def modcovar(x, order):
"""Simple and fast implementation of the covariance AR estimate
This code is 10 times faster than :func:`modcovar_marple` and more importantly
only 10 lines of code, compared to a 200 loc for :func:`modcovar_marple`
:param X: Array of complex data samples
:param int order: Order of linear prediction model
:return:
* P - Real linear prediction variance at order IP
* A - Array of complex linear prediction coefficients
.. plot::
:include-source:
:width: 80%
from spectrum import modcovar, marple_data, arma2psd, cshift
from pylab import log10, linspace, axis, plot
a, p = modcovar(marple_data, 15)
PSD = arma2psd(a)
PSD = cshift(PSD, len(PSD)/2) # switch positive and negative freq
plot(linspace(-0.5, 0.5, 4096), 10*log10(PSD/max(PSD)))
axis([-0.5,0.5,-60,0])
.. seealso:: :class:`~spectrum.modcovar.pmodcovar`
:validation: the AR parameters are the same as those returned by
a completely different function :func:`modcovar_marple`.
:References: Mathworks
"""
from spectrum import corrmtx
import scipy.linalg
X = corrmtx(x, order, 'modified')
Xc = np.matrix(X[:,1:])
X1 = np.array(X[:,0])
# Coefficients estimated via the covariance method
# Here we use lstsq rathre than solve function because Xc is not square matrix
a, residues, rank, singular_values = scipy.linalg.lstsq(-Xc, X1)
# Estimate the input white noise variance
Cz = np.dot(X1.conj().transpose(), Xc)
e = np.dot(X1.conj().transpose(), X1) + np.dot(Cz, a)
assert e.imag < 1e-4, 'wierd behaviour'
e = float(e.real) # ignore imag part that should be small
return a, e
def modcovar(x, order):
"""Simple and fast implementation of the covariance AR estimate
This code is 10 times faster than :func:`modcovar_marple` and more importantly
only 10 lines of code, compared to a 200 loc for :func:`modcovar_marple`
:param X: Array of complex data samples
:param int order: Order of linear prediction model
:return:
* P - Real linear prediction variance at order IP
* A - Array of complex linear prediction coefficients
.. plot::
:include-source:
:width: 80%
from spectrum import modcovar, marple_data, arma2psd, cshift
from pylab import log10, linspace, axis, plot
a, p = modcovar(marple_data, 15)
PSD = arma2psd(a)
PSD = cshift(PSD, len(PSD)/2) # switch positive and negative freq
plot(linspace(-0.5, 0.5, 4096), 10*log10(PSD/max(PSD)))
axis([-0.5,0.5,-60,0])
.. seealso:: :class:`~spectrum.modcovar.pmodcovar`
:validation: the AR parameters are the same as those returned by
a completely different function :func:`modcovar_marple`.
:References: Mathworks
"""
from spectrum import corrmtx
import scipy.linalg
X = corrmtx(x, order, 'modified')
Xc = np.matrix(X[:,1:])
X1 = np.array(X[:,0])
# Coefficients estimated via the covariance method
# Here we use lstsq rathre than solve function because Xc is not square matrix
a, residues, rank, singular_values = scipy.linalg.lstsq(-Xc, X1)
# Estimate the input white noise variance
Cz = np.dot(X1.conj().transpose(), Xc)
e = np.dot(X1.conj().transpose(), X1) + np.dot(Cz, a)
assert e.imag < 1e-4, 'wierd behaviour'
e = float(e.real) # ignore imag part that should be small
return a, e
def arcovar(x, order):
r"""Simple and fast implementation of the covariance AR estimate
This code is 10 times faster than :func:`arcovar_marple` and more importantly
only 10 lines of code, compared to a 200 loc for :func:`arcovar_marple`
:param array X: Array of complex data samples
:param int oder: Order of linear prediction model
:return:
* a - Array of complex forward linear prediction coefficients
* e - error
The covariance method fits a Pth order autoregressive (AR) model to the
input signal, which is assumed to be the output of
an AR system driven by white noise. This method minimizes the forward
prediction error in the least-squares sense. The output vector
contains the normalized estimate of the AR system parameters
The white noise input variance estimate is also returned.
If is the power spectral density of y(n), then:
.. math:: \frac{e}{\left| A(e^{jw}) \right|^2} = \frac{e}{\left| 1+\sum_{k-1}^P a(k)e^{-jwk}\right|^2}
Because the method characterizes the input data using an all-pole model,
the correct choice of the model order p is important.
