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090211Lecture15 - ECE 700 Digital Signal Processing/11/09...

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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 φ ( t ) and its integer translates φ ( t-n ), n integer. 0 span{ ( ), integer}. V t n n φ = Further, suppose that the basis is orthonormal. { ( ), integer} t n n φ /2 , , ( ) 2 (2 ), , . Let span{ ( ): }. k k k n k k n t t n k n V t n φ φ φ = = Z Z Define a family of scaling functions as Haar example: / 2 , 2 , 2 ( 1)2 ( ) . 0, otherwise k k k k n n t n t φ < + =
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