Haar Approximation at the kth Coarseness Level 2.2 2003/01/17 2005/09/30 19:53:32.329 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu Haar 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 ).