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3639 lines
157 KiB
Python
3639 lines
157 KiB
Python
6 years ago
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from __future__ import division, print_function, absolute_import
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import warnings
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from distutils.version import LooseVersion
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import numpy as np
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from numpy.testing import (assert_array_almost_equal,
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assert_array_equal, assert_array_less,
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assert_equal, assert_,
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assert_allclose, assert_warns)
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import pytest
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from pytest import raises as assert_raises
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from scipy._lib._numpy_compat import suppress_warnings
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from numpy import array, spacing, sin, pi, sort, sqrt
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from scipy.signal import (BadCoefficients, bessel, besselap, bilinear, buttap,
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butter, buttord, cheb1ap, cheb1ord, cheb2ap,
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cheb2ord, cheby1, cheby2, ellip, ellipap, ellipord,
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firwin, freqs_zpk, freqs, freqz, freqz_zpk,
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group_delay, iirfilter, iirnotch, iirpeak, lp2bp,
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lp2bs, lp2hp, lp2lp, normalize, sos2tf, sos2zpk,
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sosfreqz, tf2sos, tf2zpk, zpk2sos, zpk2tf,
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bilinear_zpk, lp2lp_zpk, lp2hp_zpk, lp2bp_zpk,
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lp2bs_zpk)
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from scipy.signal.filter_design import (_cplxreal, _cplxpair, _norm_factor,
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_bessel_poly, _bessel_zeros)
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try:
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import mpmath
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except ImportError:
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mpmath = None
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def mpmath_check(min_ver):
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return pytest.mark.skipif(mpmath is None or
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LooseVersion(mpmath.__version__) < LooseVersion(min_ver),
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reason="mpmath version >= %s required" % min_ver)
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class TestCplxPair(object):
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def test_trivial_input(self):
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assert_equal(_cplxpair([]).size, 0)
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assert_equal(_cplxpair(1), 1)
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def test_output_order(self):
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assert_allclose(_cplxpair([1+1j, 1-1j]), [1-1j, 1+1j])
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a = [1+1j, 1+1j, 1, 1-1j, 1-1j, 2]
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b = [1-1j, 1+1j, 1-1j, 1+1j, 1, 2]
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assert_allclose(_cplxpair(a), b)
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# points spaced around the unit circle
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z = np.exp(2j*pi*array([4, 3, 5, 2, 6, 1, 0])/7)
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z1 = np.copy(z)
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np.random.shuffle(z)
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assert_allclose(_cplxpair(z), z1)
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np.random.shuffle(z)
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assert_allclose(_cplxpair(z), z1)
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np.random.shuffle(z)
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assert_allclose(_cplxpair(z), z1)
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# Should be able to pair up all the conjugates
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x = np.random.rand(10000) + 1j * np.random.rand(10000)
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y = x.conj()
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z = np.random.rand(10000)
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x = np.concatenate((x, y, z))
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np.random.shuffle(x)
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c = _cplxpair(x)
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# Every other element of head should be conjugates:
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assert_allclose(c[0:20000:2], np.conj(c[1:20000:2]))
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# Real parts of head should be in sorted order:
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assert_allclose(c[0:20000:2].real, np.sort(c[0:20000:2].real))
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# Tail should be sorted real numbers:
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assert_allclose(c[20000:], np.sort(c[20000:]))
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def test_real_integer_input(self):
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assert_array_equal(_cplxpair([2, 0, 1]), [0, 1, 2])
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def test_tolerances(self):
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eps = spacing(1)
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assert_allclose(_cplxpair([1j, -1j, 1+1j*eps], tol=2*eps),
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[-1j, 1j, 1+1j*eps])
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# sorting close to 0
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assert_allclose(_cplxpair([-eps+1j, +eps-1j]), [-1j, +1j])
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assert_allclose(_cplxpair([+eps+1j, -eps-1j]), [-1j, +1j])
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assert_allclose(_cplxpair([+1j, -1j]), [-1j, +1j])
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def test_unmatched_conjugates(self):
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# 1+2j is unmatched
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assert_raises(ValueError, _cplxpair, [1+3j, 1-3j, 1+2j])
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# 1+2j and 1-3j are unmatched
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assert_raises(ValueError, _cplxpair, [1+3j, 1-3j, 1+2j, 1-3j])
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# 1+3j is unmatched
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assert_raises(ValueError, _cplxpair, [1+3j, 1-3j, 1+3j])
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# Not conjugates
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assert_raises(ValueError, _cplxpair, [4+5j, 4+5j])
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assert_raises(ValueError, _cplxpair, [1-7j, 1-7j])
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# No pairs
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assert_raises(ValueError, _cplxpair, [1+3j])
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assert_raises(ValueError, _cplxpair, [1-3j])
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class TestCplxReal(object):
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def test_trivial_input(self):
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assert_equal(_cplxreal([]), ([], []))
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assert_equal(_cplxreal(1), ([], [1]))
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def test_output_order(self):
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zc, zr = _cplxreal(np.roots(array([1, 0, 0, 1])))
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assert_allclose(np.append(zc, zr), [1/2 + 1j*sin(pi/3), -1])
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eps = spacing(1)
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a = [0+1j, 0-1j, eps + 1j, eps - 1j, -eps + 1j, -eps - 1j,
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1, 4, 2, 3, 0, 0,
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2+3j, 2-3j,
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1-eps + 1j, 1+2j, 1-2j, 1+eps - 1j, # sorts out of order
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3+1j, 3+1j, 3+1j, 3-1j, 3-1j, 3-1j,
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2-3j, 2+3j]
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zc, zr = _cplxreal(a)
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assert_allclose(zc, [1j, 1j, 1j, 1+1j, 1+2j, 2+3j, 2+3j, 3+1j, 3+1j,
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3+1j])
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assert_allclose(zr, [0, 0, 1, 2, 3, 4])
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z = array([1-eps + 1j, 1+2j, 1-2j, 1+eps - 1j, 1+eps+3j, 1-2*eps-3j,
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0+1j, 0-1j, 2+4j, 2-4j, 2+3j, 2-3j, 3+7j, 3-7j, 4-eps+1j,
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4+eps-2j, 4-1j, 4-eps+2j])
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zc, zr = _cplxreal(z)
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assert_allclose(zc, [1j, 1+1j, 1+2j, 1+3j, 2+3j, 2+4j, 3+7j, 4+1j,
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4+2j])
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assert_equal(zr, [])
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def test_unmatched_conjugates(self):
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# 1+2j is unmatched
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assert_raises(ValueError, _cplxreal, [1+3j, 1-3j, 1+2j])
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# 1+2j and 1-3j are unmatched
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assert_raises(ValueError, _cplxreal, [1+3j, 1-3j, 1+2j, 1-3j])
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# 1+3j is unmatched
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assert_raises(ValueError, _cplxreal, [1+3j, 1-3j, 1+3j])
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# No pairs
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assert_raises(ValueError, _cplxreal, [1+3j])
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assert_raises(ValueError, _cplxreal, [1-3j])
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def test_real_integer_input(self):
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zc, zr = _cplxreal([2, 0, 1, 4])
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assert_array_equal(zc, [])
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assert_array_equal(zr, [0, 1, 2, 4])
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class TestTf2zpk(object):
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def test_simple(self):
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z_r = np.array([0.5, -0.5])
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p_r = np.array([1.j / np.sqrt(2), -1.j / np.sqrt(2)])
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# Sort the zeros/poles so that we don't fail the test if the order
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# changes
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z_r.sort()
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p_r.sort()
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b = np.poly(z_r)
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a = np.poly(p_r)
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z, p, k = tf2zpk(b, a)
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z.sort()
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p.sort()
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assert_array_almost_equal(z, z_r)
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assert_array_almost_equal(p, p_r)
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def test_bad_filter(self):
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# Regression test for #651: better handling of badly conditioned
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# filter coefficients.
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with suppress_warnings():
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warnings.simplefilter("error", BadCoefficients)
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assert_raises(BadCoefficients, tf2zpk, [1e-15], [1.0, 1.0])
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class TestZpk2Tf(object):
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def test_identity(self):
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"""Test the identity transfer function."""
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z = []
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p = []
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k = 1.
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b, a = zpk2tf(z, p, k)
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b_r = np.array([1.]) # desired result
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a_r = np.array([1.]) # desired result
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# The test for the *type* of the return values is a regression
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# test for ticket #1095. In the case p=[], zpk2tf used to
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# return the scalar 1.0 instead of array([1.0]).
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assert_array_equal(b, b_r)
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assert_(isinstance(b, np.ndarray))
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assert_array_equal(a, a_r)
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assert_(isinstance(a, np.ndarray))
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class TestSos2Zpk(object):
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def test_basic(self):
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sos = [[1, 0, 1, 1, 0, -0.81],
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[1, 0, 0, 1, 0, +0.49]]
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z, p, k = sos2zpk(sos)
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z2 = [1j, -1j, 0, 0]
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p2 = [0.9, -0.9, 0.7j, -0.7j]
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k2 = 1
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assert_array_almost_equal(sort(z), sort(z2), decimal=4)
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assert_array_almost_equal(sort(p), sort(p2), decimal=4)
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assert_array_almost_equal(k, k2)
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sos = [[1.00000, +0.61803, 1.0000, 1.00000, +0.60515, 0.95873],
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[1.00000, -1.61803, 1.0000, 1.00000, -1.58430, 0.95873],
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[1.00000, +1.00000, 0.0000, 1.00000, +0.97915, 0.00000]]
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z, p, k = sos2zpk(sos)
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z2 = [-0.3090 + 0.9511j, -0.3090 - 0.9511j, 0.8090 + 0.5878j,
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0.8090 - 0.5878j, -1.0000 + 0.0000j, 0]
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p2 = [-0.3026 + 0.9312j, -0.3026 - 0.9312j, 0.7922 + 0.5755j,
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0.7922 - 0.5755j, -0.9791 + 0.0000j, 0]
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k2 = 1
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assert_array_almost_equal(sort(z), sort(z2), decimal=4)
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assert_array_almost_equal(sort(p), sort(p2), decimal=4)
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sos = array([[1, 2, 3, 1, 0.2, 0.3],
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[4, 5, 6, 1, 0.4, 0.5]])
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z = array([-1 - 1.41421356237310j, -1 + 1.41421356237310j,
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-0.625 - 1.05326872164704j, -0.625 + 1.05326872164704j])
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p = array([-0.2 - 0.678232998312527j, -0.2 + 0.678232998312527j,
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-0.1 - 0.538516480713450j, -0.1 + 0.538516480713450j])
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k = 4
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z2, p2, k2 = sos2zpk(sos)
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assert_allclose(_cplxpair(z2), z)
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assert_allclose(_cplxpair(p2), p)
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assert_allclose(k2, k)
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class TestSos2Tf(object):
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def test_basic(self):
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sos = [[1, 1, 1, 1, 0, -1],
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[-2, 3, 1, 1, 10, 1]]
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b, a = sos2tf(sos)
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assert_array_almost_equal(b, [-2, 1, 2, 4, 1])
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assert_array_almost_equal(a, [1, 10, 0, -10, -1])
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class TestTf2Sos(object):
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def test_basic(self):
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num = [2, 16, 44, 56, 32]
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den = [3, 3, -15, 18, -12]
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sos = tf2sos(num, den)
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sos2 = [[0.6667, 4.0000, 5.3333, 1.0000, +2.0000, -4.0000],
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[1.0000, 2.0000, 2.0000, 1.0000, -1.0000, +1.0000]]
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assert_array_almost_equal(sos, sos2, decimal=4)
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b = [1, -3, 11, -27, 18]
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a = [16, 12, 2, -4, -1]
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sos = tf2sos(b, a)
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sos2 = [[0.0625, -0.1875, 0.1250, 1.0000, -0.2500, -0.1250],
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[1.0000, +0.0000, 9.0000, 1.0000, +1.0000, +0.5000]]
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# assert_array_almost_equal(sos, sos2, decimal=4)
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class TestZpk2Sos(object):
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def test_basic(self):
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for pairing in ('nearest', 'keep_odd'):
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#
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# Cases that match octave
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#
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z = [-1, -1]
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p = [0.57149 + 0.29360j, 0.57149 - 0.29360j]
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k = 1
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sos = zpk2sos(z, p, k, pairing=pairing)
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sos2 = [[1, 2, 1, 1, -1.14298, 0.41280]] # octave & MATLAB
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assert_array_almost_equal(sos, sos2, decimal=4)
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z = [1j, -1j]
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p = [0.9, -0.9, 0.7j, -0.7j]
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k = 1
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sos = zpk2sos(z, p, k, pairing=pairing)
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sos2 = [[1, 0, 1, 1, 0, +0.49],
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[1, 0, 0, 1, 0, -0.81]] # octave
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# sos2 = [[0, 0, 1, 1, -0.9, 0],
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# [1, 0, 1, 1, 0.9, 0]] # MATLAB
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assert_array_almost_equal(sos, sos2, decimal=4)
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z = []
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p = [0.8, -0.5+0.25j, -0.5-0.25j]
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k = 1.
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sos = zpk2sos(z, p, k, pairing=pairing)
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sos2 = [[1., 0., 0., 1., 1., 0.3125],
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[1., 0., 0., 1., -0.8, 0.]] # octave, MATLAB fails
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assert_array_almost_equal(sos, sos2, decimal=4)
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z = [1., 1., 0.9j, -0.9j]
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p = [0.99+0.01j, 0.99-0.01j, 0.1+0.9j, 0.1-0.9j]
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k = 1
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sos = zpk2sos(z, p, k, pairing=pairing)
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sos2 = [[1, 0, 0.81, 1, -0.2, 0.82],
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[1, -2, 1, 1, -1.98, 0.9802]] # octave
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# sos2 = [[1, -2, 1, 1, -0.2, 0.82],
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# [1, 0, 0.81, 1, -1.98, 0.9802]] # MATLAB
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assert_array_almost_equal(sos, sos2, decimal=4)
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z = [0.9+0.1j, 0.9-0.1j, -0.9]
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p = [0.75+0.25j, 0.75-0.25j, 0.9]
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k = 1
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sos = zpk2sos(z, p, k, pairing=pairing)
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if pairing == 'keep_odd':
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sos2 = [[1, -1.8, 0.82, 1, -1.5, 0.625],
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[1, 0.9, 0, 1, -0.9, 0]] # octave; MATLAB fails
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assert_array_almost_equal(sos, sos2, decimal=4)
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else: # pairing == 'nearest'
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sos2 = [[1, 0.9, 0, 1, -1.5, 0.625],
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[1, -1.8, 0.82, 1, -0.9, 0]] # our algorithm
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assert_array_almost_equal(sos, sos2, decimal=4)
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#
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# Cases that differ from octave:
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#
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z = [-0.3090 + 0.9511j, -0.3090 - 0.9511j, 0.8090 + 0.5878j,
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+0.8090 - 0.5878j, -1.0000 + 0.0000j]
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p = [-0.3026 + 0.9312j, -0.3026 - 0.9312j, 0.7922 + 0.5755j,
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+0.7922 - 0.5755j, -0.9791 + 0.0000j]
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k = 1
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sos = zpk2sos(z, p, k, pairing=pairing)
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# sos2 = [[1, 0.618, 1, 1, 0.6052, 0.95870],
|
||
|
# [1, -1.618, 1, 1, -1.5844, 0.95878],
|
||
|
# [1, 1, 0, 1, 0.9791, 0]] # octave, MATLAB fails
|
||
|
sos2 = [[1, 1, 0, 1, +0.97915, 0],
|
||
|
[1, 0.61803, 1, 1, +0.60515, 0.95873],
|
||
|
[1, -1.61803, 1, 1, -1.58430, 0.95873]]
|
||
|
assert_array_almost_equal(sos, sos2, decimal=4)
|
||
|
|
||
|
z = [-1 - 1.4142j, -1 + 1.4142j,
|
||
|
-0.625 - 1.0533j, -0.625 + 1.0533j]
|
||
|
p = [-0.2 - 0.6782j, -0.2 + 0.6782j,
|
||
|
-0.1 - 0.5385j, -0.1 + 0.5385j]
|
||
|
k = 4
|
||
|
sos = zpk2sos(z, p, k, pairing=pairing)
|
||
|
sos2 = [[4, 8, 12, 1, 0.2, 0.3],
|
||
|
[1, 1.25, 1.5, 1, 0.4, 0.5]] # MATLAB
|
||
|
# sos2 = [[4, 8, 12, 1, 0.4, 0.5],
|
||
|
# [1, 1.25, 1.5, 1, 0.2, 0.3]] # octave
|
||
|
assert_allclose(sos, sos2, rtol=1e-4, atol=1e-4)
|
||
|
|
||
|
z = []
|
||
|
p = [0.2, -0.5+0.25j, -0.5-0.25j]
|
||
|
k = 1.
|
||
|
sos = zpk2sos(z, p, k, pairing=pairing)
|
||
|
sos2 = [[1., 0., 0., 1., -0.2, 0.],
|
||
|
[1., 0., 0., 1., 1., 0.3125]]
|
||
|
# sos2 = [[1., 0., 0., 1., 1., 0.3125],
|
||
|
# [1., 0., 0., 1., -0.2, 0]] # octave, MATLAB fails
|
||
|
assert_array_almost_equal(sos, sos2, decimal=4)
|
||
|
|
||
|
# The next two examples are adapted from Leland B. Jackson,
|
||
|
# "Digital Filters and Signal Processing (1995) p.400:
|
||
|
# http://books.google.com/books?id=VZ8uabI1pNMC&lpg=PA400&ots=gRD9pi8Jua&dq=Pole%2Fzero%20pairing%20for%20minimum%20roundoff%20noise%20in%20BSF.&pg=PA400#v=onepage&q=Pole%2Fzero%20pairing%20for%20minimum%20roundoff%20noise%20in%20BSF.&f=false
|
||
|
|
||
|
deg2rad = np.pi / 180.
|
||
|
k = 1.
|
||
|
|
||
|
# first example
|
||
|
thetas = [22.5, 45, 77.5]
|
||
|
mags = [0.8, 0.6, 0.9]
|
||
|
z = np.array([np.exp(theta * deg2rad * 1j) for theta in thetas])
|
||
|
z = np.concatenate((z, np.conj(z)))
|
||
|
p = np.array([mag * np.exp(theta * deg2rad * 1j)
|
||
|
for theta, mag in zip(thetas, mags)])
|
||
|
p = np.concatenate((p, np.conj(p)))
|
||
|
sos = zpk2sos(z, p, k)
|
||
|
# sos2 = [[1, -0.43288, 1, 1, -0.38959, 0.81], # octave,
|
||
|
# [1, -1.41421, 1, 1, -0.84853, 0.36], # MATLAB fails
|
||
|
# [1, -1.84776, 1, 1, -1.47821, 0.64]]
|
||
|
# Note that pole-zero pairing matches, but ordering is different
|
||
|
sos2 = [[1, -1.41421, 1, 1, -0.84853, 0.36],
|
||
|
[1, -1.84776, 1, 1, -1.47821, 0.64],
|
||
|
[1, -0.43288, 1, 1, -0.38959, 0.81]]
|
||
|
assert_array_almost_equal(sos, sos2, decimal=4)
|
||
|
|
||
|
# second example
|
||
|
z = np.array([np.exp(theta * deg2rad * 1j)
|
||
|
for theta in (85., 10.)])
|
||
|
z = np.concatenate((z, np.conj(z), [1, -1]))
|
||
|
sos = zpk2sos(z, p, k)
|
||
|
|
||
|
# sos2 = [[1, -0.17431, 1, 1, -0.38959, 0.81], # octave "wrong",
|
||
|
# [1, -1.96962, 1, 1, -0.84853, 0.36], # MATLAB fails
|
||
|
# [1, 0, -1, 1, -1.47821, 0.64000]]
|
||
|
# Our pole-zero pairing matches the text, Octave does not
|
||
|
sos2 = [[1, 0, -1, 1, -0.84853, 0.36],
|
||
|
[1, -1.96962, 1, 1, -1.47821, 0.64],
|
||
|
[1, -0.17431, 1, 1, -0.38959, 0.81]]
|
||
|
assert_array_almost_equal(sos, sos2, decimal=4)
|
||
|
|
||
|
|
||
|
class TestFreqs(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
_, h = freqs([1.0], [1.0], worN=8)
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
def test_output(self):
|
||
|
# 1st order low-pass filter: H(s) = 1 / (s + 1)
|
||
|
w = [0.1, 1, 10, 100]
|
||
|
num = [1]
|
||
|
den = [1, 1]
|
||
|
w, H = freqs(num, den, worN=w)
|
||
|
s = w * 1j
|
||
|
expected = 1 / (s + 1)
|
||
|
assert_array_almost_equal(H.real, expected.real)
|
||
|
assert_array_almost_equal(H.imag, expected.imag)
|
||
|
|
||
|
def test_freq_range(self):
|
||
|
# Test that freqresp() finds a reasonable frequency range.
|
||
|
# 1st order low-pass filter: H(s) = 1 / (s + 1)
|
||
|
# Expected range is from 0.01 to 10.
|
||
|
num = [1]
|
||
|
den = [1, 1]
|
||
|
n = 10
|
||
|
expected_w = np.logspace(-2, 1, n)
|
||
|
w, H = freqs(num, den, worN=n)
|
||
|
assert_array_almost_equal(w, expected_w)
|
||
|
|
||
|
def test_plot(self):
|
||
|
|
||
|
def plot(w, h):
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
assert_raises(ZeroDivisionError, freqs, [1.0], [1.0], worN=8,
|
||
|
plot=lambda w, h: 1 / 0)
|
||
|
freqs([1.0], [1.0], worN=8, plot=plot)
|
||
|
|
||
|
def test_backward_compat(self):
|
||
|
# For backward compatibility, test if None act as a wrapper for default
|
||
|
w1, h1 = freqs([1.0], [1.0])
|
||
|
w2, h2 = freqs([1.0], [1.0], None)
|
||
|
assert_array_almost_equal(w1, w2)
|
||
|
assert_array_almost_equal(h1, h2)
|
||
|
|
||
|
def test_w_or_N_types(self):
|
||
|
# Measure at 8 equally-spaced points
|
||
|
for N in (8, np.int8(8), np.int16(8), np.int32(8), np.int64(8),
|
||
|
np.array(8)):
|
||
|
w, h = freqs([1.0], [1.0], worN=N)
|
||
|
assert_equal(len(w), 8)
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
# Measure at frequency 8 rad/sec
|
||
|
for w in (8.0, 8.0+0j):
|
||
|
w_out, h = freqs([1.0], [1.0], worN=w)
|
||
|
assert_array_almost_equal(w_out, [8])
|
||
|
assert_array_almost_equal(h, [1])
|
||
|
|
||
|
|
||
|
class TestFreqs_zpk(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
_, h = freqs_zpk([1.0], [1.0], [1.0], worN=8)
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
def test_output(self):
|
||
|
# 1st order low-pass filter: H(s) = 1 / (s + 1)
|
||
|
w = [0.1, 1, 10, 100]
|
||
|
z = []
|
||
|
p = [-1]
|
||
|
k = 1
|
||
|
w, H = freqs_zpk(z, p, k, worN=w)
|
||
|
s = w * 1j
|
||
|
expected = 1 / (s + 1)
|
||
|
assert_array_almost_equal(H.real, expected.real)
|
||
|
assert_array_almost_equal(H.imag, expected.imag)
|
||
|
|
||
|
def test_freq_range(self):
|
||
|
# Test that freqresp() finds a reasonable frequency range.
|
||
|
# 1st order low-pass filter: H(s) = 1 / (s + 1)
|
||
|
# Expected range is from 0.01 to 10.
|
||
|
z = []
|
||
|
p = [-1]
|
||
|
k = 1
|
||
|
n = 10
|
||
|
expected_w = np.logspace(-2, 1, n)
|
||
|
w, H = freqs_zpk(z, p, k, worN=n)
|
||
|
assert_array_almost_equal(w, expected_w)
|
||
|
|
||
|
def test_vs_freqs(self):
|
||
|
b, a = cheby1(4, 5, 100, analog=True, output='ba')
|
||
|
z, p, k = cheby1(4, 5, 100, analog=True, output='zpk')
|
||
|
|
||
|
w1, h1 = freqs(b, a)
|
||
|
w2, h2 = freqs_zpk(z, p, k)
|
||
|
assert_allclose(w1, w2)
|
||
|
assert_allclose(h1, h2, rtol=1e-6)
|
||
|
|
||
|
def test_backward_compat(self):
|
||
|
# For backward compatibility, test if None act as a wrapper for default
|
||
|
w1, h1 = freqs_zpk([1.0], [1.0], [1.0])
|
||
|
w2, h2 = freqs_zpk([1.0], [1.0], [1.0], None)
|
||
|
assert_array_almost_equal(w1, w2)
|
||
|
assert_array_almost_equal(h1, h2)
|
||
|
|
||
|
def test_w_or_N_types(self):
|
||
|
# Measure at 8 equally-spaced points
|
||
|
for N in (8, np.int8(8), np.int16(8), np.int32(8), np.int64(8),
|
||
|
np.array(8)):
|
||
|
w, h = freqs_zpk([], [], 1, worN=N)
|
||
|
assert_equal(len(w), 8)
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
# Measure at frequency 8 rad/sec
|
||
|
for w in (8.0, 8.0+0j):
|
||
|
w_out, h = freqs_zpk([], [], 1, worN=w)
|
||
|
assert_array_almost_equal(w_out, [8])
|
||
|
assert_array_almost_equal(h, [1])
|
||
|
|
||
|
|
||
|
class TestFreqz(object):
|
||
|
|
||
|
def test_ticket1441(self):
|
||
|
"""Regression test for ticket 1441."""
|
||
|
# Because freqz previously used arange instead of linspace,
|
||
|
# when N was large, it would return one more point than
|
||
|
# requested.
|
||
|
N = 100000
|
||
|
w, h = freqz([1.0], worN=N)
|
||
|
assert_equal(w.shape, (N,))
|
||
|
|
||
|
def test_basic(self):
|
||
|
w, h = freqz([1.0], worN=8)
|
||
|
assert_array_almost_equal(w, np.pi * np.arange(8) / 8.)
