5.5. Tools and classes ¶

Contents

Tools and classes
- Classes
- Correlation
- Tools
- datasets
- Linear Algebra Tools
  - cholesky
  - eigen
  - levinson
  - toeplitz
  - linalg
- Waveforms

5.5.1. Classes ¶

This module provides the Base class for PSDs

class Spectrum(data, data_y=None, sampling=1.0, detrend=None, scale_by_freq=True, NFFT=None)¶

Base class for all Spectrum classes

All PSD classes should inherits from this class to store common attributes such as the input data or sampling frequency. An instance is created as follows:

>>> p = Spectrum(data, sampling=1024)
>>> p.data
>>> p.sampling

The input parameters are:

Parameters:

Parameters:	data (array) – input data (list or numpy.array) data_y (array) – input data required to perform cross-PSD only. sampling (float) – sampling frequency of the input `data` detrend (str) – detrend method ([None,’mean’]) to apply on the input data before computing the PSD. See `detrend`. scale_by_freq (bool) – divide the final PSD by $2\pi/df$ NFFT* (int) – total length of the final data sets (padded with zero if needed; default is 4096)

data (array) – input data (list or numpy.array)
data_y (array) – input data required to perform cross-PSD only.
sampling (float) – sampling frequency of the input data
detrend (str) – detrend method ([None,’mean’]) to apply on the input data before computing the PSD. See detrend.
scale_by_freq (bool) – divide the final PSD by $2*\pi/df$
NFFT (int) – total length of the final data sets (padded with zero if needed; default is 4096)

The input parameters are available as attributes. Additional attributes such as N (the data length), df (the frequency step are set (see constructor documentation for a complete list).

You can populate manually the psd attribute but you should respect the following convention:

if the input data is real, the PSD is assumed to be one-sided (odd length)

if the input data is complex, the PSD is assumed to be two-sided (even length).

When psd is set, sides is reset to its default value, NFFT and df are updated.

Spectrum instances have plotting utilities like plot() that take care of plotting the PSD versus the appropriate frequency range (based on sampling, NFFT and sides)

Note

the modification of some attributes (e.g., NFFT), makes the PSD obsolete. In such cases, the PSD must be re-computed before using plot() again.

At any time, you can get general information about the Spectrum instance:

>>> p = Spectrum(marple_data)
>>> print p
Spectrum summary
    Data length is 64
    PSD not yet computed
    Sampling 1.0
    freq resolution 0.015625
    datatype is complex
    sides is twosided
    scal_by_freq is True

Constructor

Attributes:

From the input parameters, the following attributes are set:

data (updates N, df, datatype)

data_y

detrend

sampling (updates df)

scale_by_freq

NFFT (reset sides, df)

The following read-only attributes are set during the initialisation:

datatype

df

N

And finally, additional read-write attributes are available:

psd: used to store the PSD data array, which size depends on sides i.e., one-sided for real data and two-sided for the complex data.

sides: if set, changed the psd.

N¶: Getter to the original data size. N is automatically updated when changing the data only.

NFFT¶

Getter/Setter to the NFFT attribute.

Parameters:

Parameters:	NFFT – a user choice for setting `NFFT`. if None, the NFFT is set to `N`, the data length. if ‘nextpow2’, computes the next power of 2 greater than or equal to the data length. if a integer is provided, it must be positive

NFFT –

a user choice for setting NFFT.

if None, the NFFT is set to N, the data length.
if ‘nextpow2’, computes the next power of 2 greater than or equal to the data length.
if a integer is provided, it must be positive

If NFFT is changed, sides is reset and df as well.

data¶: Getter/Setter for the input data. If input is a list, it is cast into a numpy.array. N, df and datatype are updated.

data_y¶: Getter/Setter to the Y-data

datatype¶: Getter to the datatype (‘real’ or ‘complex’). datatype is automatically updated when changing the data.

detrend¶

Getter/Setter to detrend:

None: do not perform any detrend.
‘mean’: remove the mean value of each segment from each segment of the data.
‘long-mean’: remove the mean value from the data before splitting it into segments.
‘linear’: remove linear trend from each segment.

df¶: Getter to step frequency. This attribute is updated as soon as data or sampling is changed

frequencies(sides=None)¶: Return the frequency vector according to sides

get_converted_psd(sides)¶

This function returns the PSD in the sides format

Parameters:	sides (str) – the PSD format in [‘onesided’, ‘twosided’, ‘centerdc’]
Returns:	the expected PSD.

from spectrum import *
p = pcovar(marple_data, 15)
p()
centerdc_psd = p.get_converted_psd('centerdc')

Note

this function does not change the object, in particular, it does not change the psd attribute. If you want to change the psd on the fly, change the attribute sides.

method¶

plot(filename=None, norm=False, ylim=None, sides=None, **kargs)¶

a simple plotting routine to plot the PSD versus frequency.

