def test_polyvalfromroots(self):
# check exception for broadcasting x values over root array with
# too few dimensions
assert_raises(ValueError, poly.polyvalfromroots,
[1], [1], tensor=False)
# check empty input
assert_equal(poly.polyvalfromroots([], [1]).size, 0)
assert_(poly.polyvalfromroots([], [1]).shape == (0,))
# check empty input + multidimensional roots
assert_equal(poly.polyvalfromroots([], [[1] * 5]).size, 0)
assert_(poly.polyvalfromroots([], [[1] * 5]).shape == (5, 0))
# check scalar input
assert_equal(poly.polyvalfromroots(1, 1), 0)
assert_(poly.polyvalfromroots(1, np.ones((3, 3))).shape == (3,))
# check normal input)
x = np.linspace(-1, 1)
y = [x ** i for i in range(5)]
for i in range(1, 5):
tgt = y[i]
res = poly.polyvalfromroots(x, [0] * i)
assert_almost_equal(res, tgt)
tgt = x * (x - 1) * (x + 1)
res = poly.polyvalfromroots(x, [-1, 0, 1])
assert_almost_equal(res, tgt)
# check that shape is preserved
for i in range(3):
dims = [2] * i
x = np.zeros(dims)
assert_equal(poly.polyvalfromroots(x, [1]).shape, dims)
assert_equal(poly.polyvalfromroots(x, [1, 0]).shape, dims)
assert_equal(poly.polyvalfromroots(x, [1, 0, 0]).shape, dims)
# check compatibility with factorization
ptest = [15, 2, -16, -2, 1]
r = poly.polyroots(ptest)
x = np.linspace(-1, 1)
assert_almost_equal(poly.polyval(x, ptest),
poly.polyvalfromroots(x, r))
# check multidimensional arrays of roots and values
# check tensor=False
rshape = (3, 5)
x = np.arange(-3, 2)
r = np.random.randint(-5, 5, size=rshape)
res = poly.polyvalfromroots(x, r, tensor=False)
tgt = np.empty(r.shape[1:])
for ii in range(tgt.size):
tgt[ii] = poly.polyvalfromroots(x[ii], r[:, ii])
assert_equal(res, tgt)
# check tensor=True
x = np.vstack([x, 2 * x])
res = poly.polyvalfromroots(x, r, tensor=True)
tgt = np.empty(r.shape[1:] + x.shape)
for ii in range(r.shape[1]):
for jj in range(x.shape[0]):
tgt[ii, jj, :] = poly.polyvalfromroots(x[jj], r[:, ii])
assert_equal(res, tgt)
def Q(racines, i):
temp = racines.copy()
x_t = temp.pop(i)
coef = 1 / nppol.polyvalfromroots(x_t, temp)
return (coef * nppol.polyfromroots(temp))
def freqz_zpk(z, p, k, worN=None, whole=False):
r"""
Compute the frequency response of a digital filter in ZPK form.
Given the Zeros, Poles and Gain of a digital filter, compute its frequency
response::
:math:`H(z)=k \prod_i (z - Z[i]) / \prod_j (z - P[j])`
where :math:`k` is the `gain`, :math:`Z` are the `zeros` and :math:`P` are
the `poles`.
Parameters
----------
z : array_like
Zeroes of a linear filter
p : array_like
Poles of a linear filter
k : scalar
Gain of a linear filter
worN : {None, int, array_like}, optional
If None (default), then compute at 512 frequencies equally spaced
around the unit circle.
If a single integer, then compute at that many frequencies.
If an array_like, compute the response at the frequencies given (in
radians/sample).
whole : bool, optional
Normally, frequencies are computed from 0 to the Nyquist frequency,
pi radians/sample (upper-half of unit-circle). If `whole` is True,
compute frequencies from 0 to 2*pi radians/sample.
Returns
-------
w : ndarray
The normalized frequencies at which `h` was computed, in
radians/sample.
h : ndarray
The frequency response.
