Symmetric Structures for Odd-Order Maximally Flat and Weighted-Least-Squares Variable Fractional-Delay Filters

Our previous work has shown that coefficient symmetry can be exploited for designing high-accuracy and low-complexity even-order variable fractional-delay (VFD) filters. The objectives of this paper are twofold; One is to derive a coefficient symmetry for odd-order VFD filters, and then formulate a closed-form weighted-least-squares (WLS) design. Exploiting the coefficient symmetry along with different-order subfilters reduces the number of independent VFD filter coefficients by more than 50% , and thus reduces the hardware cost and the number of multipliers by more than 50%. Another objective is to briefly derive the maximally flat (MF) VFD filters from Nth-degree polynomial interpolation, and then present two kinds of transformation matrices for transforming causal odd-order MF VFD filters into symmetric structures such that the filter complexity can be reduced. As a result, both WLS and MF VFD filters can be implemented as the Farrow structure with significantly reduced complexity, which facilitates high-speed VFD filtering. Various examples are given to illustrate the effectiveness of the symmetry-based design and implementations.