Parameters:	filename (str) – save the figure into a file norm – False by default. If True, the PSD is normalised. ylim – readjust the y range . sides – if not provided, `sides` is used. See `sides` for details. kargs – any optional argument accepted by `pylab.plot()`.

from spectrum import *
p = Periodogram(marple_data)
p()    # runs the psd estimate
p.plot(norm=True, marker='o')

[hires.png, pdf]

power()¶

Return the power contained in the PSD

if scale_by_freq is False, the power is:

$P = N \sum_{k=1}^{N} P_{xx}(k)$

else, it is

$P = \sum_{k=1}^{N} P_{xx}(k) \frac{df}{2\pi}$

Todo

check these equations

psd¶

Getter/Setter to psd

Parameters:	psd (array) – the array must be in agreement with the onesided/twosided convention: if the data in real, the psd must be onesided. If the data is complex, the psd must be twosided.

When you set this attribute, several attributes are set:

sides is set to onesided if datatype is real and twosided if datatype is complex.

NFFT is set to len(psd) if sides=onesided and (len(psd)-1)*2 if sides=twosided.

range.N is set to NFFT, which update df.

range¶: Read only attribute to a Range object.

run()¶

sampling¶: Getter/Setter to sampling frequency Updates the df automatically.

scale()¶

scale_by_freq¶: scale the PSD by $2*\pi/df$

sides¶

Getter/Setter to the sides attributes.

It can be ‘onesided’, ‘twosided’, ‘centerdc’. This setter changes psd to reflect the user argument choice.

If the datatype is complex, sides cannot be one-sided.

class FourierSpectrum(data, sampling=1.0, window='hanning', NFFT=None, detrend=None, scale_by_freq=True, lag=-1)¶

Spectrum based on Fourier transform.

This class inherits attributes and methods from Spectrum. It is used by children class Periodogram, pcorrelogram and Welch PSD estimates.

The parameters are those used by Spectrum

Parameters:

Parameters:	data (array) – Input data (list or numpy.array) data_y – input data required to perform cross-PSD only. sampling (float) – sampling frequency of the input `data` detrend (str) – detrend method ([None,’mean’]) to apply on the input data before computing the PSD. See `detrend`. scale_by_freq (bool) – Divide the final PSD by $2\pi/df$ NFFT* (int) – total length of the data given to the FFT

data (array) – Input data (list or numpy.array)
data_y – input data required to perform cross-PSD only.
sampling (float) – sampling frequency of the input data
detrend (str) – detrend method ([None,’mean’]) to apply on the input data before computing the PSD. See detrend.
scale_by_freq (bool) – Divide the final PSD by $2*\pi/df$
NFFT (int) – total length of the data given to the FFT

In addition you need specific parameters such as:

Parameters:	window (str) – a tapering window. See `Window`. lag (int) – to be used by the `pcorrelogram` methods only.

This class has dedicated PSDs methods such as speriodogram(), which are equivalent to children class such as Periodogram.

from spectrum import datasets
from spectrum import FourierSpectrum
s = FourierSpectrum(datasets.data_cosine(), lag=32, sampling=1024, NFFT=512)
s.periodogram()
s.plot(label='periodogram')
#s.correlogram()
s.plot(label='correlogram')
from pylab import legend, xlim
legend()
xlim(0,512)

[hires.png, pdf]

Constructor

See the class documentation for the parameters.

Additional attributes to those inherited from

Spectrum are:

lag, a lag used to compute the autocorrelation
window, the tapering window to be used

lag¶: Getter/Setter used by the correlogram when autocorrelation estimates are required.

periodogram()¶

An alias to Periodogram

The parameters are extracted from the attributes. Relevant attributes ares window, attr:sampling, attr:NFFT, attr:scale_by_freq, detrend.

from spectrum import datasets
from spectrum import FourierSpectrum
s = FourierSpectrum(datasets.data_cosine(), sampling=1024, NFFT=512)
s.periodogram()
s.plot()

[hires.png, pdf]

window¶: Tapering window to be applied

class ParametricSpectrum(data, sampling=1.0, ar_order=None, ma_order=None, lag=-1, NFFT=None, detrend=None, scale_by_freq=True)¶

Spectrum based on Fourier transform.

This class inherits attributes and methods from Spectrum. It is used by children class Periodogram, pcorrelogram and Welch PSD estimates. The parameters are those used by Spectrum.

