Multiscale system theory

A system theory based on the homogeneous dyadic tree as a possible foundation for a multiscale system theory and multiscale statistical signal processing is developed. Multiscale representations, including wavelet transforms, homogeneous trees, shift operations, and transfer functions, are discussed. It is shown that the homogeneous tree possesses strange geometric properties that have the following consequence: the double role played by the classical z-transform, namely, encoding transfer function and defining stationarity, is split between two different objects-the shifts to encode transfer functions (these are not isometries) and the translations to define stationarity (these are not easily expressed by shifts). Two system theories are sketched that emphasize each of these two different objects. Finally, a notion of stationary stochastic processes is introduced