
|
# Test actual values from other implementations
|
||
|
|
# M = 2: sqrt(2) / 2
|
||
|
|
# M = 4: 0.518562710536, 0.855039598640
|
||
|
|
# M = 6: 0.436168993154, 0.707106781187, 0.899864772847
|
||
|
|
# Ref:https://github.com/scipy/scipy/pull/4747#issuecomment-172849418
|
||
|
|
assert_allclose(windows.kaiser_bessel_derived(2, beta=np.pi / 2)[:1],
|
||
|
|
np.sqrt(2) / 2)
|
||
|
|
|
||
|
|
assert_allclose(windows.kaiser_bessel_derived(4, beta=np.pi / 2)[:2],
|
||
|
|
[0.518562710536, 0.855039598640])
|
||
|
|
|
||
|
|
assert_allclose(windows.kaiser_bessel_derived(6, beta=np.pi / 2)[:3],
|
||
|
|
[0.436168993154, 0.707106781187, 0.899864772847])
|
||
|
|
|
||
|
|
def test_exceptions(self):
|
||
|
|
M = 100
|
||
|
|
# Assert ValueError for odd window length
|
||
|
|
msg = ("Kaiser-Bessel Derived windows are only defined for even "
|
||
|
|
"number of points")
|
||
|
|
with assert_raises(ValueError, match=msg):
|
||
|
|
windows.kaiser_bessel_derived(M + 1, beta=4.)
|
||
|
|
|
||
|
|
# Assert ValueError for non-symmetric setting
|
||
|
|
msg = ("Kaiser-Bessel Derived windows are only defined for "
|
||
|
|
"symmetric shapes")
|
||
|
|
with assert_raises(ValueError, match=msg):
|
||
|
|
windows.kaiser_bessel_derived(M + 1, beta=4., sym=False)
|
||
|
|
|
||
|
|
|
||
|
|
class TestNuttall:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
assert_allclose(windows.nuttall(6, sym=False),
|
||
|
|
[0.0003628, 0.0613345, 0.5292298, 1.0, 0.5292298,
|
||
|
|
0.0613345])
|
||
|
|
assert_allclose(windows.nuttall(7, sym=False),
|
||
|
|
[0.0003628, 0.03777576895352025, 0.3427276199688195,
|
||
|
|
0.8918518610776603, 0.8918518610776603,
|
||
|
|
0.3427276199688196, 0.0377757689535203])
|
||
|
|
assert_allclose(windows.nuttall(6),
|
||
|
|
[0.0003628, 0.1105152530498718, 0.7982580969501282,
|
||
|
|
0.7982580969501283, 0.1105152530498719, 0.0003628])
|
||
|
|
assert_allclose(windows.nuttall(7, True),
|
||
|
|
[0.0003628, 0.0613345, 0.5292298, 1.0, 0.5292298,
|
||
|
|
0.0613345, 0.0003628])
|
||
|
|
|
||
|
|
|
||
|
|
class TestParzen:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
assert_allclose(windows.parzen(6),
|
||
|
|
[0.009259259259259254, 0.25, 0.8611111111111112,
|
||
|
|
0.8611111111111112, 0.25, 0.009259259259259254])
|
||
|
|
assert_allclose(windows.parzen(7, sym=True),
|
||
|
|
[0.00583090379008747, 0.1574344023323616,
|
||
|
|
0.6501457725947521, 1.0, 0.6501457725947521,
|
||
|
|
0.1574344023323616, 0.00583090379008747])
|
||
|
|
assert_allclose(windows.parzen(6, False),
|
||
|
|
[0.00583090379008747, 0.1574344023323616,
|
||
|
|
0.6501457725947521, 1.0, 0.6501457725947521,
|
||
|
|
0.1574344023323616])
|
||
|
|
|
||
|
|
|
||
|
|
class TestTriang:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
|
||
|
|
assert_allclose(windows.triang(6, True),
|
||
|
|
[1/6, 1/2, 5/6, 5/6, 1/2, 1/6])
|
||
|
|
assert_allclose(windows.triang(7),
|
||
|
|