Parameters:	data (array) – Input data (list or numpy.array) sampling (float) – sampling frequency of the input `data` detrend (str) – detrend method ([None,’mean’]) to apply on the input data before computing the PSD. See `detrend`. scale_by_freq (bool) – Divide the final PSD by $2*\pi/df$

In addition you need specific parameters such as:

Parameters:	window (str) – a tapering window. See `Window`. lag (int) – to be used by the `pcorrelogram` methods only. NFFT (int) – Total length of the data given to the FFT

This class has dedicated PSDs methods such as periodogram(), which are equivalent to children class such as Periodogram.

from spectrum import datasets
from spectrum import ParametricSpectrum
data = datasets.data_cosine(N=1024)
s = ParametricSpectrum(data, ar_order=4, ma_order=4, sampling=1024, NFFT=512, lag=10)
#s.parma()
#s.plot(sides='onesided')
#s.plot(sides='twosided')

[source code]

Exception occurred rendering plot.

Constructor

See the class documentation for the parameters.

Additional attributes to those inherited from Spectrum:

ar_order, the ar order of the PSD estimates
ma_order, the ar order of the PSD estimates

ar¶

ar_order¶

ma¶

ma_order¶

plot_reflection()¶

reflection¶: self.reflection = None elif method == ‘aryule’:

from spectrum import aryule ar, v, coeff = aryule(self.data, self.ar_order) self.ar = ar self.rho = v self.reflection = coeff

rho¶

5.5.2. Correlation ¶

Correlation module

Provides two correlation functions. CORRELATION() is slower than xcorr(). However, the output is as expected by some other functions. Ultimately, it should be replaced by xcorr().

For real data, the behaviour of the 2 functions is identical. However, for complex data, xcorr returns a 2-sides correlation.

`CORRELATION`(x[, y, maxlags, norm])	Correlation function
`xcorr`(x[, y, maxlags, norm])	Cross-correlation using numpy.correlate

CORRELATION(x, y=None, maxlags=None, norm='unbiased')¶

Correlation function

This function should give the same results as xcorr() but it returns the positive lags only. Moreover the algorithm does not use FFT as compared to other algorithms.

Parameters:

Parameters:	x (array) – first data array of length N y (array) – second data array of length N. If not specified, computes the autocorrelation. maxlags (int) – compute cross correlation between [0:maxlags] when maxlags is not specified, the range of lags is [0:maxlags]. norm (str) – normalisation in [‘biased’, ‘unbiased’, None, ‘coeff’] biased correlation=raw/N, unbiased correlation=raw/(N-\|lag\|) coeff correlation=raw/(rms(x).rms(y))/N None correlation=raw
Returns:	a numpy.array correlation sequence, r[1,N] a float for the zero-lag correlation, r[0]

x (array) – first data array of length N
y (array) – second data array of length N. If not specified, computes the autocorrelation.
maxlags (int) – compute cross correlation between [0:maxlags] when maxlags is not specified, the range of lags is [0:maxlags].
norm (str) –
normalisation in [‘biased’, ‘unbiased’, None, ‘coeff’]
- biased correlation=raw/N,
- unbiased correlation=raw/(N-|lag|)
- coeff correlation=raw/(rms(x).rms(y))/N
- None correlation=raw

Returns:

a numpy.array correlation sequence, r[1,N]
a float for the zero-lag correlation, r[0]

The unbiased correlation has the form:

$\hat{r}_{xx} = \frac{1}{N-m}T \sum_{n=0}^{N-m-1} x[n+m]x^*[n] T$

The biased correlation differs by the front factor only:

$\check{r}_{xx} = \frac{1}{N}T \sum_{n=0}^{N-m-1} x[n+m]x^*[n] T$

with $0\leq m\leq N-1$ .

>>> from spectrum import *
>>> x = [1,2,3,4,5]
>>> res = CORRELATION(x,x, maxlags=0, norm='biased')
>>> res[0]
11.0

Note

this function should be replaced by xcorr().

See also

xcorr()

xcorr(x, y=None, maxlags=None, norm='biased')¶

Cross-correlation using numpy.correlate

Estimates the cross-correlation (and autocorrelation) sequence of a random process of length N. By default, there is no normalisation and the output sequence of the cross-correlation has a length 2*N+1.

Parameters:	x (array) – first data array of length N y (array) – second data array of length N. If not specified, computes the autocorrelation. maxlags (int) – compute cross correlation between [-maxlags:maxlags] when maxlags is not specified, the range of lags is [-N+1:N-1]. option (str) – normalisation in [‘biased’, ‘unbiased’, None, ‘coeff’]

The true cross-correlation sequence is

$r_{xy}[m] = E(x[n+m].y^*[n]) = E(x[n].y^*[n-m])$

However, in practice, only a finite segment of one realization of the infinite-length random process is available.

The correlation is estimated using numpy.correlate(x,y,’full’). Normalisation is handled by this function using the following cases:

‘biased’: Biased estimate of the cross-correlation function

‘unbiased’: Unbiased estimate of the cross-correlation function

‘coeff’: Normalizes the sequence so the autocorrelations at zero

lag is 1.0.

