A simple and unified proof of dyadic shift invariance and the extension to cyclic shift invariance

A simple and unified proof of the dyadic shift invariance and an extension to cyclic shift invariance are presented. First, the concept of the dyadic shift invariance (DSI) and cyclic shift invariant (CSI) functions is proposed. Basic properties of the DSI and CSI functions are considered. Then it is shown that the Walsh-Hadamard transform (WHT) and discrete Fourier transform (DFT) are, in fact, special cases of the DSI and CSI functions, respectively. Many properties of the WHT and DFT can then be obtained easily from DSI and CSI points of view. The proposed unified approach is simple and rigorous. It is also shown that the properties of the WHT and DFT are the consequence of the basic principles of the DSI and CSI functions