See Also
--------
freqs : Compute the frequency response of an analog filter in TF form
freqs_zpk : Compute the frequency response of an analog filter in ZPK form
freqz : Compute the frequency response of a digital filter in TF form
Notes
-----
.. versionadded: 0.19.0
Examples
--------
>>> from scipy import signal
>>> z, p, k = signal.butter(4, 0.2, output='zpk')
>>> w, h = signal.freqz_zpk(z, p, k)
>>> import matplotlib.pyplot as plt
>>> fig = plt.figure()
>>> plt.title('Digital filter frequency response')
>>> ax1 = fig.add_subplot(111)
>>> plt.plot(w, 20 * np.log10(abs(h)), 'b')
>>> plt.ylabel('Amplitude [dB]', color='b')
>>> plt.xlabel('Frequency [rad/sample]')
>>> ax2 = ax1.twinx()
>>> angles = np.unwrap(np.angle(h))
>>> plt.plot(w, angles, 'g')
>>> plt.ylabel('Angle (radians)', color='g')
>>> plt.grid()
>>> plt.axis('tight')
>>> plt.show()
"""
z, p = map(atleast_1d, (z, p))
if whole:
lastpoint = 2 * pi
else:
lastpoint = pi
if worN is None:
N = 512
w = numpy.linspace(0, lastpoint, N, endpoint=False)
elif isinstance(worN, int):
N = worN
w = numpy.linspace(0, lastpoint, N, endpoint=False)
else:
w = worN
w = atleast_1d(w)
zm1 = exp(1j * w)
h = k * polyvalfromroots(zm1, z) / polyvalfromroots(zm1, p)
return w, h
def test_polyvalfromroots(self):
# check exception for broadcasting x values over root array with
# too few dimensions
assert_raises(ValueError, poly.polyvalfromroots,
[1], [1], tensor=False)
# check empty input
assert_equal(poly.polyvalfromroots([], [1]).size, 0)
assert_(poly.polyvalfromroots([], [1]).shape == (0,))
# check empty input + multidimensional roots
assert_equal(poly.polyvalfromroots([], [[1] * 5]).size, 0)
assert_(poly.polyvalfromroots([], [[1] * 5]).shape == (5, 0))
# check scalar input
assert_equal(poly.polyvalfromroots(1, 1), 0)
assert_(poly.polyvalfromroots(1, np.ones((3, 3))).shape == (3,))
# check normal input)
x = np.linspace(-1, 1)
y = [x**i for i in range(5)]
for i in range(1, 5):
tgt = y[i]
res = poly.polyvalfromroots(x, [0]*i)
assert_almost_equal(res, tgt)
tgt = x*(x - 1)*(x + 1)
res = poly.polyvalfromroots(x, [-1, 0, 1])
assert_almost_equal(res, tgt)
# check that shape is preserved
for i in range(3):
dims = [2]*i
x = np.zeros(dims)
assert_equal(poly.polyvalfromroots(x, [1]).shape, dims)
assert_equal(poly.polyvalfromroots(x, [1, 0]).shape, dims)
assert_equal(poly.polyvalfromroots(x, [1, 0, 0]).shape, dims)
# check compatibility with factorization
ptest = [15, 2, -16, -2, 1]
r = poly.polyroots(ptest)
x = np.linspace(-1, 1)
assert_almost_equal(poly.polyval(x, ptest),
poly.polyvalfromroots(x, r))
# check multidimensional arrays of roots and values
# check tensor=False
rshape = (3, 5)
x = np.arange(-3, 2)
r = np.random.randint(-5, 5, size=rshape)
res = poly.polyvalfromroots(x, r, tensor=False)
tgt = np.empty(r.shape[1:])
for ii in range(tgt.size):
tgt[ii] = poly.polyvalfromroots(x[ii], r[:, ii])
assert_equal(res, tgt)
# check tensor=True
x = np.vstack([x, 2*x])
res = poly.polyvalfromroots(x, r, tensor=True)
tgt = np.empty(r.shape[1:] + x.shape)
for ii in range(r.shape[1]):
for jj in range(x.shape[0]):
tgt[ii, jj, :] = poly.polyvalfromroots(x[jj], r[:, ii])
assert_equal(res, tgt)
def freqs_zpk(z, p, k, worN=None):
"""
Compute frequency response of analog filter.
Given the zeros `z`, poles `p`, and gain `k` of a filter, compute its
frequency response::
(jw-z[0]) * (jw-z[1]) * ... * (jw-z[-1])
H(w) = k * ----------------------------------------
(jw-p[0]) * (jw-p[1]) * ... * (jw-p[-1])
Parameters
----------
z : array_like
Zeroes of a linear filter
p : array_like
Poles of a linear filter
k : scalar
Gain of a linear filter
worN : {None, int, array_like}, optional
If None, then compute at 200 frequencies around the interesting parts
of the response curve (determined by pole-zero locations). If a single
integer, then compute at that many frequencies. Otherwise, compute the
response at the angular frequencies (e.g. rad/s) given in `worN`.
Returns
-------
w : ndarray
The angular frequencies at which `h` was computed.
h : ndarray
The frequency response.
See Also
--------
freqs : Compute the frequency response of an analog filter in TF form
freqz : Compute the frequency response of a digital filter in TF form
freqz_zpk : Compute the frequency response of a digital filter in ZPK form
Notes
-----
.. versionadded: 0.19.0
Examples
--------
>>> from scipy.signal import freqs_zpk, iirfilter
>>> z, p, k = iirfilter(4, [1, 10], 1, 60, analog=True, ftype='cheby1',
... output='zpk')
>>> w, h = freqs_zpk(z, p, k, worN=np.logspace(-1, 2, 1000))
>>> import matplotlib.pyplot as plt
>>> plt.semilogx(w, 20 * np.log10(abs(h)))
>>> plt.xlabel('Frequency')
>>> plt.ylabel('Amplitude response [dB]')
>>> plt.grid()
>>> plt.show()
"""
k = np.asarray(k)
if k.size > 1:
raise ValueError('k must be a single scalar gain')
if worN is None:
w = findfreqs(z, p, 200, kind='zp')
elif isinstance(worN, int):
N = worN
w = findfreqs(z, p, N, kind='zp')
else:
w = worN
w = atleast_1d(w)
s = 1j * w
num = polyvalfromroots(s, z)
den = polyvalfromroots(s, p)
h = k * num / den
return w, h