Returns:	a numpy.array containing the cross-correlation sequence (length 2*N-1) lags vector

Note

If x and y are not the same length, the shorter vector is zero-padded to the length of the longer vector.

Examples

>>> from spectrum import *
>>> x = [1,2,3,4,5]
>>> c, l = xcorr(x,x, maxlags=0, norm='biased')
>>> c
array([ 11.])

See also

CORRELATION().

5.5.3. Tools ¶

Tools module

`db2mag`(xdb)	Convert decibels (dB) to magnitude
`db2pow`(xdb)	Convert decibels (dB) to power
`mag2db`(x)	Convert magnitude to decibels (dB)
`nextpow2`(x)	returns the smallest power of two that is greater than or equal to the
`pow2db`(x)	returns the corresponding decibel (dB) value for a power value x.
`onesided_2_twosided`(data)	Convert a two-sided PSD to a one-sided PSD
`twosided_2_onesided`(data)	Convert a one-sided PSD to a twosided PSD
`centerdc_2_twosided`(data)	Convert a center-dc PSD to a twosided PSD
`twosided_2_centerdc`(data)	Convert a two-sided PSD to a center-dc PSD

Code author: Thomas Cokelaer, 2011

centerdc_2_twosided(data)¶: Convert a center-dc PSD to a twosided PSD

cshift(data, offset)¶

Circular shift to the right (within an array) by a given offset

Parameters:	data (array) – input data (list or numpy.array) offset (int) – shift the array with the offset

>>> from spectrum import cshift
>>> cshift([0, 1, 2, 3, -2, -1], 2)
array([-2, -1,  0,  1,  2,  3])

db2mag(xdb)¶

Convert decibels (dB) to magnitude

>>> db2mag(-20)
0.1

See also

pow2db()

db2pow(xdb)¶

Convert decibels (dB) to power

>>> p = db2pow(-10)
>>> p
0.1

See also

pow2db()

fftshift(x)¶

wrapper to numpy.fft.fftshift

>>> from spectrum import fftshift
>>> x = [100, 2, 3, 4, 5]
>>> fftshift(x)
array([  4,   5, 100,   2,   3])

mag2db(x)¶

Convert magnitude to decibels (dB)

The relationship between magnitude and decibels is:

$X_{dB} = 20 * \log_{10}(x)$

>>> mag2db(0.1)
-20.0

See also

db2mag()

nextpow2(x)¶

returns the smallest power of two that is greater than or equal to the absolute value of x.

This function is useful for optimizing FFT operations, which are most efficient when sequence length is an exact power of two.

Example :

>>> x = [255, 256, 257]
>>> nextpow2(x)
array([8, 8, 9])

onesided_2_twosided(data)¶

Convert a two-sided PSD to a one-sided PSD

In order to keep the power in the twosided PSD the same as in the onesided version, the twosided values are 2 times lower than the input data (except for the zero-lag and N-lag values).

>>> twosided_2_onesided([10, 4, 6, 8])
array([ 10.,   2.,   3.,   3., 2., 8.])

pow2db(x)¶

returns the corresponding decibel (dB) value for a power value x.

The relationship between power and decibels is:

$X_{dB} = 10 * \log_{10}(x)$

>>> x = pow2db(0.1)
>>> x
-10.0

twosided(data)¶

return a twosided vector with non-duplication of the first element

>>> from spectrum import twosided
>>> a = [1,2,3]
>>> twosided(a)
array([3, 2, 1, 2, 3])

twosided_2_centerdc(data)¶: Convert a two-sided PSD to a center-dc PSD

twosided_2_onesided(data)¶

Convert a one-sided PSD to a twosided PSD

In order to keep the power in the onesided PSD the same as in the twosided version, the onesided values are twice as much as in the input data (except for the zero-lag value).

>>> twosided_2_onesided([10, 2,3,3,2,8])
array([ 10.,   4.,   6.,   8.])

5.5.4. datasets ¶

The datasets module provides data sets to test the Spectrum functionalities.

`data_cosine`([N, A, sampling, freq])	Return a noisy cosine at a given frequency.
`marple_data`	list() -> new empty list
`TimeSeries`(data[, sampling])	A simple Base Class for various data sets.

Code author: Thomas Cokelaer 2011

Reference:	[Marple]

class TimeSeries(data, sampling=1)¶

A simple Base Class for various data sets.