[1/4, 1/2, 3/4, 1, 3/4, 1/2, 1/4])
|
||
|
|
assert_allclose(windows.triang(6, sym=False),
|
||
|
|
[1/4, 1/2, 3/4, 1, 3/4, 1/2])
|
||
|
|
|
||
|
|
|
||
|
|
tukey_data = {
|
||
|
|
(4, 0.5, True): array([0.0, 1.0, 1.0, 0.0]),
|
||
|
|
(4, 0.9, True): array([0.0, 0.84312081893436686,
|
||
|
|
0.84312081893436686, 0.0]),
|
||
|
|
(4, 1.0, True): array([0.0, 0.75, 0.75, 0.0]),
|
||
|
|
(4, 0.5, False): array([0.0, 1.0, 1.0, 1.0]),
|
||
|
|
(4, 0.9, False): array([0.0, 0.58682408883346526,
|
||
|
|
1.0, 0.58682408883346526]),
|
||
|
|
(4, 1.0, False): array([0.0, 0.5, 1.0, 0.5]),
|
||
|
|
(5, 0.0, True): array([1.0, 1.0, 1.0, 1.0, 1.0]),
|
||
|
|
(5, 0.8, True): array([0.0, 0.69134171618254492,
|
||
|
|
1.0, 0.69134171618254492, 0.0]),
|
||
|
|
(5, 1.0, True): array([0.0, 0.5, 1.0, 0.5, 0.0]),
|
||
|
|
|
||
|
|
(6, 0): [1, 1, 1, 1, 1, 1],
|
||
|
|
(7, 0): [1, 1, 1, 1, 1, 1, 1],
|
||
|
|
(6, .25): [0, 1, 1, 1, 1, 0],
|
||
|
|
(7, .25): [0, 1, 1, 1, 1, 1, 0],
|
||
|
|
(6,): [0, 0.9045084971874737, 1.0, 1.0, 0.9045084971874735, 0],
|
||
|
|
(7,): [0, 0.75, 1.0, 1.0, 1.0, 0.75, 0],
|
||
|
|
(6, .75): [0, 0.5522642316338269, 1.0, 1.0, 0.5522642316338267, 0],
|
||
|
|
(7, .75): [0, 0.4131759111665348, 0.9698463103929542, 1.0,
|
||
|
|
0.9698463103929542, 0.4131759111665347, 0],
|
||
|
|
(6, 1): [0, 0.3454915028125263, 0.9045084971874737, 0.9045084971874737,
|
||
|
|
0.3454915028125263, 0],
|
||
|
|
(7, 1): [0, 0.25, 0.75, 1.0, 0.75, 0.25, 0],
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
class TestTukey:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
# Test against hardcoded data
|
||
|
|
for k, v in tukey_data.items():
|
||
|
|
if v is None:
|
||
|
|
assert_raises(ValueError, windows.tukey, *k)
|
||
|
|
else:
|
||
|
|
win = windows.tukey(*k)
|
||
|
|
assert_allclose(win, v, rtol=1e-15, atol=1e-15)
|
||
|
|
|
||
|
|
def test_extremes(self):
|
||
|
|
# Test extremes of alpha correspond to boxcar and hann
|
||
|
|
tuk0 = windows.tukey(100, 0)
|
||
|
|
box0 = windows.boxcar(100)
|
||
|
|
assert_array_almost_equal(tuk0, box0)
|
||
|
|
|
||
|
|
tuk1 = windows.tukey(100, 1)
|
||
|
|
han1 = windows.hann(100)
|
||
|
|
assert_array_almost_equal(tuk1, han1)
|
||
|
|
|
||
|
|
|
||
|
|
dpss_data = {
|
||
|
|
# All values from MATLAB:
|
||
|
|
# * taper[1] of (3, 1.4, 3) sign-flipped
|
||
|
|
# * taper[3] of (5, 1.5, 5) sign-flipped
|
||
|
|
(4, 0.1, 2): ([[0.497943898, 0.502047681, 0.502047681, 0.497943898], [0.670487993, 0.224601537, -0.224601537, -0.670487993]], [0.197961815, 0.002035474]), # noqa: E501
|
||
|
|
(3, 1.4, 3): ([[0.410233151, 0.814504464, 0.410233151], [0.707106781, 0.0, -0.707106781], [0.575941629, -0.580157287, 0.575941629]], [0.999998093, 0.998067480, 0.801934426]), # noqa: E501
|
||
|
|
