An Introduction to the Mathematics of Linear Predictive Filtering as Applied to Speech Analysis and Synthesis
MASSACHUSETTS INST OF TECH LEXINGTON LINCOLN LAB
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The note presents a tutorial survey of the mathematics that is used in the study of linear predictive filtering as applied to the analysis and synthesis of speech. Speech is modelled as the output of an all-pole filter that is driven by either a periodic pulse train or white noise. A minimum mean- squared-error technique for estimating the coefficients of this filter from speech data is presented. This technique leads to a set of equations for the coefficient estimates which can be solved by a computationally efficient recursive technique known as Levinsons method. The filter derived by the above mentioned technique can be realized by any standard technique however, a particularly interesting realization is in terms of a digital simulation of a non-uniform acoustic tube. It is shown that any stable all-pole filter can be realized as an acoustic tube and, moreover, that the Levinson recursion produces as a by-product exactly the reflection coefficients needed for such a realization. The report concludes by showing how the classical theory of orthogonal polynomials can be applied to the speech analysissynthesis problem and used to derive many of the results obtained above by other means.
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