>>> data = [1, 2, 3, 4, 3, 2, 1, 0 ]
>>> ts = TimeSeries(data, sampling=1)
>>> ts.plot()
>>> ts.dt
1.0

Parameters:	data (array) – input data (list or numpy.array) sampling – the sampling frequency of the data (default 1Hz)

plot(**kargs)¶: Plot the data set, using the sampling information to set the x-axis correctly.

data_cosine(N=1024, A=0.1, sampling=1024.0, freq=200)¶

Return a noisy cosine at a given frequency.

Parameters:	N – the final data size A – the strength of the noise sampling (float) – sampling frequency of the input `data`. freq (float) – the frequency $f_0$ of the cosine.

$x[t] = cos(2\pi t * f_0) + A w[t]$

where w[t] is a white noise of variance 1.

>>> a = data_cosine(N=1024, sampling=1024, A=0.5, freq=100)

marple_data = [(1.349839091+2.011167288j), (-2.117270231+0.817693591j), (-1.786421657-1.291698933j), (1.162236333-1.482598066j), (1.641072035+0.372950256j), (0.072213709+1.828492761j), (-1.564284801+0.824533045j), (-1.080565453-1.869776845j), (0.92712909-1.743406534j), (1.891979456+0.972347319j), (-0.105391249+1.602209687j), (-1.618367076+0.63751328j), (-0.945704579-1.079569221j), (1.135566235-1.692269921j), (1.855816245+0.986030221j), (-1.032083511+1.414613724j), (-1.571600199+0.089229003j), (-0.243143231-1.444692016j), (0.838980973-0.985756695j), (1.516003132+0.928058863j), (0.257979959+1.170676708j), (-2.057927608+0.343388647j), (-0.578682184-1.441192508j), (1.584011555-1.011150956j), (0.614114344+1.508176208j), (-0.710567117+1.130144477j), (-1.100205779-0.584209621j), (0.150702029-1.217450142j), (0.748856127-0.804411888j), (0.795235813+1.114466429j), (-0.071512341+1.017092347j), (-1.732939839-0.283070654j), (0.404945314-0.78170836j), (1.293794155-0.352723092j), (-0.119905084+0.905150294j), (-0.522588372+0.437393665j), (-0.974838495-0.670074046j), (0.275279552-0.509659231j), (0.854210198-0.008278057j), (0.289598197+0.50623399j), (-0.283553183+0.250371397j), (-0.359602571-0.135261074j), (0.102775671-0.466086507j), (-0.00972265+0.030377999j), (0.185930878+0.8088696j), (-0.243692726-0.200126961j), (-0.270986766-0.460243553j), (0.399368525+0.249096692j), (-0.250714004-0.36299023j), (0.419116348-0.389185309j), (-0.050458215+0.702862442j), (-0.395043731+0.140808776j), (0.746575892-0.126762003j), (-0.55907619+0.523169816j), (-0.34438926-0.913451135j), (0.733228028-0.006237417j), (-0.480273813+0.509469569j), (0.033316225+0.087501869j), (-0.32122913-0.254548967j), (-0.063007891-0.499800682j), (1.239739418-0.013479125j), (0.083303742+0.673984587j), (-0.762731433+0.40897125j), (-0.895898521-0.364855707j)]¶: 64-complex data length from Marple reference [Marple]

5.5.5. Linear Algebra Tools ¶

5.5.5.1. cholesky ¶

Cholesky methods

CHOLESKY(A, B[, method]) Solve linear system AX=B using CHOLESKY method.

Code author: Thomas Cokelaer, 2011

CHOLESKY(A, B, method='scipy')¶

Solve linear system AX=B using CHOLESKY method.

Parameters:

Parameters:	A – an input Hermitian matrix B – an array method (str) – a choice of method in [numpy, scipy, numpy_solver] numpy_solver relies entirely on numpy.solver (no cholesky decomposition) numpy relies on the numpy.linalg.cholesky for the decomposition and numpy.linalg.solve for the inversion. scipy uses scipy.linalg.cholesky for the decomposition and scipy.linalg.cho_solve for the inversion.

A – an input Hermitian matrix
B – an array
method (str) –
a choice of method in [numpy, scipy, numpy_solver]
- numpy_solver relies entirely on numpy.solver (no cholesky decomposition)
- numpy relies on the numpy.linalg.cholesky for the decomposition and numpy.linalg.solve for the inversion.
- scipy uses scipy.linalg.cholesky for the decomposition and scipy.linalg.cho_solve for the inversion.