(5, 1.5, 5): ([[0.1745071052, 0.4956749177, 0.669109327, 0.495674917, 0.174507105], [0.4399493348, 0.553574369, 0.0, -0.553574369, -0.439949334], [0.631452756, 0.073280238, -0.437943884, 0.073280238, 0.631452756], [0.553574369, -0.439949334, 0.0, 0.439949334, -0.553574369], [0.266110290, -0.498935248, 0.600414741, -0.498935248, 0.266110290147157]], [0.999728571, 0.983706916, 0.768457889, 0.234159338, 0.013947282907567]), # noqa: E501
|
||
|
|
(100, 2, 4): ([[0.0030914414, 0.0041266922, 0.005315076, 0.006665149, 0.008184854, 0.0098814158, 0.011761239, 0.013829809, 0.016091597, 0.018549973, 0.02120712, 0.02406396, 0.027120092, 0.030373728, 0.033821651, 0.037459181, 0.041280145, 0.045276872, 0.049440192, 0.053759447, 0.058222524, 0.062815894, 0.067524661, 0.072332638, 0.077222418, 0.082175473, 0.087172252, 0.092192299, 0.097214376, 0.1022166, 0.10717657, 0.11207154, 0.11687856, 0.12157463, 0.12613686, 0.13054266, 0.13476986, 0.13879691, 0.14260302, 0.14616832, 0.14947401, 0.1525025, 0.15523755, 0.15766438, 0.15976981, 0.16154233, 0.16297223, 0.16405162, 0.16477455, 0.16513702, 0.16513702, 0.16477455, 0.16405162, 0.16297223, 0.16154233, 0.15976981, 0.15766438, 0.15523755, 0.1525025, 0.14947401, 0.14616832, 0.14260302, 0.13879691, 0.13476986, 0.13054266, 0.12613686, 0.12157463, 0.11687856, 0.11207154, 0.10717657, 0.1022166, 0.097214376, 0.092192299, 0.087172252, 0.082175473, 0.077222418, 0.072332638, 0.067524661, 0.062815894, 0.058222524, 0.053759447, 0.049440192, 0.045276872, 0.041280145, 0.037459181, 0.033821651, 0.030373728, 0.027120092, 0.02406396, 0.02120712, 0.018549973, 0.016091597, 0.013829809, 0.011761239, 0.0098814158, 0.008184854, 0.006665149, 0.005315076, 0.0041266922, 0.0030914414], [0.018064449, 0.022040342, 0.026325013, 0.030905288, 0.035764398, 0.040881982, 0.046234148, 0.051793558, 0.057529559, 0.063408356, 0.069393216, 0.075444716, 0.081521022, 0.087578202, 0.093570567, 0.099451049, 0.10517159, 0.11068356, 0.11593818, 0.12088699, 0.12548227, 0.12967752, 0.1334279, 0.13669069, 0.13942569, 0.1415957, 0.14316686, 0.14410905, 0.14439626, 0.14400686, 0.14292389, 0.1411353, 0.13863416, 0.13541876, 0.13149274, 0.12686516, 0.12155045, 0.1155684, 0.10894403, 0.10170748, 0.093893752, 0.08554251, 0.076697768, 0.067407559, 0.057723559, 0.04770068, 0.037396627, 0.026871428, 0.016186944, 0.0054063557, -0.0054063557, -0.016186944, -0.026871428, -0.037396627, -0.04770068, -0.057723559, -0.067407559, -0.076697768, -0.08554251, -0.093893752, -0.10170748, -0.10894403, -0.1155684, -0.12155045, -0.12686516, -0.13149274, -0.13541876, -0.13863416, -0.1411353, -0.14292389, -0.14400686, -0.14439626, -0.14410905, -0.14316686, -0.1415957, -0.13942569, -0.13669069, -0.1334279, -0.12967752, -0.12548227, -0.12088699, -0.11593818, -0.11068356, -0.10517159, -0.099451049, -0.093570567, -0.087578202, -0.081521022, -0.075444716, -0.069393216, -0.063408356, -0.057529559, -0.051793558, -0.046234148, -0.040881982, -0.035764398, -0.030905288, -0.026325013, -0.022040342, -0.018064449], [0.064817553, 0.072567801, 0.080292992, 0.087918235, 0.095367076, 0.10256232, 