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Merge pull request #68 from SuperKogito/fix-lpcc
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fix LPCC init and LPC error computation
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@SuperKogito
SuperKogito authored May 29, 2024
2 parents af577d7 + 426d2db commit a57ebf5
Showing 1 changed file with 51 additions and 54 deletions.
105 changes: 51 additions & 54 deletions spafe/features/lpc.py
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Original file line number Diff line number Diff line change
Expand Up @@ -36,49 +36,6 @@ def __lpc_helper(frame, order):
Returns:
- (numpy.ndarray) : linear prediction coefficents (lpc coefficents: a).
- (numpy.ndarray) : the error term is the square root of the squared prediction error (e**2).
Note:
The premis of linear predictive analysis is that the nth sample can be estimated by a
linear combination of the previous p samples:
.. math::
xp[n] = -a[1] * x[n-1] - ... -a[k] * x[n-k] ... - a[p] * x[n-p] = - \\sum_{k=1}^{p+1} a_{k} . x[n-k]
where xp is the predicted signal. a_{1},.., a_{p} are known as the predictor
coefficents and p is called the model order and n is the sample index.
Based on the previous equation, we can estimate the prediction error as follows [Ucl-brain]_:
.. math::
e[n] = x[n] - xp[n] \\implies x[n] = e[n] - \\sum_{k=1}^{p+1} a_{k} . x[n-k]
The unknown here are the LP coefficients a, hence we need to minimize e to find those.
We can further rewrite the previous equations for all samples [Collomb]_:
.. math::
E = \\sum_{i=1}^{N} (x[i] - (-\\sum_{k=1}^{p+1} a_{k} . x[i-k])) \\text{for x\\in[1,p]}
All the previous steps can be presented in a matrix, which is a toeplitz matrix: R.A = 0
_ _
-r[1] = r[0] r[1] ... r[p-1] a[1]
: : : : :
: : : _ * :
-r[p] = r[p-1] r[p-2] ... r[0] a[p]
To solve this, one can use the Levinson-Durbin, which is a well-known
algorithm to solve the Hermitian toeplitz with respect to a. Using the
special symmetry in the matrix, the inversion can be done in O(p^2)
instead of O(p^3).
References:
.. [Darconis] : Draconi, Replacing Levinson implementation in scikits.talkbox,
Stackoverflow, https://stackoverflow.com/a/43457190/6939324
.. [Cournapeau] : David Cournapeau D. talkbox, https://github.com/cournape/talkbox
.. [Menares] : Menares E. F. M., ML-experiments, https://github.com/erickfmm/ML-experiments
.. [Collomb] : Collomb C. Linear Prediction and Levinson-Durbin Algorithm, 03.02.2009,
<https://www.academia.edu/8479430/Linear_Prediction_and_Levinson-Durbin_Algorithm_Contents>
.. [Ucl-brain] : Ucl psychology and language sciences, Faculty of brain Sciences, Unit 8 linear prediction
<https://www.phon.ucl.ac.uk/courses/spsci/dsp/lpc.html>
"""
p = order + 1
r = np.zeros(p, frame.dtype)
Expand Down Expand Up @@ -128,6 +85,49 @@ def lpc(
Architecture of linear prediction components extraction algorithm.
The premis of linear predictive analysis is that the nth sample can be estimated by a
linear combination of the previous p samples:
.. math::
xp[n] = -a[1] * x[n-1] - ... -a[k] * x[n-k] ... - a[p] * x[n-p] = - \\sum_{k=1}^{p+1} a_{k} . x[n-k]
where xp is the predicted signal. a_{1},.., a_{p} are known as the predictor
coefficents and p is called the model order and n is the sample index.
Based on the previous equation, we can estimate the prediction error as follows [Ucl-brain]_:
.. math::
e[n] = x[n] - xp[n] \\implies x[n] = e[n] - \\sum_{k=1}^{p+1} a_{k} . x[n-k]
The unknown here are the LP coefficients a, hence we need to minimize e to find those.
We can further rewrite the previous equations for all samples [Collomb]_:
.. math::
E = \\sum_{i=1}^{N} (x[i] - (-\\sum_{k=1}^{p+1} a_{k} . x[i-k])) \\text{for x\\in[1,p]}
All the previous steps can be presented in a matrix, which is a toeplitz matrix: R.A = 0
_ _
-r[1] = r[0] r[1] ... r[p-1] a[1]
: : : : :
: : : _ * :
-r[p] = r[p-1] r[p-2] ... r[0] a[p]
To solve this, one can use the Levinson-Durbin, which is a well-known
algorithm to solve the Hermitian toeplitz with respect to a. Using the
special symmetry in the matrix, the inversion can be done in O(p^2)
instead of O(p^3).
References:
.. [Darconis] : Draconi, Replacing Levinson implementation in scikits.talkbox,
Stackoverflow, https://stackoverflow.com/a/43457190/6939324
.. [Cournapeau] : David Cournapeau D. talkbox, https://github.com/cournape/talkbox
.. [Menares] : Menares E. F. M., ML-experiments, https://github.com/erickfmm/ML-experiments
.. [Collomb] : Collomb C. Linear Prediction and Levinson-Durbin Algorithm, 03.02.2009,
<https://www.academia.edu/8479430/Linear_Prediction_and_Levinson-Durbin_Algorithm_Contents>
.. [Ucl-brain] : Ucl psychology and language sciences, Faculty of brain Sciences, Unit 8 linear prediction
<https://www.phon.ucl.ac.uk/courses/spsci/dsp/lpc.html>
Examples
.. plot::
Expand Down Expand Up @@ -175,7 +175,7 @@ def lpc(
a_mat[i, :] = a
e_vec[i] = e

return np.array(a_mat), np.sqrt(e_vec)
return np.array(a_mat), np.array(e_vec)


def lpc2lpcc(a, e, nceps):
Expand All @@ -196,7 +196,7 @@ def lpc2lpcc(a, e, nceps):
C_{m}=\\left\\{\\begin{array}{l}
log_{e}(p), & \\text{if } m = 0 \\\\
a_{m} + \\sum_{k=1}^{m-1} \\frac{k}{m} C_{m} a_{m-k} , & \\text{if } 1 < m < p \\\\
a_{m} + \\sum_{k=1}^{m-1} \\frac{k}{m} C_{m} a_{m-k} , & \\text{if } 1 <= m <= p \\\\
\\sum_{k=m-p}^{m-1} \\frac{k}{m} C_{m} a_{m-k} , & \\text{if } m > p \\end{array}\\right.
References:
Expand All @@ -207,19 +207,16 @@ def lpc2lpcc(a, e, nceps):
SpringerBriefs in Electrical and Computer Engineering. doi:10.1007/978-3-319-17163-0
"""
p = len(a)
c = [0 for i in range(nceps)]
c = [1] * nceps

c[0] = np.log(zero_handling(e))
c[1:p] = [
a[m] + sum([(k / m) * c[k] * a[m - k] for k in range(1, m)])
for m in range(1, p)
]

for m in range(1, p):
c[m] = a[m] + sum([(k / m) * c[k] * a[m - k] for k in range(1, m)])

if nceps > p:
c[p:nceps] = [
sum([(k / m) * c[k] * a[m - k] for k in range(m - p, m)])
for m in range(p, nceps)
]
for m in range(p + 1, nceps):
c[m] = sum([(k / m) * c[k] * a[m - k] for k in range(m - p, m)])

return c

Expand Down

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