090209Lecture14

090209Lecture14 - ECE 700 Digital Signal Processing/09/09 ECE700WI2009 1 Orthogonality Principle ECE700WI2009 Scaling function Start with a mother

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ECE 700 WI 2009 ECE 700 Digital Signal Processing Lecture 14 02/09/09 1
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ECE 700 WI 2009 Orthogonality Principle
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ECE 700 WI 2009 3 Scaling function Start with a mother scaling function φ ( t ) in L 2 . Let V 0 be the vector space spanned by φ ( t ) and its integer translates φ ( t-n ), n integer. 0 span{ ( ), integer}. Vt n n φ = Further, suppose that the basis is orthonormal. { ( ), integer} tn n Haar example: 1, 0 1 () . 0, otherwise t t ≤< = 1 1 ( t ) t 1 2 0 1 1 , , ( ) 0 , . tt d t d t tm n m φφ −∞ == = −− = ∫∫ ( t n ) t 1 n 1 m ( t m )
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ECE 700 WI 2009 4 Family of scaling functions /2 , , () 2 (2 ) , , . Let span{ ( ) : }. kk kn n tt n k n Vt n φφ φ −− = −∈ =∈ 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 5 Haar family of scaling functions Note that in this example, for any k , φ k,n ( t ) are orthonormal (see rows above).
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ECE 700 WI 2009 6 Example functions from Haar V 0 and V 1
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ECE 700 WI 2009 7 Nested subspaces Assume that the mother scaling function is such that the V k ’s form a nested
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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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090209Lecture14 - ECE 700 Digital Signal Processing/09/09 ECE700WI2009 1 Orthogonality Principle ECE700WI2009 Scaling function Start with a mother

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