A state space approach to the analysis of frames and wavelets
Xiaofang Chen · 2011
Frames exist in source coding, robust transmission and signal reconstruction, etc.Frames introduce redundancies, yielding more flexibility in designs.Mathematically, a frame of a vector space with finite inner product allows each element in the vector space to be written as a linear combination of elements in the frame.The frame elements may be dependent or redundant.From linear system theory point of view, frames can be represented by linear operators, which may be associated with state space representations.Such frames may be modeled as linear time-invariant systems or linear time-varying systems with state space representations.Frames are often associated with wavelet frames, which possess certain structures.One typical wavelet frame is realized in a discrete wavelet transform (DWT).The DWT partitions an input signal into several bands, where each band is in certain vector space.An inverse DWT (IDWT) reconstructs the input signal from signals in all bands.The IDWT system is an inverse or pseudo-inverse system of the DWT system.Typically the DWT employs an analysis tree-structure multirate filter bank (FB), which may be regarded as the adjoint of the pre-frame operator of the underlying frame.The IDWT may be regarded as the pre-frame operator (synthesis operator) of the underlying dual frame.Some IDWTs of certain wavelets such as Butterworth wavelets, Haar wavelets, Daubechies wavelets, Spline wavelets, employ synthesis tree-structure multirate FBs.State space realizations of such wavelet frames and dual frames can be represented in terms of state space matrices of the