Symmetric orthogonal filters and wavelets with linear-phase moments

In this paper we study symmetric orthogonal filters with linear-phase moments, which are of interest in wavelet analysis and its applications. We investigate relations and connections among the linear-phase moments, sum rules, and symmetry of an orthogonal filter. As one of the results, we show that if a real-valued orthogonal filter aa is symmetric about a point, then aa has sum rules of order mm if and only if it has linear-phase moments of order 2m2m. These connections among the linear-phase moments, sum rules, and symmetry help us to reduce the computational complexity of constructing symmetric real-valued orthogonal filters, and to understand better symmetric complex-valued orthogonal filters with linear-phase moments. To illustrate the results in the paper, we provide many examples of univariate symmetric orthogonal filters with linear-phase moments. In particular, we obtain an example of symmetric real-valued 4-orthogonal filters whose associated orthogonal 4-refinable function lies in C2(R)C2(R).

► We study the linear-phase moment, sum rule, and symmetry of a complex orthogonal filter. ► We prove the relation lpm(a)=2sr(a)lpm(a)=2sr(a) for any real-valued symmetric M-orthogonal filter aa. ► We give an algorithm for constructing symmetric orthogonal filters with linear-phase moments. ► We present many examples of symmetric M-orthogonal filters with linear-phase moments. ► We obtain a symmetric real-valued 4-orthogonal filter with a C2C2 refinable function.