.. plot::
:width: 80%
:include-source:
from spectrum import arcovar, marple_data, arma2psd
from pylab import plot, log10, linspace, axis
ar_values, error = arcovar(marple_data, 15)
psd = arma2psd(ar_values, sides='centerdc')
plot(linspace(-0.5, 0.5, len(psd)), 10*log10(psd/max(psd)))
axis([-0.5, 0.5, -60, 0])
.. seealso:: :class:`pcovar`
:validation: the AR parameters are the same as those returned by
a completely different function :func:`arcovar_marple`.
:References: [Mathworks]_
"""
from spectrum import corrmtx
import scipy.linalg
X = corrmtx(x, order, 'covariance')
Xc = np.matrix(X[:, 1:])
X1 = np.array(X[:, 0])
# Coefficients estimated via the covariance method
# Here we use lstsq rathre than solve function because Xc is not square
# matrix
a, _residues, _rank, _singular_values = scipy.linalg.lstsq(-Xc, X1)
# Estimate the input white noise variance
Cz = np.dot(X1.conj().transpose(), Xc)
e = np.dot(X1.conj().transpose(), X1) + np.dot(Cz, a)
assert e.imag < 1e-4, 'wierd behaviour'
e = float(e.real) # ignore imag part that should be small
return a, e
def arcovar(x, order):
r"""Simple and fast implementation of the covariance AR estimate
This code is 10 times faster than :func:`arcovar_marple` and more importantly
only 10 lines of code, compared to a 200 loc for :func:`arcovar_marple`
:param array X: Array of complex data samples
:param int oder: Order of linear prediction model
:return:
* a - Array of complex forward linear prediction coefficients
* e - error
The covariance method fits a Pth order autoregressive (AR) model to the
input signal, which is assumed to be the output of
an AR system driven by white noise. This method minimizes the forward
prediction error in the least-squares sense. The output vector
contains the normalized estimate of the AR system parameters
The white noise input variance estimate is also returned.
If is the power spectral density of y(n), then:
.. math:: \frac{e}{\left| A(e^{jw}) \right|^2} = \frac{e}{\left| 1+\sum_{k-1}^P a(k)e^{-jwk}\right|^2}
Because the method characterizes the input data using an all-pole model,
the correct choice of the model order p is important.
.. plot::
:width: 80%
:include-source:
from spectrum import arcovar, marple_data, arma2psd
from pylab import plot, log10, linspace, axis
ar_values, error = arcovar(marple_data, 15)
psd = arma2psd(ar_values, sides='centerdc')
plot(linspace(-0.5, 0.5, len(psd)), 10*log10(psd/max(psd)))
axis([-0.5, 0.5, -60, 0])
.. seealso:: :class:`pcovar`
:validation: the AR parameters are the same as those returned by
a completely different function :func:`arcovar_marple`.
:References: [Mathworks]_
"""
from spectrum import corrmtx
import scipy.linalg
