# 12.1 The haar system as an example of dwt

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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 .

$\phi (t)=\begin{cases}1 & \text{if 0\le t< 1}\\ 0 & \text{otherwise}\end{cases}$

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

${\phi }_{k,n}(t)=\forall k, n, k\in \mathbb{Z}n\in \mathbb{Z}\colon 2^{-\left(\frac{k}{2}\right)}\phi (2^{-k}t-n)=2^{-\left(\frac{k}{2}\right)}\phi (\frac{1}{2^{k}}(t-n2^{k}))$

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

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