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
w, h = freqz([1.0], worN=9)
|
||
|
assert_array_almost_equal(w, np.pi * np.arange(9) / 9.)
|
||
|
assert_array_almost_equal(h, np.ones(9))
|
||
|
|
||
|
for a in [1, np.ones(2)]:
|
||
|
w, h = freqz(np.ones(2), a, worN=0)
|
||
|
assert_equal(w.shape, (0,))
|
||
|
assert_equal(h.shape, (0,))
|
||
|
assert_equal(h.dtype, np.dtype('complex128'))
|
||
|
|
||
|
t = np.linspace(0, 1, 4, endpoint=False)
|
||
|
for b, a, h_whole in zip(
|
||
|
([1., 0, 0, 0], np.sin(2 * np.pi * t)),
|
||
|
([1., 0, 0, 0], [0.5, 0, 0, 0]),
|
||
|
([1., 1., 1., 1.], [0, -4j, 0, 4j])):
|
||
|
w, h = freqz(b, a, worN=4, whole=True)
|
||
|
expected_w = np.linspace(0, 2 * np.pi, 4, endpoint=False)
|
||
|
assert_array_almost_equal(w, expected_w)
|
||
|
assert_array_almost_equal(h, h_whole)
|
||
|
# simultaneously check int-like support
|
||
|
w, h = freqz(b, a, worN=np.int32(4), whole=True)
|
||
|
assert_array_almost_equal(w, expected_w)
|
||
|
assert_array_almost_equal(h, h_whole)
|
||
|
w, h = freqz(b, a, worN=w, whole=True)
|
||
|
assert_array_almost_equal(w, expected_w)
|
||
|
assert_array_almost_equal(h, h_whole)
|
||
|
|
||
|
def test_basic_whole(self):
|
||
|
w, h = freqz([1.0], worN=8, whole=True)
|
||
|
assert_array_almost_equal(w, 2 * np.pi * np.arange(8.0) / 8)
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
def test_plot(self):
|
||
|
|
||
|
def plot(w, h):
|
||
|
assert_array_almost_equal(w, np.pi * np.arange(8.0) / 8)
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
assert_raises(ZeroDivisionError, freqz, [1.0], worN=8,
|
||
|
plot=lambda w, h: 1 / 0)
|
||
|
freqz([1.0], worN=8, plot=plot)
|
||
|
|
||
|
def test_fft_wrapping(self):
|
||
|
# Some simple real FIR filters
|
||
|
bs = list() # filters
|
||
|
as_ = list()
|
||
|
hs_whole = list()
|
||
|
hs_half = list()
|
||
|
# 3 taps
|
||
|
t = np.linspace(0, 1, 3, endpoint=False)
|
||
|
bs.append(np.sin(2 * np.pi * t))
|
||
|
as_.append(3.)
|
||
|
hs_whole.append([0, -0.5j, 0.5j])
|
||
|
hs_half.append([0, np.sqrt(1./12.), -0.5j])
|
||
|
# 4 taps
|
||
|
t = np.linspace(0, 1, 4, endpoint=False)
|
||
|
bs.append(np.sin(2 * np.pi * t))
|
||
|
as_.append(0.5)
|
||
|
hs_whole.append([0, -4j, 0, 4j])
|
||
|
hs_half.append([0, np.sqrt(8), -4j, -np.sqrt(8)])
|
||
|
del t
|
||
|
for ii, b in enumerate(bs):
|
||
|
# whole
|
||
|
a = as_[ii]
|
||
|
expected_w = np.linspace(0, 2 * np.pi, len(b), endpoint=False)
|
||
|
w, h = freqz(b, a, worN=expected_w, whole=True) # polyval
|
||
|
err_msg = 'b = %s, a=%s' % (b, a)
|
||
|
assert_array_almost_equal(w, expected_w, err_msg=err_msg)
|
||
|
assert_array_almost_equal(h, hs_whole[ii], err_msg=err_msg)
|
||
|
w, h = freqz(b, a, worN=len(b), whole=True) # FFT
|
||
|
assert_array_almost_equal(w, expected_w, err_msg=err_msg)
|
||
|
assert_array_almost_equal(h, hs_whole[ii], err_msg=err_msg)
|
||
|
# non-whole
|
||
|
expected_w = np.linspace(0, np.pi, len(b), endpoint=False)
|
||
|
w, h = freqz(b, a, worN=expected_w, whole=False) # polyval
|
||
|
assert_array_almost_equal(w, expected_w, err_msg=err_msg)
|
||
|
assert_array_almost_equal(h, hs_half[ii], err_msg=err_msg)
|
||
|
w, h = freqz(b, a, worN=len(b), whole=False) # FFT
|
||
|
assert_array_almost_equal(w, expected_w, err_msg=err_msg)
|
||
|
assert_array_almost_equal(h, hs_half[ii], err_msg=err_msg)
|
||
|
|
||
|
# some random FIR filters (real + complex)
|
||
|
# assume polyval is accurate
|
||
|
rng = np.random.RandomState(0)
|
||
|
for ii in range(2, 10): # number of taps
|
||
|
b = rng.randn(ii)
|
||
|
for kk in range(2):
|
||
|
a = rng.randn(1) if kk == 0 else rng.randn(3)
|
||
|
for jj in range(2):
|
||
|
if jj == 1:
|
||
|
b = b + rng.randn(ii) * 1j
|
||
|
# whole
|
||
|
expected_w = np.linspace(0, 2 * np.pi, ii, endpoint=False)
|
||
|
w, expected_h = freqz(b, a, worN=expected_w, whole=True)
|
||
|
assert_array_almost_equal(w, expected_w)
|
||
|
w, h = freqz(b, a, worN=ii, whole=True)
|
||
|
assert_array_almost_equal(w, expected_w)
|
||
|
assert_array_almost_equal(h, expected_h)
|
||
|
# half
|
||
|
expected_w = np.linspace(0, np.pi, ii, endpoint=False)
|
||
|
w, expected_h = freqz(b, a, worN=expected_w, whole=False)
|
||
|
assert_array_almost_equal(w, expected_w)
|
||
|
w, h = freqz(b, a, worN=ii, whole=False)
|
||
|
assert_array_almost_equal(w, expected_w)
|
||
|
assert_array_almost_equal(h, expected_h)
|
||
|
|
||
|
def test_broadcasting1(self):
|
||
|
# Test broadcasting with worN an integer or a 1-D array,
|
||
|
# b and a are n-dimensional arrays.
|
||
|
np.random.seed(123)
|
||
|
b = np.random.rand(3, 5, 1)
|
||
|
a = np.random.rand(2, 1)
|
||
|
for whole in [False, True]:
|
||
|
# Test with worN being integers (one fast for FFT and one not),
|
||
|
# a 1-D array, and an empty array.
|
||
|
for worN in [16, 17, np.linspace(0, 1, 10), np.array([])]:
|
||
|
w, h = freqz(b, a, worN=worN, whole=whole)
|
||
|
for k in range(b.shape[1]):
|
||
|
bk = b[:, k, 0]
|
||
|
ak = a[:, 0]
|
||
|
ww, hh = freqz(bk, ak, worN=worN, whole=whole)
|
||
|
assert_allclose(ww, w)
|
||
|
assert_allclose(hh, h[k])
|
||
|
|
||
|
def test_broadcasting2(self):
|
||
|
# Test broadcasting with worN an integer or a 1-D array,
|
||
|
# b is an n-dimensional array, and a is left at the default value.
|
||
|
np.random.seed(123)
|
||
|
b = np.random.rand(3, 5, 1)
|
||
|
for whole in [False, True]:
|
||
|
for worN in [16, 17, np.linspace(0, 1, 10)]:
|
||
|
w, h = freqz(b, worN=worN, whole=whole)
|
||
|
for k in range(b.shape[1]):
|
||
|
bk = b[:, k, 0]
|
||
|
ww, hh = freqz(bk, worN=worN, whole=whole)
|
||
|
assert_allclose(ww, w)
|
||
|
assert_allclose(hh, h[k])
|
||
|
|
||
|
def test_broadcasting3(self):
|
||
|
# Test broadcasting where b.shape[-1] is the same length
|
||
|
# as worN, and a is left at the default value.
|
||
|
np.random.seed(123)
|
||
|
N = 16
|
||
|
b = np.random.rand(3, N)
|
||
|
for whole in [False, True]:
|
||
|
for worN in [N, np.linspace(0, 1, N)]:
|
||
|
w, h = freqz(b, worN=worN, whole=whole)
|
||
|
assert_equal(w.size, N)
|
||
|
for k in range(N):
|
||
|
bk = b[:, k]
|
||
|
ww, hh = freqz(bk, worN=w[k], whole=whole)
|
||
|
assert_allclose(ww, w[k])
|
||
|
assert_allclose(hh, h[k])
|
||
|
|
||
|
def test_broadcasting4(self):
|
||
|
# Test broadcasting with worN a 2-D array.
|
||
|
np.random.seed(123)
|
||
|
b = np.random.rand(4, 2, 1, 1)
|
||
|
a = np.random.rand(5, 2, 1, 1)
|
||
|
for whole in [False, True]:
|
||
|
for worN in [np.random.rand(6, 7), np.empty((6, 0))]:
|
||
|
w, h = freqz(b, a, worN=worN, whole=whole)
|
||
|
assert_allclose(w, worN, rtol=1e-14)
|
||
|
assert_equal(h.shape, (2,) + worN.shape)
|
||
|
for k in range(2):
|
||
|
ww, hh = freqz(b[:, k, 0, 0], a[:, k, 0, 0],
|
||
|
worN=worN.ravel(),
|
||
|
whole=whole)
|
||
|
assert_allclose(ww, worN.ravel(), rtol=1e-14)
|
||
|
assert_allclose(hh, h[k, :, :].ravel())
|
||
|
|
||
|
def test_backward_compat(self):
|
||
|
# For backward compatibility, test if None act as a wrapper for default
|
||
|
w1, h1 = freqz([1.0], 1)
|
||
|
w2, h2 = freqz([1.0], 1, None)
|
||
|
assert_array_almost_equal(w1, w2)
|
||
|
assert_array_almost_equal(h1, h2)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
fs = 900
|
||
|
b = [0.039479155677484369, 0.11843746703245311, 0.11843746703245311,
|
||
|
0.039479155677484369]
|
||
|
a = [1.0, -1.3199152021838287, 0.80341991081938424,
|
||
|
-0.16767146321568049]
|
||
|
|
||
|
# N = None, whole=False
|
||
|
w1, h1 = freqz(b, a, fs=fs)
|
||
|
w2, h2 = freqz(b, a)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs/2, 512, endpoint=False))
|
||
|
|
||
|
# N = None, whole=True
|
||
|
w1, h1 = freqz(b, a, whole=True, fs=fs)
|
||
|
w2, h2 = freqz(b, a, whole=True)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs, 512, endpoint=False))
|
||
|
|
||
|
# N = 5, whole=False
|
||
|
w1, h1 = freqz(b, a, 5, fs=fs)
|
||
|
w2, h2 = freqz(b, a, 5)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs/2, 5, endpoint=False))
|
||
|
|
||
|
# N = 5, whole=True
|
||
|
w1, h1 = freqz(b, a, 5, whole=True, fs=fs)
|
||
|
w2, h2 = freqz(b, a, 5, whole=True)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs, 5, endpoint=False))
|
||
|
|
||
|
# w is an array_like
|
||
|
for w in ([123], (123,), np.array([123]), (50, 123, 230),
|
||
|
np.array([50, 123, 230])):
|
||
|
w1, h1 = freqz(b, a, w, fs=fs)
|
||
|
w2, h2 = freqz(b, a, 2*pi*np.array(w)/fs)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w, w1)
|
||
|
|
||
|
def test_w_or_N_types(self):
|
||
|
# Measure at 7 (polyval) or 8 (fft) equally-spaced points
|
||
|
for N in (7, np.int8(7), np.int16(7), np.int32(7), np.int64(7),
|
||
|
np.array(7),
|
||
|
8, np.int8(8), np.int16(8), np.int32(8), np.int64(8),
|
||
|
np.array(8)):
|
||
|
|
||
|
w, h = freqz([1.0], worN=N)
|
||
|
assert_array_almost_equal(w, np.pi * np.arange(N) / N)
|
||
|
assert_array_almost_equal(h, np.ones(N))
|
||
|
|
||
|
w, h = freqz([1.0], worN=N, fs=100)
|
||
|
assert_array_almost_equal(w, np.linspace(0, 50, N, endpoint=False))
|
||
|
assert_array_almost_equal(h, np.ones(N))
|
||
|
|
||
|
# Measure at frequency 8 Hz
|
||
|
for w in (8.0, 8.0+0j):
|
||
|
# Only makes sense when fs is specified
|
||
|
w_out, h = freqz([1.0], worN=w, fs=100)
|
||
|
assert_array_almost_equal(w_out, [8])
|
||
|
assert_array_almost_equal(h, [1])
|
||
|
|
||
|
|
||
|
class TestSOSFreqz(object):
|
||
|
|
||
|
def test_sosfreqz_basic(self):
|
||
|
# Compare the results of freqz and sosfreqz for a low order
|
||
|
# Butterworth filter.
|
||
|
|
||
|
N = 500
|
||
|
|
||
|
b, a = butter(4, 0.2)
|
||
|
sos = butter(4, 0.2, output='sos')
|
||
|
w, h = freqz(b, a, worN=N)
|
||
|
w2, h2 = sosfreqz(sos, worN=N)
|
||
|
assert_equal(w2, w)
|
||
|
assert_allclose(h2, h, rtol=1e-10, atol=1e-14)
|
||
|
|
||
|
b, a = ellip(3, 1, 30, (0.2, 0.3), btype='bandpass')
|
||
|
sos = ellip(3, 1, 30, (0.2, 0.3), btype='bandpass', output='sos')
|
||
|
w, h = freqz(b, a, worN=N)
|
||
|
w2, h2 = sosfreqz(sos, worN=N)
|
||
|
assert_equal(w2, w)
|
||
|
assert_allclose(h2, h, rtol=1e-10, atol=1e-14)
|
||
|
# must have at least one section
|
||
|
assert_raises(ValueError, sosfreqz, sos[:0])
|
||
|
|
||
|
def test_sosfrez_design(self):
|
||
|
# Compare sosfreqz output against expected values for different
|
||
|
# filter types
|
||
|
|
||
|
# from cheb2ord
|
||
|
N, Wn = cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 60)
|
||
|
sos = cheby2(N, 60, Wn, 'stop', output='sos')
|
||
|
w, h = sosfreqz(sos)
|
||
|
h = np.abs(h)
|
||
|
w /= np.pi
|
||
|
assert_allclose(20 * np.log10(h[w <= 0.1]), 0, atol=3.01)
|
||
|
assert_allclose(20 * np.log10(h[w >= 0.6]), 0., atol=3.01)
|
||
|
assert_allclose(h[(w >= 0.2) & (w <= 0.5)], 0., atol=1e-3) # <= -60 dB
|
||
|
|
||
|
N, Wn = cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 150)
|
||
|
sos = cheby2(N, 150, Wn, 'stop', output='sos')
|
||
|
w, h = sosfreqz(sos)
|
||
|
dB = 20*np.log10(np.abs(h))
|
||
|
w /= np.pi
|
||
|
assert_allclose(dB[w <= 0.1], 0, atol=3.01)
|
||
|
assert_allclose(dB[w >= 0.6], 0., atol=3.01)
|
||
|
assert_array_less(dB[(w >= 0.2) & (w <= 0.5)], -149.9)
|
||
|
|
||
|
# from cheb1ord
|
||
|
N, Wn = cheb1ord(0.2, 0.3, 3, 40)
|
||
|
sos = cheby1(N, 3, Wn, 'low', output='sos')
|
||
|
w, h = sosfreqz(sos)
|
||
|
h = np.abs(h)
|
||
|
w /= np.pi
|
||
|
assert_allclose(20 * np.log10(h[w <= 0.2]), 0, atol=3.01)
|
||
|
assert_allclose(h[w >= 0.3], 0., atol=1e-2) # <= -40 dB
|
||
|
|
||
|
N, Wn = cheb1ord(0.2, 0.3, 1, 150)
|
||
|
sos = cheby1(N, 1, Wn, 'low', output='sos')
|
||
|
w, h = sosfreqz(sos)
|
||
|
dB = 20*np.log10(np.abs(h))
|
||
|
w /= np.pi
|
||
|
assert_allclose(dB[w <= 0.2], 0, atol=1.01)
|
||
|
assert_array_less(dB[w >= 0.3], -149.9)
|
||
|
|
||
|
# adapted from ellipord
|
||
|
N, Wn = ellipord(0.3, 0.2, 3, 60)
|
||
|
sos = ellip(N, 0.3, 60, Wn, 'high', output='sos')
|
||
|
w, h = sosfreqz(sos)
|
||
|
h = np.abs(h)
|
||
|
w /= np.pi
|
||
|
assert_allclose(20 * np.log10(h[w >= 0.3]), 0, atol=3.01)
|
||
|
assert_allclose(h[w <= 0.1], 0., atol=1.5e-3) # <= -60 dB (approx)
|
||
|
|
||
|
# adapted from buttord
|
||
|
N, Wn = buttord([0.2, 0.5], [0.14, 0.6], 3, 40)
|
||
|
sos = butter(N, Wn, 'band', output='sos')
|
||
|
w, h = sosfreqz(sos)
|
||
|
h = np.abs(h)
|
||
|
w /= np.pi
|
||
|
assert_allclose(h[w <= 0.14], 0., atol=1e-2) # <= -40 dB
|
||
|
assert_allclose(h[w >= 0.6], 0., atol=1e-2) # <= -40 dB
|
||
|
assert_allclose(20 * np.log10(h[(w >= 0.2) & (w <= 0.5)]),
|
||
|
0, atol=3.01)
|
||
|
|
||
|
N, Wn = buttord([0.2, 0.5], [0.14, 0.6], 3, 100)
|
||
|
sos = butter(N, Wn, 'band', output='sos')
|
||
|
w, h = sosfreqz(sos)
|
||
|
dB = 20*np.log10(np.maximum(np.abs(h), 1e-10))
|
||
|
w /= np.pi
|
||
|
assert_array_less(dB[(w > 0) & (w <= 0.14)], -99.9)
|
||
|
assert_array_less(dB[w >= 0.6], -99.9)
|
||
|
assert_allclose(dB[(w >= 0.2) & (w <= 0.5)], 0, atol=3.01)
|
||
|
|
||
|
@pytest.mark.xfail
|
||
|
def test_sosfreqz_design_ellip(self):
|
||
|
N, Wn = ellipord(0.3, 0.1, 3, 60)
|
||
|
sos = ellip(N, 0.3, 60, Wn, 'high', output='sos')
|
||
|
w, h = sosfreqz(sos)
|
||
|
h = np.abs(h)
|
||
|
w /= np.pi
|
||
|
assert_allclose(20 * np.log10(h[w >= 0.3]), 0, atol=3.01)
|
||
|
assert_allclose(h[w <= 0.1], 0., atol=1.5e-3) # <= -60 dB (approx)
|
||
|
|
||
|
N, Wn = ellipord(0.3, 0.2, .5, 150)
|
||
|
sos = ellip(N, .5, 150, Wn, 'high', output='sos')
|
||
|
w, h = sosfreqz(sos)
|
||
|
dB = 20*np.log10(np.maximum(np.abs(h), 1e-10))
|
||
|
w /= np.pi
|
||
|
assert_allclose(dB[w >= 0.3], 0, atol=.55)
|
||
|
# this is not great (147 instead of 150, could be ellip[ord] problem?)
|
||
|
assert_array_less(dB[(w > 0) & (w <= 0.25)], -147)
|
||
|
|
||
|
@mpmath_check("0.10")
|
||
|
def test_sos_freqz_against_mp(self):
|
||
|
# Compare the result of sosfreqz applied to a high order Butterworth
|
||
|
# filter against the result computed using mpmath. (signal.freqz fails
|
||
|
# miserably with such high order filters.)
|
||
|
from . import mpsig
|
||
|
N = 500
|
||
|
order = 25
|
||
|
Wn = 0.15
|
||
|
with mpmath.workdps(80):
|
||
|
z_mp, p_mp, k_mp = mpsig.butter_lp(order, Wn)
|
||
|
w_mp, h_mp = mpsig.zpkfreqz(z_mp, p_mp, k_mp, N)
|
||
|
w_mp = np.array([float(x) for x in w_mp])
|
||
|
h_mp = np.array([complex(x) for x in h_mp])
|
||
|
|
||
|
sos = butter(order, Wn, output='sos')
|
||
|
w, h = sosfreqz(sos, worN=N)
|
||
|
assert_allclose(w, w_mp, rtol=1e-12, atol=1e-14)
|
||
|
assert_allclose(h, h_mp, rtol=1e-12, atol=1e-14)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
fs = 900
|
||
|
sos = [[0.03934683014103762, 0.07869366028207524, 0.03934683014103762,
|
||
|
1.0, -0.37256600288916636, 0.0],
|
||
|
[1.0, 1.0, 0.0, 1.0, -0.9495739996946778, 0.45125966317124144]]
|
||
|
|
||
|
# N = None, whole=False
|
||
|
w1, h1 = sosfreqz(sos, fs=fs)
|
||
|
w2, h2 = sosfreqz(sos)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs/2, 512, endpoint=False))
|
||
|
|
||
|
# N = None, whole=True
|
||
|
w1, h1 = sosfreqz(sos, whole=True, fs=fs)
|
||
|
w2, h2 = sosfreqz(sos, whole=True)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs, 512, endpoint=False))
|
||
|
|
||
|
# N = 5, whole=False
|
||
|
w1, h1 = sosfreqz(sos, 5, fs=fs)
|
||
|
w2, h2 = sosfreqz(sos, 5)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs/2, 5, endpoint=False))
|
||
|
|
||
|
# N = 5, whole=True
|
||
|
w1, h1 = sosfreqz(sos, 5, whole=True, fs=fs)
|
||
|
w2, h2 = sosfreqz(sos, 5, whole=True)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs, 5, endpoint=False))
|
||
|
|
||
|
# w is an array_like
|
||
|
for w in ([123], (123,), np.array([123]), (50, 123, 230),
|
||
|
np.array([50, 123, 230])):
|
||
|
w1, h1 = sosfreqz(sos, w, fs=fs)
|
||
|
w2, h2 = sosfreqz(sos, 2*pi*np.array(w)/fs)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w, w1)
|
||
|
|
||
|
def test_w_or_N_types(self):
|
||
|
# Measure at 7 (polyval) or 8 (fft) equally-spaced points
|
||
|
for N in (7, np.int8(7), np.int16(7), np.int32(7), np.int64(7),
|
||
|
np.array(7),
|
||
|
8, np.int8(8), np.int16(8), np.int32(8), np.int64(8),
|
||
|
np.array(8)):
|
||
|
|
||
|
w, h = sosfreqz([1, 0, 0, 1, 0, 0], worN=N)
|
||
|
assert_array_almost_equal(w, np.pi * np.arange(N) / N)
|
||
|
assert_array_almost_equal(h, np.ones(N))
|
||
|
|
||
|
w, h = sosfreqz([1, 0, 0, 1, 0, 0], worN=N, fs=100)
|
||
|
assert_array_almost_equal(w, np.linspace(0, 50, N, endpoint=False))
|
||
|
assert_array_almost_equal(h, np.ones(N))
|
||
|
|
||
|
# Measure at frequency 8 Hz
|
||
|
for w in (8.0, 8.0+0j):
|
||
|
# Only makes sense when fs is specified
|
||
|
w_out, h = sosfreqz([1, 0, 0, 1, 0, 0], worN=w, fs=100)
|
||
|
assert_array_almost_equal(w_out, [8])
|
||
|
assert_array_almost_equal(h, [1])
|
||
|
|
||
|
|
||
|
class TestFreqz_zpk(object):
|
||
|
|
||
|
def test_ticket1441(self):
|
||
|
"""Regression test for ticket 1441."""
|
||
|
# Because freqz previously used arange instead of linspace,
|
||
|
# when N was large, it would return one more point than
|
||
|
# requested.
|
||
|
N = 100000
|
||
|
w, h = freqz_zpk([0.5], [0.5], 1.0, worN=N)
|
||
|
assert_equal(w.shape, (N,))
|
||
|
|
||
|
def test_basic(self):
|
||
|
w, h = freqz_zpk([0.5], [0.5], 1.0, worN=8)
|
||
|
assert_array_almost_equal(w, np.pi * np.arange(8.0) / 8)
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
def test_basic_whole(self):
|
||
|
w, h = freqz_zpk([0.5], [0.5], 1.0, worN=8, whole=True)
|
||
|
assert_array_almost_equal(w, 2 * np.pi * np.arange(8.0) / 8)
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
def test_vs_freqz(self):
|
||
|
b, a = cheby1(4, 5, 0.5, analog=False, output='ba')
|
||
|
z, p, k = cheby1(4, 5, 0.5, analog=False, output='zpk')
|
||
|
|
||
|
w1, h1 = freqz(b, a)
|
||
|
w2, h2 = freqz_zpk(z, p, k)
|
||
|
assert_allclose(w1, w2)
|
||
|
assert_allclose(h1, h2, rtol=1e-6)
|
||
|
|
||
|
def test_backward_compat(self):
|
||
|
# For backward compatibility, test if None act as a wrapper for default
|
||
|
w1, h1 = freqz_zpk([0.5], [0.5], 1.0)
|
||
|
w2, h2 = freqz_zpk([0.5], [0.5], 1.0, None)
|
||
|
assert_array_almost_equal(w1, w2)
|
||
|
assert_array_almost_equal(h1, h2)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
fs = 900
|
||
|
z = [-1, -1, -1]
|
||
|
p = [0.4747869998473389+0.4752230717749344j, 0.37256600288916636,
|
||
|
0.4747869998473389-0.4752230717749344j]
|
||
|
k = 0.03934683014103762
|
||
|
|
||
|
# N = None, whole=False
|
||
|
w1, h1 = freqz_zpk(z, p, k, whole=False, fs=fs)
|
||
|
w2, h2 = freqz_zpk(z, p, k, whole=False)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs/2, 512, endpoint=False))
|
||
|
|
||
|
# N = None, whole=True
|
||
|
w1, h1 = freqz_zpk(z, p, k, whole=True, fs=fs)
|
||
|
w2, h2 = freqz_zpk(z, p, k, whole=True)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs, 512, endpoint=False))
|
||
|
|
||
|
# N = 5, whole=False
|
||
|
w1, h1 = freqz_zpk(z, p, k, 5, fs=fs)
|
||
|
w2, h2 = freqz_zpk(z, p, k, 5)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs/2, 5, endpoint=False))
|
||
|
|
||
|
# N = 5, whole=True
|
||
|
w1, h1 = freqz_zpk(z, p, k, 5, whole=True, fs=fs)
|
||
|
w2, h2 = freqz_zpk(z, p, k, 5, whole=True)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w1, np.linspace(0, fs, 5, endpoint=False))
|
||
|
|
||
|
# w is an array_like
|
||
|
for w in ([123], (123,), np.array([123]), (50, 123, 230),
|
||
|
np.array([50, 123, 230])):
|
||
|
w1, h1 = freqz_zpk(z, p, k, w, fs=fs)
|
||
|
w2, h2 = freqz_zpk(z, p, k, 2*pi*np.array(w)/fs)
|
||
|
assert_allclose(h1, h2)
|
||
|
assert_allclose(w, w1)
|
||
|
|
||
|
def test_w_or_N_types(self):
|
||
|
# Measure at 8 equally-spaced points
|
||
|
for N in (8, np.int8(8), np.int16(8), np.int32(8), np.int64(8),
|
||
|
np.array(8)):
|
||
|
|
||
|
w, h = freqz_zpk([], [], 1, worN=N)
|
||
|
assert_array_almost_equal(w, np.pi * np.arange(8) / 8.)
