lazy_lpc
Module¶
Linear Predictive Coding (LPC) module
Summary of module contents:
Name | Description |
---|---|
ParCorError | Error when trying to find the partial correlation coefficients (reflection coefficients) and there’s no way to find them. |
toeplitz | Find the toeplitz matrix as a list of lists given its first line/column. |
levinson_durbin | Solve the Yule-Walker linear system of equations. |
lpc | This is a StrategyDict instance object called lpc .
Strategies stored: 5. |
parcor | Find the partial correlation coefficients (PARCOR), or reflection coefficients, relative to the lattice implementation of a given LTI FIR LinearFilter with a constant denominator (i.e., LPC analysis filter, or any filter without feedback). |
parcor_stable | Tests whether the given filter is stable or not by using the partial correlation coefficients (reflection coefficients) of the given filter. |
lsf | Find the Line Spectral Frequencies (LSF) from a given FIR filter. |
lsf_stable | Tests whether the given filter is stable or not by using the Line Spectral Frequencies (LSF) of the given filter. Needs NumPy. |
-
exception
ParCorError
[source]¶ Bases:
ZeroDivisionError
Error when trying to find the partial correlation coefficients (reflection coefficients) and there’s no way to find them.
-
levinson_durbin
(acdata, order=None)[source]¶ Solve the Yule-Walker linear system of equations.
They’re given by:
\[R . a = r\]where \(R\) is a simmetric Toeplitz matrix where each element are lags from the given autocorrelation list. \(R\) and \(r\) are defined (Python indexing starts with zero and slices don’t include the last element):
\[ \begin{align}\begin{aligned}R[i][j] = acdata[abs(j - i)]\\r = acdata[1 : order + 1]\end{aligned}\end{align} \]Parameters: - acdata – Autocorrelation lag list, commonly the
acorr
function output. - order – The order of the resulting ZFilter object. Defaults to
len(acdata) - 1
.
Returns: A FIR filter, as a ZFilter object. The mean squared error over the given data (variance of the white noise) is in its “error” attribute.
See also
- Examples:
>>> data = [2, 2, 0, 0, -1, -1, 0, 0, 1, 1] >>> acdata = acorr(data) >>> acdata [12, 6, 0, -3, -6, -3, 0, 2, 4, 2] >>> ldfilt = levinson_durbin(acorr(data), 3) >>> ldfilt 1 - 0.625 * z^-1 + 0.25 * z^-2 + 0.125 * z^-3 >>> ldfilt.error # Squared! See lpc for more information about this 7.875
The Levinson-Durbin algorithm used to solve the equations needs \(O(order^2)\) floating point operations.
- acdata – Autocorrelation lag list, commonly the
-
parcor
(fir_filt)[source]¶ Find the partial correlation coefficients (PARCOR), or reflection coefficients, relative to the lattice implementation of a given LTI FIR LinearFilter with a constant denominator (i.e., LPC analysis filter, or any filter without feedback).
Parameters: fir_filt – A ZFilter object, causal, LTI and with a constant denominator. Returns: A generator that results in each partial correlation coefficient from iterative decomposition, reversing the Levinson-Durbin algorithm. - Examples:
>>> filt = levinson_durbin([1, 2, 3, 4, 5, 3, 2, 1]) >>> filt 1 - 0.275 * z^-1 - 0.275 * z^-2 - 0.4125 * z^-3 + 1.5 * z^-4 - 0.9125 * z^-5 - 0.275 * z^-6 - 0.275 * z^-7 >>> round(filt.error, 4) 1.9125 >>> k_generator = parcor(filt) >>> k_generator <generator object parcor at ...> >>> [round(k, 7) for k in k_generator] [-0.275, -0.3793103, -1.4166667, -0.2, -0.25, -0.3333333, -2.0]
See also
levinson_durbin
- Levinson-Durbin algorithm for solving Yule-Walker equations (Toeplitz matrix linear system).
-
parcor_stable
(filt)[source]¶ Tests whether the given filter is stable or not by using the partial correlation coefficients (reflection coefficients) of the given filter.
Parameters: filt – A LTI filter as a LinearFilter object. Returns: A boolean that is true only when all correlation coefficients are inside the unit circle. Critical stability (i.e., when outer coefficient has magnitude equals to one) is seem as an instability, and returns False. See also
parcor
- Partial correlation coefficients generator.
lsf_stable
- Tests filter stability with Line Spectral Frequencies (LSF) values.
-
lsf
(fir_filt)[source]¶ Find the Line Spectral Frequencies (LSF) from a given FIR filter.
Parameters: filt – A LTI FIR filter as a LinearFilter object. Returns: A tuple with all LSFs in rad/sample, alternating from the forward prediction and backward prediction filters, starting with the lowest LSF value.
-
lsf_stable
(filt)[source]¶ Tests whether the given filter is stable or not by using the Line Spectral Frequencies (LSF) of the given filter. Needs NumPy.
Parameters: filt – A LTI filter as a LinearFilter object. Returns: A boolean that is true only when the LSF values from forward and backward prediction filters alternates. Critical stability (both forward and backward filters has the same LSF value) is seem as an instability, and returns False. See also
lsf
- Gets the Line Spectral Frequencies from a filter. Needs NumPy.
parcor_stable
- Tests filter stability with partial correlation coefficients (reflection coefficients).
lazy_lpc.lpc
StrategyDict¶
This is a StrategyDict instance object called
lpc
. Strategies stored: 5.
Strategy lpc.autocor (Default).
Aliases available are lpc.acorr
, lpc.autocorrelation
, lpc.auto_correlation
.
