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Haar Approximation at the kth Coarseness Level

Module by: Phil Schniter

It is instructive to consider the approximation of signal xt∈ ℒ 2 x t ℒ 2 at coarseness-level-kk of the Haar system. For the Haar case, projection of xt∈ ℒ 2 x t ℒ 2 onto V k V k is accomplished using the basis coefficients
c k , n =∫-∞∞ φ k , n txtdt=∫n2kn+12k2-k2xtdt 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=-∞∞∫n2kn+12k2-k2xtdt φ k , n t=∑n=-∞∞12k∫n2kn+12kxtdt2k2 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∫n2kn+12kxtdt=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 1).
haarapprox_Vk.png
Figure 1

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