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
w, h = freqz_zpk([], [], 1, worN=N, fs=100)
|
||
|
assert_array_almost_equal(w, np.linspace(0, 50, 8, endpoint=False))
|
||
|
assert_array_almost_equal(h, np.ones(8))
|
||
|
|
||
|
# Measure at frequency 8 Hz
|
||
|
for w in (8.0, 8.0+0j):
|
||
|
# Only makes sense when fs is specified
|
||
|
w_out, h = freqz_zpk([], [], 1, worN=w, fs=100)
|
||
|
assert_array_almost_equal(w_out, [8])
|
||
|
assert_array_almost_equal(h, [1])
|
||
|
|
||
|
|
||
|
class TestNormalize(object):
|
||
|
|
||
|
def test_allclose(self):
|
||
|
"""Test for false positive on allclose in normalize() in
|
||
|
filter_design.py"""
|
||
|
# Test to make sure the allclose call within signal.normalize does not
|
||
|
# choose false positives. Then check against a known output from MATLAB
|
||
|
# to make sure the fix doesn't break anything.
|
||
|
|
||
|
# These are the coefficients returned from
|
||
|
# `[b,a] = cheby1(8, 0.5, 0.048)'
|
||
|
# in MATLAB. There are at least 15 significant figures in each
|
||
|
# coefficient, so it makes sense to test for errors on the order of
|
||
|
# 1e-13 (this can always be relaxed if different platforms have
|
||
|
# different rounding errors)
|
||
|
b_matlab = np.array([2.150733144728282e-11, 1.720586515782626e-10,
|
||
|
6.022052805239190e-10, 1.204410561047838e-09,
|
||
|
1.505513201309798e-09, 1.204410561047838e-09,
|
||
|
6.022052805239190e-10, 1.720586515782626e-10,
|
||
|
2.150733144728282e-11])
|
||
|
a_matlab = np.array([1.000000000000000e+00, -7.782402035027959e+00,
|
||
|
2.654354569747454e+01, -5.182182531666387e+01,
|
||
|
6.334127355102684e+01, -4.963358186631157e+01,
|
||
|
2.434862182949389e+01, -6.836925348604676e+00,
|
||
|
8.412934944449140e-01])
|
||
|
|
||
|
# This is the input to signal.normalize after passing through the
|
||
|
# equivalent steps in signal.iirfilter as was done for MATLAB
|
||
|
b_norm_in = np.array([1.5543135865293012e-06, 1.2434508692234413e-05,
|
||
|
4.3520780422820447e-05, 8.7041560845640893e-05,
|
||
|
1.0880195105705122e-04, 8.7041560845640975e-05,
|
||
|
4.3520780422820447e-05, 1.2434508692234413e-05,
|
||
|
1.5543135865293012e-06])
|
||
|
a_norm_in = np.array([7.2269025909127173e+04, -5.6242661430467968e+05,
|
||
|
1.9182761917308895e+06, -3.7451128364682454e+06,
|
||
|
4.5776121393762771e+06, -3.5869706138592605e+06,
|
||
|
1.7596511818472347e+06, -4.9409793515707983e+05,
|
||
|
6.0799461347219651e+04])
|
||
|
|
||
|
b_output, a_output = normalize(b_norm_in, a_norm_in)
|
||
|
|
||
|
# The test on b works for decimal=14 but the one for a does not. For
|
||
|
# the sake of consistency, both of these are decimal=13. If something
|
||
|
# breaks on another platform, it is probably fine to relax this lower.
|
||
|
assert_array_almost_equal(b_matlab, b_output, decimal=13)
|
||
|
assert_array_almost_equal(a_matlab, a_output, decimal=13)
|
||
|
|
||
|
def test_errors(self):
|
||
|
"""Test the error cases."""
|
||
|
# all zero denominator
|
||
|
assert_raises(ValueError, normalize, [1, 2], 0)
|
||
|
|
||
|
# denominator not 1 dimensional
|
||
|
assert_raises(ValueError, normalize, [1, 2], [[1]])
|
||
|
|
||
|
# numerator too many dimensions
|
||
|
assert_raises(ValueError, normalize, [[[1, 2]]], 1)
|
||
|
|
||
|
|
||
|
class TestLp2lp(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
b = [1]
|
||
|
a = [1, np.sqrt(2), 1]
|
||
|
b_lp, a_lp = lp2lp(b, a, 0.38574256627112119)
|
||
|
assert_array_almost_equal(b_lp, [0.1488], decimal=4)
|
||
|
assert_array_almost_equal(a_lp, [1, 0.5455, 0.1488], decimal=4)
|
||
|
|
||
|
|
||
|
class TestLp2hp(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
b = [0.25059432325190018]
|
||
|
a = [1, 0.59724041654134863, 0.92834805757524175, 0.25059432325190018]
|
||
|
b_hp, a_hp = lp2hp(b, a, 2*np.pi*5000)
|
||
|
assert_allclose(b_hp, [1, 0, 0, 0])
|
||
|
assert_allclose(a_hp, [1, 1.1638e5, 2.3522e9, 1.2373e14], rtol=1e-4)
|
||
|
|
||
|
|
||
|
class TestLp2bp(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
b = [1]
|
||
|
a = [1, 2, 2, 1]
|
||
|
b_bp, a_bp = lp2bp(b, a, 2*np.pi*4000, 2*np.pi*2000)
|
||
|
assert_allclose(b_bp, [1.9844e12, 0, 0, 0], rtol=1e-6)
|
||
|
assert_allclose(a_bp, [1, 2.5133e4, 2.2108e9, 3.3735e13,
|
||
|
1.3965e18, 1.0028e22, 2.5202e26], rtol=1e-4)
|
||
|
|
||
|
|
||
|
class TestLp2bs(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
b = [1]
|
||
|
a = [1, 1]
|
||
|
b_bs, a_bs = lp2bs(b, a, 0.41722257286366754, 0.18460575326152251)
|
||
|
assert_array_almost_equal(b_bs, [1, 0, 0.17407], decimal=5)
|
||
|
assert_array_almost_equal(a_bs, [1, 0.18461, 0.17407], decimal=5)
|
||
|
|
||
|
|
||
|
class TestBilinear(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
b = [0.14879732743343033]
|
||
|
a = [1, 0.54552236880522209, 0.14879732743343033]
|
||
|
b_z, a_z = bilinear(b, a, 0.5)
|
||
|
assert_array_almost_equal(b_z, [0.087821, 0.17564, 0.087821],
|
||
|
decimal=5)
|
||
|
assert_array_almost_equal(a_z, [1, -1.0048, 0.35606], decimal=4)
|
||
|
|
||
|
b = [1, 0, 0.17407467530697837]
|
||
|
a = [1, 0.18460575326152251, 0.17407467530697837]
|
||
|
b_z, a_z = bilinear(b, a, 0.5)
|
||
|
assert_array_almost_equal(b_z, [0.86413, -1.2158, 0.86413],
|
||
|
decimal=4)
|
||
|
assert_array_almost_equal(a_z, [1, -1.2158, 0.72826],
|
||
|
decimal=4)
|
||
|
|
||
|
|
||
|
class TestLp2lp_zpk(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
z = []
|
||
|
p = [(-1+1j)/np.sqrt(2), (-1-1j)/np.sqrt(2)]
|
||
|
k = 1
|
||
|
z_lp, p_lp, k_lp = lp2lp_zpk(z, p, k, 5)
|
||
|
assert_array_equal(z_lp, [])
|
||
|
assert_allclose(sort(p_lp), sort(p)*5)
|
||
|
assert_allclose(k_lp, 25)
|
||
|
|
||
|
# Pseudo-Chebyshev with both poles and zeros
|
||
|
z = [-2j, +2j]
|
||
|
p = [-0.75, -0.5-0.5j, -0.5+0.5j]
|
||
|
k = 3
|
||
|
z_lp, p_lp, k_lp = lp2lp_zpk(z, p, k, 20)
|
||
|
assert_allclose(sort(z_lp), sort([-40j, +40j]))
|
||
|
assert_allclose(sort(p_lp), sort([-15, -10-10j, -10+10j]))
|
||
|
assert_allclose(k_lp, 60)
|
||
|
|
||
|
|
||
|
class TestLp2hp_zpk(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
z = []
|
||
|
p = [(-1+1j)/np.sqrt(2), (-1-1j)/np.sqrt(2)]
|
||
|
k = 1
|
||
|
|
||
|
z_hp, p_hp, k_hp = lp2hp_zpk(z, p, k, 5)
|
||
|
assert_array_equal(z_hp, [0, 0])
|
||
|
assert_allclose(sort(p_hp), sort(p)*5)
|
||
|
assert_allclose(k_hp, 1)
|
||
|
|
||
|
z = [-2j, +2j]
|
||
|
p = [-0.75, -0.5-0.5j, -0.5+0.5j]
|
||
|
k = 3
|
||
|
z_hp, p_hp, k_hp = lp2hp_zpk(z, p, k, 6)
|
||
|
assert_allclose(sort(z_hp), sort([-3j, 0, +3j]))
|
||
|
assert_allclose(sort(p_hp), sort([-8, -6-6j, -6+6j]))
|
||
|
assert_allclose(k_hp, 32)
|
||
|
|
||
|
|
||
|
class TestLp2bp_zpk(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
z = [-2j, +2j]
|
||
|
p = [-0.75, -0.5-0.5j, -0.5+0.5j]
|
||
|
k = 3
|
||
|
z_bp, p_bp, k_bp = lp2bp_zpk(z, p, k, 15, 8)
|
||
|
assert_allclose(sort(z_bp), sort([-25j, -9j, 0, +9j, +25j]))
|
||
|
assert_allclose(sort(p_bp), sort([-3 + 6j*sqrt(6),
|
||
|
-3 - 6j*sqrt(6),
|
||
|
+2j+sqrt(-8j-225)-2,
|
||
|
-2j+sqrt(+8j-225)-2,
|
||
|
+2j-sqrt(-8j-225)-2,
|
||
|
-2j-sqrt(+8j-225)-2, ]))
|
||
|
assert_allclose(k_bp, 24)
|
||
|
|
||
|
|
||
|
class TestLp2bs_zpk(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
z = [-2j, +2j]
|
||
|
p = [-0.75, -0.5-0.5j, -0.5+0.5j]
|
||
|
k = 3
|
||
|
|
||
|
z_bs, p_bs, k_bs = lp2bs_zpk(z, p, k, 35, 12)
|
||
|
|
||
|
assert_allclose(sort(z_bs), sort([+35j, -35j,
|
||
|
+3j+sqrt(1234)*1j,
|
||
|
-3j+sqrt(1234)*1j,
|
||
|
+3j-sqrt(1234)*1j,
|
||
|
-3j-sqrt(1234)*1j]))
|
||
|
assert_allclose(sort(p_bs), sort([+3j*sqrt(129) - 8,
|
||
|
-3j*sqrt(129) - 8,
|
||
|
(-6 + 6j) - sqrt(-1225 - 72j),
|
||
|
(-6 - 6j) - sqrt(-1225 + 72j),
|
||
|
(-6 + 6j) + sqrt(-1225 - 72j),
|
||
|
(-6 - 6j) + sqrt(-1225 + 72j), ]))
|
||
|
assert_allclose(k_bs, 32)
|
||
|
|
||
|
|
||
|
class TestBilinear_zpk(object):
|
||
|
|
||
|
def test_basic(self):
|
||
|
z = [-2j, +2j]
|
||
|
p = [-0.75, -0.5-0.5j, -0.5+0.5j]
|
||
|
k = 3
|
||
|
|
||
|
z_d, p_d, k_d = bilinear_zpk(z, p, k, 10)
|
||
|
|
||
|
assert_allclose(sort(z_d), sort([(20-2j)/(20+2j), (20+2j)/(20-2j),
|
||
|
-1]))
|
||
|
assert_allclose(sort(p_d), sort([77/83,
|
||
|
(1j/2 + 39/2) / (41/2 - 1j/2),
|
||
|
(39/2 - 1j/2) / (1j/2 + 41/2), ]))
|
||
|
assert_allclose(k_d, 9696/69803)
|
||
|
|
||
|
|
||
|
class TestPrototypeType(object):
|
||
|
|
||
|
def test_output_type(self):
|
||
|
# Prototypes should consistently output arrays, not lists
|
||
|
# https://github.com/scipy/scipy/pull/441
|
||
|
for func in (buttap,
|
||
|
besselap,
|
||
|
lambda N: cheb1ap(N, 1),
|
||
|
lambda N: cheb2ap(N, 20),
|
||
|
lambda N: ellipap(N, 1, 20)):
|
||
|
for N in range(7):
|
||
|
z, p, k = func(N)
|
||
|
assert_(isinstance(z, np.ndarray))
|
||
|
assert_(isinstance(p, np.ndarray))
|
||
|
|
||
|
|
||
|
def dB(x):
|
||
|
# Return magnitude in decibels, avoiding divide-by-zero warnings
|
||
|
# (and deal with some "not less-ordered" errors when -inf shows up)
|
||
|
return 20 * np.log10(np.maximum(np.abs(x), np.finfo(np.float64).tiny))
|
||
|
|
||
|
|
||
|
class TestButtord(object):
|
||
|
|
||
|
def test_lowpass(self):
|
||
|
wp = 0.2
|
||
|
ws = 0.3
|
||
|
rp = 3
|
||
|
rs = 60
|
||
|
N, Wn = buttord(wp, ws, rp, rs, False)
|
||
|
b, a = butter(N, Wn, 'lowpass', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp, dB(h[w <= wp]))
|
||
|
assert_array_less(dB(h[ws <= w]), -rs)
|
||
|
|
||
|
assert_equal(N, 16)
|
||
|
assert_allclose(Wn, 2.0002776782743284e-01, rtol=1e-15)
|
||
|
|
||
|
def test_highpass(self):
|
||
|
wp = 0.3
|
||
|
ws = 0.2
|
||
|
rp = 3
|
||
|
rs = 70
|
||
|
N, Wn = buttord(wp, ws, rp, rs, False)
|
||
|
b, a = butter(N, Wn, 'highpass', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp, dB(h[wp <= w]))
|
||
|
assert_array_less(dB(h[w <= ws]), -rs)
|
||
|
|
||
|
assert_equal(N, 18)
|
||
|
assert_allclose(Wn, 2.9996603079132672e-01, rtol=1e-15)
|
||
|
|
||
|
def test_bandpass(self):
|
||
|
wp = [0.2, 0.5]
|
||
|
ws = [0.1, 0.6]
|
||
|
rp = 3
|
||
|
rs = 80
|
||
|
N, Wn = buttord(wp, ws, rp, rs, False)
|
||
|
b, a = butter(N, Wn, 'bandpass', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_and(wp[0] <= w, w <= wp[1])]))
|
||
|
assert_array_less(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 18)
|
||
|
assert_allclose(Wn, [1.9998742411409134e-01, 5.0002139595676276e-01],
|
||
|
rtol=1e-15)
|
||
|
|
||
|
def test_bandstop(self):
|
||
|
wp = [0.1, 0.6]
|
||
|
ws = [0.2, 0.5]
|
||
|
rp = 3
|
||
|
rs = 90
|
||
|
N, Wn = buttord(wp, ws, rp, rs, False)
|
||
|
b, a = butter(N, Wn, 'bandstop', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp,
|
||
|
dB(h[np.logical_or(w <= wp[0], wp[1] <= w)]))
|
||
|
assert_array_less(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]),
|
||
|
-rs)
|
||
|
|
||
|
assert_equal(N, 20)
|
||
|
assert_allclose(Wn, [1.4759432329294042e-01, 5.9997365985276407e-01],
|
||
|
rtol=1e-6)
|
||
|
|
||
|
def test_analog(self):
|
||
|
wp = 200
|
||
|
ws = 600
|
||
|
rp = 3
|
||
|
rs = 60
|
||
|
N, Wn = buttord(wp, ws, rp, rs, True)
|
||
|
b, a = butter(N, Wn, 'lowpass', True)
|
||
|
w, h = freqs(b, a)
|
||
|
assert_array_less(-rp, dB(h[w <= wp]))
|
||
|
assert_array_less(dB(h[ws <= w]), -rs)
|
||
|
|
||
|
assert_equal(N, 7)
|
||
|
assert_allclose(Wn, 2.0006785355671877e+02, rtol=1e-15)
|
||
|
|
||
|
n, Wn = buttord(1, 550/450, 1, 26, analog=True)
|
||
|
assert_equal(n, 19)
|
||
|
assert_allclose(Wn, 1.0361980524629517, rtol=1e-15)
|
||
|
|
||
|
assert_equal(buttord(1, 1.2, 1, 80, analog=True)[0], 55)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
wp = [4410, 11025]
|
||
|
ws = [2205, 13230]
|
||
|
rp = 3
|
||
|
rs = 80
|
||
|
fs = 44100
|
||
|
N, Wn = buttord(wp, ws, rp, rs, False, fs=fs)
|
||
|
b, a = butter(N, Wn, 'bandpass', False, fs=fs)
|
||
|
w, h = freqz(b, a, fs=fs)
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_and(wp[0] <= w, w <= wp[1])]))
|
||
|
assert_array_less(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 18)
|
||
|
assert_allclose(Wn, [4409.722701715714, 11025.47178084662],
|
||
|
rtol=1e-15)
|
||
|
|
||
|
|
||
|
class TestCheb1ord(object):
|
||
|
|
||
|
def test_lowpass(self):
|
||
|
wp = 0.2
|
||
|
ws = 0.3
|
||
|
rp = 3
|
||
|
rs = 60
|
||
|
N, Wn = cheb1ord(wp, ws, rp, rs, False)
|
||
|
b, a = cheby1(N, rp, Wn, 'low', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1, dB(h[w <= wp]))
|
||
|
assert_array_less(dB(h[ws <= w]), -rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 8)
|
||
|
assert_allclose(Wn, 0.2, rtol=1e-15)
|
||
|
|
||
|
def test_highpass(self):
|
||
|
wp = 0.3
|
||
|
ws = 0.2
|
||
|
rp = 3
|
||
|
rs = 70
|
||
|
N, Wn = cheb1ord(wp, ws, rp, rs, False)
|
||
|
b, a = cheby1(N, rp, Wn, 'high', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1, dB(h[wp <= w]))
|
||
|
assert_array_less(dB(h[w <= ws]), -rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 9)
|
||
|
assert_allclose(Wn, 0.3, rtol=1e-15)
|
||
|
|
||
|
def test_bandpass(self):
|
||
|
wp = [0.2, 0.5]
|
||
|
ws = [0.1, 0.6]
|
||
|
rp = 3
|
||
|
rs = 80
|
||
|
N, Wn = cheb1ord(wp, ws, rp, rs, False)
|
||
|
b, a = cheby1(N, rp, Wn, 'band', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_and(wp[0] <= w, w <= wp[1])]))
|
||
|
assert_array_less(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 9)
|
||
|
assert_allclose(Wn, [0.2, 0.5], rtol=1e-15)
|
||
|
|
||
|
def test_bandstop(self):
|
||
|
wp = [0.1, 0.6]
|
||
|
ws = [0.2, 0.5]
|
||
|
rp = 3
|
||
|
rs = 90
|
||
|
N, Wn = cheb1ord(wp, ws, rp, rs, False)
|
||
|
b, a = cheby1(N, rp, Wn, 'stop', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_or(w <= wp[0], wp[1] <= w)]))
|
||
|
assert_array_less(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 10)
|
||
|
assert_allclose(Wn, [0.14758232569947785, 0.6], rtol=1e-5)
|
||
|
|
||
|
def test_analog(self):
|
||
|
wp = 700
|
||
|
ws = 100
|
||
|
rp = 3
|
||
|
rs = 70
|
||
|
N, Wn = cheb1ord(wp, ws, rp, rs, True)
|
||
|
b, a = cheby1(N, rp, Wn, 'high', True)
|
||
|
w, h = freqs(b, a)
|
||
|
assert_array_less(-rp - 0.1, dB(h[wp <= w]))
|
||
|
assert_array_less(dB(h[w <= ws]), -rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 4)
|
||
|
assert_allclose(Wn, 700, rtol=1e-15)
|
||
|
|
||
|
assert_equal(cheb1ord(1, 1.2, 1, 80, analog=True)[0], 17)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
wp = 4800
|
||
|
ws = 7200
|
||
|
rp = 3
|
||
|
rs = 60
|
||
|
fs = 48000
|
||
|
N, Wn = cheb1ord(wp, ws, rp, rs, False, fs=fs)
|
||
|
b, a = cheby1(N, rp, Wn, 'low', False, fs=fs)
|
||
|
w, h = freqz(b, a, fs=fs)
|
||
|
