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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 .

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).

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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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