090211Lecture15

090211Lecture15 - ECE 700 Digital Signal Processing Lecture...

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ECE 700 WI 2009 ECE 700 Digital Signal Processing Lecture 15 02/11/09 1
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ECE 700 WI 2009 2 Orthonormal scaling functions Consider a mother scaling function φ ( t ) in L 2 . Let V 0 be the vector space spanned by ( t ) and its integer translates ( t-n ), n integer. 0 span{ ( ), integer}. Vt n n = Further, suppose that the basis is orthonormal. { ( ), integer} tn n /2 , , () 2 (2 ) , , . Let span{ ( ): }. kk kn n tt n k n n φφ −− = −∈ =∈ Z Z Define a family of scaling functions as Haar example: , 2, 2 (1 ) 2 () . 0, otherwise k nt n t ≤< + =
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ECE 700 WI 2009 3 Nested sequence of subspaces The V k ’s form a nested sequence of subspaces i.e. 210 1 2 VV V −− ⊂⊂ "" Haar example: /2 ,2 (1 ) / 2 1 1 1, 2 , (2 )2 (2 1)2 () , 0, otherwise 2, 2 ( 1 ) 2 . 0, otherwise kk k kn k nt n t n t φ −+ + + + ≤< + = + = Level k Level k+1
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ECE 700 WI 2009 4 Best approximation in V k The best approximation of x ( t ) belonging to L 2 in the subspace V k is the orthogonal projection of x ( t ) in V k . ,, () , , (). kk n k n n kn x tc t ct x t φ =−∞ = =
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ECE 700 WI 2009 5 Haar Example
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ECE 700 WI 2009 6 The scaling equation Consider the two subspaces, Any vector in V 1 can be expressed as a linear combination of a basis for V 0 . So, 10 . VV 1,0 0, () [ ] 1 []( ) 2 2 2 [ ] (2 ) . n n n n th n t t hn t n n t n φφ =−∞ = ⇒= This is known as the scaling equation or the two-scale equation. It relates φ ( t ) at two different scale factors, and is thus different from a differential or difference equation.
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This note was uploaded on 04/18/2009 for the course ECE 700 taught by Professor Un during the Winter '09 term at Ohio State.

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090211Lecture15 - ECE 700 Digital Signal Processing Lecture...

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