On Closed Form Calculation of Line Spectral Frequencies (LSF)

The mathematical theory of closed form functions for calculating LSFs on the basis of generating functions is presented. Exploiting recurrence relationships in the series expansion of Chebyshev polynomials of the first kind makes it possible to bootstrap iterative LSF-search from a set of characteristic polynomial zeros. The theoretical analysis is based on decomposition of sequences into symmetric and anti-symmetric polynomials defined as a series expansion of reduced Chebyshev polynomials of the first kind. Two variants of closed form functions are presented — each characterised by using a recurrence relationship in Chebyshev polynomials. The first exploits the well known three terms recurrence relationships of Chebyshev polynomials. The second hitherto unused recurrence properties of Chebyshev coefficients defining a set of coefficients and zeros used for bootstrapping calculation of LSFs. The theory is tested using bootstrapped calculation of zeros and by evaluating the complexity of the closed form function. The results of the lower complexity calculations show that real axis zeros are within a given iteration tolerance when compared to results of a standard root-finder.

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