Savitzky-Golay Smoothing for Multivariate Cyclic Measurement Data

This paper extends Savitzky-Golay smoothing to cyclic multivariate data. An algebraic framework is presented that permits the synthesis and analysis of polynomial approximations. A new generalized recurrence relationship is derived, which can be used to synthesize orthogonal polynomials from a set of arbitrary nodes lying in the complex plane. It is proved that any weighted polynomial whose weighting function is positive definite can be synthesized from the Gram polynomials using Cholesky factorization. The linear operator required for cyclic local polynomial approximation are circulant; this enables efficient computation via the FFT and ensures that the Gibbs error is invariant to the position of the discontinuity in the data. The new algebraic framework enables the direct computation of the frequency response of the filter from the rows of the linear operator matrices.