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Sunday, 9 February 2014

LPCs using the Burg method in Julia

This post will discuss the Burg method of computing LPCs. Other methods include the Autocorrelation method, the Covariance method and the Modified Covariance method.

The burg method builds on the Modified Covariance (MCOV) method. The prediction error is defined in exactly the same way, however it may be that the MCOV method produces an unstable filter because it uses unconstrained optimisation to obtain the LPC coefficients. The burg method minimises the prediction error under the Levinson constraint:

\[ a_{p,k} = a_{p-1,k} + a_{p,p}a_{p-1,p-k}\]

Lets define forward and backward prediction error as:

\[f_p = \sum_{k=0}^p a_{p,k}x(n+p-k) \]

and

\[b_p = \sum_{k=0}^p a_{p,k}x(n+k) \]

Minimisation of the prediction error power with respect to \(a_{p,p}\) leads to

\[ a_{p,p} = -\dfrac{2 \sum_{n=0}^{N-p-1} b_{p-1}(n) f_{p-1}(n+1)}{\sum_{n=0}^{N-p-1} \left(b_{p-1}^2(n) + f_{p-1}^2(n+1)\right)} \]

This leads to a recursive equation described below.

Initialise:


\[ P_0 = \dfrac{1}{N}\sum_{n=0}^{N-1}x^2(n) \] \[ f_0(n) = b_0(n) = x(n), \qquad n = 0,1,\ldots,N-1\]

For \(p = 1,2,\ldots,L\):


\[ a_{p,p} = -\dfrac{2 \sum_{n=0}^{N-p-1} b_{p-1}(n) f_{p-1}(n+1)}{\sum_{n=0}^{N-p-1} \left(b_{p-1}^2(n) + f_{p-1}^2(n+1)\right)} \] \[ a_{p,k} = a_{p-1,k} + a_{p,p}a_{p-1,p-k}, \qquad k=1,2,\ldots,p-1\] \[ P_p = P_{p-1}(1-a_{p,p}^2) \] \[ f_p(n) = f_{p-1}(n+1) + a_{p,p}b_{p-1}(n)\] \[ b_p(n) = b_{p-1}(n) + a_{p,p}f_{p-1}(n+1)\]

Where \(G_L^2 = P_L \). A julia implementation of this algorithm:

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