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Haar Scaling Function and Wavelet

Module by: Charlet Reedstrom, Phil Schniter. E-mail the authors

Summary: (Blank Abstract)

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Consider the scaling function φt={1  if  0t<10  if  else φ t 1 0 t 1 0 else

Define V 0 =closurespan φtn nZ V 0 closure span φ t n n where the overbar denotes closure which is the union of a set and its limit points.

Figure 1
Figure 1 (square_scaling_fn.png)

Any f 0 t V 0 f 0 t V 0 can be written as f 0 t= n = a n φtn f 0 t n a n φ t n . V 0 V 0 contains functions constant over n n+1 n n 1 .

Figure 2
Figure 2 (function_representation.png)

Now define φ k , n t=2k2φ2ktn φ k , n t 2 k 2 φ 2 k t n Note: φ k , n =1 φ k , n 1

Figure 3
Figure 3 (scaling_functions.png)
Figure 4: Haar Scaling Functions that Span V j V j
Figure 4 (haar_scaling_functions.png)

Now define V k =closurespan φ k , n t nZ V k closure span φ k , n t n . Then note that V 2 V 1 V 0 V -1 V 2 V 1 V 0 V -1 coarse fine coarse fine

Figure 5
Figure 5 (vspaces.png)

Can show:

  1. V =0 V 0 ... contains no signals.
  2. V =L2 V L 2 ... contains all signals.
  3. φ k , n t nZ φ k , n t n form an orthonormal basis for V k V k .

So what does the projection of ft L 2 f t L 2 onto V k V k look like?

Figure 6
Figure 6 (projection.png)

Say: f k t= m m a k , m φ k , m t Say: f k t m m a k , m φ k , m t

Know: n :( φ k , n t,ft f k t=0)( φ k , n t,ft= φ k , n , f k t) Know: n φ k , n t f t f k t 0 φ k , n t f t φ k , n f k t
(1)

n : φ k , n t,ft= m m a k , m ( φ k , n t, φ k , m t) n φ k , n t f t m m a k , m φ k , n t φ k , m t n : φ k , n t,ft= a k , n n φ k , n t f t a k , n Thus:

a k , n = φ k , n t¯ftd t a k , n t φ k , n t f t
(2)

a k , n =n2k(n+1)2kft2k2d t a k , n t n 2 k n 1 2 k f t 2 k 2 Since:

Figure 7
Figure 7 (Qknt.png)

and

f n t= n =12kn2k(n+1)2kftd t (2k2 φ k , n t) f n t n 1 2 k t n 2 k n 1 2 k f t 2 k 2 φ k , n t
(3)

Where

  • 12kn2k(n+1)2kftd t 1 2 k t n 2 k n 1 2 k f t is the average value of ft f t in the interval
  • 2k2 φ k , n t 2 k 2 φ k , n t is height 1

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