Docstring starts with:
Find the Linear Predictive Coding (LPC) coefficients as a ZFilter object, the analysis whitening filter. This implementation uses the autocorrelation method, using the Levinson-Durbin algorithm or Numpy pseudo-inverse for linear system solving, when needed.
Strategy lpc.covar.
Aliases available are lpc.cov
, lpc.covariance
, lpc.ncovar
, lpc.ncov
, lpc.ncovariance
.
Docstring starts with:
Find the Linear Predictive Coding (LPC) coefficients as a ZFilter object, the analysis whitening filter. This implementation uses the covariance method, assuming a zero-mean stochastic process, using numpy.linalg.pinv as a linear system solver.
Strategy lpc.kautocor.
Aliases available are lpc.kacorr
, lpc.kautocorrelation
, lpc.kauto_correlation
.
Docstring starts with:
Find the Linear Predictive Coding (LPC) coefficients as a ZFilter object, the analysis whitening filter. This implementation uses the autocorrelation method, using the Levinson-Durbin algorithm.
Strategy lpc.kcovar.
Aliases available are lpc.kcov
, lpc.kcovariance
.
Docstring starts with:
Find the Linear Predictive Coding (LPC) coefficients as a ZFilter object, the analysis whitening filter. This implementation is based on the covariance method, assuming a zero-mean stochastic process, finding the coefficients iteratively and greedily like the lattice implementation in Levinson-Durbin algorithm, although the lag matrix found from the given block don’t have to be toeplitz. Slow, but this strategy don’t need NumPy.
Strategy lpc.nautocor.
Aliases available are lpc.nacorr
, lpc.nautocorrelation
, lpc.nauto_correlation
.
Docstring starts with:
Find the Linear Predictive Coding (LPC) coefficients as a ZFilter object, the analysis whitening filter. This implementation uses the autocorrelation method, using numpy.linalg.pinv as a linear system solver.
Note
This docstring is self-generated, see the StrategyDict class and the strategies docs for more details.
-
covar
(blk, order=None)¶ Find the Linear Predictive Coding (LPC) coefficients as a ZFilter object, the analysis whitening filter. This implementation uses the covariance method, assuming a zero-mean stochastic process, using numpy.linalg.pinv as a linear system solver.
-
autocor
(blk, order=None)¶ Find the Linear Predictive Coding (LPC) coefficients as a ZFilter object, the analysis whitening filter. This implementation uses the autocorrelation method, using the Levinson-Durbin algorithm or Numpy pseudo-inverse for linear system solving, when needed.
Parameters: - blk – An iterable with well-defined length. Don’t use this function with Stream objects!
- order – The order of the resulting ZFilter object. Defaults to
len(blk) - 1
.
Returns: A FIR filter, as a ZFilter object. The mean squared error over the given block is in its “error” attribute.
Hint
See
lpc.kautocor
example, which should apply equally for this strategy.See also
levinson_durbin
- Levinson-Durbin algorithm for solving Yule-Walker equations (Toeplitz matrix linear system).
lpc.nautocor
- LPC coefficients from linear system solved with Numpy pseudo-inverse.
lpc.kautocor
- LPC coefficients obtained with Levinson-Durbin algorithm.
-
kcovar
(blk, order=None)¶ Find the Linear Predictive Coding (LPC) coefficients as a ZFilter object, the analysis whitening filter. This implementation is based on the covariance method, assuming a zero-mean stochastic process, finding the coefficients iteratively and greedily like the lattice implementation in Levinson-Durbin algorithm, although the lag matrix found from the given block don’t have to be toeplitz. Slow, but this strategy don’t need NumPy.
-
kautocor
(blk, order=None)¶ Find the Linear Predictive Coding (LPC) coefficients as a ZFilter object, the analysis whitening filter. This implementation uses the autocorrelation method, using the Levinson-Durbin algorithm.
Parameters: - blk – An iterable with well-defined length. Don’t use this function with Stream objects!
- order – The order of the resulting ZFilter object. Defaults to
len(blk) - 1
.
Returns: A FIR filter, as a ZFilter object. The mean squared error over the given block is in its “error” attribute.
- Examples:
>>> data = [-1, 0, 1, 0] * 4 >>> len(data) # Small data 16 >>> filt = lpc.kautocor(data, 2) >>> filt # The analysis filter 1 + 0.875 * z^-2 >>> filt.numerator # List of coefficients [1, 0.0, 0.875] >>> filt.error # Prediction error (squared!) 1.875
See also
levinson_durbin
- Levinson-Durbin algorithm for solving Yule-Walker equations (Toeplitz matrix linear system).
lpc.autocor
- LPC coefficients by using one of the autocorrelation method strategies.
lpc.nautocor
- LPC coefficients from linear system solved with Numpy pseudo-inverse.
-
nautocor
(blk, order=None)¶ Find the Linear Predictive Coding (LPC) coefficients as a ZFilter object, the analysis whitening filter. This implementation uses the autocorrelation method, using numpy.linalg.pinv as a linear system solver.
Parameters: - blk – An iterable with well-defined length. Don’t use this function with Stream objects!
- order – The order of the resulting ZFilter object. Defaults to
len(blk) - 1
.
Returns: A FIR filter, as a ZFilter object. The mean squared error over the given block is in its “error” attribute.
Hint
See
lpc.kautocor
example, which should apply equally for this strategy.See also
lpc.autocor
- LPC coefficients by using one of the autocorrelation method strategies.
lpc.kautocor
- LPC coefficients obtained with Levinson-Durbin algorithm.