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

c k , n t φ k , n t x t t n 2 k n 1 2 k 2 k 2 x t
giving the approximation
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
where 1 2 k t n 2 k n 1 2 k x t average value of x(t) in interval 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 2 k and approximating the function by a constant over that interval (see ).

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Source:  OpenStax, Dspa. OpenStax CNX. May 18, 2010 Download for free at http://cnx.org/content/col10599/1.5
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