0.10942687, 0.1158846, 0.12186124, 0.12728523, 0.13208858, 0.13620771, 0.13958427, 0.14216587, 0.14390678, 0.14476863, 0.1447209, 0.14374148, 0.14181704, 0.13894336, 0.13512554, 0.13037812, 0.1247251, 0.11819984, 0.11084487, 0.10271159, 0.093859853, 0.084357497, 0.074279719, 0.063708406, 0.052731374, 0.041441525, 0.029935953, 0.018314987, 0.0066811877, -0.0048616765, -0.016209689, -0.027259848, -0.037911124, -0.048065512, -0.05762905, -0.066512804, -0.0746338, -0.081915903, -0.088290621, -0.09369783, -0.098086416, -0.10141482, -0.10365146, -0.10477512, -0.10477512, -0.10365146, -0.10141482, -0.098086416, -0.09369783, -0.088290621, -0.081915903, -0.0746338, -0.066512804, -0.05762905, -0.048065512, -0.037911124, -0.027259848, -0.016209689, -0.0048616765, 0.0066811877, 0.018314987, 0.029935953, 0.041441525, 0.052731374, 0.063708406, 0.074279719, 0.084357497, 0.093859853, 0.10271159, 0.11084487, 0.11819984, 0.1247251, 0.13037812, 0.13512554, 0.13894336, 0.14181704, 0.14374148, 0.1447209, 0.14476863, 0.14390678, 0.14216587, 0.13958427, 0.13620771, 0.13208858, 0.12728523, 0.12186124, 0.1158846, 0.10942687, 0.10256232, 0.095367076, 0.087918235, 0.080292992, 0.072567801, 0.064817553], [0.14985551, 0.15512305, 0.15931467, 0.16236806, 0.16423291, 0.16487165, 0.16426009, 0.1623879, 0.1592589, 0.15489114, 0.14931693, 0.14258255, 0.13474785, 0.1258857, 0.11608124, 0.10543095, 0.094041635, 0.082029213, 0.069517411, 0.056636348, 0.043521
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
class TestDPSS:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
# Test against hardcoded data
|
||
|
|
for k, v in dpss_data.items():
|
||
|
|
win, ratios = windows.dpss(*k, return_ratios=True)
|
||
|
|
assert_allclose(win, v[0], atol=1e-7, err_msg=k)
|
||
|
|
assert_allclose(ratios, v[1], rtol=1e-5, atol=1e-7, err_msg=k)
|
||
|
|
|
||
|
|
def test_unity(self):
|
||
|
|
# Test unity value handling (gh-2221)
|
||
|
|
for M in range(1, 21):
|
||
|
|
# corrected w/approximation (default)
|
||
|
|
win = windows.dpss(M, M / 2.1)
|
||
|
|
expected = M % 2 # one for odd, none for even
|
||
|
|
assert_equal(np.isclose(win, 1.).sum(), expected,
|
||
|
|
err_msg=f'{win}')
|
||
|
|
# corrected w/subsample delay (slower)
|
||
|
|
win_sub = windows.dpss(M, M / 2.1, norm='subsample')
|
||
|
|
if M > 2:
|
||
|
|
# @M=2 the subsample doesn't do anything
|
||
|
|
assert_equal(np.isclose(win_sub, 1.).sum(), expected,
|
||
|
|
err_msg=f'{win_sub}')
|
||
|
|
assert_allclose(win, win_sub, rtol=0.03) # within 3%
|
||
|
|
# not the same, l2-norm
|
||
|
|
win_2 = windows.dpss(M, M / 2.1, norm=2)
|
||
|
|
expected = 1 if M == 1 else 0
|
||
|
|
assert_equal(np.isclose(win_2, 1.).sum(), expected,
|
||
|
|
err_msg=f'{win_2}')
|
||
|
|
|
||
|
|
def test_extremes(self):
|
||
|
|
# Test extremes of alpha
|
||
|
|
lam = windows.dpss(31, 6, 4, return_ratios=True)[1]
|
||
|
|
assert_array_almost_equal(lam, 1.)