X = corrmtx(x, order, 'covariance')
Xc = np.matrix(X[:, 1:])
X1 = np.array(X[:, 0])
# Coefficients estimated via the covariance method
# Here we use lstsq rathre than solve function because Xc is not square
# matrix
a, _residues, _rank, _singular_values = scipy.linalg.lstsq(-Xc, X1)
# Estimate the input white noise variance
Cz = np.dot(X1.conj().transpose(), Xc)
e = np.dot(X1.conj().transpose(), X1) + np.dot(Cz, a)
assert e.imag < 1e-4, 'wierd behaviour'
e = float(e.real) # ignore imag part that should be small
return a, e
def test_corrmtx():
C = corrmtx([1, 2, 3, 4, 5, 6, 7, 8 + 1j], 2, method='modified')
assert_array_almost_equal, C, numpy.array([[3. + 0.j, 2. + 0.j, 1. + 0.j],
[4. + 0.j, 3. + 0.j, 2. + 0.j],
[5. + 0.j, 4. + 0.j, 3. + 0.j],
[6. + 0.j, 5. + 0.j, 4. + 0.j],
[7. + 0.j, 6. + 0.j, 5. + 0.j],
[8. + 1.j, 7. + 0.j, 6. + 0.j],
[1. - 0.j, 2. - 0.j, 3. - 0.j],
[2. - 0.j, 3. - 0.j, 4. - 0.j],
[3. - 0.j, 4. - 0.j, 5. - 0.j],
[4. - 0.j, 5. - 0.j, 6. - 0.j],
[5. - 0.j, 6. - 0.j, 7. - 0.j],
[6. - 0.j, 7. - 0.j, 8. - 1.j]])
C = corrmtx([1, 2, 3, 4, 5, 6, 7, 8 + 1j], 3, method='autocorrelation')
assert_array_almost_equal(
C,
numpy.array([[1. + 0.j, 0. + 0.j, 0. + 0.j, 0. + 0.j],
[2. + 0.j, 1. + 0.j, 0. + 0.j, 0. + 0.j],
[3. + 0.j, 2. + 0.j, 1. + 0.j, 0. + 0.j],
[4. + 0.j, 3. + 0.j, 2. + 0.j, 1. + 0.j],
[5. + 0.j, 4. + 0.j, 3. + 0.j, 2. + 0.j],
[6. + 0.j, 5. + 0.j, 4. + 0.j, 3. + 0.j],
[7. + 0.j, 6. + 0.j, 5. + 0.j, 4. + 0.j],
[8. + 1.j, 7. + 0.j, 6. + 0.j, 5. + 0.j],
[0. + 0.j, 8. + 1.j, 7. + 0.j, 6. + 0.j],
[0. + 0.j, 0. + 0.j, 8. + 1.j, 7. + 0.j],
[0. + 0.j, 0. + 0.j, 0. + 0.j, 8. + 1.j]]))
C = corrmtx([1, 2, 3, 4, 5, 6, 7, 8 + 1j], 3, method='covariance')
assert_array_almost_equal(
C,
numpy.array([[4. + 0.j, 3. + 0.j, 2. + 0.j, 1. + 0.j],
[5. + 0.j, 4. + 0.j, 3. + 0.j, 2. + 0.j],
[6. + 0.j, 5. + 0.j, 4. + 0.j, 3. + 0.j],
[7. + 0.j, 6. + 0.j, 5. + 0.j, 4. + 0.j],
[8. + 1.j, 7. + 0.j, 6. + 0.j, 5. + 0.j]]))
C = corrmtx([1, 2, 3, 4, 5, 6, 7, 8 + 1j], 3, method='postwindowed')
assert_array_almost_equal(
C,
numpy.array([[4. + 0.j, 3. + 0.j, 2. + 0.j, 1. + 0.j],
[5. + 0.j, 4. + 0.j, 3. + 0.j, 2. + 0.j],
[6. + 0.j, 5. + 0.j, 4. + 0.j, 3. + 0.j],
[7. + 0.j, 6. + 0.j, 5. + 0.j, 4. + 0.j],
[8. + 1.j, 7. + 0.j, 6. + 0.j, 5. + 0.j],
[0. + 0.j, 8. + 1.j, 7. + 0.j, 6. + 0.j],
[0. + 0.j, 0. + 0.j, 8. + 1.j, 7. + 0.j],
[0. + 0.j, 0. + 0.j, 0. + 0.j, 8. + 1.j]]))
C = corrmtx([1, 2, 3, 4, 5, 6, 7, 8 + 1j], 3, method='prewindowed')
assert_array_almost_equal(
C,
numpy.array([[1. + 0.j, 0. + 0.j, 0. + 0.j, 0. + 0.j],
[2. + 0.j, 1. + 0.j, 0. + 0.j, 0. + 0.j],
[3. + 0.j, 2. + 0.j, 1. + 0.j, 0. + 0.j],
[4. + 0.j, 3. + 0.j, 2. + 0.j, 1. + 0.j],