assert_array_less(-rp - 0.1, dB(h[w <= wp]))
|
||
|
assert_array_less(dB(h[ws <= w]), -rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 8)
|
||
|
assert_allclose(Wn, 4800, rtol=1e-15)
|
||
|
|
||
|
|
||
|
class TestCheb2ord(object):
|
||
|
|
||
|
def test_lowpass(self):
|
||
|
wp = 0.2
|
||
|
ws = 0.3
|
||
|
rp = 3
|
||
|
rs = 60
|
||
|
N, Wn = cheb2ord(wp, ws, rp, rs, False)
|
||
|
b, a = cheby2(N, rs, Wn, 'lp', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1, dB(h[w <= wp]))
|
||
|
assert_array_less(dB(h[ws <= w]), -rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 8)
|
||
|
assert_allclose(Wn, 0.28647639976553163, rtol=1e-15)
|
||
|
|
||
|
def test_highpass(self):
|
||
|
wp = 0.3
|
||
|
ws = 0.2
|
||
|
rp = 3
|
||
|
rs = 70
|
||
|
N, Wn = cheb2ord(wp, ws, rp, rs, False)
|
||
|
b, a = cheby2(N, rs, Wn, 'hp', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1, dB(h[wp <= w]))
|
||
|
assert_array_less(dB(h[w <= ws]), -rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 9)
|
||
|
assert_allclose(Wn, 0.20697492182903282, rtol=1e-15)
|
||
|
|
||
|
def test_bandpass(self):
|
||
|
wp = [0.2, 0.5]
|
||
|
ws = [0.1, 0.6]
|
||
|
rp = 3
|
||
|
rs = 80
|
||
|
N, Wn = cheb2ord(wp, ws, rp, rs, False)
|
||
|
b, a = cheby2(N, rs, Wn, 'bp', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_and(wp[0] <= w, w <= wp[1])]))
|
||
|
assert_array_less(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 9)
|
||
|
assert_allclose(Wn, [0.14876937565923479, 0.59748447842351482],
|
||
|
rtol=1e-15)
|
||
|
|
||
|
def test_bandstop(self):
|
||
|
wp = [0.1, 0.6]
|
||
|
ws = [0.2, 0.5]
|
||
|
rp = 3
|
||
|
rs = 90
|
||
|
N, Wn = cheb2ord(wp, ws, rp, rs, False)
|
||
|
b, a = cheby2(N, rs, Wn, 'bs', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_or(w <= wp[0], wp[1] <= w)]))
|
||
|
assert_array_less(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 10)
|
||
|
assert_allclose(Wn, [0.19926249974781743, 0.50125246585567362],
|
||
|
rtol=1e-6)
|
||
|
|
||
|
def test_analog(self):
|
||
|
wp = [20, 50]
|
||
|
ws = [10, 60]
|
||
|
rp = 3
|
||
|
rs = 80
|
||
|
N, Wn = cheb2ord(wp, ws, rp, rs, True)
|
||
|
b, a = cheby2(N, rs, Wn, 'bp', True)
|
||
|
w, h = freqs(b, a)
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_and(wp[0] <= w, w <= wp[1])]))
|
||
|
assert_array_less(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 11)
|
||
|
assert_allclose(Wn, [1.673740595370124e+01, 5.974641487254268e+01],
|
||
|
rtol=1e-15)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
wp = 150
|
||
|
ws = 100
|
||
|
rp = 3
|
||
|
rs = 70
|
||
|
fs = 1000
|
||
|
N, Wn = cheb2ord(wp, ws, rp, rs, False, fs=fs)
|
||
|
b, a = cheby2(N, rs, Wn, 'hp', False, fs=fs)
|
||
|
w, h = freqz(b, a, fs=fs)
|
||
|
assert_array_less(-rp - 0.1, dB(h[wp <= w]))
|
||
|
assert_array_less(dB(h[w <= ws]), -rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 9)
|
||
|
assert_allclose(Wn, 103.4874609145164, rtol=1e-15)
|
||
|
|
||
|
|
||
|
class TestEllipord(object):
|
||
|
|
||
|
def test_lowpass(self):
|
||
|
wp = 0.2
|
||
|
ws = 0.3
|
||
|
rp = 3
|
||
|
rs = 60
|
||
|
N, Wn = ellipord(wp, ws, rp, rs, False)
|
||
|
b, a = ellip(N, rp, rs, Wn, 'lp', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1, dB(h[w <= wp]))
|
||
|
assert_array_less(dB(h[ws <= w]), -rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 5)
|
||
|
assert_allclose(Wn, 0.2, rtol=1e-15)
|
||
|
|
||
|
def test_highpass(self):
|
||
|
wp = 0.3
|
||
|
ws = 0.2
|
||
|
rp = 3
|
||
|
rs = 70
|
||
|
N, Wn = ellipord(wp, ws, rp, rs, False)
|
||
|
b, a = ellip(N, rp, rs, Wn, 'hp', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1, dB(h[wp <= w]))
|
||
|
assert_array_less(dB(h[w <= ws]), -rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 6)
|
||
|
assert_allclose(Wn, 0.3, rtol=1e-15)
|
||
|
|
||
|
def test_bandpass(self):
|
||
|
wp = [0.2, 0.5]
|
||
|
ws = [0.1, 0.6]
|
||
|
rp = 3
|
||
|
rs = 80
|
||
|
N, Wn = ellipord(wp, ws, rp, rs, False)
|
||
|
b, a = ellip(N, rp, rs, Wn, 'bp', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_and(wp[0] <= w, w <= wp[1])]))
|
||
|
assert_array_less(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 6)
|
||
|
assert_allclose(Wn, [0.2, 0.5], rtol=1e-15)
|
||
|
|
||
|
def test_bandstop(self):
|
||
|
wp = [0.1, 0.6]
|
||
|
ws = [0.2, 0.5]
|
||
|
rp = 3
|
||
|
rs = 90
|
||
|
N, Wn = ellipord(wp, ws, rp, rs, False)
|
||
|
b, a = ellip(N, rp, rs, Wn, 'bs', False)
|
||
|
w, h = freqz(b, a)
|
||
|
w /= np.pi
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_or(w <= wp[0], wp[1] <= w)]))
|
||
|
assert_array_less(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 7)
|
||
|
assert_allclose(Wn, [0.14758232794342988, 0.6], rtol=1e-5)
|
||
|
|
||
|
def test_analog(self):
|
||
|
wp = [1000, 6000]
|
||
|
ws = [2000, 5000]
|
||
|
rp = 3
|
||
|
rs = 90
|
||
|
N, Wn = ellipord(wp, ws, rp, rs, True)
|
||
|
b, a = ellip(N, rp, rs, Wn, 'bs', True)
|
||
|
w, h = freqs(b, a)
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_or(w <= wp[0], wp[1] <= w)]))
|
||
|
assert_array_less(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 8)
|
||
|
assert_allclose(Wn, [1666.6666, 6000])
|
||
|
|
||
|
assert_equal(ellipord(1, 1.2, 1, 80, analog=True)[0], 9)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
wp = [400, 2400]
|
||
|
ws = [800, 2000]
|
||
|
rp = 3
|
||
|
rs = 90
|
||
|
fs = 8000
|
||
|
N, Wn = ellipord(wp, ws, rp, rs, False, fs=fs)
|
||
|
b, a = ellip(N, rp, rs, Wn, 'bs', False, fs=fs)
|
||
|
w, h = freqz(b, a, fs=fs)
|
||
|
assert_array_less(-rp - 0.1,
|
||
|
dB(h[np.logical_or(w <= wp[0], wp[1] <= w)]))
|
||
|
assert_array_less(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]),
|
||
|
-rs + 0.1)
|
||
|
|
||
|
assert_equal(N, 7)
|
||
|
assert_allclose(Wn, [590.3293117737195, 2400], rtol=1e-5)
|
||
|
|
||
|
|
||
|
class TestBessel(object):
|
||
|
|
||
|
def test_degenerate(self):
|
||
|
for norm in ('delay', 'phase', 'mag'):
|
||
|
# 0-order filter is just a passthrough
|
||
|
b, a = bessel(0, 1, analog=True, norm=norm)
|
||
|
assert_array_equal(b, [1])
|
||
|
assert_array_equal(a, [1])
|
||
|
|
||
|
# 1-order filter is same for all types
|
||
|
b, a = bessel(1, 1, analog=True, norm=norm)
|
||
|
assert_allclose(b, [1], rtol=1e-15)
|
||
|
assert_allclose(a, [1, 1], rtol=1e-15)
|
||
|
|
||
|
z, p, k = bessel(1, 0.3, analog=True, output='zpk', norm=norm)
|
||
|
assert_array_equal(z, [])
|
||
|
assert_allclose(p, [-0.3], rtol=1e-14)
|
||
|
assert_allclose(k, 0.3, rtol=1e-14)
|
||
|
|
||
|
def test_high_order(self):
|
||
|
# high even order, 'phase'
|
||
|
z, p, k = bessel(24, 100, analog=True, output='zpk')
|
||
|
z2 = []
|
||
|
p2 = [
|
||
|
-9.055312334014323e+01 + 4.844005815403969e+00j,
|
||
|
-8.983105162681878e+01 + 1.454056170018573e+01j,
|
||
|
-8.837357994162065e+01 + 2.426335240122282e+01j,
|
||
|
-8.615278316179575e+01 + 3.403202098404543e+01j,
|
||
|
-8.312326467067703e+01 + 4.386985940217900e+01j,
|
||
|
-7.921695461084202e+01 + 5.380628489700191e+01j,
|
||
|
-7.433392285433246e+01 + 6.388084216250878e+01j,
|
||
|
-6.832565803501586e+01 + 7.415032695116071e+01j,
|
||
|
-6.096221567378025e+01 + 8.470292433074425e+01j,
|
||
|
-5.185914574820616e+01 + 9.569048385258847e+01j,
|
||
|
-4.027853855197555e+01 + 1.074195196518679e+02j,
|
||
|
-2.433481337524861e+01 + 1.207298683731973e+02j,
|
||
|
]
|
||
|
k2 = 9.999999999999989e+47
|
||
|
assert_array_equal(z, z2)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(np.union1d(p2, np.conj(p2)), key=np.imag))
|
||
|
assert_allclose(k, k2, rtol=1e-14)
|
||
|
|
||
|
# high odd order, 'phase'
|
||
|
z, p, k = bessel(23, 1000, analog=True, output='zpk')
|
||
|
z2 = []
|
||
|
p2 = [
|
||
|
-2.497697202208956e+02 + 1.202813187870698e+03j,
|
||
|
-4.126986617510172e+02 + 1.065328794475509e+03j,
|
||
|
-5.304922463809596e+02 + 9.439760364018479e+02j,
|
||
|
-9.027564978975828e+02 + 1.010534334242318e+02j,
|
||
|
-8.909283244406079e+02 + 2.023024699647598e+02j,
|
||
|
-8.709469394347836e+02 + 3.039581994804637e+02j,
|
||
|
-8.423805948131370e+02 + 4.062657947488952e+02j,
|
||
|
-8.045561642249877e+02 + 5.095305912401127e+02j,
|
||
|
-7.564660146766259e+02 + 6.141594859516342e+02j,
|
||
|
-6.965966033906477e+02 + 7.207341374730186e+02j,
|
||
|
-6.225903228776276e+02 + 8.301558302815096e+02j,
|
||
|
-9.066732476324988e+02]
|
||
|
k2 = 9.999999999999983e+68
|
||
|
assert_array_equal(z, z2)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(np.union1d(p2, np.conj(p2)), key=np.imag))
|
||
|
assert_allclose(k, k2, rtol=1e-14)
|
||
|
|
||
|
# high even order, 'delay' (Orchard 1965 "The Roots of the
|
||
|
# Maximally Flat-Delay Polynomials" Table 1)
|
||
|
z, p, k = bessel(31, 1, analog=True, output='zpk', norm='delay')
|
||
|
p2 = [-20.876706,
|
||
|
-20.826543 + 1.735732j,
|
||
|
-20.675502 + 3.473320j,
|
||
|
-20.421895 + 5.214702j,
|
||
|
-20.062802 + 6.961982j,
|
||
|
-19.593895 + 8.717546j,
|
||
|
-19.009148 + 10.484195j,
|
||
|
-18.300400 + 12.265351j,
|
||
|
-17.456663 + 14.065350j,
|
||
|
-16.463032 + 15.889910j,
|
||
|
-15.298849 + 17.746914j,
|
||
|
-13.934466 + 19.647827j,
|
||
|
-12.324914 + 21.610519j,
|
||
|
-10.395893 + 23.665701j,
|
||
|
- 8.005600 + 25.875019j,
|
||
|
- 4.792045 + 28.406037j,
|
||
|
]
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(np.union1d(p2, np.conj(p2)), key=np.imag))
|
||
|
|
||
|
# high odd order, 'delay'
|
||
|
z, p, k = bessel(30, 1, analog=True, output='zpk', norm='delay')
|
||
|
p2 = [-20.201029 + 0.867750j,
|
||
|
-20.097257 + 2.604235j,
|
||
|
-19.888485 + 4.343721j,
|
||
|
-19.572188 + 6.088363j,
|
||
|
-19.144380 + 7.840570j,
|
||
|
-18.599342 + 9.603147j,
|
||
|
-17.929195 + 11.379494j,
|
||
|
-17.123228 + 13.173901j,
|
||
|
-16.166808 + 14.992008j,
|
||
|
-15.039580 + 16.841580j,
|
||
|
-13.712245 + 18.733902j,
|
||
|
-12.140295 + 20.686563j,
|
||
|
-10.250119 + 22.729808j,
|
||
|
- 7.901170 + 24.924391j,
|
||
|
- 4.734679 + 27.435615j,
|
||
|
]
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(np.union1d(p2, np.conj(p2)), key=np.imag))
|
||
|
|
||
|
def test_refs(self):
|
||
|
# Compare to http://www.crbond.com/papers/bsf2.pdf
|
||
|
# "Delay Normalized Bessel Polynomial Coefficients"
|
||
|
bond_b = 10395
|
||
|
bond_a = [1, 21, 210, 1260, 4725, 10395, 10395]
|
||
|
b, a = bessel(6, 1, norm='delay', analog=True)
|
||
|
assert_allclose(bond_b, b)
|
||
|
assert_allclose(bond_a, a)
|
||
|
|
||
|
# "Delay Normalized Bessel Pole Locations"
|
||
|
bond_poles = {
|
||
|
1: [-1.0000000000],
|
||
|
2: [-1.5000000000 + 0.8660254038j],
|
||
|
3: [-1.8389073227 + 1.7543809598j, -2.3221853546],
|
||
|
4: [-2.1037893972 + 2.6574180419j, -2.8962106028 + 0.8672341289j],
|
||
|
5: [-2.3246743032 + 3.5710229203j, -3.3519563992 + 1.7426614162j,
|
||
|
-3.6467385953],
|
||
|
6: [-2.5159322478 + 4.4926729537j, -3.7357083563 + 2.6262723114j,
|
||
|
-4.2483593959 + 0.8675096732j],
|
||
|
7: [-2.6856768789 + 5.4206941307j, -4.0701391636 + 3.5171740477j,
|
||
|
-4.7582905282 + 1.7392860611j, -4.9717868585],
|
||
|
8: [-2.8389839489 + 6.3539112986j, -4.3682892172 + 4.4144425005j,
|
||
|
-5.2048407906 + 2.6161751526j, -5.5878860433 + 0.8676144454j],
|
||
|
9: [-2.9792607982 + 7.2914636883j, -4.6384398872 + 5.3172716754j,
|
||
|
-5.6044218195 + 3.4981569179j, -6.1293679043 + 1.7378483835j,
|
||
|
-6.2970191817],
|
||
|
10: [-3.1089162336 + 8.2326994591j, -4.8862195669 + 6.2249854825j,
|
||
|
-5.9675283286 + 4.3849471889j, -6.6152909655 + 2.6115679208j,
|
||
|
-6.9220449054 + 0.8676651955j]
|
||
|
}
|
||
|
|
||
|
for N in range(1, 11):
|
||
|
p1 = np.sort(bond_poles[N])
|
||
|
p2 = np.sort(np.concatenate(_cplxreal(besselap(N, 'delay')[1])))
|
||
|
assert_array_almost_equal(p1, p2, decimal=10)
|
||
|
|
||
|
# "Frequency Normalized Bessel Pole Locations"
|
||
|
bond_poles = {
|
||
|
1: [-1.0000000000],
|
||
|
2: [-1.1016013306 + 0.6360098248j],
|
||
|
3: [-1.0474091610 + 0.9992644363j, -1.3226757999],
|
||
|
4: [-0.9952087644 + 1.2571057395j, -1.3700678306 + 0.4102497175j],
|
||
|
5: [-0.9576765486 + 1.4711243207j, -1.3808773259 + 0.7179095876j,
|
||
|
-1.5023162714],
|
||
|
6: [-0.9306565229 + 1.6618632689j, -1.3818580976 + 0.9714718907j,
|
||
|
-1.5714904036 + 0.3208963742j],
|
||
|
7: [-0.9098677806 + 1.8364513530j, -1.3789032168 + 1.1915667778j,
|
||
|
-1.6120387662 + 0.5892445069j, -1.6843681793],
|
||
|
8: [-0.8928697188 + 1.9983258436j, -1.3738412176 + 1.3883565759j,
|
||
|
-1.6369394181 + 0.8227956251j, -1.7574084004 + 0.2728675751j],
|
||
|
9: [-0.8783992762 + 2.1498005243j, -1.3675883098 + 1.5677337122j,
|
||
|
-1.6523964846 + 1.0313895670j, -1.8071705350 + 0.5123837306j,
|
||
|
-1.8566005012],
|
||
|
10: [-0.8657569017 + 2.2926048310j, -1.3606922784 + 1.7335057427j,
|
||
|
-1.6618102414 + 1.2211002186j, -1.8421962445 + 0.7272575978j,
|
||
|
-1.9276196914 + 0.2416234710j]
|
||
|
}
|
||
|
|
||
|
for N in range(1, 11):
|
||
|
p1 = np.sort(bond_poles[N])
|
||
|
p2 = np.sort(np.concatenate(_cplxreal(besselap(N, 'mag')[1])))
|
||
|
assert_array_almost_equal(p1, p2, decimal=10)
|
||
|
|
||
|
# Compare to http://www.rane.com/note147.html
|
||
|
# "Table 1 - Bessel Crossovers of Second, Third, and Fourth-Order"
|
||
|
a = [1, 1, 1/3]
|
||
|
b2, a2 = bessel(2, 1, norm='delay', analog=True)
|
||
|
assert_allclose(a[::-1], a2/b2)
|
||
|
|
||
|
a = [1, 1, 2/5, 1/15]
|
||
|
b2, a2 = bessel(3, 1, norm='delay', analog=True)
|
||
|
assert_allclose(a[::-1], a2/b2)
|
||
|
|
||
|
a = [1, 1, 9/21, 2/21, 1/105]
|
||
|
b2, a2 = bessel(4, 1, norm='delay', analog=True)
|
||
|
assert_allclose(a[::-1], a2/b2)
|
||
|
|
||
|
a = [1, np.sqrt(3), 1]
|
||
|
b2, a2 = bessel(2, 1, norm='phase', analog=True)
|
||
|
assert_allclose(a[::-1], a2/b2)
|
||
|
|
||
|
# TODO: Why so inaccurate? Is reference flawed?
|
||
|
a = [1, 2.481, 2.463, 1.018]
|
||
|
b2, a2 = bessel(3, 1, norm='phase', analog=True)
|
||
|
assert_array_almost_equal(a[::-1], a2/b2, decimal=1)
|
||
|
|
||
|
# TODO: Why so inaccurate? Is reference flawed?
|
||
|
a = [1, 3.240, 4.5, 3.240, 1.050]
|
||
|
b2, a2 = bessel(4, 1, norm='phase', analog=True)
|
||
|
assert_array_almost_equal(a[::-1], a2/b2, decimal=1)
|
||
|
|
||
|
# Table of -3 dB factors:
|
||
|
N, scale = 2, 1.272
|
||
|
scale2 = besselap(N, 'mag')[1] / besselap(N, 'phase')[1]
|
||
|
assert_array_almost_equal(scale, scale2, decimal=3)
|
||
|
|
||
|
# TODO: Why so inaccurate? Is reference flawed?
|
||
|
N, scale = 3, 1.413
|
||
|
scale2 = besselap(N, 'mag')[1] / besselap(N, 'phase')[1]
|
||
|
assert_array_almost_equal(scale, scale2, decimal=2)
