In this paper, the concept of the dyadic shift invariance (DSI) and cyclic shift invariant (CSI) functions are proposed. First, basic properties of the DSI and CSI functions are presented. Then, we can show that the Walsh-Hadamard transform (WHT) and discrete Fourier transform (DFT) functions 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. We will show that the properties of the WHT and DFT are the consequence of the basic principles of the DSI and CSI functions.