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7.3 Haar approximation at the kth coarseness level

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It is instructive to consider the approximation of signal x t ℒ 2 at coarseness-level- k of the Haar system. For the Haar case, projection of x t ℒ 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, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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