|
||
|
|
||
|
# TODO: Why so inaccurate? Is reference flawed?
|
||
|
N, scale = 4, 1.533
|
||
|
scale2 = besselap(N, 'mag')[1] / besselap(N, 'phase')[1]
|
||
|
assert_array_almost_equal(scale, scale2, decimal=1)
|
||
|
|
||
|
def test_hardcoded(self):
|
||
|
# Compare to values from original hardcoded implementation
|
||
|
originals = {
|
||
|
0: [],
|
||
|
1: [-1],
|
||
|
2: [-.8660254037844386467637229 + .4999999999999999999999996j],
|
||
|
3: [-.9416000265332067855971980,
|
||
|
-.7456403858480766441810907 + .7113666249728352680992154j],
|
||
|
4: [-.6572111716718829545787788 + .8301614350048733772399715j,
|
||
|
-.9047587967882449459642624 + .2709187330038746636700926j],
|
||
|
5: [-.9264420773877602247196260,
|
||
|
-.8515536193688395541722677 + .4427174639443327209850002j,
|
||
|
-.5905759446119191779319432 + .9072067564574549539291747j],
|
||
|
6: [-.9093906830472271808050953 + .1856964396793046769246397j,
|
||
|
-.7996541858328288520243325 + .5621717346937317988594118j,
|
||
|
-.5385526816693109683073792 + .9616876881954277199245657j],
|
||
|
7: [-.9194871556490290014311619,
|
||
|
-.8800029341523374639772340 + .3216652762307739398381830j,
|
||
|
-.7527355434093214462291616 + .6504696305522550699212995j,
|
||
|
-.4966917256672316755024763 + 1.002508508454420401230220j],
|
||
|
8: [-.9096831546652910216327629 + .1412437976671422927888150j,
|
||
|
-.8473250802359334320103023 + .4259017538272934994996429j,
|
||
|
-.7111381808485399250796172 + .7186517314108401705762571j,
|
||
|
-.4621740412532122027072175 + 1.034388681126901058116589j],
|
||
|
9: [-.9154957797499037686769223,
|
||
|
-.8911217017079759323183848 + .2526580934582164192308115j,
|
||
|
-.8148021112269012975514135 + .5085815689631499483745341j,
|
||
|
-.6743622686854761980403401 + .7730546212691183706919682j,
|
||
|
-.4331415561553618854685942 + 1.060073670135929666774323j],
|
||
|
10: [-.9091347320900502436826431 + .1139583137335511169927714j,
|
||
|
-.8688459641284764527921864 + .3430008233766309973110589j,
|
||
|
-.7837694413101441082655890 + .5759147538499947070009852j,
|
||
|
-.6417513866988316136190854 + .8175836167191017226233947j,
|
||
|
-.4083220732868861566219785 + 1.081274842819124562037210j],
|
||
|
11: [-.9129067244518981934637318,
|
||
|
-.8963656705721166099815744 + .2080480375071031919692341j,
|
||
|
-.8453044014712962954184557 + .4178696917801248292797448j,
|
||
|
-.7546938934722303128102142 + .6319150050721846494520941j,
|
||
|
-.6126871554915194054182909 + .8547813893314764631518509j,
|
||
|
-.3868149510055090879155425 + 1.099117466763120928733632j],
|
||
|
12: [-.9084478234140682638817772 + 95506365213450398415258360e-27j,
|
||
|
-.8802534342016826507901575 + .2871779503524226723615457j,
|
||
|
-.8217296939939077285792834 + .4810212115100676440620548j,
|
||
|
-.7276681615395159454547013 + .6792961178764694160048987j,
|
||
|
-.5866369321861477207528215 + .8863772751320727026622149j,
|
||
|
-.3679640085526312839425808 + 1.114373575641546257595657j],
|
||
|
13: [-.9110914665984182781070663,
|
||
|
-.8991314665475196220910718 + .1768342956161043620980863j,
|
||
|
-.8625094198260548711573628 + .3547413731172988997754038j,
|
||
|
-.7987460692470972510394686 + .5350752120696801938272504j,
|
||
|
-.7026234675721275653944062 + .7199611890171304131266374j,
|
||
|
-.5631559842430199266325818 + .9135900338325109684927731j,
|
||
|
-.3512792323389821669401925 + 1.127591548317705678613239j],
|
||
|
14: [-.9077932138396487614720659 + 82196399419401501888968130e-27j,
|
||
|
-.8869506674916445312089167 + .2470079178765333183201435j,
|
||
|
-.8441199160909851197897667 + .4131653825102692595237260j,
|
||
|
-.7766591387063623897344648 + .5819170677377608590492434j,
|
||
|
-.6794256425119233117869491 + .7552857305042033418417492j,
|
||
|
-.5418766775112297376541293 + .9373043683516919569183099j,
|
||
|
-.3363868224902037330610040 + 1.139172297839859991370924j],
|
||
|
15: [-.9097482363849064167228581,
|
||
|
-.9006981694176978324932918 + .1537681197278439351298882j,
|
||
|
-.8731264620834984978337843 + .3082352470564267657715883j,
|
||
|
-.8256631452587146506294553 + .4642348752734325631275134j,
|
||
|
-.7556027168970728127850416 + .6229396358758267198938604j,
|
||
|
-.6579196593110998676999362 + .7862895503722515897065645j,
|
||
|
-.5224954069658330616875186 + .9581787261092526478889345j,
|
||
|
-.3229963059766444287113517 + 1.149416154583629539665297j],
|
||
|
16: [-.9072099595087001356491337 + 72142113041117326028823950e-27j,
|
||
|
-.8911723070323647674780132 + .2167089659900576449410059j,
|
||
|
-.8584264231521330481755780 + .3621697271802065647661080j,
|
||
|
-.8074790293236003885306146 + .5092933751171800179676218j,
|
||
|
-.7356166304713115980927279 + .6591950877860393745845254j,
|
||
|
-.6379502514039066715773828 + .8137453537108761895522580j,
|
||
|
-.5047606444424766743309967 + .9767137477799090692947061j,
|
||
|
-.3108782755645387813283867 + 1.158552841199330479412225j],
|
||
|
17: [-.9087141161336397432860029,
|
||
|
-.9016273850787285964692844 + .1360267995173024591237303j,
|
||
|
-.8801100704438627158492165 + .2725347156478803885651973j,
|
||
|
-.8433414495836129204455491 + .4100759282910021624185986j,
|
||
|
-.7897644147799708220288138 + .5493724405281088674296232j,
|
||
|
-.7166893842372349049842743 + .6914936286393609433305754j,
|
||
|
-.6193710717342144521602448 + .8382497252826992979368621j,
|
||
|
-.4884629337672704194973683 + .9932971956316781632345466j,
|
||
|
-.2998489459990082015466971 + 1.166761272925668786676672j],
|
||
|
18: [-.9067004324162775554189031 + 64279241063930693839360680e-27j,
|
||
|
-.8939764278132455733032155 + .1930374640894758606940586j,
|
||
|
-.8681095503628830078317207 + .3224204925163257604931634j,
|
||
|
-.8281885016242836608829018 + .4529385697815916950149364j,
|
||
|
-.7726285030739558780127746 + .5852778162086640620016316j,
|
||
|
-.6987821445005273020051878 + .7204696509726630531663123j,
|
||
|
-.6020482668090644386627299 + .8602708961893664447167418j,
|
||
|
-.4734268069916151511140032 + 1.008234300314801077034158j,
|
||
|
-.2897592029880489845789953 + 1.174183010600059128532230j],
|
||
|
19: [-.9078934217899404528985092,
|
||
|
-.9021937639390660668922536 + .1219568381872026517578164j,
|
||
|
-.8849290585034385274001112 + .2442590757549818229026280j,
|
||
|
-.8555768765618421591093993 + .3672925896399872304734923j,
|
||
|
-.8131725551578197705476160 + .4915365035562459055630005j,
|
||
|
-.7561260971541629355231897 + .6176483917970178919174173j,
|
||
|
-.6818424412912442033411634 + .7466272357947761283262338j,
|
||
|
-.5858613321217832644813602 + .8801817131014566284786759j,
|
||
|
-.4595043449730988600785456 + 1.021768776912671221830298j,
|
||
|
-.2804866851439370027628724 + 1.180931628453291873626003j],
|
||
|
20: [-.9062570115576771146523497 + 57961780277849516990208850e-27j,
|
||
|
-.8959150941925768608568248 + .1740317175918705058595844j,
|
||
|
-.8749560316673332850673214 + .2905559296567908031706902j,
|
||
|
-.8427907479956670633544106 + .4078917326291934082132821j,
|
||
|
-.7984251191290606875799876 + .5264942388817132427317659j,
|
||
|
-.7402780309646768991232610 + .6469975237605228320268752j,
|
||
|
-.6658120544829934193890626 + .7703721701100763015154510j,
|
||
|
-.5707026806915714094398061 + .8982829066468255593407161j,
|
||
|
-.4465700698205149555701841 + 1.034097702560842962315411j,
|
||
|
-.2719299580251652601727704 + 1.187099379810885886139638j],
|
||
|
21: [-.9072262653142957028884077,
|
||
|
-.9025428073192696303995083 + .1105252572789856480992275j,
|
||
|
-.8883808106664449854431605 + .2213069215084350419975358j,
|
||
|
-.8643915813643204553970169 + .3326258512522187083009453j,
|
||
|
-.8299435470674444100273463 + .4448177739407956609694059j,
|
||
|
-.7840287980408341576100581 + .5583186348022854707564856j,
|
||
|
-.7250839687106612822281339 + .6737426063024382240549898j,
|
||
|
-.6506315378609463397807996 + .7920349342629491368548074j,
|
||
|
-.5564766488918562465935297 + .9148198405846724121600860j,
|
||
|
-.4345168906815271799687308 + 1.045382255856986531461592j,
|
||
|
-.2640041595834031147954813 + 1.192762031948052470183960j],
|
||
|
22: [-.9058702269930872551848625 + 52774908289999045189007100e-27j,
|
||
|
-.8972983138153530955952835 + .1584351912289865608659759j,
|
||
|
-.8799661455640176154025352 + .2644363039201535049656450j,
|
||
|
-.8534754036851687233084587 + .3710389319482319823405321j,
|
||
|
-.8171682088462720394344996 + .4785619492202780899653575j,
|
||
|
-.7700332930556816872932937 + .5874255426351153211965601j,
|
||
|
-.7105305456418785989070935 + .6982266265924524000098548j,
|
||
|
-.6362427683267827226840153 + .8118875040246347267248508j,
|
||
|
-.5430983056306302779658129 + .9299947824439872998916657j,
|
||
|
-.4232528745642628461715044 + 1.055755605227545931204656j,
|
||
|
-.2566376987939318038016012 + 1.197982433555213008346532j],
|
||
|
23: [-.9066732476324988168207439,
|
||
|
-.9027564979912504609412993 + .1010534335314045013252480j,
|
||
|
-.8909283242471251458653994 + .2023024699381223418195228j,
|
||
|
-.8709469395587416239596874 + .3039581993950041588888925j,
|
||
|
-.8423805948021127057054288 + .4062657948237602726779246j,
|
||
|
-.8045561642053176205623187 + .5095305912227258268309528j,
|
||
|
-.7564660146829880581478138 + .6141594859476032127216463j,
|
||
|
-.6965966033912705387505040 + .7207341374753046970247055j,
|
||
|
-.6225903228771341778273152 + .8301558302812980678845563j,
|
||
|
-.5304922463810191698502226 + .9439760364018300083750242j,
|
||
|
-.4126986617510148836149955 + 1.065328794475513585531053j,
|
||
|
-.2497697202208956030229911 + 1.202813187870697831365338j],
|
||
|
24: [-.9055312363372773709269407 + 48440066540478700874836350e-27j,
|
||
|
-.8983105104397872954053307 + .1454056133873610120105857j,
|
||
|
-.8837358034555706623131950 + .2426335234401383076544239j,
|
||
|
-.8615278304016353651120610 + .3403202112618624773397257j,
|
||
|
-.8312326466813240652679563 + .4386985933597305434577492j,
|
||
|
-.7921695462343492518845446 + .5380628490968016700338001j,
|
||
|
-.7433392285088529449175873 + .6388084216222567930378296j,
|
||
|
-.6832565803536521302816011 + .7415032695091650806797753j,
|
||
|
-.6096221567378335562589532 + .8470292433077202380020454j,
|
||
|
-.5185914574820317343536707 + .9569048385259054576937721j,
|
||
|
-.4027853855197518014786978 + 1.074195196518674765143729j,
|
||
|
-.2433481337524869675825448 + 1.207298683731972524975429j],
|
||
|
25: [-.9062073871811708652496104,
|
||
|
-.9028833390228020537142561 + 93077131185102967450643820e-27j,
|
||
|
-.8928551459883548836774529 + .1863068969804300712287138j,
|
||
|
-.8759497989677857803656239 + .2798521321771408719327250j,
|
||
|
-.8518616886554019782346493 + .3738977875907595009446142j,
|
||
|
-.8201226043936880253962552 + .4686668574656966589020580j,
|
||
|
-.7800496278186497225905443 + .5644441210349710332887354j,
|
||
|
-.7306549271849967721596735 + .6616149647357748681460822j,
|
||
|
-.6704827128029559528610523 + .7607348858167839877987008j,
|
||
|
-.5972898661335557242320528 + .8626676330388028512598538j,
|
||
|
-.5073362861078468845461362 + .9689006305344868494672405j,
|
||
|
-.3934529878191079606023847 + 1.082433927173831581956863j,
|
||
|
-.2373280669322028974199184 + 1.211476658382565356579418j],
|
||
|
}
|
||
|
for N in originals:
|
||
|
p1 = sorted(np.union1d(originals[N],
|
||
|
np.conj(originals[N])), key=np.imag)
|
||
|
p2 = sorted(besselap(N)[1], key=np.imag)
|
||
|
assert_allclose(p1, p2, rtol=1e-14)
|
||
|
|
||
|
def test_norm_phase(self):
|
||
|
# Test some orders and frequencies and see that they have the right
|
||
|
# phase at w0
|
||
|
for N in (1, 2, 3, 4, 5, 51, 72):
|
||
|
for w0 in (1, 100):
|
||
|
b, a = bessel(N, w0, analog=True, norm='phase')
|
||
|
w = np.linspace(0, w0, 100)
|
||
|
w, h = freqs(b, a, w)
|
||
|
phase = np.unwrap(np.angle(h))
|
||
|
assert_allclose(phase[[0, -1]], (0, -N*pi/4), rtol=1e-1)
|
||
|
|
||
|
def test_norm_mag(self):
|
||
|
# Test some orders and frequencies and see that they have the right
|
||
|
# mag at w0
|
||
|
for N in (1, 2, 3, 4, 5, 51, 72):
|
||
|
for w0 in (1, 100):
|
||
|
b, a = bessel(N, w0, analog=True, norm='mag')
|
||
|
w = (0, w0)
|
||
|
w, h = freqs(b, a, w)
|
||
|
mag = abs(h)
|
||
|
assert_allclose(mag, (1, 1/np.sqrt(2)))
|
||
|
|
||
|
def test_norm_delay(self):
|
||
|
# Test some orders and frequencies and see that they have the right
|
||
|
# delay at DC
|
||
|
for N in (1, 2, 3, 4, 5, 51, 72):
|
||
|
for w0 in (1, 100):
|
||
|
b, a = bessel(N, w0, analog=True, norm='delay')
|
||
|
w = np.linspace(0, 10*w0, 1000)
|
||
|
w, h = freqs(b, a, w)
|
||
|
delay = -np.diff(np.unwrap(np.angle(h)))/np.diff(w)
|
||
|
assert_allclose(delay[0], 1/w0, rtol=1e-4)
|
||
|
|
||
|
def test_norm_factor(self):
|
||
|
mpmath_values = {
|
||
|
1: 1, 2: 1.361654128716130520, 3: 1.755672368681210649,
|
||
|
4: 2.113917674904215843, 5: 2.427410702152628137,
|
||
|
6: 2.703395061202921876, 7: 2.951722147038722771,
|
||
|
8: 3.179617237510651330, 9: 3.391693138911660101,
|
||
|
10: 3.590980594569163482, 11: 3.779607416439620092,
|
||
|
12: 3.959150821144285315, 13: 4.130825499383535980,
|
||
|
14: 4.295593409533637564, 15: 4.454233021624377494,
|
||
|
16: 4.607385465472647917, 17: 4.755586548961147727,
|
||
|
18: 4.899289677284488007, 19: 5.038882681488207605,
|
||
|
20: 5.174700441742707423, 21: 5.307034531360917274,
|
||
|
22: 5.436140703250035999, 23: 5.562244783787878196,
|
||
|
24: 5.685547371295963521, 25: 5.806227623775418541,
|
||
|
50: 8.268963160013226298, 51: 8.352374541546012058,
|
||
|
}
|
||
|
for N in mpmath_values:
|
||
|
z, p, k = besselap(N, 'delay')
|
||
|
assert_allclose(mpmath_values[N], _norm_factor(p, k), rtol=1e-13)
|
||
|
|
||
|
def test_bessel_poly(self):
|
||
|
assert_array_equal(_bessel_poly(5), [945, 945, 420, 105, 15, 1])
|
||
|
assert_array_equal(_bessel_poly(4, True), [1, 10, 45, 105, 105])
|
||
|
|
||
|
def test_bessel_zeros(self):
|
||
|
assert_array_equal(_bessel_zeros(0), [])
|
||
|
|
||
|
def test_invalid(self):
|
||
|
assert_raises(ValueError, besselap, 5, 'nonsense')
|
||
|
assert_raises(ValueError, besselap, -5)
|
||
|
assert_raises(ValueError, besselap, 3.2)
|
||
|
assert_raises(ValueError, _bessel_poly, -3)
|
||
|
assert_raises(ValueError, _bessel_poly, 3.3)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
for norm in ('phase', 'mag', 'delay'):
|
||
|
for fs in (900, 900.1, 1234.567):
|
||
|
for N in (0, 1, 2, 3, 10):
|
||
|
for fc in (100, 100.1, 432.12345):
|
||
|
for btype in ('lp', 'hp'):
|
||
|
ba1 = bessel(N, fc, btype, fs=fs)
|
||
|
ba2 = bessel(N, fc/(fs/2), btype)
|
||
|
assert_allclose(ba1, ba2)
|
||
|
for fc in ((100, 200), (100.1, 200.2), (321.123, 432.123)):
|
||
|
for btype in ('bp', 'bs'):
|
||
|
ba1 = bessel(N, fc, btype, fs=fs)
|
||
|
for seq in (list, tuple, array):
|
||
|
fcnorm = seq([f/(fs/2) for f in fc])
|
||
|
ba2 = bessel(N, fcnorm, btype)
|
||
|
assert_allclose(ba1, ba2)
|
||
|
|
||
|
|
||
|
class TestButter(object):
|
||
|
|
||
|
def test_degenerate(self):
|
||
|
# 0-order filter is just a passthrough
|
||
|
b, a = butter(0, 1, analog=True)
|
||
|
assert_array_equal(b, [1])
|
||
|
assert_array_equal(a, [1])
|
||
|
|
||
|
# 1-order filter is same for all types
|
||
|
b, a = butter(1, 1, analog=True)
|
||
|
assert_array_almost_equal(b, [1])
|
||
|
assert_array_almost_equal(a, [1, 1])
|
||
|
|
||
|
z, p, k = butter(1, 0.3, output='zpk')
|
||
|
assert_array_equal(z, [-1])
|
||
|
assert_allclose(p, [3.249196962329063e-01], rtol=1e-14)
|
||
|
assert_allclose(k, 3.375401518835469e-01, rtol=1e-14)
|
||
|
|
||
|
def test_basic(self):
|
||
|
# analog s-plane
|
||
|
for N in range(25):
|
||
|
wn = 0.01
|
||
|
z, p, k = butter(N, wn, 'low', analog=True, output='zpk')
|
||
|
assert_array_almost_equal([], z)
|
||
|
assert_(len(p) == N)
|
||
|
# All poles should be at distance wn from origin
|
||
|
assert_array_almost_equal(wn, abs(p))
|
||
|
assert_(all(np.real(p) <= 0)) # No poles in right half of S-plane
|
||
|
assert_array_almost_equal(wn**N, k)
|
||
|
|
||
|
# digital z-plane
|
||
|
for N in range(25):
|
||
|
wn = 0.01
|
||
|
z, p, k = butter(N, wn, 'high', analog=False, output='zpk')
|
||
|
assert_array_equal(np.ones(N), z) # All zeros exactly at DC
|
||
|
assert_(all(np.abs(p) <= 1)) # No poles outside unit circle
|
||
|
|
||
|
b1, a1 = butter(2, 1, analog=True)
|
||
|
assert_array_almost_equal(b1, [1])
|
||
|
assert_array_almost_equal(a1, [1, np.sqrt(2), 1])
|
||
|
|
||
|
b2, a2 = butter(5, 1, analog=True)
|
||
|
assert_array_almost_equal(b2, [1])
|
||
|
assert_array_almost_equal(a2, [1, 3.2361, 5.2361,
|
||
|
5.2361, 3.2361, 1], decimal=4)
|
||
|
|
||
|
b3, a3 = butter(10, 1, analog=True)
|
||
|
assert_array_almost_equal(b3, [1])
|
||
|
assert_array_almost_equal(a3, [1, 6.3925, 20.4317, 42.8021, 64.8824,
|
||
|
74.2334, 64.8824, 42.8021, 20.4317,
|
||
|
6.3925, 1], decimal=4)
|
||
|
|
||
|
b2, a2 = butter(19, 1.0441379169150726, analog=True)
|
||
|
assert_array_almost_equal(b2, [2.2720], decimal=4)
|
||
|
assert_array_almost_equal(a2, 1.0e+004 * np.array([
|
||
|
0.0001, 0.0013, 0.0080, 0.0335, 0.1045, 0.2570,
|
||
|
0.5164, 0.8669, 1.2338, 1.5010, 1.5672, 1.4044,
|
||
|
1.0759, 0.6986, 0.3791, 0.1681, 0.0588, 0.0153,
|
||
|
0.0026, 0.0002]), decimal=0)
|
||
|
|
||
|
b, a = butter(5, 0.4)
|
||
|
assert_array_almost_equal(b, [0.0219, 0.1097, 0.2194,
|
||
|
0.2194, 0.1097, 0.0219], decimal=4)
|
||
|
assert_array_almost_equal(a, [1.0000, -0.9853, 0.9738,
|
||
|
-0.3864, 0.1112, -0.0113], decimal=4)
|
||
|
|
||
|
def test_highpass(self):
|
||
|
# highpass, high even order
|
||
|
z, p, k = butter(28, 0.43, 'high', output='zpk')
|
||
|
z2 = np.ones(28)
|
||
|
p2 = [
|
||
|
2.068257195514592e-01 + 9.238294351481734e-01j,
|
||
|
2.068257195514592e-01 - 9.238294351481734e-01j,
|
||
|
1.874933103892023e-01 + 8.269455076775277e-01j,
|
||
|
1.874933103892023e-01 - 8.269455076775277e-01j,
|
||
|
1.717435567330153e-01 + 7.383078571194629e-01j,
|
||
|
1.717435567330153e-01 - 7.383078571194629e-01j,
|
||
|
1.588266870755982e-01 + 6.564623730651094e-01j,
|
||
|
1.588266870755982e-01 - 6.564623730651094e-01j,
|
||
|
1.481881532502603e-01 + 5.802343458081779e-01j,
|
||
|
1.481881532502603e-01 - 5.802343458081779e-01j,
|
||
|
1.394122576319697e-01 + 5.086609000582009e-01j,
|
||
|
1.394122576319697e-01 - 5.086609000582009e-01j,
|
||
|
1.321840881809715e-01 + 4.409411734716436e-01j,
|
||
|
1.321840881809715e-01 - 4.409411734716436e-01j,
|
||
|
1.262633413354405e-01 + 3.763990035551881e-01j,
|
||
|
1.262633413354405e-01 - 3.763990035551881e-01j,
|
||
|
1.214660449478046e-01 + 3.144545234797277e-01j,
|
||
|
1.214660449478046e-01 - 3.144545234797277e-01j,
|
||
|
1.104868766650320e-01 + 2.771505404367791e-02j,
|
||
|
1.104868766650320e-01 - 2.771505404367791e-02j,
|
||
|
1.111768629525075e-01 + 8.331369153155753e-02j,
|
||
|
1.111768629525075e-01 - 8.331369153155753e-02j,
|
||
|
1.125740630842972e-01 + 1.394219509611784e-01j,
|
||
|
1.125740630842972e-01 - 1.394219509611784e-01j,
|
||
|
1.147138487992747e-01 + 1.963932363793666e-01j,
|
||
|
1.147138487992747e-01 - 1.963932363793666e-01j,
|
||
|
1.176516491045901e-01 + 2.546021573417188e-01j,
|
||
|
1.176516491045901e-01 - 2.546021573417188e-01j,
|
||
|
]
|
||
|
k2 = 1.446671081817286e-06
|
||
|
assert_array_equal(z, z2)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(p2, key=np.imag), rtol=1e-7)
|
||
|
assert_allclose(k, k2, rtol=1e-10)
|
||
|
|
||
|
# highpass, high odd order
|
||
|
z, p, k = butter(27, 0.56, 'high', output='zpk')
|
||
|
z2 = np.ones(27)
|
||
|
p2 = [
|
||
|
-1.772572785680147e-01 + 9.276431102995948e-01j,
|
||
|
-1.772572785680147e-01 - 9.276431102995948e-01j,
|
||
|
-1.600766565322114e-01 + 8.264026279893268e-01j,
|
||
|
-1.600766565322114e-01 - 8.264026279893268e-01j,
|
||
|
-1.461948419016121e-01 + 7.341841939120078e-01j,
|
||
|
-1.461948419016121e-01 - 7.341841939120078e-01j,
|
||
|
-1.348975284762046e-01 + 6.493235066053785e-01j,
|
||
|
