12.1 The haar system as an example of dwt

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 .

From the mother scaling function, we define a family of shifted and stretched scaling functions $\{{\phi }_{k,n}(t)\}$ according to and

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 $\{{\phi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ is orthonormal for each $k$ ( i.e. , along each row of figures).