Coefficient-Symmetries for Implementing Arbitrary-Order Lagrange-Type Variable Fractional-Delay Digital Filters

In this paper, we first derive an explicit formula for expressing the coefficients of an arbitrary-order Lagrange-type variable fractional-delay (VFD) digital filter as polynomials of the VFD parameter , and then develop some useful symmetries for both even- and odd-order Lagrange-type VFD filter coefficients. The coefficient-symmetries facilitate the evaluations of VFD filter coefficients as well as variable frequency responses with reduced computational complexity. More importantly, the coefficient-symmetries can be exploited for efficiently implementing both even- and odd-order Lagrange-type VFD filters as Farrow structure and a more efficient one called even-odd structure such that the subfilters have symmetric or anti-symmetric coefficients, which saves the storage for VFD filter coefficients and reduces the number of multiplications required in VFD filtering process by almost 50%. Therefore, exploiting the coefficient-symmetries not only speeds up the VFD filtering, but also reduces implementation cost.