|
||
|
|
lam = windows.dpss(31, 7, 4, return_ratios=True)[1]
|
||
|
|
assert_array_almost_equal(lam, 1.)
|
||
|
|
lam = windows.dpss(31, 8, 4, return_ratios=True)[1]
|
||
|
|
assert_array_almost_equal(lam, 1.)
|
||
|
|
|
||
|
|
def test_degenerate(self):
|
||
|
|
# Test failures
|
||
|
|
assert_raises(ValueError, windows.dpss, 4, 1.5, -1) # Bad Kmax
|
||
|
|
assert_raises(ValueError, windows.dpss, 4, 1.5, -5)
|
||
|
|
assert_raises(TypeError, windows.dpss, 4, 1.5, 1.1)
|
||
|
|
assert_raises(ValueError, windows.dpss, 3, 1.5, 3) # NW must be < N/2.
|
||
|
|
assert_raises(ValueError, windows.dpss, 3, -1, 3) # NW must be pos
|
||
|
|
assert_raises(ValueError, windows.dpss, 3, 0, 3)
|
||
|
|
assert_raises(ValueError, windows.dpss, -1, 1, 3) # negative M
|
||
|
|
|
||
|
|
|
||
|
|
class TestLanczos:
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
# Analytical results:
|
||
|
|
# sinc(x) = sinc(-x)
|
||
|
|
# sinc(pi) = 0, sinc(0) = 1
|
||
|
|
# Hand computation on WolframAlpha:
|
||
|
|
# sinc(2 pi / 3) = 0.413496672
|
||
|
|
# sinc(pi / 3) = 0.826993343
|
||
|
|
# sinc(3 pi / 5) = 0.504551152
|
||
|
|
# sinc(pi / 5) = 0.935489284
|
||
|
|
assert_allclose(windows.lanczos(6, sym=False),
|
||
|
|
[0., 0.413496672,
|
||
|
|
0.826993343, 1., 0.826993343,
|
||
|
|
0.413496672],
|
||
|
|
atol=1e-9)
|
||
|
|
assert_allclose(windows.lanczos(6),
|
||
|
|
[0., 0.504551152,
|
||
|
|
0.935489284, 0.935489284,
|
||
|
|
0.504551152, 0.],
|
||
|
|
atol=1e-9)
|
||
|
|
assert_allclose(windows.lanczos(7, sym=True),
|
||
|
|
[0., 0.413496672,
|
||
|
|
0.826993343, 1., 0.826993343,
|
||
|
|
0.413496672, 0.],
|
||
|
|
atol=1e-9)
|
||
|
|
|
||
|
|
def test_array_size(self):
|
||
|
|
for n in [0, 10, 11]:
|
||
|
|
assert_equal(len(windows.lanczos(n, sym=False)), n)
|
||
|
|
assert_equal(len(windows.lanczos(n, sym=True)), n)
|
||
|
|
|
||
|
|
|
||
|
|
class TestGetWindow:
|
||
|
|
|
||
|
|
def test_boxcar(self):
|
||
|
|
w = windows.get_window('boxcar', 12)
|
||
|
|
assert_array_equal(w, np.ones_like(w))
|
||
|
|
|
||
|
|
# window is a tuple of len 1
|
||
|
|
w = windows.get_window(('boxcar',), 16)
|
||
|
|
assert_array_equal(w, np.ones_like(w))
|
||
|
|
|
||
|
|
def test_cheb_odd(self):
|
||
|
|
with suppress_warnings() as sup:
|
||
|
|
sup.filter(UserWarning, "This window is not suitable")
|
||
|
|
w = windows.get_window(('chebwin', -40), 53, fftbins=False)
|
||
|
|
assert_array_almost_equal(w, cheb_odd_true, decimal=4)
|
||
|
|
|
||
|
|
def test_cheb_even(self):
|
||
|
|
with suppress_warnings() as sup:
|
||
|
|
sup.filter(UserWarning, "This window is not suitable")
|
||
|
|
w = windows.get_window(('chebwin', 40), 54, fftbins=False)
|
||
|
|
assert_array_almost_equal(w, cheb_even_true, decimal=4)
|
||
|
|
|
||
|
|
def test_dpss(self):
|
||
|
|
win1 = windows.get_window(('dpss', 3), 64, fftbins=False)
