def pascal(n):
"""Return Pascal matrix
:param int n: size of the matrix
.. doctest::
>>> from spectrum import pascal
>>> pascal(6)
array([[ 1., 1., 1., 1., 1., 1.],
[ 1., 2., 3., 4., 5., 6.],
[ 1., 3., 6., 10., 15., 21.],
[ 1., 4., 10., 20., 35., 56.],
[ 1., 5., 15., 35., 70., 126.],
[ 1., 6., 21., 56., 126., 252.]])
.. todo:: use the symmetric property to improve computational time if needed
"""
errors.is_positive_integer(n)
result = numpy.zeros((n, n))
#fill the first row and column
for i in range(0, n):
result[i, 0] = 1
result[0, i] = 1
if n > 1:
for i in range(1, n):
for j in range(1, n):
result[i, j] = result[i - 1, j] + result[i, j - 1]
return result
def pascal(n):
"""Return Pascal matrix
:param int n: size of the matrix
>>> pascal(6)
array([[ 1., 1., 1., 1., 1., 1.],
[ 1., 2., 3., 4., 5., 6.],
[ 1., 3., 6., 10., 15., 21.],
[ 1., 4., 10., 20., 35., 56.],
[ 1., 5., 15., 35., 70., 126.],
[ 1., 6., 21., 56., 126., 252.]])
.. todo:: use the symmetric property to improve computational time if needed
"""
errors.is_positive_integer(n)
result = numpy.zeros((n, n))
#fill the first row and column
for i in range(0, n):
result[i, 0] = 1
result[0, i] = 1
if n > 1:
for i in range(1, n):
for j in range(1, n):
result[i, j] = result[i-1, j] + result[i, j-1]
return result
def minvar(X, order, sampling=1., NFFT=default_NFFT):
r"""Minimum Variance Spectral Estimation (MV)
This function computes the minimum variance spectral estimate using
the Musicus procedure. The Burg algorithm from :func:`~spectrum.burg.arburg`
is used for the estimation of the autoregressive parameters.
The MV spectral estimator is given by:
.. math:: P_{MV}(f) = \frac{T}{e^H(f) R^{-1}_p e(f)}
where :math:`R^{-1}_p` is the inverse of the estimated autocorrelation
matrix (Toeplitz) and :math:`e(f)` is the complex sinusoid vector.
:param X: Array of complex or real data samples (length N)
:param int order: Dimension of correlation matrix (AR order = order - 1 )
:param float T: Sample interval (PSD scaling)
:param int NFFT: length of the final PSD
:return:
* PSD - Power spectral density values (two-sided)
* AR - AR coefficients (Burg algorithm)
* k - Reflection coefficients (Burg algorithm)
.. note:: The MV spectral estimator is not a true PSD function because the
area under the MV estimate does not represent the total power in the
measured process. MV minimises the variance of the output of a narrowband
filter and adpats itself to the spectral content of the input data
at each frequency.
:Example: The following example computes a PSD estimate using :func:`minvar`
The output PSD is transformed to a ``centerdc`` PSD and plotted.
.. plot::
:width: 80%
:include-source:
from spectrum import *
from pylab import plot, log10, linspace, xlim
psd, A, k = minvar(marple_data, 15)
psd = twosided_2_centerdc(psd) # switch positive and negative freq
f = linspace(-0.5, 0.5, len(psd))
plot(f, 10 * log10(psd/max(psd)))
xlim(-0.5, 0.5 )
.. seealso::
* External functions used are :meth:`~spectrum.burg.arburg`
and numpy.fft.fft
* :class:`pminvar`, a Class dedicated to MV method.
:Reference: [Marple]_
"""
errors.is_positive_integer(order)
errors.is_positive_integer(NFFT)
psi = np.zeros(NFFT, dtype=complex)
# First, we need to compute the AR values (note that order-1)
A, P, k = arburg(X, order - 1)
# add the order 0
A = np.insert(A, 0, 1. + 0j)
# We cannot compare the output with those of MARPLE in a precise way.
# Indeed the burg algorithm is only single precision in fortram code
# So, the AR values are slightly differnt.
# The followign values are those from Marple
"""A[1] = 2.62284255-0.701703191j
A[2] = 4.97930574-2.32781982j
A[3] = 6.78445101-5.02477741j
A[4] =7.85207081-8.01284409j
A[5] =7.39412165-10.7684202j
A[6] =6.03175116-12.7067814j
A[7] =3.80106878-13.6808891j
A[8] =1.48207295-13.2265558j
A[9] =-0.644280195-11.4574194j
A[10] =-2.02386642-8.53268814j
A[11] =-2.32437634-5.25636244j
A[12] =-1.75356281-2.46820402j
A[13] =-0.888899028-0.781434655j
A[14] =-0.287197977-0.0918145925j
P = 0.00636525545
"""
# if we use exactly the same AR coeff and P from Marple Burg output, then
# we can compare the following code. This has been done and reveals that
# the FFT in marple is also slightly different (precision) from this one.
# However, the results are sufficiently close (when NFFT is small) that
# we are confident the following code is correct.
# Compute the psi coefficients
for K in range(0, order):
SUM = 0.