Description

When a matrix is square and Hermitian (symmetric with lower part being the complex conjugate of the upper one), then the usual triangular factorization takes on the special form:

$A = R R^H$

where $R$ is a lower triangular matrix with nonzero real principal diagonal element. The input matrix can be made of complex data. Then, the inversion to find $x$ is made as follows:

$Ry = B$

and

$Rx = y$

>>> import numpy
>>> from spectrum import CHOLESKY
>>> A = numpy.array([[ 2.0+0.j ,  0.5-0.5j, -0.2+0.1j],
...    [ 0.5+0.5j,  1.0+0.j ,  0.3-0.2j],
...    [-0.2-0.1j,  0.3+0.2j,  0.5+0.j ]])
>>> B = numpy.array([ 1.0+3.j ,  2.0-1.j ,  0.5+0.8j])
>>> CHOLESKY(A, B)
array([ 0.95945946+5.25675676j,  4.41891892-7.04054054j,
       -5.13513514+6.35135135j])

5.5.5.2. eigen ¶

MINEIGVAL(T0, T, TOL)¶

Finds the minimum eigenvalue of a Hermitian Toeplitz matrix

The classical power method is used together with a fast Toeplitz equation solution routine. The eigenvector is normalized to unit length.

Parameters:

Parameters:	T0 – Scalar corresponding to real matrix element t(0) T – Array of M complex matrix elements t(1),...,t(M) C from the left column of the Toeplitz matrix TOL – Real scalar tolerance; routine exits when [ EVAL(k) - EVAL(k-1) ]/EVAL(k-1) < TOL , where the index k denotes the iteration number.
Returns:	EVAL - Real scalar denoting the minimum eigenvalue of matrix EVEC - Array of M complex eigenvector elements associated

T0 – Scalar corresponding to real matrix element t(0)
T – Array of M complex matrix elements t(1),...,t(M) C from the left column of the Toeplitz matrix
TOL – Real scalar tolerance; routine exits when [ EVAL(k) - EVAL(k-1) ]/EVAL(k-1) < TOL , where the index k denotes the iteration number.

Returns:

EVAL - Real scalar denoting the minimum eigenvalue of matrix
EVEC - Array of M complex eigenvector elements associated

Note

External array T must be dimensioned >= M
array EVEC must be >= M+1
Internal array E must be dimensioned >= M+1 .
dependencies
- spectrum.toeplitz.HERMTOEP()

5.5.5.3. levinson ¶

Levinson module

LEVINSON(r[, order, allow_singularity]) Levinson-Durbin recursion.

Code author: Thomas Cokelaer, 2011

LEVINSON(r, order=None, allow_singularity=False)¶

Levinson-Durbin recursion.

Find the coefficients of a length(r)-1 order autoregressive linear process

Parameters:

Parameters:	r – autocorrelation sequence of length N + 1 (first element being the zero-lag autocorrelation) order – requested order of the autoregressive coefficients. default is N. allow_singularity – false by default. Other implementations may be True (e.g., octave)
Returns:	the N+1 autoregressive coefficients $A=(1, a_1...a_N)$ the prediction errors the N reflections coefficients values

r – autocorrelation sequence of length N + 1 (first element being the zero-lag autocorrelation)
order – requested order of the autoregressive coefficients. default is N.
allow_singularity – false by default. Other implementations may be True (e.g., octave)

Returns:

the N+1 autoregressive coefficients $A=(1, a_1...a_N)$
the prediction errors
the N reflections coefficients values

This algorithm solves the set of complex linear simultaneous equations using Levinson algorithm.

$\bold{T}_M \left( \begin{array}{c} 1 \\ \bold{a}_M \end{array} \right) = \left( \begin{array}{c} \rho_M \\ \bold{0}_M \end{array} \right)$

where $\bold{T}_M$ is a Hermitian Toeplitz matrix with elements $T_0, T_1, \dots ,T_M$ .

Note

Solving this equations by Gaussian elimination would require $M^3$ operations whereas the levinson algorithm requires $M^2+M$ additions and $M^2+M$ multiplications.

This is equivalent to solve the following symmetric Toeplitz system of linear equations

$\left( \begin{array}{cccc} r_1 & r_2^* & \dots & r_{n}^*\\ r_2 & r_1^* & \dots & r_{n-1}^*\\ \dots & \dots & \dots & \dots\\ r_n & \dots & r_2 & r_1 \end{array} \right) \left( \begin{array}{cccc} a_2\\ a_3 \\ \dots \\ a_{N+1} \end{array} \right) = \left( \begin{array}{cccc} -r_2\\ -r_3 \\ \dots \\ -r_{N+1} \end{array} \right)$

where $r = (r_1 ... r_{N+1})$ is the input autocorrelation vector, and $r_i^*$ denotes the complex conjugate of $r_i$ . The input r is typically a vector of autocorrelation coefficients where lag 0 is the first element $r_1$ .

