Explicit construction of symmetric orthogonal wavelet frames in [formula omitted

Abstract

Recently, some researchers propose the concept of orthogonal wavelet frames, which are useful for multiple access communication systems. In this article, we first give two explicit algorithms for constructing paraunitary symmetric matrices ( p.s.m. for short), whose entries are symmetric or antisymmetric Laurent polynomials. We also give two algorithms for constructing orthogonal wavelet frames from existing tight or dual wavelet frames in L 2 ( R s ) . The constructed orthogonal wavelet frames are also tight or dual ones. Furthermore, based on the constructed p.s.m. and the existing symmetric tight (dual) wavelet frames, we can obtain symmetric orthogonal ( s.o. for short) tight (dual) wavelet frames in L 2 ( R s ) . From the constructed s.o. wavelet frames in L 2 ( R s ) , we can obtain s.o. wavelet frames in L 2 ( R m ) by the projection method, where m ≤ s . To illustrate our results, we construct s.o. wavelet frames in L 2 ( R ) and L 2 ( R 2 ) from the quadratic B-spline B 3 ( x ) . Especially, in Example 2, we obtain nonseparable tight 2 I 2 -wavelet frames in L 2 ( R 2 ) from a separable tight 2 I 2 -wavelet frame constructed by tensor product.