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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 .
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
$k$ (
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