The Haar System as an Example of DWT 2.10 2002/01/07 2005/10/05 15:28:46.647 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu Discrete Wavelet Transform Haar scaling function wavelet (Blank Abstract) The Haar basis is perhaps the simplest example of a DWT basis, and we will frequently refer to it in our DWT development. Keep in mind, however, that the Haar basis is only an example; there are many other ways of constructing a DWT decomposition. For the Haar case, the mother scaling function is defined by and . φ t 1 0 t 1 0

From the mother scaling function, we define a family of shifted and stretched scaling functions φ k , n t according to and φ k , n t k n k n 2 k 2 φ 2 k t n 2 k 2 φ 1 2 k t n 2 k

which are illustrated in for various k and n. makes clear the principle that incrementing n by one shifts the pulse one place to the right. Observe from that φ k , n t n is orthonormal for each k (i.e., along each row of figures).