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The Haar System as an Example of DWT

Module by: Phil Schniter. E-mail the author

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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 Equation 1 and Figure 1.

φt=1if0≤t<10otherwise φ t 1 0 t 1 0 (1)

Figure 1
Figure 1 (scalingfn.png)

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

φ k , n t=∀k,n,k∈ℤn∈ℤ:2-k2φ2-kt−n=2-k2φ12kt−n2k 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 (2)

Figure 2
Figure 2 (shiftandstretchfn.png)

which are illustrated in Figure 3 for various kk and nn. Equation 2 makes clear the principle that incrementing nn by one shifts the pulse one place to the right. Observe from Figure 3 that { φ k , n t|n∈ℤ} φ k , n t n is orthonormal for each kk (i.e., along each row of figures).

Figure 3
Figure 3 (haarscalingfn.png)

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