-1.348975284762046e-01 - 6.493235066053785e-01j,
|
||
|
-1.256628210712206e-01 + 5.704921366889227e-01j,
|
||
|
-1.256628210712206e-01 - 5.704921366889227e-01j,
|
||
|
-1.181038235962314e-01 + 4.966120551231630e-01j,
|
||
|
-1.181038235962314e-01 - 4.966120551231630e-01j,
|
||
|
-1.119304913239356e-01 + 4.267938916403775e-01j,
|
||
|
-1.119304913239356e-01 - 4.267938916403775e-01j,
|
||
|
-1.069237739782691e-01 + 3.602914879527338e-01j,
|
||
|
-1.069237739782691e-01 - 3.602914879527338e-01j,
|
||
|
-1.029178030691416e-01 + 2.964677964142126e-01j,
|
||
|
-1.029178030691416e-01 - 2.964677964142126e-01j,
|
||
|
-9.978747500816100e-02 + 2.347687643085738e-01j,
|
||
|
-9.978747500816100e-02 - 2.347687643085738e-01j,
|
||
|
-9.743974496324025e-02 + 1.747028739092479e-01j,
|
||
|
-9.743974496324025e-02 - 1.747028739092479e-01j,
|
||
|
-9.580754551625957e-02 + 1.158246860771989e-01j,
|
||
|
-9.580754551625957e-02 - 1.158246860771989e-01j,
|
||
|
-9.484562207782568e-02 + 5.772118357151691e-02j,
|
||
|
-9.484562207782568e-02 - 5.772118357151691e-02j,
|
||
|
-9.452783117928215e-02
|
||
|
]
|
||
|
k2 = 9.585686688851069e-09
|
||
|
assert_array_equal(z, z2)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(p2, key=np.imag), rtol=1e-8)
|
||
|
assert_allclose(k, k2)
|
||
|
|
||
|
def test_bandpass(self):
|
||
|
z, p, k = butter(8, [0.25, 0.33], 'band', output='zpk')
|
||
|
z2 = [1, 1, 1, 1, 1, 1, 1, 1,
|
||
|
-1, -1, -1, -1, -1, -1, -1, -1]
|
||
|
p2 = [
|
||
|
4.979909925436156e-01 + 8.367609424799387e-01j,
|
||
|
4.979909925436156e-01 - 8.367609424799387e-01j,
|
||
|
4.913338722555539e-01 + 7.866774509868817e-01j,
|
||
|
4.913338722555539e-01 - 7.866774509868817e-01j,
|
||
|
5.035229361778706e-01 + 7.401147376726750e-01j,
|
||
|
5.035229361778706e-01 - 7.401147376726750e-01j,
|
||
|
5.307617160406101e-01 + 7.029184459442954e-01j,
|
||
|
5.307617160406101e-01 - 7.029184459442954e-01j,
|
||
|
5.680556159453138e-01 + 6.788228792952775e-01j,
|
||
|
5.680556159453138e-01 - 6.788228792952775e-01j,
|
||
|
6.100962560818854e-01 + 6.693849403338664e-01j,
|
||
|
6.100962560818854e-01 - 6.693849403338664e-01j,
|
||
|
6.904694312740631e-01 + 6.930501690145245e-01j,
|
||
|
6.904694312740631e-01 - 6.930501690145245e-01j,
|
||
|
6.521767004237027e-01 + 6.744414640183752e-01j,
|
||
|
6.521767004237027e-01 - 6.744414640183752e-01j,
|
||
|
]
|
||
|
k2 = 3.398854055800844e-08
|
||
|
assert_array_equal(z, z2)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(p2, key=np.imag), rtol=1e-13)
|
||
|
assert_allclose(k, k2, rtol=1e-13)
|
||
|
|
||
|
# bandpass analog
|
||
|
z, p, k = butter(4, [90.5, 110.5], 'bp', analog=True, output='zpk')
|
||
|
z2 = np.zeros(4)
|
||
|
p2 = [
|
||
|
-4.179137760733086e+00 + 1.095935899082837e+02j,
|
||
|
-4.179137760733086e+00 - 1.095935899082837e+02j,
|
||
|
-9.593598668443835e+00 + 1.034745398029734e+02j,
|
||
|
-9.593598668443835e+00 - 1.034745398029734e+02j,
|
||
|
-8.883991981781929e+00 + 9.582087115567160e+01j,
|
||
|
-8.883991981781929e+00 - 9.582087115567160e+01j,
|
||
|
-3.474530886568715e+00 + 9.111599925805801e+01j,
|
||
|
-3.474530886568715e+00 - 9.111599925805801e+01j,
|
||
|
]
|
||
|
k2 = 1.600000000000001e+05
|
||
|
assert_array_equal(z, z2)
|
||
|
assert_allclose(sorted(p, key=np.imag), sorted(p2, key=np.imag))
|
||
|
assert_allclose(k, k2, rtol=1e-15)
|
||
|
|
||
|
def test_bandstop(self):
|
||
|
z, p, k = butter(7, [0.45, 0.56], 'stop', output='zpk')
|
||
|
z2 = [-1.594474531383421e-02 + 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 - 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 + 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 - 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 + 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 - 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 + 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 - 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 + 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 - 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 + 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 - 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 + 9.998728744679880e-01j,
|
||
|
-1.594474531383421e-02 - 9.998728744679880e-01j]
|
||
|
p2 = [-1.766850742887729e-01 + 9.466951258673900e-01j,
|
||
|
-1.766850742887729e-01 - 9.466951258673900e-01j,
|
||
|
1.467897662432886e-01 + 9.515917126462422e-01j,
|
||
|
1.467897662432886e-01 - 9.515917126462422e-01j,
|
||
|
-1.370083529426906e-01 + 8.880376681273993e-01j,
|
||
|
-1.370083529426906e-01 - 8.880376681273993e-01j,
|
||
|
1.086774544701390e-01 + 8.915240810704319e-01j,
|
||
|
1.086774544701390e-01 - 8.915240810704319e-01j,
|
||
|
-7.982704457700891e-02 + 8.506056315273435e-01j,
|
||
|
-7.982704457700891e-02 - 8.506056315273435e-01j,
|
||
|
5.238812787110331e-02 + 8.524011102699969e-01j,
|
||
|
5.238812787110331e-02 - 8.524011102699969e-01j,
|
||
|
-1.357545000491310e-02 + 8.382287744986582e-01j,
|
||
|
-1.357545000491310e-02 - 8.382287744986582e-01j]
|
||
|
k2 = 4.577122512960063e-01
|
||
|
assert_allclose(sorted(z, key=np.imag), sorted(z2, key=np.imag))
|
||
|
assert_allclose(sorted(p, key=np.imag), sorted(p2, key=np.imag))
|
||
|
assert_allclose(k, k2, rtol=1e-14)
|
||
|
|
||
|
def test_ba_output(self):
|
||
|
b, a = butter(4, [100, 300], 'bandpass', analog=True)
|
||
|
b2 = [1.6e+09, 0, 0, 0, 0]
|
||
|
a2 = [1.000000000000000e+00, 5.226251859505511e+02,
|
||
|
2.565685424949238e+05, 6.794127417357160e+07,
|
||
|
1.519411254969542e+10, 2.038238225207147e+12,
|
||
|
2.309116882454312e+14, 1.411088002066486e+16,
|
||
|
8.099999999999991e+17]
|
||
|
assert_allclose(b, b2, rtol=1e-14)
|
||
|
assert_allclose(a, a2, rtol=1e-14)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
for fs in (900, 900.1, 1234.567):
|
||
|
for N in (0, 1, 2, 3, 10):
|
||
|
for fc in (100, 100.1, 432.12345):
|
||
|
for btype in ('lp', 'hp'):
|
||
|
ba1 = butter(N, fc, btype, fs=fs)
|
||
|
ba2 = butter(N, fc/(fs/2), btype)
|
||
|
assert_allclose(ba1, ba2)
|
||
|
for fc in ((100, 200), (100.1, 200.2), (321.123, 432.123)):
|
||
|
for btype in ('bp', 'bs'):
|
||
|
ba1 = butter(N, fc, btype, fs=fs)
|
||
|
for seq in (list, tuple, array):
|
||
|
fcnorm = seq([f/(fs/2) for f in fc])
|
||
|
ba2 = butter(N, fcnorm, btype)
|
||
|
assert_allclose(ba1, ba2)
|
||
|
|
||
|
|
||
|
class TestCheby1(object):
|
||
|
|
||
|
def test_degenerate(self):
|
||
|
# 0-order filter is just a passthrough
|
||
|
# Even-order filters have DC gain of -rp dB
|
||
|
b, a = cheby1(0, 10*np.log10(2), 1, analog=True)
|
||
|
assert_array_almost_equal(b, [1/np.sqrt(2)])
|
||
|
assert_array_equal(a, [1])
|
||
|
|
||
|
# 1-order filter is same for all types
|
||
|
b, a = cheby1(1, 10*np.log10(2), 1, analog=True)
|
||
|
assert_array_almost_equal(b, [1])
|
||
|
assert_array_almost_equal(a, [1, 1])
|
||
|
|
||
|
z, p, k = cheby1(1, 0.1, 0.3, output='zpk')
|
||
|
assert_array_equal(z, [-1])
|
||
|
assert_allclose(p, [-5.390126972799615e-01], rtol=1e-14)
|
||
|
assert_allclose(k, 7.695063486399808e-01, rtol=1e-14)
|
||
|
|
||
|
def test_basic(self):
|
||
|
for N in range(25):
|
||
|
wn = 0.01
|
||
|
z, p, k = cheby1(N, 1, wn, 'low', analog=True, output='zpk')
|
||
|
assert_array_almost_equal([], z)
|
||
|
assert_(len(p) == N)
|
||
|
assert_(all(np.real(p) <= 0)) # No poles in right half of S-plane
|
||
|
|
||
|
for N in range(25):
|
||
|
wn = 0.01
|
||
|
z, p, k = cheby1(N, 1, wn, 'high', analog=False, output='zpk')
|
||
|
assert_array_equal(np.ones(N), z) # All zeros exactly at DC
|
||
|
assert_(all(np.abs(p) <= 1)) # No poles outside unit circle
|
||
|
|
||
|
# Same test as TestNormalize
|
||
|
b, a = cheby1(8, 0.5, 0.048)
|
||
|
assert_array_almost_equal(b, [
|
||
|
2.150733144728282e-11, 1.720586515782626e-10,
|
||
|
6.022052805239190e-10, 1.204410561047838e-09,
|
||
|
1.505513201309798e-09, 1.204410561047838e-09,
|
||
|
6.022052805239190e-10, 1.720586515782626e-10,
|
||
|
2.150733144728282e-11], decimal=14)
|
||
|
assert_array_almost_equal(a, [
|
||
|
1.000000000000000e+00, -7.782402035027959e+00,
|
||
|
2.654354569747454e+01, -5.182182531666387e+01,
|
||
|
6.334127355102684e+01, -4.963358186631157e+01,
|
||
|
2.434862182949389e+01, -6.836925348604676e+00,
|
||
|
8.412934944449140e-01], decimal=14)
|
||
|
|
||
|
b, a = cheby1(4, 1, [0.4, 0.7], btype='band')
|
||
|
assert_array_almost_equal(b, [0.0084, 0, -0.0335, 0, 0.0502, 0,
|
||
|
-0.0335, 0, 0.0084], decimal=4)
|
||
|
assert_array_almost_equal(a, [1.0, 1.1191, 2.862, 2.2986, 3.4137,
|
||
|
1.8653, 1.8982, 0.5676, 0.4103],
|
||
|
decimal=4)
|
||
|
|
||
|
b2, a2 = cheby1(5, 3, 1, analog=True)
|
||
|
assert_array_almost_equal(b2, [0.0626], decimal=4)
|
||
|
assert_array_almost_equal(a2, [1, 0.5745, 1.4150, 0.5489, 0.4080,
|
||
|
0.0626], decimal=4)
|
||
|
|
||
|
b, a = cheby1(8, 0.5, 0.1)
|
||
|
assert_array_almost_equal(b, 1.0e-006 * np.array([
|
||
|
0.00703924326028, 0.05631394608227, 0.19709881128793,
|
||
|
0.39419762257586, 0.49274702821983, 0.39419762257586,
|
||
|
0.19709881128793, 0.05631394608227, 0.00703924326028]),
|
||
|
decimal=13)
|
||
|
assert_array_almost_equal(a, [
|
||
|
1.00000000000000, -7.44912258934158, 24.46749067762108,
|
||
|
-46.27560200466141, 55.11160187999928, -42.31640010161038,
|
||
|
20.45543300484147, -5.69110270561444, 0.69770374759022],
|
||
|
decimal=13)
|
||
|
|
||
|
b, a = cheby1(8, 0.5, 0.25)
|
||
|
assert_array_almost_equal(b, 1.0e-003 * np.array([
|
||
|
0.00895261138923, 0.07162089111382, 0.25067311889837,
|
||
|
0.50134623779673, 0.62668279724591, 0.50134623779673,
|
||
|
0.25067311889837, 0.07162089111382, 0.00895261138923]),
|
||
|
decimal=13)
|
||
|
assert_array_almost_equal(a, [1.00000000000000, -5.97529229188545,
|
||
|
16.58122329202101, -27.71423273542923,
|
||
|
30.39509758355313, -22.34729670426879,
|
||
|
10.74509800434910, -3.08924633697497,
|
||
|
0.40707685889802], decimal=13)
|
||
|
|
||
|
def test_highpass(self):
|
||
|
# high even order
|
||
|
z, p, k = cheby1(24, 0.7, 0.2, 'high', output='zpk')
|
||
|
z2 = np.ones(24)
|
||
|
p2 = [-6.136558509657073e-01 + 2.700091504942893e-01j,
|
||
|
-6.136558509657073e-01 - 2.700091504942893e-01j,
|
||
|
-3.303348340927516e-01 + 6.659400861114254e-01j,
|
||
|
-3.303348340927516e-01 - 6.659400861114254e-01j,
|
||
|
8.779713780557169e-03 + 8.223108447483040e-01j,
|
||
|
8.779713780557169e-03 - 8.223108447483040e-01j,
|
||
|
2.742361123006911e-01 + 8.356666951611864e-01j,
|
||
|
2.742361123006911e-01 - 8.356666951611864e-01j,
|
||
|
4.562984557158206e-01 + 7.954276912303594e-01j,
|
||
|
4.562984557158206e-01 - 7.954276912303594e-01j,
|
||
|
5.777335494123628e-01 + 7.435821817961783e-01j,
|
||
|
5.777335494123628e-01 - 7.435821817961783e-01j,
|
||
|
6.593260977749194e-01 + 6.955390907990932e-01j,
|
||
|
6.593260977749194e-01 - 6.955390907990932e-01j,
|
||
|
7.149590948466562e-01 + 6.559437858502012e-01j,
|
||
|
7.149590948466562e-01 - 6.559437858502012e-01j,
|
||
|
7.532432388188739e-01 + 6.256158042292060e-01j,
|
||
|
7.532432388188739e-01 - 6.256158042292060e-01j,
|
||
|
7.794365244268271e-01 + 6.042099234813333e-01j,
|
||
|
7.794365244268271e-01 - 6.042099234813333e-01j,
|
||
|
7.967253874772997e-01 + 5.911966597313203e-01j,
|
||
|
7.967253874772997e-01 - 5.911966597313203e-01j,
|
||
|
8.069756417293870e-01 + 5.862214589217275e-01j,
|
||
|
8.069756417293870e-01 - 5.862214589217275e-01j]
|
||
|
k2 = 6.190427617192018e-04
|
||
|
assert_array_equal(z, z2)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(p2, key=np.imag), rtol=1e-10)
|
||
|
assert_allclose(k, k2, rtol=1e-10)
|
||
|
|
||
|
# high odd order
|
||
|
z, p, k = cheby1(23, 0.8, 0.3, 'high', output='zpk')
|
||
|
z2 = np.ones(23)
|
||
|
p2 = [-7.676400532011010e-01,
|
||
|
-6.754621070166477e-01 + 3.970502605619561e-01j,
|
||
|
-6.754621070166477e-01 - 3.970502605619561e-01j,
|
||
|
-4.528880018446727e-01 + 6.844061483786332e-01j,
|
||
|
-4.528880018446727e-01 - 6.844061483786332e-01j,
|
||
|
-1.986009130216447e-01 + 8.382285942941594e-01j,
|
||
|
-1.986009130216447e-01 - 8.382285942941594e-01j,
|
||
|
2.504673931532608e-02 + 8.958137635794080e-01j,
|
||
|
2.504673931532608e-02 - 8.958137635794080e-01j,
|
||
|
2.001089429976469e-01 + 9.010678290791480e-01j,
|
||
|
2.001089429976469e-01 - 9.010678290791480e-01j,
|
||
|
3.302410157191755e-01 + 8.835444665962544e-01j,
|
||
|
3.302410157191755e-01 - 8.835444665962544e-01j,
|
||
|
4.246662537333661e-01 + 8.594054226449009e-01j,
|
||
|
4.246662537333661e-01 - 8.594054226449009e-01j,
|
||
|
4.919620928120296e-01 + 8.366772762965786e-01j,
|
||
|
4.919620928120296e-01 - 8.366772762965786e-01j,
|
||
|
5.385746917494749e-01 + 8.191616180796720e-01j,
|
||
|
5.385746917494749e-01 - 8.191616180796720e-01j,
|
||
|
5.855636993537203e-01 + 8.060680937701062e-01j,
|
||
|
5.855636993537203e-01 - 8.060680937701062e-01j,
|
||
|
5.688812849391721e-01 + 8.086497795114683e-01j,
|
||
|
5.688812849391721e-01 - 8.086497795114683e-01j]
|
||
|
k2 = 1.941697029206324e-05
|
||
|
assert_array_equal(z, z2)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(p2, key=np.imag), rtol=1e-10)
|
||
|
assert_allclose(k, k2, rtol=1e-10)
|
||
|
|
||
|
z, p, k = cheby1(10, 1, 1000, 'high', analog=True, output='zpk')
|
||
|
z2 = np.zeros(10)
|
||
|
p2 = [-3.144743169501551e+03 + 3.511680029092744e+03j,
|
||
|
-3.144743169501551e+03 - 3.511680029092744e+03j,
|
||
|
-5.633065604514602e+02 + 2.023615191183945e+03j,
|
||
|
-5.633065604514602e+02 - 2.023615191183945e+03j,
|
||
|
-1.946412183352025e+02 + 1.372309454274755e+03j,
|
||
|
-1.946412183352025e+02 - 1.372309454274755e+03j,
|
||
|
-7.987162953085479e+01 + 1.105207708045358e+03j,
|
||
|
-7.987162953085479e+01 - 1.105207708045358e+03j,
|
||
|
-2.250315039031946e+01 + 1.001723931471477e+03j,
|
||
|
-2.250315039031946e+01 - 1.001723931471477e+03j]
|
||
|
k2 = 8.912509381337453e-01
|
||
|
assert_array_equal(z, z2)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(p2, key=np.imag), rtol=1e-13)
|
||
|
assert_allclose(k, k2, rtol=1e-15)
|
||
|
|
||
|
def test_bandpass(self):
|
||
|
z, p, k = cheby1(8, 1, [0.3, 0.4], 'bp', output='zpk')
|
||
|
z2 = [1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1]
|
||
|
p2 = [3.077784854851463e-01 + 9.453307017592942e-01j,
|
||
|
3.077784854851463e-01 - 9.453307017592942e-01j,
|
||
|
3.280567400654425e-01 + 9.272377218689016e-01j,
|
||
|
3.280567400654425e-01 - 9.272377218689016e-01j,
|
||
|
3.677912763284301e-01 + 9.038008865279966e-01j,
|
||
|
3.677912763284301e-01 - 9.038008865279966e-01j,
|
||
|
4.194425632520948e-01 + 8.769407159656157e-01j,
|
||
|
4.194425632520948e-01 - 8.769407159656157e-01j,
|
||
|
4.740921994669189e-01 + 8.496508528630974e-01j,
|
||
|
4.740921994669189e-01 - 8.496508528630974e-01j,
|
||
|
5.234866481897429e-01 + 8.259608422808477e-01j,
|
||
|
5.234866481897429e-01 - 8.259608422808477e-01j,
|
||
|
5.844717632289875e-01 + 8.052901363500210e-01j,
|
||
|
5.844717632289875e-01 - 8.052901363500210e-01j,
|
||
|
5.615189063336070e-01 + 8.100667803850766e-01j,
|
||
|
5.615189063336070e-01 - 8.100667803850766e-01j]
|
||
|
k2 = 5.007028718074307e-09
|
||
|
assert_array_equal(z, z2)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(p2, key=np.imag), rtol=1e-13)
|
||
|
assert_allclose(k, k2, rtol=1e-13)
|
||
|
|
||
|
def test_bandstop(self):
|
||
|
z, p, k = cheby1(7, 1, [0.5, 0.6], 'stop', output='zpk')
|
||
|
z2 = [-1.583844403245361e-01 + 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 - 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 + 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 - 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 + 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 - 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 + 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 - 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 + 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 - 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 + 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 - 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 + 9.873775210440450e-01j,
|
||
|
-1.583844403245361e-01 - 9.873775210440450e-01j]
|
||
|
p2 = [-8.942974551472813e-02 + 3.482480481185926e-01j,
|
||
|
-8.942974551472813e-02 - 3.482480481185926e-01j,
|
||
|
1.293775154041798e-01 + 8.753499858081858e-01j,
|
||
|
1.293775154041798e-01 - 8.753499858081858e-01j,
|
||
|
3.399741945062013e-02 + 9.690316022705607e-01j,
|
||
|
3.399741945062013e-02 - 9.690316022705607e-01j,
|
||
|
4.167225522796539e-04 + 9.927338161087488e-01j,
|
||
|
4.167225522796539e-04 - 9.927338161087488e-01j,
|
||
|
-3.912966549550960e-01 + 8.046122859255742e-01j,
|
||
|
-3.912966549550960e-01 - 8.046122859255742e-01j,
|
||
|
-3.307805547127368e-01 + 9.133455018206508e-01j,
|
||
|
-3.307805547127368e-01 - 9.133455018206508e-01j,
|
||
|
-3.072658345097743e-01 + 9.443589759799366e-01j,
|
||
|
-3.072658345097743e-01 - 9.443589759799366e-01j]
|
||
|
k2 = 3.619438310405028e-01
|
||
|
assert_allclose(sorted(z, key=np.imag),
|
||
|
sorted(z2, key=np.imag), rtol=1e-13)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(p2, key=np.imag), rtol=1e-13)
|
||
|
assert_allclose(k, k2, rtol=1e-15)
|
||
|
|
||
|
def test_ba_output(self):
|
||
|
# with transfer function conversion, without digital conversion
|
||
|
b, a = cheby1(5, 0.9, [210, 310], 'stop', analog=True)
|
||
|
b2 = [1.000000000000006e+00, 0,
|
||
|
3.255000000000020e+05, 0,
|
||
|
4.238010000000026e+10, 0,
|
||
|
2.758944510000017e+15, 0,
|
||
|
8.980364380050052e+19, 0,
|
||
|
1.169243442282517e+24
|
||
|
]
|
||
|
a2 = [1.000000000000000e+00, 4.630555945694342e+02,
|
||
|
4.039266454794788e+05, 1.338060988610237e+08,
|
||
|
5.844333551294591e+10, 1.357346371637638e+13,
|
||
|
3.804661141892782e+15, 5.670715850340080e+17,
|
||
|
1.114411200988328e+20, 8.316815934908471e+21,
|
||
|
1.169243442282517e+24
|
||
|
]
|
||
|
assert_allclose(b, b2, rtol=1e-14)
|
||
|
assert_allclose(a, a2, rtol=1e-14)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
for fs in (900, 900.1, 1234.567):
|
||
|
for N in (0, 1, 2, 3, 10):
|
||
|
for fc in (100, 100.1, 432.12345):
|
||
|
for btype in ('lp', 'hp'):
|
||
|
ba1 = cheby1(N, 1, fc, btype, fs=fs)
|
||
|
ba2 = cheby1(N, 1, fc/(fs/2), btype)
|
||
|
assert_allclose(ba1, ba2)
|
||
|
for fc in ((100, 200), (100.1, 200.2), (321.123, 432.123)):
|
||
|
for btype in ('bp', 'bs'):
|
||
|
ba1 = cheby1(N, 1, fc, btype, fs=fs)
|
||
|
for seq in (list, tuple, array):
|
||
|
fcnorm = seq([f/(fs/2) for f in fc])
|
||
|
ba2 = cheby1(N, 1, fcnorm, btype)
|
||
|
assert_allclose(ba1, ba2)
|
||
|
|
||
|
|
||
|
class TestCheby2(object):
|
||
|
|
||
|
def test_degenerate(self):
|
||
|
# 0-order filter is just a passthrough
|
||
|
# Stopband ripple factor doesn't matter
|
||
|
b, a = cheby2(0, 123.456, 1, analog=True)
|
||
|
assert_array_equal(b, [1])
|
||
|
assert_array_equal(a, [1])
|
||
|
|
||
|
# 1-order filter is same for all types
|
||
|
b, a = cheby2(1, 10*np.log10(2), 1, analog=True)
|
||
|
assert_array_almost_equal(b, [1])
|
||
|
assert_array_almost_equal(a, [1, 1])
|
||
|
|
||
|
z, p, k = cheby2(1, 50, 0.3, output='zpk')
|
||
|
assert_array_equal(z, [-1])
|
||
|
assert_allclose(p, [9.967826460175649e-01], rtol=1e-14)
|
||
|
assert_allclose(k, 1.608676991217512e-03, rtol=1e-14)
|
||
|
|
||
|
def test_basic(self):
|
||
|
for N in range(25):
|
||
|
wn = 0.01
|
||
|
z, p, k = cheby2(N, 40, wn, 'low', analog=True, output='zpk')
|
||
|
assert_(len(p) == N)
|
||
|
assert_(all(np.real(p) <= 0)) # No poles in right half of S-plane
|
||
|
|
||
|
for N in range(25):
|
||
|
wn = 0.01
|
||
|
z, p, k = cheby2(N, 40, wn, 'high', analog=False, output='zpk')
|
||
|
assert_(all(np.abs(p) <= 1)) # No poles outside unit circle
|
||
|
|
||
|
B, A = cheby2(18, 100, 0.5)
|
||
|
assert_array_almost_equal(B, [
|
||
|
0.00167583914216, 0.01249479541868, 0.05282702120282,
|
||
|
0.15939804265706, 0.37690207631117, 0.73227013789108,
|
||
|
1.20191856962356, 1.69522872823393, 2.07598674519837,
|
||
