Levinson Durbin Recursion

The matrix given in Equation 12.24 can of course be solved by any matrix inversion technique. Such techniques can be slow however (usually of the order of where p is the dimensionality of the matrix) and hence faster techniques have been developed to find the values of ak from the autocorrelation functions R k). In particular, it can be shown that the Levinson-Durbin recursion technique can solve Equation 12.28 in p time. For our purposes, analysis speed is really not an issue, and so we will forgo a detailed discussion of this and other related techniques. However, a brief overview of the technique is interesting in that it sheds light on the relationship between linear prediction and the all-pole tube model discussed in Chapter 11. 

The Levinson-Durbin recursion operates by considering a set of initialisation conditions, and using these to find the coefficients of a first order polynomial (that is, just a single term in z which minimise the mean squared error. From this we find the coefficients of a second order polynomial, third order and so on, using the previous polynomial and minimisation of the error, until we have reached a polynomial of the required filter order P. 

Recall that the key equation in the Levinson-Durbin recursion is the step that relates a predictor coefficient aj to the quantity k and the previous predictor coefficient ay i given in Equation 12.30. 

The significance of all this can be seen if we compare Equation 12.35 with Equation 11.27. Recall that Equation 11.27 was used to find the transfer function for the tube model from the reflection coefficients, in the special case where losses only occurred at the lips. The equations are in fact identical if we set r/c = —kt. This means that as a by product of Levinson-Durbin recursion, we can in fact easily determine the reflection coefficients which would give rise to the tube model having the same transfer function to that of the LP model. This is a particularly nice result not only have we shown that the tube model is all-pole and that the LP model is all pole we have now in fact shown that the two models are in fact equivalent and hence we can find the various parameters of the tube directly from LP analysis. 

To estimate this conditional correlation measure, a backward recursion (backward forecast) is applied, the so-called Durbin-Levinson-algorithm. Again, if the PACF value for a specific lag exceeds a critical bound, a significant AR-like correlation is to be... 

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