|
||
|
|
win2 = windows.dpss(64, 3)
|
||
|
|
assert_array_almost_equal(win1, win2, decimal=4)
|
||
|
|
|
||
|
|
def test_kaiser_float(self):
|
||
|
|
win1 = windows.get_window(7.2, 64)
|
||
|
|
win2 = windows.kaiser(64, 7.2, False)
|
||
|
|
assert_allclose(win1, win2)
|
||
|
|
|
||
|
|
def test_invalid_inputs(self):
|
||
|
|
# Window is not a float, tuple, or string
|
||
|
|
assert_raises(ValueError, windows.get_window, set('hann'), 8)
|
||
|
|
|
||
|
|
# Unknown window type error
|
||
|
|
assert_raises(ValueError, windows.get_window, 'broken', 4)
|
||
|
|
|
||
|
|
def test_array_as_window(self):
|
||
|
|
# github issue 3603
|
||
|
|
osfactor = 128
|
||
|
|
sig = np.arange(128)
|
||
|
|
|
||
|
|
win = windows.get_window(('kaiser', 8.0), osfactor // 2)
|
||
|
|
with assert_raises(ValueError, match='must have the same length'):
|
||
|
|
resample(sig, len(sig) * osfactor, window=win)
|
||
|
|
|
||
|
|
def test_general_cosine(self):
|
||
|
|
assert_allclose(get_window(('general_cosine', [0.5, 0.3, 0.2]), 4),
|
||
|
|
[0.4, 0.3, 1, 0.3])
|
||
|
|
assert_allclose(get_window(('general_cosine', [0.5, 0.3, 0.2]), 4,
|
||
|
|
fftbins=False),
|
||
|
|
[0.4, 0.55, 0.55, 0.4])
|
||
|
|
|
||
|
|
def test_general_hamming(self):
|
||
|
|
assert_allclose(get_window(('general_hamming', 0.7), 5),
|
||
|
|
[0.4, 0.6072949, 0.9427051, 0.9427051, 0.6072949])
|
||
|
|
assert_allclose(get_window(('general_hamming', 0.7), 5, fftbins=False),
|
||
|
|
[0.4, 0.7, 1.0, 0.7, 0.4])
|
||
|
|
|
||
|
|
def test_lanczos(self):
|
||
|
|
assert_allclose(get_window('lanczos', 6),
|
||
|
|
[0., 0.413496672, 0.826993343, 1., 0.826993343,
|
||
|
|
0.413496672], atol=1e-9)
|
||
|
|
assert_allclose(get_window('lanczos', 6, fftbins=False),
|
||
|
|
[0., 0.504551152, 0.935489284, 0.935489284,
|
||
|
|
0.504551152, 0.], atol=1e-9)
|
||
|
|
assert_allclose(get_window('lanczos', 6), get_window('sinc', 6))
|
||
|
|
|
||
|
|
|
||
|
|
def test_windowfunc_basics():
|
||
|
|
for window_name, params in window_funcs:
|
||
|
|
window = getattr(windows, window_name)
|
||
|
|
with suppress_warnings() as sup:
|
||
|
|
sup.filter(UserWarning, "This window is not suitable")
|
||
|
|
# Check symmetry for odd and even lengths
|
||
|
|
w1 = window(8, *params, sym=True)
|
||
|
|
w2 = window(7, *params, sym=False)
|
||
|
|
assert_array_almost_equal(w1[:-1], w2)
|
||
|
|
|
||
|
|
w1 = window(9, *params, sym=True)
|
||
|
|
w2 = window(8, *params, sym=False)
|
||
|
|
assert_array_almost_equal(w1[:-1], w2)
|
||
|
|
|
||
|
|
# Check that functions run and output lengths are correct
|
||
|
|
assert_equal(len(window(6, *params, sym=True)), 6)
|
||
|
|
assert_equal(len(window(6, *params, sym=False)), 6)
|
||
|
|
assert_equal(len(window(7, *params, sym=True)), 7)
|
||
|
|
assert_equal(len(window(7, *params, sym=False)), 7)