MK = order - K
# Correlate the autoregressive parameters
for I in range(0, order - K):
SUM = SUM + float(MK - 2 *
I) * A[I].conjugate() * A[I + K] # Eq. (12.25)
SUM = SUM / P
if K != 0:
psi[NFFT - K] = SUM.conjugate()
psi[K] = SUM
# Compute FFT of denominator
psi = fft(psi, NFFT)
# Invert the psi terms at this point to get PSD values
PSD = sampling / np.real(psi)
return PSD, A, k
def minvar(X, order, sampling=1., NFFT=default_NFFT):
r"""Minimum Variance Spectral Estimation (MV)
This function computes the minimum variance spectral estimate using
the Musicus procedure. The Burg algorithm from :func:`~spectrum.burg.arburg`
is used for the estimation of the autoregressive parameters.
The MV spectral estimator is given by:
.. math:: P_{MV}(f) = \frac{T}{e^H(f) R^{-1}_p e(f)}
where :math:`R^{-1}_p` is the inverse of the estimated autocorrelation
matrix (Toeplitz) and :math:`e(f)` is the complex sinusoid vector.
:param X: Array of complex or real data samples (length N)
:param int order: Dimension of correlation matrix (AR order = order - 1 )
:param float T: Sample interval (PSD scaling)
:param int NFFT: length of the final PSD
:return:
* PSD - Power spectral density values (two-sided)
* AR - AR coefficients (Burg algorithm)
* k - Reflection coefficients (Burg algorithm)
.. note:: The MV spectral estimator is not a true PSD function because the
area under the MV estimate does not represent the total power in the
measured process. MV minimises the variance of the output of a narrowband
filter and adpats itself to the spectral content of the input data
at each frequency.
:Example: The following example computes a PSD estimate using :func:`minvar`
The output PSD is transformed to a ``centerdc`` PSD and plotted.
.. plot::
:width: 80%
:include-source:
from spectrum import *
from pylab import plot, log10, linspace, xlim
psd, A, k = minvar(marple_data, 15)
psd = twosided_2_centerdc(psd) # switch positive and negative freq
f = linspace(-0.5, 0.5, len(psd))
plot(f, 10 * log10(psd/max(psd)))
xlim(-0.5, 0.5 )
.. seealso::
* External functions used are :meth:`~spectrum.burg.arburg`
and numpy.fft.fft
* :class:`pminvar`, a Class dedicated to MV method.
:Reference: [Marple]_
"""
errors.is_positive_integer(order)
errors.is_positive_integer(NFFT)
psi = np.zeros(NFFT, dtype=complex)
# First, we need to compute the AR values (note that order-1)
A, P, k = arburg (X, order - 1)
# add the order 0
A = np.insert(A, 0, 1.+0j)
# We cannot compare the output with those of MARPLE in a precise way.
# Indeed the burg algorithm is only single precision in fortram code
# So, the AR values are slightly differnt.
# The followign values are those from Marple
"""A[1] = 2.62284255-0.701703191j
A[2] = 4.97930574-2.32781982j
A[3] = 6.78445101-5.02477741j
A[4] =7.85207081-8.01284409j
A[5] =7.39412165-10.7684202j
A[6] =6.03175116-12.7067814j
A[7] =3.80106878-13.6808891j
A[8] =1.48207295-13.2265558j
A[9] =-0.644280195-11.4574194j
A[10] =-2.02386642-8.53268814j
A[11] =-2.32437634-5.25636244j
A[12] =-1.75356281-2.46820402j
A[13] =-0.888899028-0.781434655j
A[14] =-0.287197977-0.0918145925j
P = 0.00636525545
"""
# if we use exactly the same AR coeff and P from Marple Burg output, then
# we can compare the following code. This has been done and reveals that
# the FFT in marple is also slightly different (precision) from this one.
# However, the results are sufficiently close (when NFFT is small) that
# we are confident the following code is correct.
# Compute the psi coefficients
for K in range(0, order):
SUM = 0.
MK = order-K
# Correlate the autoregressive parameters
for I in range(0, order - K):
SUM = SUM + float(MK-2*I) * A[I].conjugate()*A[I+K] # Eq. (12.25)
SUM = SUM/P
if K != 0:
psi[NFFT-K] = SUM.conjugate()
psi[K] = SUM
# Compute FFT of denominator
psi = fft(psi, NFFT)
# Invert the psi terms at this point to get PSD values
PSD = sampling / np.real(psi)
return PSD, A, k