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# Haar Scaling Function and Wavelet

Module by: Phil Schniter. E-mail the author

Summary: (Blank Abstract)

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## 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

φt={1  if  0t<10  otherwise   φ t 1 0 t 1 0
(1)

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

φ k , n t=k,n,kZnZ:2k2φ2ktn=2k2φ12k(tn2k) 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)

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 nZ φ k , n t n is orthonormal for each kk (i.e., along each row of figures).

## A Hierarchy of Detail

Given a mother scaling function φt 2 φ t 2 — the choice of which will be discussed later — let us construct scaling functions at "coarseness-level-k" and "shift-n" as follows: φ k , n t=2k2φ2ktn . φ k , n t 2 k 2 φ 2 k t n . Let us then use V k V k to denote the subspace defined by linear combinations of scaling functions at the kkth level: V k =span φ k , n t nZ . V k span φ k , n t n . In the Haar system, for example, V 0 V 0 and V 1 V 1 consist of signals with the characteristics of x 0 t x 0 t and x 1 t x 1 t illustrated in Figure 4, respectively.

We will be careful to choose a scaling function φt φ t which ensures that the following nesting property is satisfied: V 2 V 1 V 0 V -1 V -2 V 2 V 1 V 0 V -1 V -2 coarse detailed coarse detailed In other words, any signal in V k V k can be constructed as a linear combination of more detailed signals in V k 1 V k 1 . (The Haar system gives proof that at least one such φt φ t exists.) The nesting property can be depicted using the set-theoretic diagram, Figure 5, where V 1 V 1 is represented by the contents of the largest egg (which includes the smaller two eggs), V 0 V 0 is represented by the contents of the medium-sized egg (which includes the smallest egg), and V 1 V 1 is represented by the contents of the smallest egg.

Going further, we will assume that φt φ t is designed to yield the following three important properties:

1. φ k , n t nZ φ k , n t n constitutes an orthonormal basis for V k V k ,
2. V =0 V 0 (contains no signals). 1
3. V = 2 V 2 (contains all signals).
Because φ k , n t nZ φ k , n t n is an orthonormal basis, the best (in 2 2 norm) approximation of xt 2 x t 2 at coarseness-level-k is given by the orthogonal projection, Figure 6
x k t=n= c k , n φ k , n t x k t n c k , n φ k , n t
(3)
c k , n = φ k , n t,xt c k , n φ k , n t x t
(4)

We will soon derive conditions on the scaling function φt φ t which ensure that the properties above are satisfied.

## Haar Approximation at the kth Coarseness Level

It is instructive to consider the approximation of signal xt 2 x t 2 at coarseness-level-k of the Haar system. For the Haar case, projection of xt 2 x t 2 onto V k V k is accomplished using the basis coeficients

c k , n = φ k , n txtdt=n2k(n+1)2k2k2xtdt c k , n t φ k , n t x t t n 2 k n 1 2 k 2 k 2 x t
(5)
giving the approximation
x k t=n= c k , n φ k , n t=n=n2k(n+1)2k2k2xtdt φ k , n t=n=12kn2k(n+1)2kxtdt(2k2 φ k , n t) x k t n c k , n φ k , n t n t n 2 k n 1 2 k 2 k 2 x t φ k , n t n 1 2 k t n 2 k n 1 2 k x t 2 k 2 φ k , n t
(6)
where 12kn2k(n+1)2kxtdt=average value of x(t) in interval 1 2 k t n 2 k n 1 2 k x t average value of x(t) in interval k:2k2 φ k , n t=height=1 k 2 k 2 φ k , n t height 1 This corresponds to taking the average value of the signal in each interval of width 2k 2 k and approximating the function by a constant over that interval (see Figure 7).

## Footnotes

1. While at first glance it might seem that V V should contain non-zero constant signals (e.g., xt=a x t a for aR a ), the only constant signal in 2 2 , the space of square-integrable signals, is the zero signal.

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