|
2.21972389625291, 2.07598674519838, 1.69522872823395,
|
||
|
1.20191856962359, 0.73227013789110, 0.37690207631118,
|
||
|
0.15939804265707, 0.05282702120282, 0.01249479541868,
|
||
|
0.00167583914216], decimal=13)
|
||
|
assert_array_almost_equal(A, [
|
||
|
1.00000000000000, -0.27631970006174, 3.19751214254060,
|
||
|
-0.15685969461355, 4.13926117356269, 0.60689917820044,
|
||
|
2.95082770636540, 0.89016501910416, 1.32135245849798,
|
||
|
0.51502467236824, 0.38906643866660, 0.15367372690642,
|
||
|
0.07255803834919, 0.02422454070134, 0.00756108751837,
|
||
|
0.00179848550988, 0.00033713574499, 0.00004258794833,
|
||
|
0.00000281030149], decimal=13)
|
||
|
|
||
|
def test_highpass(self):
|
||
|
# high even order
|
||
|
z, p, k = cheby2(26, 60, 0.3, 'high', output='zpk')
|
||
|
z2 = [9.981088955489852e-01 + 6.147058341984388e-02j,
|
||
|
9.981088955489852e-01 - 6.147058341984388e-02j,
|
||
|
9.832702870387426e-01 + 1.821525257215483e-01j,
|
||
|
9.832702870387426e-01 - 1.821525257215483e-01j,
|
||
|
9.550760158089112e-01 + 2.963609353922882e-01j,
|
||
|
9.550760158089112e-01 - 2.963609353922882e-01j,
|
||
|
9.162054748821922e-01 + 4.007087817803773e-01j,
|
||
|
9.162054748821922e-01 - 4.007087817803773e-01j,
|
||
|
8.700619897368064e-01 + 4.929423232136168e-01j,
|
||
|
8.700619897368064e-01 - 4.929423232136168e-01j,
|
||
|
5.889791753434985e-01 + 8.081482110427953e-01j,
|
||
|
5.889791753434985e-01 - 8.081482110427953e-01j,
|
||
|
5.984900456570295e-01 + 8.011302423760501e-01j,
|
||
|
5.984900456570295e-01 - 8.011302423760501e-01j,
|
||
|
6.172880888914629e-01 + 7.867371958365343e-01j,
|
||
|
6.172880888914629e-01 - 7.867371958365343e-01j,
|
||
|
6.448899971038180e-01 + 7.642754030030161e-01j,
|
||
|
6.448899971038180e-01 - 7.642754030030161e-01j,
|
||
|
6.804845629637927e-01 + 7.327624168637228e-01j,
|
||
|
6.804845629637927e-01 - 7.327624168637228e-01j,
|
||
|
8.202619107108660e-01 + 5.719881098737678e-01j,
|
||
|
8.202619107108660e-01 - 5.719881098737678e-01j,
|
||
|
7.228410452536148e-01 + 6.910143437705678e-01j,
|
||
|
7.228410452536148e-01 - 6.910143437705678e-01j,
|
||
|
7.702121399578629e-01 + 6.377877856007792e-01j,
|
||
|
7.702121399578629e-01 - 6.377877856007792e-01j]
|
||
|
p2 = [7.365546198286450e-01 + 4.842085129329526e-02j,
|
||
|
7.365546198286450e-01 - 4.842085129329526e-02j,
|
||
|
7.292038510962885e-01 + 1.442201672097581e-01j,
|
||
|
7.292038510962885e-01 - 1.442201672097581e-01j,
|
||
|
7.151293788040354e-01 + 2.369925800458584e-01j,
|
||
|
7.151293788040354e-01 - 2.369925800458584e-01j,
|
||
|
6.955051820787286e-01 + 3.250341363856910e-01j,
|
||
|
6.955051820787286e-01 - 3.250341363856910e-01j,
|
||
|
6.719122956045220e-01 + 4.070475750638047e-01j,
|
||
|
6.719122956045220e-01 - 4.070475750638047e-01j,
|
||
|
6.461722130611300e-01 + 4.821965916689270e-01j,
|
||
|
6.461722130611300e-01 - 4.821965916689270e-01j,
|
||
|
5.528045062872224e-01 + 8.162920513838372e-01j,
|
||
|
5.528045062872224e-01 - 8.162920513838372e-01j,
|
||
|
5.464847782492791e-01 + 7.869899955967304e-01j,
|
||
|
5.464847782492791e-01 - 7.869899955967304e-01j,
|
||
|
5.488033111260949e-01 + 7.520442354055579e-01j,
|
||
|
5.488033111260949e-01 - 7.520442354055579e-01j,
|
||
|
6.201874719022955e-01 + 5.500894392527353e-01j,
|
||
|
6.201874719022955e-01 - 5.500894392527353e-01j,
|
||
|
5.586478152536709e-01 + 7.112676877332921e-01j,
|
||
|
5.586478152536709e-01 - 7.112676877332921e-01j,
|
||
|
5.958145844148228e-01 + 6.107074340842115e-01j,
|
||
|
5.958145844148228e-01 - 6.107074340842115e-01j,
|
||
|
5.747812938519067e-01 + 6.643001536914696e-01j,
|
||
|
5.747812938519067e-01 - 6.643001536914696e-01j]
|
||
|
k2 = 9.932997786497189e-02
|
||
|
assert_allclose(sorted(z, key=np.angle),
|
||
|
sorted(z2, key=np.angle), rtol=1e-13)
|
||
|
assert_allclose(sorted(p, key=np.angle),
|
||
|
sorted(p2, key=np.angle), rtol=1e-12)
|
||
|
assert_allclose(k, k2, rtol=1e-11)
|
||
|
|
||
|
# high odd order
|
||
|
z, p, k = cheby2(25, 80, 0.5, 'high', output='zpk')
|
||
|
z2 = [9.690690376586687e-01 + 2.467897896011971e-01j,
|
||
|
9.690690376586687e-01 - 2.467897896011971e-01j,
|
||
|
9.999999999999492e-01,
|
||
|
8.835111277191199e-01 + 4.684101698261429e-01j,
|
||
|
8.835111277191199e-01 - 4.684101698261429e-01j,
|
||
|
7.613142857900539e-01 + 6.483830335935022e-01j,
|
||
|
7.613142857900539e-01 - 6.483830335935022e-01j,
|
||
|
6.232625173626231e-01 + 7.820126817709752e-01j,
|
||
|
6.232625173626231e-01 - 7.820126817709752e-01j,
|
||
|
4.864456563413621e-01 + 8.737108351316745e-01j,
|
||
|
4.864456563413621e-01 - 8.737108351316745e-01j,
|
||
|
3.618368136816749e-01 + 9.322414495530347e-01j,
|
||
|
3.618368136816749e-01 - 9.322414495530347e-01j,
|
||
|
2.549486883466794e-01 + 9.669545833752675e-01j,
|
||
|
2.549486883466794e-01 - 9.669545833752675e-01j,
|
||
|
1.676175432109457e-01 + 9.858520980390212e-01j,
|
||
|
1.676175432109457e-01 - 9.858520980390212e-01j,
|
||
|
1.975218468277521e-03 + 9.999980492540941e-01j,
|
||
|
1.975218468277521e-03 - 9.999980492540941e-01j,
|
||
|
1.786959496651858e-02 + 9.998403260399917e-01j,
|
||
|
1.786959496651858e-02 - 9.998403260399917e-01j,
|
||
|
9.967933660557139e-02 + 9.950196127985684e-01j,
|
||
|
9.967933660557139e-02 - 9.950196127985684e-01j,
|
||
|
5.013970951219547e-02 + 9.987422137518890e-01j,
|
||
|
5.013970951219547e-02 - 9.987422137518890e-01j]
|
||
|
p2 = [4.218866331906864e-01,
|
||
|
4.120110200127552e-01 + 1.361290593621978e-01j,
|
||
|
4.120110200127552e-01 - 1.361290593621978e-01j,
|
||
|
3.835890113632530e-01 + 2.664910809911026e-01j,
|
||
|
3.835890113632530e-01 - 2.664910809911026e-01j,
|
||
|
3.399195570456499e-01 + 3.863983538639875e-01j,
|
||
|
3.399195570456499e-01 - 3.863983538639875e-01j,
|
||
|
2.855977834508353e-01 + 4.929444399540688e-01j,
|
||
|
2.855977834508353e-01 - 4.929444399540688e-01j,
|
||
|
2.255765441339322e-01 + 5.851631870205766e-01j,
|
||
|
2.255765441339322e-01 - 5.851631870205766e-01j,
|
||
|
1.644087535815792e-01 + 6.637356937277153e-01j,
|
||
|
1.644087535815792e-01 - 6.637356937277153e-01j,
|
||
|
-7.293633845273095e-02 + 9.739218252516307e-01j,
|
||
|
-7.293633845273095e-02 - 9.739218252516307e-01j,
|
||
|
1.058259206358626e-01 + 7.304739464862978e-01j,
|
||
|
1.058259206358626e-01 - 7.304739464862978e-01j,
|
||
|
-5.703971947785402e-02 + 9.291057542169088e-01j,
|
||
|
-5.703971947785402e-02 - 9.291057542169088e-01j,
|
||
|
5.263875132656864e-02 + 7.877974334424453e-01j,
|
||
|
5.263875132656864e-02 - 7.877974334424453e-01j,
|
||
|
-3.007943405982616e-02 + 8.846331716180016e-01j,
|
||
|
-3.007943405982616e-02 - 8.846331716180016e-01j,
|
||
|
6.857277464483946e-03 + 8.383275456264492e-01j,
|
||
|
6.857277464483946e-03 - 8.383275456264492e-01j]
|
||
|
k2 = 6.507068761705037e-03
|
||
|
assert_allclose(sorted(z, key=np.angle),
|
||
|
sorted(z2, key=np.angle), rtol=1e-13)
|
||
|
assert_allclose(sorted(p, key=np.angle),
|
||
|
sorted(p2, key=np.angle), rtol=1e-12)
|
||
|
assert_allclose(k, k2, rtol=1e-11)
|
||
|
|
||
|
def test_bandpass(self):
|
||
|
z, p, k = cheby2(9, 40, [0.07, 0.2], 'pass', output='zpk')
|
||
|
z2 = [-9.999999999999999e-01,
|
||
|
3.676588029658514e-01 + 9.299607543341383e-01j,
|
||
|
3.676588029658514e-01 - 9.299607543341383e-01j,
|
||
|
7.009689684982283e-01 + 7.131917730894889e-01j,
|
||
|
7.009689684982283e-01 - 7.131917730894889e-01j,
|
||
|
7.815697973765858e-01 + 6.238178033919218e-01j,
|
||
|
7.815697973765858e-01 - 6.238178033919218e-01j,
|
||
|
8.063793628819866e-01 + 5.913986160941200e-01j,
|
||
|
8.063793628819866e-01 - 5.913986160941200e-01j,
|
||
|
1.000000000000001e+00,
|
||
|
9.944493019920448e-01 + 1.052168511576739e-01j,
|
||
|
9.944493019920448e-01 - 1.052168511576739e-01j,
|
||
|
9.854674703367308e-01 + 1.698642543566085e-01j,
|
||
|
9.854674703367308e-01 - 1.698642543566085e-01j,
|
||
|
9.762751735919308e-01 + 2.165335665157851e-01j,
|
||
|
9.762751735919308e-01 - 2.165335665157851e-01j,
|
||
|
9.792277171575134e-01 + 2.027636011479496e-01j,
|
||
|
9.792277171575134e-01 - 2.027636011479496e-01j]
|
||
|
p2 = [8.143803410489621e-01 + 5.411056063397541e-01j,
|
||
|
8.143803410489621e-01 - 5.411056063397541e-01j,
|
||
|
7.650769827887418e-01 + 5.195412242095543e-01j,
|
||
|
7.650769827887418e-01 - 5.195412242095543e-01j,
|
||
|
6.096241204063443e-01 + 3.568440484659796e-01j,
|
||
|
6.096241204063443e-01 - 3.568440484659796e-01j,
|
||
|
6.918192770246239e-01 + 4.770463577106911e-01j,
|
||
|
6.918192770246239e-01 - 4.770463577106911e-01j,
|
||
|
6.986241085779207e-01 + 1.146512226180060e-01j,
|
||
|
6.986241085779207e-01 - 1.146512226180060e-01j,
|
||
|
8.654645923909734e-01 + 1.604208797063147e-01j,
|
||
|
8.654645923909734e-01 - 1.604208797063147e-01j,
|
||
|
9.164831670444591e-01 + 1.969181049384918e-01j,
|
||
|
9.164831670444591e-01 - 1.969181049384918e-01j,
|
||
|
9.630425777594550e-01 + 2.317513360702271e-01j,
|
||
|
9.630425777594550e-01 - 2.317513360702271e-01j,
|
||
|
9.438104703725529e-01 + 2.193509900269860e-01j,
|
||
|
9.438104703725529e-01 - 2.193509900269860e-01j]
|
||
|
k2 = 9.345352824659604e-03
|
||
|
assert_allclose(sorted(z, key=np.angle),
|
||
|
sorted(z2, key=np.angle), rtol=1e-13)
|
||
|
assert_allclose(sorted(p, key=np.angle),
|
||
|
sorted(p2, key=np.angle), rtol=1e-13)
|
||
|
assert_allclose(k, k2, rtol=1e-11)
|
||
|
|
||
|
def test_bandstop(self):
|
||
|
z, p, k = cheby2(6, 55, [0.1, 0.9], 'stop', output='zpk')
|
||
|
z2 = [6.230544895101009e-01 + 7.821784343111114e-01j,
|
||
|
6.230544895101009e-01 - 7.821784343111114e-01j,
|
||
|
9.086608545660115e-01 + 4.175349702471991e-01j,
|
||
|
9.086608545660115e-01 - 4.175349702471991e-01j,
|
||
|
9.478129721465802e-01 + 3.188268649763867e-01j,
|
||
|
9.478129721465802e-01 - 3.188268649763867e-01j,
|
||
|
-6.230544895100982e-01 + 7.821784343111109e-01j,
|
||
|
-6.230544895100982e-01 - 7.821784343111109e-01j,
|
||
|
-9.086608545660116e-01 + 4.175349702472088e-01j,
|
||
|
-9.086608545660116e-01 - 4.175349702472088e-01j,
|
||
|
-9.478129721465784e-01 + 3.188268649763897e-01j,
|
||
|
-9.478129721465784e-01 - 3.188268649763897e-01j]
|
||
|
p2 = [-9.464094036167638e-01 + 1.720048695084344e-01j,
|
||
|
-9.464094036167638e-01 - 1.720048695084344e-01j,
|
||
|
-8.715844103386737e-01 + 1.370665039509297e-01j,
|
||
|
-8.715844103386737e-01 - 1.370665039509297e-01j,
|
||
|
-8.078751204586425e-01 + 5.729329866682983e-02j,
|
||
|
-8.078751204586425e-01 - 5.729329866682983e-02j,
|
||
|
9.464094036167665e-01 + 1.720048695084332e-01j,
|
||
|
9.464094036167665e-01 - 1.720048695084332e-01j,
|
||
|
8.078751204586447e-01 + 5.729329866683007e-02j,
|
||
|
8.078751204586447e-01 - 5.729329866683007e-02j,
|
||
|
8.715844103386721e-01 + 1.370665039509331e-01j,
|
||
|
8.715844103386721e-01 - 1.370665039509331e-01j]
|
||
|
k2 = 2.917823332763358e-03
|
||
|
assert_allclose(sorted(z, key=np.angle),
|
||
|
sorted(z2, key=np.angle), rtol=1e-13)
|
||
|
assert_allclose(sorted(p, key=np.angle),
|
||
|
sorted(p2, key=np.angle), rtol=1e-13)
|
||
|
assert_allclose(k, k2, rtol=1e-11)
|
||
|
|
||
|
def test_ba_output(self):
|
||
|
# with transfer function conversion, without digital conversion
|
||
|
b, a = cheby2(5, 20, [2010, 2100], 'stop', True)
|
||
|
b2 = [1.000000000000000e+00, 0, # Matlab: 6.683253076978249e-12,
|
||
|
2.111512500000000e+07, 0, # Matlab: 1.134325604589552e-04,
|
||
|
1.782966433781250e+14, 0, # Matlab: 7.216787944356781e+02,
|
||
|
7.525901316990656e+20, 0, # Matlab: 2.039829265789886e+09,
|
||
|
1.587960565565748e+27, 0, # Matlab: 2.161236218626134e+15,
|
||
|
1.339913493808585e+33]
|
||
|
a2 = [1.000000000000000e+00, 1.849550755473371e+02,
|
||
|
2.113222918998538e+07, 3.125114149732283e+09,
|
||
|
1.785133457155609e+14, 1.979158697776348e+16,
|
||
|
7.535048322653831e+20, 5.567966191263037e+22,
|
||
|
1.589246884221346e+27, 5.871210648525566e+28,
|
||
|
1.339913493808590e+33]
|
||
|
assert_allclose(b, b2, rtol=1e-14)
|
||
|
assert_allclose(a, a2, rtol=1e-14)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
for fs in (900, 900.1, 1234.567):
|
||
|
for N in (0, 1, 2, 3, 10):
|
||
|
for fc in (100, 100.1, 432.12345):
|
||
|
for btype in ('lp', 'hp'):
|
||
|
ba1 = cheby2(N, 20, fc, btype, fs=fs)
|
||
|
ba2 = cheby2(N, 20, fc/(fs/2), btype)
|
||
|
assert_allclose(ba1, ba2)
|
||
|
for fc in ((100, 200), (100.1, 200.2), (321.123, 432.123)):
|
||
|
for btype in ('bp', 'bs'):
|
||
|
ba1 = cheby2(N, 20, fc, btype, fs=fs)
|
||
|
for seq in (list, tuple, array):
|
||
|
fcnorm = seq([f/(fs/2) for f in fc])
|
||
|
ba2 = cheby2(N, 20, fcnorm, btype)
|
||
|
assert_allclose(ba1, ba2)
|
||
|
|
||
|
class TestEllip(object):
|
||
|
|
||
|
def test_degenerate(self):
|
||
|
# 0-order filter is just a passthrough
|
||
|
# Even-order filters have DC gain of -rp dB
|
||
|
# Stopband ripple factor doesn't matter
|
||
|
b, a = ellip(0, 10*np.log10(2), 123.456, 1, analog=True)
|
||
|
assert_array_almost_equal(b, [1/np.sqrt(2)])
|
||
|
assert_array_equal(a, [1])
|
||
|
|
||
|
# 1-order filter is same for all types
|
||
|
b, a = ellip(1, 10*np.log10(2), 1, 1, analog=True)
|
||
|
assert_array_almost_equal(b, [1])
|
||
|
assert_array_almost_equal(a, [1, 1])
|
||
|
|
||
|
z, p, k = ellip(1, 1, 55, 0.3, output='zpk')
|
||
|
assert_allclose(z, [-9.999999999999998e-01], rtol=1e-14)
|
||
|
assert_allclose(p, [-6.660721153525525e-04], rtol=1e-10)
|
||
|
assert_allclose(k, 5.003330360576763e-01, rtol=1e-14)
|
||
|
|
||
|
def test_basic(self):
|
||
|
for N in range(25):
|
||
|
wn = 0.01
|
||
|
z, p, k = ellip(N, 1, 40, wn, 'low', analog=True, output='zpk')
|
||
|
assert_(len(p) == N)
|
||
|
assert_(all(np.real(p) <= 0)) # No poles in right half of S-plane
|
||
|
|
||
|
for N in range(25):
|
||
|
wn = 0.01
|
||
|
z, p, k = ellip(N, 1, 40, wn, 'high', analog=False, output='zpk')
|
||
|
assert_(all(np.abs(p) <= 1)) # No poles outside unit circle
|
||
|
|
||
|
b3, a3 = ellip(5, 3, 26, 1, analog=True)
|
||
|
assert_array_almost_equal(b3, [0.1420, 0, 0.3764, 0,
|
||
|
0.2409], decimal=4)
|
||
|
assert_array_almost_equal(a3, [1, 0.5686, 1.8061, 0.8017, 0.8012,
|
||
|
0.2409], decimal=4)
|
||
|
|
||
|
b, a = ellip(3, 1, 60, [0.4, 0.7], 'stop')
|
||
|
assert_array_almost_equal(b, [0.3310, 0.3469, 1.1042, 0.7044, 1.1042,
|
||
|
0.3469, 0.3310], decimal=4)
|
||
|
assert_array_almost_equal(a, [1.0000, 0.6973, 1.1441, 0.5878, 0.7323,
|
||
|
0.1131, -0.0060], decimal=4)
|
||
|
|
||
|
def test_highpass(self):
|
||
|
# high even order
|
||
|
z, p, k = ellip(24, 1, 80, 0.3, 'high', output='zpk')
|
||
|
z2 = [9.761875332501075e-01 + 2.169283290099910e-01j,
|
||
|
9.761875332501075e-01 - 2.169283290099910e-01j,
|
||
|
8.413503353963494e-01 + 5.404901600661900e-01j,
|
||
|
8.413503353963494e-01 - 5.404901600661900e-01j,
|
||
|
7.160082576305009e-01 + 6.980918098681732e-01j,
|
||
|
7.160082576305009e-01 - 6.980918098681732e-01j,
|
||
|
6.456533638965329e-01 + 7.636306264739803e-01j,
|
||
|
6.456533638965329e-01 - 7.636306264739803e-01j,
|
||
|
6.127321820971366e-01 + 7.902906256703928e-01j,
|
||
|
6.127321820971366e-01 - 7.902906256703928e-01j,
|
||
|
5.983607817490196e-01 + 8.012267936512676e-01j,
|
||
|
5.983607817490196e-01 - 8.012267936512676e-01j,
|
||
|
5.922577552594799e-01 + 8.057485658286990e-01j,
|
||
|
5.922577552594799e-01 - 8.057485658286990e-01j,
|
||
|
5.896952092563588e-01 + 8.076258788449631e-01j,
|
||
|
5.896952092563588e-01 - 8.076258788449631e-01j,
|
||
|
5.886248765538837e-01 + 8.084063054565607e-01j,
|
||
|
5.886248765538837e-01 - 8.084063054565607e-01j,
|
||
|
5.881802711123132e-01 + 8.087298490066037e-01j,
|
||
|
5.881802711123132e-01 - 8.087298490066037e-01j,
|
||
|
5.879995719101164e-01 + 8.088612386766461e-01j,
|
||
|
5.879995719101164e-01 - 8.088612386766461e-01j,
|
||
|
5.879354086709576e-01 + 8.089078780868164e-01j,
|
||
|
5.879354086709576e-01 - 8.089078780868164e-01j]
|
||
|
p2 = [-3.184805259081650e-01 + 4.206951906775851e-01j,
|
||
|
-3.184805259081650e-01 - 4.206951906775851e-01j,
|
||
|
1.417279173459985e-01 + 7.903955262836452e-01j,
|
||
|
1.417279173459985e-01 - 7.903955262836452e-01j,
|
||
|
4.042881216964651e-01 + 8.309042239116594e-01j,
|
||
|
4.042881216964651e-01 - 8.309042239116594e-01j,
|
||
|
5.128964442789670e-01 + 8.229563236799665e-01j,
|
||
|
5.128964442789670e-01 - 8.229563236799665e-01j,
|
||
|
5.569614712822724e-01 + 8.155957702908510e-01j,
|
||
|
5.569614712822724e-01 - 8.155957702908510e-01j,
|
||
|
5.750478870161392e-01 + 8.118633973883931e-01j,
|
||
|
5.750478870161392e-01 - 8.118633973883931e-01j,
|
||
|
5.825314018170804e-01 + 8.101960910679270e-01j,
|
||
|
5.825314018170804e-01 - 8.101960910679270e-01j,
|
||
|
5.856397379751872e-01 + 8.094825218722543e-01j,
|
||
|
5.856397379751872e-01 - 8.094825218722543e-01j,
|
||
|
5.869326035251949e-01 + 8.091827531557583e-01j,
|
||
|
5.869326035251949e-01 - 8.091827531557583e-01j,
|
||
|
5.874697218855733e-01 + 8.090593298213502e-01j,
|
||
|
5.874697218855733e-01 - 8.090593298213502e-01j,
|
||
|
5.876904783532237e-01 + 8.090127161018823e-01j,
|
||
|
5.876904783532237e-01 - 8.090127161018823e-01j,
|
||
|
5.877753105317594e-01 + 8.090050577978136e-01j,
|
||
|
5.877753105317594e-01 - 8.090050577978136e-01j]
|
||
|
k2 = 4.918081266957108e-02
|
||
|
assert_allclose(sorted(z, key=np.angle),
|
||
|
sorted(z2, key=np.angle), rtol=1e-4)
|
||
|
assert_allclose(sorted(p, key=np.angle),
|
||
|
sorted(p2, key=np.angle), rtol=1e-4)
|
||
|
assert_allclose(k, k2, rtol=1e-3)
|
||
|
|
||
|
# high odd order
|
||
|
z, p, k = ellip(23, 1, 70, 0.5, 'high', output='zpk')
|
||
|
z2 = [9.999999999998661e-01,
|
||
|
6.603717261750994e-01 + 7.509388678638675e-01j,
|
||
|
6.603717261750994e-01 - 7.509388678638675e-01j,
|
||
|
2.788635267510325e-01 + 9.603307416968041e-01j,
|
||
|
2.788635267510325e-01 - 9.603307416968041e-01j,
|
||
|
1.070215532544218e-01 + 9.942567008268131e-01j,
|
||
|
1.070215532544218e-01 - 9.942567008268131e-01j,
|
||
|
4.049427369978163e-02 + 9.991797705105507e-01j,
|
||
|
4.049427369978163e-02 - 9.991797705105507e-01j,
|
||
|
1.531059368627931e-02 + 9.998827859909265e-01j,
|
||
|
1.531059368627931e-02 - 9.998827859909265e-01j,
|
||
|
5.808061438534933e-03 + 9.999831330689181e-01j,
|
||
|
5.808061438534933e-03 - 9.999831330689181e-01j,
|
||
|
2.224277847754599e-03 + 9.999975262909676e-01j,
|
||
|
2.224277847754599e-03 - 9.999975262909676e-01j,
|
||
|
8.731857107534554e-04 + 9.999996187732845e-01j,
|
||
|
8.731857107534554e-04 - 9.999996187732845e-01j,
|
||
|
3.649057346914968e-04 + 9.999999334218996e-01j,
|
||
|
3.649057346914968e-04 - 9.999999334218996e-01j,
|
||
|
1.765538109802615e-04 + 9.999999844143768e-01j,
|
||
|
1.765538109802615e-04 - 9.999999844143768e-01j,
|
||
|
1.143655290967426e-04 + 9.999999934602630e-01j,
|
||
|
1.143655290967426e-04 - 9.999999934602630e-01j]
|
||
|
p2 = [-6.322017026545028e-01,
|
||
|
-4.648423756662754e-01 + 5.852407464440732e-01j,
|
||
|
-4.648423756662754e-01 - 5.852407464440732e-01j,
|
||
|
-2.249233374627773e-01 + 8.577853017985717e-01j,
|
||
|
-2.249233374627773e-01 - 8.577853017985717e-01j,
|
||
|
-9.234137570557621e-02 + 9.506548198678851e-01j,
|
||
|
-9.234137570557621e-02 - 9.506548198678851e-01j,
|
||
|
-3.585663561241373e-02 + 9.821494736043981e-01j,
|
||
|
-3.585663561241373e-02 - 9.821494736043981e-01j,
|
||
|
-1.363917242312723e-02 + 9.933844128330656e-01j,
|
||
|
-1.363917242312723e-02 - 9.933844128330656e-01j,
|
||
|
-5.131505238923029e-03 + 9.975221173308673e-01j,
|
||
|
-5.131505238923029e-03 - 9.975221173308673e-01j,
|
||
|
-1.904937999259502e-03 + 9.990680819857982e-01j,
|
||
|
-1.904937999259502e-03 - 9.990680819857982e-01j,
|
||
|
-6.859439885466834e-04 + 9.996492201426826e-01j,
|
||
|
-6.859439885466834e-04 - 9.996492201426826e-01j,
|
||
|
-2.269936267937089e-04 + 9.998686250679161e-01j,
|
||
|
-2.269936267937089e-04 - 9.998686250679161e-01j,
|
||
|
-5.687071588789117e-05 + 9.999527573294513e-01j,
|
||
|
-5.687071588789117e-05 - 9.999527573294513e-01j,
|
||
|
-6.948417068525226e-07 + 9.999882737700173e-01j,
|
||
|