>>> import numpy; from spectrum import LEVINSON
>>> T = numpy.array([3., -2+0.5j, .7-1j])
>>> a, e, k = LEVINSON(T)

levdown(anxt, enxt=None)¶

One step backward Levinson recursion

Parameters:

Parameters:	anxt – enxt –
Returns:	acur the P’th order prediction polynomial based on the P+1’th order prediction polynomial, anxt. ecur the the P’th order prediction error based on the P+1’th order prediction error, enxt.

anxt –
enxt –

Returns:

acur the P’th order prediction polynomial based on the P+1’th order prediction polynomial, anxt.
ecur the the P’th order prediction error based on the P+1’th order prediction error, enxt.

levup(acur, knxt, ecur=None)¶

LEVUP One step forward Levinson recursion

Parameters:

Parameters:	acur – knxt –
Returns:	anxt the P+1’th order prediction polynomial based on the P’th order prediction polynomial, acur, and the P+1’th order reflection coefficient, Knxt. enxt the P+1’th order prediction prediction error, based on the P’th order prediction error, ecur.
References :	Stoica R. Moses, Introduction to Spectral Analysis Prentice Hall, N.J., 1997, Chapter 3.

acur –
knxt –

Returns:

anxt the P+1’th order prediction polynomial based on the P’th order prediction polynomial, acur, and the P+1’th order reflection coefficient, Knxt.
enxt the P+1’th order prediction prediction error, based on the P’th order prediction error, ecur.

References :

Stoica R. Moses, Introduction to Spectral Analysis Prentice Hall, N.J., 1997, Chapter 3.

rlevinson(a, efinal)¶

computes the autocorrelation coefficients, R based on the prediction polynomial A and the final prediction error Efinal, using the stepdown algorithm.

Works for real or complex data

Parameters:

Parameters:	a – efinal –
Returns:	R, the autocorrelation U prediction coefficient kr reflection coefficients e errors

a –
efinal –

Returns:

R, the autocorrelation
U prediction coefficient
kr reflection coefficients
e errors

A should be a minimum phase polynomial and A(1) is assumed to be unity.

Returns:	(P+1) by (P+1) upper triangular matrix, U, that holds the i’th order prediction polynomials Ai, i=1:P, where P is the order of the input polynomial, A. [ 1 a1(1)* a2(2)* ..... aP(P) * ] [ 0 1 a2(1)* ..... aP(P-1)* ] U = [ .................................] [ 0 0 0 ..... 1 ]

from which the i’th order prediction polynomial can be extracted using Ai=U(i+1:-1:1,i+1)’. The first row of U contains the conjugates of the reflection coefficients, and the K’s may be extracted using, K=conj(U(1,2:end)).

Todo

remove the conjugate when data is real data, clean up the code test and doc.

5.5.5.4. toeplitz ¶

Toeplitz module

These functions are not yet used by other functions, which explains the lack of documentation, test, examples.

Nevertheless, they can be used for production.

HERMTOEP(T0, T, Z) solve Tx=Z by a variation of Levinson algorithm where T

Code author: Thomas Cokelaer, 2011

HERMTOEP(T0, T, Z)¶

solve Tx=Z by a variation of Levinson algorithm where T is a complex hermitian toeplitz matrix

Parameters:	T0 – zero lag value T – r1 to rN
Returns:	X

used by eigen PSD method

5.5.5.5. linalg ¶

Linear algebra tools

`csvd`(A)	SVD decomposition using numpy.linalg.svd
`corrmtx`(x_input, m[, method])	Correlation matrix
`pascal`(n)	Return Pascal matrix

Code author: Thomas Cokelaer 2011

pascal(n)¶

Return Pascal matrix

Parameters:	n (int) – 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

csvd(A)¶

SVD decomposition using numpy.linalg.svd

Parameters:	A – a M by N matrix
Returns:	U, a M by M matrix S the N eigen values V a N by N matrix

See numpy.linalg.svd() for a detailed documentation.

Should return the same as in [Marple] , CSVD routine.

U, S, V = numpy.linalg.svd(A)
U, S, V = cvsd(A)

corrmtx(x_input, m, method='autocorrelation')¶

Correlation matrix

This function is used by PSD estimator functions. It generates the correlation matrix from a correlation data set and a maximum lag.

Parameters:	x (array) – autocorrelation samples (1D) m (int) – the maximum lag

Depending on the choice of the method, the correlation matrix has different sizes, but the number of rows is always m+1.

Method can be :

‘autocorrelation’: (default) X is the (n+m)-by-(m+1) rectangular Toeplitz matrix derived using prewindowed and postwindowed data.
‘prewindowed’: X is the n-by-(m+1) rectangular Toeplitz matrix derived using prewindowed data only.
‘postwindowed’: X is the n-by-(m+1) rectangular Toeplitz matrix that derived using postwindowed data only.
‘covariance’: X is the (n-m)-by-(m+1) rectangular Toeplitz matrix derived using nonwindowed data.
‘modified’: X is the 2(n-m)-by-(m+1) modified rectangular Toeplitz matrix that generates an autocorrelation estimate for the length n data vector x, derived using forward and backward prediction error estimates.