[5. + 0.j, 4. + 0.j, 3. + 0.j, 2. + 0.j],
[6. + 0.j, 5. + 0.j, 4. + 0.j, 3. + 0.j],
[7. + 0.j, 6. + 0.j, 5. + 0.j, 4. + 0.j],
[8. + 1.j, 7. + 0.j, 6. + 0.j, 5. + 0.j]]))
for method in [
'prewindowed', 'postwindowed', 'autocorrelation', 'modified',
'covariance'
]:
corrmtx([1, 2, 3, 4, 5, 6, 7, 8], 3, method=method)
try:
corrmtx([1, 2, 3, 4, 5, 6, 7, 8], 3, method='dummy')
assert False
except:
assert True
def test_corrmtx():
C = corrmtx([1,2,3,4,5,6,7,8+1j], 2, method='modified')
assert_array_almost_equal, C, numpy.array([[ 3.+0.j, 2.+0.j, 1.+0.j],
[ 4.+0.j, 3.+0.j, 2.+0.j],
[ 5.+0.j, 4.+0.j, 3.+0.j],
[ 6.+0.j, 5.+0.j, 4.+0.j],
[ 7.+0.j, 6.+0.j, 5.+0.j],
[ 8.+1.j, 7.+0.j, 6.+0.j],
[ 1.-0.j, 2.-0.j, 3.-0.j],
[ 2.-0.j, 3.-0.j, 4.-0.j],
[ 3.-0.j, 4.-0.j, 5.-0.j],
[ 4.-0.j, 5.-0.j, 6.-0.j],
[ 5.-0.j, 6.-0.j, 7.-0.j],
[ 6.-0.j, 7.-0.j, 8.-1.j]])
C = corrmtx([1,2,3,4,5,6,7,8+1j], 3, method='autocorrelation')
assert_array_almost_equal(C, numpy.array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[ 2.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],
[ 3.+0.j, 2.+0.j, 1.+0.j, 0.+0.j],
[ 4.+0.j, 3.+0.j, 2.+0.j, 1.+0.j],
[ 5.+0.j, 4.+0.j, 3.+0.j, 2.+0.j],
[ 6.+0.j, 5.+0.j, 4.+0.j, 3.+0.j],
[ 7.+0.j, 6.+0.j, 5.+0.j, 4.+0.j],
[ 8.+1.j, 7.+0.j, 6.+0.j, 5.+0.j],
[ 0.+0.j, 8.+1.j, 7.+0.j, 6.+0.j],
[ 0.+0.j, 0.+0.j, 8.+1.j, 7.+0.j],
[ 0.+0.j, 0.+0.j, 0.+0.j, 8.+1.j]]))
C = corrmtx([1,2,3,4,5,6,7,8+1j], 3, method='covariance')
assert_array_almost_equal(C, numpy.array([[ 4.+0.j, 3.+0.j, 2.+0.j, 1.+0.j],
[ 5.+0.j, 4.+0.j, 3.+0.j, 2.+0.j],
[ 6.+0.j, 5.+0.j, 4.+0.j, 3.+0.j],
[ 7.+0.j, 6.+0.j, 5.+0.j, 4.+0.j],
[ 8.+1.j, 7.+0.j, 6.+0.j, 5.+0.j]]))
C = corrmtx([1,2,3,4,5,6,7,8+1j], 3, method='postwindowed')
assert_array_almost_equal(C, numpy.array([[ 4.+0.j, 3.+0.j, 2.+0.j, 1.+0.j],
[ 5.+0.j, 4.+0.j, 3.+0.j, 2.+0.j],
[ 6.+0.j, 5.+0.j, 4.+0.j, 3.+0.j],
[ 7.+0.j, 6.+0.j, 5.+0.j, 4.+0.j],
[ 8.+1.j, 7.+0.j, 6.+0.j, 5.+0.j],
[ 0.+0.j, 8.+1.j, 7.+0.j, 6.+0.j],
[ 0.+0.j, 0.+0.j, 8.+1.j, 7.+0.j],
[ 0.+0.j, 0.+0.j, 0.+0.j, 8.+1.j]]))
C = corrmtx([1,2,3,4,5,6,7,8+1j], 3, method='prewindowed')
assert_array_almost_equal(C, numpy.array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[ 2.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],
[ 3.+0.j, 2.+0.j, 1.+0.j, 0.+0.j],
[ 4.+0.j, 3.+0.j, 2.+0.j, 1.+0.j],
[ 5.+0.j, 4.+0.j, 3.+0.j, 2.+0.j],
[ 6.+0.j, 5.+0.j, 4.+0.j, 3.+0.j],
[ 7.+0.j, 6.+0.j, 5.+0.j, 4.+0.j],
[ 8.+1.j, 7.+0.j, 6.+0.j, 5.+0.j]]))
for method in ['prewindowed','postwindowed','autocorrelation','modified','covariance']:
corrmtx([1,2,3,4,5,6,7,8], 3, method=method)
try:
corrmtx([1,2,3,4,5,6,7,8], 3, method='dummy')
assert False
except:
assert True