|
||
|
|
|
||
|
|
# Check invalid lengths
|
||
|
|
assert_raises(ValueError, window, 5.5, *params)
|
||
|
|
assert_raises(ValueError, window, -7, *params)
|
||
|
|
|
||
|
|
# Check degenerate cases
|
||
|
|
assert_array_equal(window(0, *params, sym=True), [])
|
||
|
|
assert_array_equal(window(0, *params, sym=False), [])
|
||
|
|
assert_array_equal(window(1, *params, sym=True), [1])
|
||
|
|
assert_array_equal(window(1, *params, sym=False), [1])
|
||
|
|
|
||
|
|
# Check dtype
|
||
|
|
assert_(window(0, *params, sym=True).dtype == 'float')
|
||
|
|
assert_(window(0, *params, sym=False).dtype == 'float')
|
||
|
|
assert_(window(1, *params, sym=True).dtype == 'float')
|
||
|
|
assert_(window(1, *params, sym=False).dtype == 'float')
|
||
|
|
assert_(window(6, *params, sym=True).dtype == 'float')
|
||
|
|
assert_(window(6, *params, sym=False).dtype == 'float')
|
||
|
|
|
||
|
|
# Check normalization
|
||
|
|
assert_array_less(window(10, *params, sym=True), 1.01)
|
||
|
|
assert_array_less(window(10, *params, sym=False), 1.01)
|
||
|
|
assert_array_less(window(9, *params, sym=True), 1.01)
|
||
|
|
assert_array_less(window(9, *params, sym=False), 1.01)
|
||
|
|
|
||
|
|
# Check that DFT-even spectrum is purely real for odd and even
|
||
|
|
assert_allclose(fft(window(10, *params, sym=False)).imag,
|
||
|
|
0, atol=1e-14)
|
||
|
|
assert_allclose(fft(window(11, *params, sym=False)).imag,
|
||
|
|
0, atol=1e-14)
|
||
|
|
|
||
|
|
|
||
|
|
def test_needs_params():
|
||
|
|
for winstr in ['kaiser', 'ksr', 'kaiser_bessel_derived', 'kbd',
|
||
|
|
'gaussian', 'gauss', 'gss',
|
||
|
|
'general gaussian', 'general_gaussian',
|
||
|
|
'general gauss', 'general_gauss', 'ggs',
|
||
|
|
'dss', 'dpss', 'general cosine', 'general_cosine',
|
||
|
|
'chebwin', 'cheb', 'general hamming', 'general_hamming',
|
||
|
|
]:
|
||
|
|
assert_raises(ValueError, get_window, winstr, 7)
|
||
|
|
|
||
|
|
|
||
|
|
def test_not_needs_params():
|
||
|
|
for winstr in ['barthann',
|
||
|
|
'bartlett',
|
||
|
|
'blackman',
|
||
|
|
'blackmanharris',
|
||
|
|
'bohman',
|
||
|
|
'boxcar',
|
||
|
|
'cosine',
|
||
|
|
'flattop',
|
||
|
|
'hamming',
|
||
|
|
'nuttall',
|
||
|
|
'parzen',
|
||
|
|
'taylor',
|
||
|
|
'exponential',
|
||
|
|
'poisson',
|
||
|
|
'tukey',
|
||
|
|
'tuk',
|
||
|
|
'triangle',
|
||
|
|
'lanczos',
|
||
|
|
'sinc',
|
||
|
|
]:
|
||
|
|
win = get_window(winstr, 7)
|
||
|
|
assert_equal(len(win), 7)
|
||
|
|
|
||
|
|
|
||
|
|
def test_symmetric():
|
||
|
|
|
||
|
|
for win in [windows.lanczos]:
|
||
|
|
# Even sampling points
|
||
|
|
w = win(4096)
|
||
|
|
error = np.max(np.abs(w-np.flip(w)))
|
||
|
|
assert_equal(error, 0.0)
|
||
|
|
|
||
|
|
# Odd sampling points
|
||
|
|
w = win(4097)
|
||
|
|
error = np.max(np.abs(w-np.flip(w)))
|
||
|
|
assert_equal(error, 0.0)
|