-6.948417068525226e-07 - 9.999882737700173e-01j]
|
||
|
k2 = 1.220910020289434e-02
|
||
|
assert_allclose(sorted(z, key=np.angle),
|
||
|
sorted(z2, key=np.angle), rtol=1e-4)
|
||
|
assert_allclose(sorted(p, key=np.angle),
|
||
|
sorted(p2, key=np.angle), rtol=1e-4)
|
||
|
assert_allclose(k, k2, rtol=1e-3)
|
||
|
|
||
|
def test_bandpass(self):
|
||
|
z, p, k = ellip(7, 1, 40, [0.07, 0.2], 'pass', output='zpk')
|
||
|
z2 = [-9.999999999999991e-01,
|
||
|
6.856610961780020e-01 + 7.279209168501619e-01j,
|
||
|
6.856610961780020e-01 - 7.279209168501619e-01j,
|
||
|
7.850346167691289e-01 + 6.194518952058737e-01j,
|
||
|
7.850346167691289e-01 - 6.194518952058737e-01j,
|
||
|
7.999038743173071e-01 + 6.001281461922627e-01j,
|
||
|
7.999038743173071e-01 - 6.001281461922627e-01j,
|
||
|
9.999999999999999e-01,
|
||
|
9.862938983554124e-01 + 1.649980183725925e-01j,
|
||
|
9.862938983554124e-01 - 1.649980183725925e-01j,
|
||
|
9.788558330548762e-01 + 2.045513580850601e-01j,
|
||
|
9.788558330548762e-01 - 2.045513580850601e-01j,
|
||
|
9.771155231720003e-01 + 2.127093189691258e-01j,
|
||
|
9.771155231720003e-01 - 2.127093189691258e-01j]
|
||
|
p2 = [8.063992755498643e-01 + 5.858071374778874e-01j,
|
||
|
8.063992755498643e-01 - 5.858071374778874e-01j,
|
||
|
8.050395347071724e-01 + 5.639097428109795e-01j,
|
||
|
8.050395347071724e-01 - 5.639097428109795e-01j,
|
||
|
8.113124936559144e-01 + 4.855241143973142e-01j,
|
||
|
8.113124936559144e-01 - 4.855241143973142e-01j,
|
||
|
8.665595314082394e-01 + 3.334049560919331e-01j,
|
||
|
8.665595314082394e-01 - 3.334049560919331e-01j,
|
||
|
9.412369011968871e-01 + 2.457616651325908e-01j,
|
||
|
9.412369011968871e-01 - 2.457616651325908e-01j,
|
||
|
9.679465190411238e-01 + 2.228772501848216e-01j,
|
||
|
9.679465190411238e-01 - 2.228772501848216e-01j,
|
||
|
9.747235066273385e-01 + 2.178937926146544e-01j,
|
||
|
9.747235066273385e-01 - 2.178937926146544e-01j]
|
||
|
k2 = 8.354782670263239e-03
|
||
|
assert_allclose(sorted(z, key=np.angle),
|
||
|
sorted(z2, key=np.angle), rtol=1e-4)
|
||
|
assert_allclose(sorted(p, key=np.angle),
|
||
|
sorted(p2, key=np.angle), rtol=1e-4)
|
||
|
assert_allclose(k, k2, rtol=1e-3)
|
||
|
|
||
|
z, p, k = ellip(5, 1, 75, [90.5, 110.5], 'pass', True, 'zpk')
|
||
|
z2 = [-5.583607317695175e-14 + 1.433755965989225e+02j,
|
||
|
-5.583607317695175e-14 - 1.433755965989225e+02j,
|
||
|
5.740106416459296e-14 + 1.261678754570291e+02j,
|
||
|
5.740106416459296e-14 - 1.261678754570291e+02j,
|
||
|
-2.199676239638652e-14 + 6.974861996895196e+01j,
|
||
|
-2.199676239638652e-14 - 6.974861996895196e+01j,
|
||
|
-3.372595657044283e-14 + 7.926145989044531e+01j,
|
||
|
-3.372595657044283e-14 - 7.926145989044531e+01j,
|
||
|
0]
|
||
|
p2 = [-8.814960004852743e-01 + 1.104124501436066e+02j,
|
||
|
-8.814960004852743e-01 - 1.104124501436066e+02j,
|
||
|
-2.477372459140184e+00 + 1.065638954516534e+02j,
|
||
|
-2.477372459140184e+00 - 1.065638954516534e+02j,
|
||
|
-3.072156842945799e+00 + 9.995404870405324e+01j,
|
||
|
-3.072156842945799e+00 - 9.995404870405324e+01j,
|
||
|
-2.180456023925693e+00 + 9.379206865455268e+01j,
|
||
|
-2.180456023925693e+00 - 9.379206865455268e+01j,
|
||
|
-7.230484977485752e-01 + 9.056598800801140e+01j,
|
||
|
-7.230484977485752e-01 - 9.056598800801140e+01j]
|
||
|
k2 = 3.774571622827070e-02
|
||
|
assert_allclose(sorted(z, key=np.imag),
|
||
|
sorted(z2, key=np.imag), rtol=1e-4)
|
||
|
assert_allclose(sorted(p, key=np.imag),
|
||
|
sorted(p2, key=np.imag), rtol=1e-6)
|
||
|
assert_allclose(k, k2, rtol=1e-3)
|
||
|
|
||
|
def test_bandstop(self):
|
||
|
z, p, k = ellip(8, 1, 65, [0.2, 0.4], 'stop', output='zpk')
|
||
|
z2 = [3.528578094286510e-01 + 9.356769561794296e-01j,
|
||
|
3.528578094286510e-01 - 9.356769561794296e-01j,
|
||
|
3.769716042264783e-01 + 9.262248159096587e-01j,
|
||
|
3.769716042264783e-01 - 9.262248159096587e-01j,
|
||
|
4.406101783111199e-01 + 8.976985411420985e-01j,
|
||
|
4.406101783111199e-01 - 8.976985411420985e-01j,
|
||
|
5.539386470258847e-01 + 8.325574907062760e-01j,
|
||
|
5.539386470258847e-01 - 8.325574907062760e-01j,
|
||
|
6.748464963023645e-01 + 7.379581332490555e-01j,
|
||
|
6.748464963023645e-01 - 7.379581332490555e-01j,
|
||
|
7.489887970285254e-01 + 6.625826604475596e-01j,
|
||
|
7.489887970285254e-01 - 6.625826604475596e-01j,
|
||
|
7.913118471618432e-01 + 6.114127579150699e-01j,
|
||
|
7.913118471618432e-01 - 6.114127579150699e-01j,
|
||
|
7.806804740916381e-01 + 6.249303940216475e-01j,
|
||
|
7.806804740916381e-01 - 6.249303940216475e-01j]
|
||
|
|
||
|
p2 = [-1.025299146693730e-01 + 5.662682444754943e-01j,
|
||
|
-1.025299146693730e-01 - 5.662682444754943e-01j,
|
||
|
1.698463595163031e-01 + 8.926678667070186e-01j,
|
||
|
1.698463595163031e-01 - 8.926678667070186e-01j,
|
||
|
2.750532687820631e-01 + 9.351020170094005e-01j,
|
||
|
2.750532687820631e-01 - 9.351020170094005e-01j,
|
||
|
3.070095178909486e-01 + 9.457373499553291e-01j,
|
||
|
3.070095178909486e-01 - 9.457373499553291e-01j,
|
||
|
7.695332312152288e-01 + 2.792567212705257e-01j,
|
||
|
7.695332312152288e-01 - 2.792567212705257e-01j,
|
||
|
8.083818999225620e-01 + 4.990723496863960e-01j,
|
||
|
8.083818999225620e-01 - 4.990723496863960e-01j,
|
||
|
8.066158014414928e-01 + 5.649811440393374e-01j,
|
||
|
8.066158014414928e-01 - 5.649811440393374e-01j,
|
||
|
8.062787978834571e-01 + 5.855780880424964e-01j,
|
||
|
8.062787978834571e-01 - 5.855780880424964e-01j]
|
||
|
k2 = 2.068622545291259e-01
|
||
|
assert_allclose(sorted(z, key=np.angle),
|
||
|
sorted(z2, key=np.angle), rtol=1e-6)
|
||
|
assert_allclose(sorted(p, key=np.angle),
|
||
|
sorted(p2, key=np.angle), rtol=1e-5)
|
||
|
assert_allclose(k, k2, rtol=1e-5)
|
||
|
|
||
|
def test_ba_output(self):
|
||
|
# with transfer function conversion, without digital conversion
|
||
|
b, a = ellip(5, 1, 40, [201, 240], 'stop', True)
|
||
|
b2 = [
|
||
|
1.000000000000000e+00, 0, # Matlab: 1.743506051190569e-13,
|
||
|
2.426561778314366e+05, 0, # Matlab: 3.459426536825722e-08,
|
||
|
2.348218683400168e+10, 0, # Matlab: 2.559179747299313e-03,
|
||
|
1.132780692872241e+15, 0, # Matlab: 8.363229375535731e+01,
|
||
|
2.724038554089566e+19, 0, # Matlab: 1.018700994113120e+06,
|
||
|
2.612380874940186e+23
|
||
|
]
|
||
|
a2 = [
|
||
|
1.000000000000000e+00, 1.337266601804649e+02,
|
||
|
2.486725353510667e+05, 2.628059713728125e+07,
|
||
|
2.436169536928770e+10, 1.913554568577315e+12,
|
||
|
1.175208184614438e+15, 6.115751452473410e+16,
|
||
|
2.791577695211466e+19, 7.241811142725384e+20,
|
||
|
2.612380874940182e+23
|
||
|
]
|
||
|
assert_allclose(b, b2, rtol=1e-6)
|
||
|
assert_allclose(a, a2, rtol=1e-4)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
for fs in (900, 900.1, 1234.567):
|
||
|
for N in (0, 1, 2, 3, 10):
|
||
|
for fc in (100, 100.1, 432.12345):
|
||
|
for btype in ('lp', 'hp'):
|
||
|
ba1 = ellip(N, 1, 20, fc, btype, fs=fs)
|
||
|
ba2 = ellip(N, 1, 20, fc/(fs/2), btype)
|
||
|
assert_allclose(ba1, ba2)
|
||
|
for fc in ((100, 200), (100.1, 200.2), (321.123, 432.123)):
|
||
|
for btype in ('bp', 'bs'):
|
||
|
ba1 = ellip(N, 1, 20, fc, btype, fs=fs)
|
||
|
for seq in (list, tuple, array):
|
||
|
fcnorm = seq([f/(fs/2) for f in fc])
|
||
|
ba2 = ellip(N, 1, 20, fcnorm, btype)
|
||
|
assert_allclose(ba1, ba2)
|
||
|
|
||
|
|
||
|
def test_sos_consistency():
|
||
|
# Consistency checks of output='sos' for the specialized IIR filter
|
||
|
# design functions.
|
||
|
design_funcs = [(bessel, (0.1,)),
|
||
|
(butter, (0.1,)),
|
||
|
(cheby1, (45.0, 0.1)),
|
||
|
(cheby2, (0.087, 0.1)),
|
||
|
(ellip, (0.087, 45, 0.1))]
|
||
|
for func, args in design_funcs:
|
||
|
name = func.__name__
|
||
|
|
||
|
b, a = func(2, *args, output='ba')
|
||
|
sos = func(2, *args, output='sos')
|
||
|
assert_allclose(sos, [np.hstack((b, a))], err_msg="%s(2,...)" % name)
|
||
|
|
||
|
zpk = func(3, *args, output='zpk')
|
||
|
sos = func(3, *args, output='sos')
|
||
|
assert_allclose(sos, zpk2sos(*zpk), err_msg="%s(3,...)" % name)
|
||
|
|
||
|
zpk = func(4, *args, output='zpk')
|
||
|
sos = func(4, *args, output='sos')
|
||
|
assert_allclose(sos, zpk2sos(*zpk), err_msg="%s(4,...)" % name)
|
||
|
|
||
|
|
||
|
class TestIIRNotch(object):
|
||
|
|
||
|
def test_ba_output(self):
|
||
|
# Compare coeficients with Matlab ones
|
||
|
# for the equivalent input:
|
||
|
b, a = iirnotch(0.06, 30)
|
||
|
b2 = [
|
||
|
9.9686824e-01, -1.9584219e+00,
|
||
|
9.9686824e-01
|
||
|
]
|
||
|
a2 = [
|
||
|
1.0000000e+00, -1.9584219e+00,
|
||
|
9.9373647e-01
|
||
|
]
|
||
|
|
||
|
assert_allclose(b, b2, rtol=1e-8)
|
||
|
assert_allclose(a, a2, rtol=1e-8)
|
||
|
|
||
|
def test_frequency_response(self):
|
||
|
# Get filter coeficients
|
||
|
b, a = iirnotch(0.3, 30)
|
||
|
|
||
|
# Get frequency response
|
||
|
w, h = freqz(b, a, 1000)
|
||
|
|
||
|
# Pick 5 point
|
||
|
p = [200, # w0 = 0.200
|
||
|
295, # w0 = 0.295
|
||
|
300, # w0 = 0.300
|
||
|
305, # w0 = 0.305
|
||
|
400] # w0 = 0.400
|
||
|
|
||
|
# Get frequency response correspondent to each of those points
|
||
|
hp = h[p]
|
||
|
|
||
|
# Check if the frequency response fulfill the specifications:
|
||
|
# hp[0] and hp[4] correspond to frequencies distant from
|
||
|
# w0 = 0.3 and should be close to 1
|
||
|
assert_allclose(abs(hp[0]), 1, rtol=1e-2)
|
||
|
assert_allclose(abs(hp[4]), 1, rtol=1e-2)
|
||
|
|
||
|
# hp[1] and hp[3] correspond to frequencies approximately
|
||
|
# on the edges of the passband and should be close to -3dB
|
||
|
assert_allclose(abs(hp[1]), 1/np.sqrt(2), rtol=1e-2)
|
||
|
assert_allclose(abs(hp[3]), 1/np.sqrt(2), rtol=1e-2)
|
||
|
|
||
|
# hp[2] correspond to the frequency that should be removed
|
||
|
# the frequency response should be very close to 0
|
||
|
assert_allclose(abs(hp[2]), 0, atol=1e-10)
|
||
|
|
||
|
def test_errors(self):
|
||
|
# Exception should be raised if w0 > 1 or w0 <0
|
||
|
assert_raises(ValueError, iirnotch, w0=2, Q=30)
|
||
|
assert_raises(ValueError, iirnotch, w0=-1, Q=30)
|
||
|
|
||
|
# Exception should be raised if any of the parameters
|
||
|
# are not float (or cannot be converted to one)
|
||
|
assert_raises(ValueError, iirnotch, w0="blabla", Q=30)
|
||
|
assert_raises(TypeError, iirnotch, w0=-1, Q=[1, 2, 3])
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
# Get filter coeficients
|
||
|
b, a = iirnotch(1500, 30, fs=10000)
|
||
|
|
||
|
# Get frequency response
|
||
|
w, h = freqz(b, a, 1000, fs=10000)
|
||
|
|
||
|
# Pick 5 point
|
||
|
p = [200, # w0 = 1000
|
||
|
295, # w0 = 1475
|
||
|
300, # w0 = 1500
|
||
|
305, # w0 = 1525
|
||
|
400] # w0 = 2000
|
||
|
|
||
|
# Get frequency response correspondent to each of those points
|
||
|
hp = h[p]
|
||
|
|
||
|
# Check if the frequency response fulfill the specifications:
|
||
|
# hp[0] and hp[4] correspond to frequencies distant from
|
||
|
# w0 = 1500 and should be close to 1
|
||
|
assert_allclose(abs(hp[0]), 1, rtol=1e-2)
|
||
|
assert_allclose(abs(hp[4]), 1, rtol=1e-2)
|
||
|
|
||
|
# hp[1] and hp[3] correspond to frequencies approximately
|
||
|
# on the edges of the passband and should be close to -3dB
|
||
|
assert_allclose(abs(hp[1]), 1/np.sqrt(2), rtol=1e-2)
|
||
|
assert_allclose(abs(hp[3]), 1/np.sqrt(2), rtol=1e-2)
|
||
|
|
||
|
# hp[2] correspond to the frequency that should be removed
|
||
|
# the frequency response should be very close to 0
|
||
|
assert_allclose(abs(hp[2]), 0, atol=1e-10)
|
||
|
|
||
|
|
||
|
class TestIIRPeak(object):
|
||
|
|
||
|
def test_ba_output(self):
|
||
|
# Compare coeficients with Matlab ones
|
||
|
# for the equivalent input:
|
||
|
b, a = iirpeak(0.06, 30)
|
||
|
b2 = [
|
||
|
3.131764229e-03, 0,
|
||
|
-3.131764229e-03
|
||
|
]
|
||
|
a2 = [
|
||
|
1.0000000e+00, -1.958421917e+00,
|
||
|
9.9373647e-01
|
||
|
]
|
||
|
assert_allclose(b, b2, rtol=1e-8)
|
||
|
assert_allclose(a, a2, rtol=1e-8)
|
||
|
|
||
|
def test_frequency_response(self):
|
||
|
# Get filter coeficients
|
||
|
b, a = iirpeak(0.3, 30)
|
||
|
|
||
|
# Get frequency response
|
||
|
w, h = freqz(b, a, 1000)
|
||
|
|
||
|
# Pick 5 point
|
||
|
p = [30, # w0 = 0.030
|
||
|
295, # w0 = 0.295
|
||
|
300, # w0 = 0.300
|
||
|
305, # w0 = 0.305
|
||
|
800] # w0 = 0.800
|
||
|
|
||
|
# Get frequency response correspondent to each of those points
|
||
|
hp = h[p]
|
||
|
|
||
|
# Check if the frequency response fulfill the specifications:
|
||
|
# hp[0] and hp[4] correspond to frequencies distant from
|
||
|
# w0 = 0.3 and should be close to 0
|
||
|
assert_allclose(abs(hp[0]), 0, atol=1e-2)
|
||
|
assert_allclose(abs(hp[4]), 0, atol=1e-2)
|
||
|
|
||
|
# hp[1] and hp[3] correspond to frequencies approximately
|
||
|
# on the edges of the passband and should be close to 10**(-3/20)
|
||
|
assert_allclose(abs(hp[1]), 1/np.sqrt(2), rtol=1e-2)
|
||
|
assert_allclose(abs(hp[3]), 1/np.sqrt(2), rtol=1e-2)
|
||
|
|
||
|
# hp[2] correspond to the frequency that should be retained and
|
||
|
# the frequency response should be very close to 1
|
||
|
assert_allclose(abs(hp[2]), 1, rtol=1e-10)
|
||
|
|
||
|
def test_errors(self):
|
||
|
# Exception should be raised if w0 > 1 or w0 <0
|
||
|
assert_raises(ValueError, iirpeak, w0=2, Q=30)
|
||
|
assert_raises(ValueError, iirpeak, w0=-1, Q=30)
|
||
|
|
||
|
# Exception should be raised if any of the parameters
|
||
|
# are not float (or cannot be converted to one)
|
||
|
assert_raises(ValueError, iirpeak, w0="blabla", Q=30)
|
||
|
assert_raises(TypeError, iirpeak, w0=-1, Q=[1, 2, 3])
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
# Get filter coeficients
|
||
|
b, a = iirpeak(1200, 30, fs=8000)
|
||
|
|
||
|
# Get frequency response
|
||
|
w, h = freqz(b, a, 1000, fs=8000)
|
||
|
|
||
|
# Pick 5 point
|
||
|
p = [30, # w0 = 120
|
||
|
295, # w0 = 1180
|
||
|
300, # w0 = 1200
|
||
|
305, # w0 = 1220
|
||
|
800] # w0 = 3200
|
||
|
|
||
|
# Get frequency response correspondent to each of those points
|
||
|
hp = h[p]
|
||
|
|
||
|
# Check if the frequency response fulfill the specifications:
|
||
|
# hp[0] and hp[4] correspond to frequencies distant from
|
||
|
# w0 = 1200 and should be close to 0
|
||
|
assert_allclose(abs(hp[0]), 0, atol=1e-2)
|
||
|
assert_allclose(abs(hp[4]), 0, atol=1e-2)
|
||
|
|
||
|
# hp[1] and hp[3] correspond to frequencies approximately
|
||
|
# on the edges of the passband and should be close to 10**(-3/20)
|
||
|
assert_allclose(abs(hp[1]), 1/np.sqrt(2), rtol=1e-2)
|
||
|
assert_allclose(abs(hp[3]), 1/np.sqrt(2), rtol=1e-2)
|
||
|
|
||
|
# hp[2] correspond to the frequency that should be retained and
|
||
|
# the frequency response should be very close to 1
|
||
|
assert_allclose(abs(hp[2]), 1, rtol=1e-10)
|
||
|
|
||
|
|
||
|
class TestIIRFilter(object):
|
||
|
|
||
|
def test_symmetry(self):
|
||
|
# All built-in IIR filters are real, so should have perfectly
|
||
|
# symmetrical poles and zeros. Then ba representation (using
|
||
|
# numpy.poly) will be purely real instead of having negligible
|
||
|
# imaginary parts.
|
||
|
for N in np.arange(1, 26):
|
||
|
for ftype in ('butter', 'bessel', 'cheby1', 'cheby2', 'ellip'):
|
||
|
z, p, k = iirfilter(N, 1.1, 1, 20, 'low', analog=True,
|
||
|
ftype=ftype, output='zpk')
|
||
|
assert_array_equal(sorted(z), sorted(z.conj()))
|
||
|
assert_array_equal(sorted(p), sorted(p.conj()))
|
||
|
assert_equal(k, np.real(k))
|
||
|
|
||
|
b, a = iirfilter(N, 1.1, 1, 20, 'low', analog=True,
|
||
|
ftype=ftype, output='ba')
|
||
|
assert_(issubclass(b.dtype.type, np.floating))
|
||
|
assert_(issubclass(a.dtype.type, np.floating))
|
||
|
|
||
|
def test_int_inputs(self):
|
||
|
# Using integer frequency arguments and large N should not produce
|
||
|
# np.ints that wraparound to negative numbers
|
||
|
k = iirfilter(24, 100, btype='low', analog=True, ftype='bessel',
|
||
|
output='zpk')[2]
|
||
|
k2 = 9.999999999999989e+47
|
||
|
assert_allclose(k, k2)
|
||
|
|
||
|
def test_invalid_wn_size(self):
|
||
|
# low and high have 1 Wn, band and stop have 2 Wn
|
||
|
assert_raises(ValueError, iirfilter, 1, [0.1, 0.9], btype='low')
|
||
|
assert_raises(ValueError, iirfilter, 1, [0.2, 0.5], btype='high')
|
||
|
assert_raises(ValueError, iirfilter, 1, 0.2, btype='bp')
|
||
|
assert_raises(ValueError, iirfilter, 1, 400, btype='bs', analog=True)
|
||
|
|
||
|
def test_invalid_wn_range(self):
|
||
|
# For digital filters, 0 <= Wn <= 1
|
||
|
assert_raises(ValueError, iirfilter, 1, 2, btype='low')
|
||
|
assert_raises(ValueError, iirfilter, 1, [0.5, 1], btype='band')
|
||
|
assert_raises(ValueError, iirfilter, 1, [0., 0.5], btype='band')
|
||
|
assert_raises(ValueError, iirfilter, 1, -1, btype='high')
|
||
|
assert_raises(ValueError, iirfilter, 1, [1, 2], btype='band')
|
||
|
assert_raises(ValueError, iirfilter, 1, [10, 20], btype='stop')
|
||
|
|
||
|
|
||
|
class TestGroupDelay(object):
|
||
|
def test_identity_filter(self):
|
||
|
w, gd = group_delay((1, 1))
|
||
|
assert_array_almost_equal(w, pi * np.arange(512) / 512)
|
||
|
assert_array_almost_equal(gd, np.zeros(512))
|
||
|
w, gd = group_delay((1, 1), whole=True)
|
||
|
assert_array_almost_equal(w, 2 * pi * np.arange(512) / 512)
|
||
|
assert_array_almost_equal(gd, np.zeros(512))
|
||
|
|
||
|
def test_fir(self):
|
||
|
# Let's design linear phase FIR and check that the group delay
|
||
|
# is constant.
|
||
|
N = 100
|
||
|
b = firwin(N + 1, 0.1)
|
||
|
w, gd = group_delay((b, 1))
|
||
|
assert_allclose(gd, 0.5 * N)
|
||
|
|
||
|
def test_iir(self):
|
||
|
# Let's design Butterworth filter and test the group delay at
|
||
|
# some points against MATLAB answer.
|
||
|
b, a = butter(4, 0.1)
|
||
|
w = np.linspace(0, pi, num=10, endpoint=False)
|
||
|
w, gd = group_delay((b, a), w=w)
|
||
|
matlab_gd = np.array([8.249313898506037, 11.958947880907104,
|
||
|
2.452325615326005, 1.048918665702008,
|
||
|
0.611382575635897, 0.418293269460578,
|
||
|
0.317932917836572, 0.261371844762525,
|
||
|
0.229038045801298, 0.212185774208521])
|
||
|
assert_array_almost_equal(gd, matlab_gd)
|
||
|
|
||
|
def test_singular(self):
|
||
|
# Let's create a filter with zeros and poles on the unit circle and
|
||
|
# check if warning is raised and the group delay is set to zero at
|
||
|
# these frequencies.
|
||
|
z1 = np.exp(1j * 0.1 * pi)
|
||
|
z2 = np.exp(1j * 0.25 * pi)
|
||
|
p1 = np.exp(1j * 0.5 * pi)
|
||
|
p2 = np.exp(1j * 0.8 * pi)
|
||
|
b = np.convolve([1, -z1], [1, -z2])
|
||
|
a = np.convolve([1, -p1], [1, -p2])
|
||
|
w = np.array([0.1 * pi, 0.25 * pi, -0.5 * pi, -0.8 * pi])
|
||
|
|
||
|
w, gd = assert_warns(UserWarning, group_delay, (b, a), w=w)
|
||
|
assert_allclose(gd, 0)
|
||
|
|
||
|
def test_backward_compat(self):
|
||
|
# For backward compatibility, test if None act as a wrapper for default
|
||
|
w1, gd1 = group_delay((1, 1))
|
||
|
w2, gd2 = group_delay((1, 1), None)
|
||
|
assert_array_almost_equal(w1, w2)
|
||
|
assert_array_almost_equal(gd1, gd2)
|
||
|
|
||
|
def test_fs_param(self):
|
||
|
# Let's design Butterworth filter and test the group delay at
|
||
|
# some points against the normalized frequency answer.
|
||
|
b, a = butter(4, 4800, fs=96000)
|
||
|
w = np.linspace(0, 96000/2, num=10, endpoint=False)
|
||
|
w, gd = group_delay((b, a), w=w, fs=96000)
|
||
|
norm_gd = np.array([8.249313898506037, 11.958947880907104,
|
||
|
2.452325615326005, 1.048918665702008,
|
||
|
0.611382575635897, 0.418293269460578,
|
||
|
0.317932917836572, 0.261371844762525,
|
||
|
0.229038045801298, 0.212185774208521])
|
||
|
assert_array_almost_equal(gd, norm_gd)
|
||
|
|
||
|
def test_w_or_N_types(self):
|
||
|
# Measure at 8 equally-spaced points
|
||
|
for N in (8, np.int8(8), np.int16(8), np.int32(8), np.int64(8),
|
||
|
np.array(8)):
|
||
|
w, gd = group_delay((1, 1), N)
|
||
|
assert_array_almost_equal(w, pi * np.arange(8) / 8)
|
||
|
assert_array_almost_equal(gd, np.zeros(8))
|
||
|
|
||
|
# Measure at frequency 8 rad/sec
|
||
|
for w in (8.0, 8.0+0j):
|
||
|
w_out, gd = group_delay((1, 1), w)
|
||
|
assert_array_almost_equal(w_out, [8])
|
||
|
assert_array_almost_equal(gd, [0])
|