Returns:	the autocorrelation matrix R, the (m+1)-by-(m+1) autocorrelation matrix estimate `R= X'*X`.

Algorithm details:

The autocorrelation matrix is a $(N+p) \times (p+1)$ rectangular Toeplilz data matrix:

$X_p = \begin{pmatrix}L_p\\T_p\\Up\end{pmatrix}$

where the lower triangular $p \times (p+1)$ matrix $L_p$ is

$L_p = \begin{pmatrix} x[1] & \cdots & 0 & 0 \\ \vdots & \ddots & \vdots & \vdots \\ x[p] & \cdots & x[1] & 0 \end{pmatrix}$

where the rectangular $(N-p) \times (p+1)$ matrix $T_p$ is

$T_p = \begin{pmatrix} x[p+1] & \cdots & x[1] \\ \vdots & \ddots & \vdots \\ x[N-p] & \cdots & x[p+1] \\ \vdots & \ddots & \vdots \\ x[N] & \cdots & x[N-p] \end{pmatrix}$

and where the upper triangular $p \times (p+1)$ matrix $U_p$ is

$U_p = \begin{pmatrix} 0 & x[N] & \cdots & x[N-p+1] \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & x[N] \end{pmatrix}$

From this definition, the prewindowed matrix is

$X_p = \begin{pmatrix}L_p\\T_p\end{pmatrix}$

the postwindowed matrix is

$X_p = \begin{pmatrix}T_p\\U_p\end{pmatrix}$

the covariance matrix is:

$X_p = \begin{pmatrix}T_p\end{pmatrix}$

and the modified covariance matrix is:

$X_p = \begin{pmatrix}T_p\\T_p^*\end{pmatrix}$

5.5.6. Waveforms ¶

morlet(lb, ub, n)¶

Generate the Morlet waveform

The Morlet waveform is defined as follows:

$w[x] = \cos{5x} \exp^{-x^2/2}$

Parameters:	lb – lower bound ub – upper bound n (int) – waveform data samples

from spectrum import *
from pylab import *
plot(morlet(0,10,100))

[hires.png, pdf]

chirp(t, f0=0.0, t1=1.0, f1=100.0, form='linear', phase=0)¶

Evaluate a chirp signal at time t.

A chirp signal is a frequency swept cosine wave.

$a = \pi (f_1 - f_0) / t_1$

$b = 2 \pi f_0$

$y = \cos\left( \pi\frac{f_1-f_0}{t_1} t^2 + 2\pi f_0 t + \rm{phase} \right)$

Parameters:	t (array) – times at which to evaluate the chirp signal f0 (float) – frequency at time t=0 (Hz) t1 (float) – time t1 f1 (float) – frequency at time t=t1 (Hz) form (str) – shape of frequency sweep in [‘linear’, ‘quadratic’, ‘logarithmic’] phase (float) – phase shift at t=0

The parameter form can be:

‘linear’ $f(t) = (f_1-f_0)(t/t_1) + f_0$

‘quadratic’ $f(t) = (f_1-f_0)(t/t_1)^2 + f_0$

‘logarithmic’ $f(t) = (f_1-f_0)^{(t/t_1)} + f_0$

Example:

from spectrum import *
from pylab import *
t = linspace(0, 1, 1000)
y = chirp(t, form='linear')
plot(y)
hold(True)
y = chirp(t, form='quadratic')
plot(y, 'r')

[hires.png, pdf]

mexican(lb, ub, n)¶

Generate the mexican hat wavelet

The Mexican wavelet is:

$w[x] = \cos{5x} \exp^{-x^2/2}$

Parameters:	lb – lower bound ub – upper bound n (int) – waveform data samples
Returns:	the waveform

from spectrum import *
from pylab import *
plot(mexican(0, 10, 100))

[hires.png, pdf]

meyeraux(x)¶

Compute the Meyer auxiliary function

The Meyer function is

$y = 35 x^4-84 x^5+70 x^6-20 x^7$

Parameters:	x (array) –
Returns:	the waveform

from spectrum import *
from pylab import *
t = linspace(0, 1, 1000)
plot(t, meyeraux(t))

[hires.png, pdf]

Front page|Spectrum - Spectral Analysis in Python (0.5.2)

5.5. Tools and classes ¶

5.5.1. Classes ¶

5.5.2. Correlation ¶

5.5.3. Tools ¶

5.5.4. datasets ¶

5.5.5. Linear Algebra Tools ¶

5.5.5.1. cholesky ¶

5.5.5.2. eigen ¶

5.5.5.3. levinson ¶

5.5.5.4. toeplitz ¶

5.5.5.5. linalg ¶

5.5.6. Waveforms ¶

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