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Course by: Janko Calic. E-mail the author

# Haar Approximation at the kth Coarseness Level

Module by: Phil Schniter. E-mail the author

It is instructive to consider the approximation of signal xt 2 x t 2 at coarseness-level-kk 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 coefficients

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
(1)
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
(2)
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 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).

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