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Haar Approximation at the kth Coarseness Level

Module by: Phil Schniter. E-mail the author

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

c k , n =∫−∞∞ φ k , n ⁢t⁢x⁢tdt=∫n⁢2k(n+1)⁢2k2−k2⁢x⁢tdt c k , n t φ k , n t x t t n 2 k n 1 2 k 2 k 2 x t
(1)
giving the approximation
x k ⁢t=∑n=−∞∞ c k , n ⁢ φ k , n ⁢t=∑n=−∞∞∫n⁢2k(n+1)⁢2k2−k2⁢x⁢tdt⁢ φ k , n ⁢t=∑n=−∞∞12k⁢∫n⁢2k(n+1)⁢2kx⁢tdt⁢(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
(2)
where 12k⁢∫n⁢2k(n+1)⁢2kx⁢tdt=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 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 1).

Figure 1
Figure 1 (haarapprox_Vk.png)

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