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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 φ⁢t={1  if  0≤t<10  if  else φ t 1 0 t 1 0 else

Define V 0 =closure⁢span⁢ φ⁢t−n n∈Z 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 ⁢φ⁢t−n 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=2−k2⁢φ⁢2−k⁢t−n 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 =closure⁢span⁢ φ k , n ⁢t n∈Z 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 n∈Z φ k , n t n form an orthonormal basis for V k V k .

So what does the projection of f⁢t∈ 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,f⁢t− f k ⁢t〉=0)⇒(〈 φ k , n ⁢t,f⁢t〉=〈 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,f⁢t〉=∑ 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,f⁢t〉= a k , n n φ k , n t f t a k , n Thus:

a k , n =∫−∞∞ φ k , n ⁢t¯⁢f⁢td t a k , n t φ k , n t f t
(2)

a k , n =∫n⁢2k(n+1)⁢2kf⁢t⁢2−k2d 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 =−∞∞12k⁢∫n⁢2k(n+1)⁢2kf⁢td t ⁢(2−k2⁢ 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

  • 12k⁢∫n⁢2k(n+1)⁢2kf⁢td t 1 2 k t n 2 k n 1 2 k f t is the average value of f⁢t f t in the interval
  • 2−k2⁢ φ k , n ⁢t 2 k 2 φ k , n t is height 1

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