090202Lecture11

090202Lecture11 - ECE 700 ECE 700 WI 2009 Digital Signal...

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Unformatted text preview: ECE 700 ECE 700 WI 2009 Digital Signal Processing Lecture 11 02/02/09 1 2 Channel modulation matrices 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 G z G z H z H z Y z X z G z G z H z H z Y z X z- = ----- ECE 700 WI 2009 ( ) ( ) 1 1 1 [ ( )] ( ) ( ) ( ) 2 ( ) is called the Analysis Modulation Matrix and ( ) is called the Sy m T m H z G z m m H z G z dhhhDhhh¡ dhhhDhhh¡ nthesis Modulation Matrix. 1 ( ) 1 ( ) 1 1 ( ) ( ) 2 , ( ) ( ) det( ( )) det( ( )) ( ) ( ) ( ) ( ). l m m H z G z z H z G z H z H z H z H z H z H z-- = -- =--- For perfect reconstruction, Y ( z ) = z- l X ( z ), need PR FIR filterbank Suppose that we want all four filters H ( z ), H 1 ( z ), G ( z ) and G 1 ( z ) to be real-coefficient FIR filters. This is possible if ( ) ( ) 1 det( ( )) , in which case ( ) ( ) 2 . ) ) m k l k H z cz H z G z z z z--- =- = - ECE 700 WI 2009 1 ( ) ( ) H z G z c-- Choosing l = k , we get a real, causal FIR PR filterbank if 1 ( ) 1 1 ( ), ( ) : causal, real-coeff. FIR filters det( ( )) 2 ( ) ( ) 2 ( ) ( ) m k H z H z H z cz G z H z c G z H z c- = =- - =- Will lead to orthogonal and bi- orthogonal filterbanks. Orthogonal PR FIR filterbanks The FIR PR conditions allow us to choose suitable H ( z ) and H 1 ( z ) to meet the PR conditions from the previous slide. For orthogonal PR FIR filterbanks, we make the choices: 1 1 ( 1) 1 ( ) is of even length (odd order 1) ( ) ( ) ( ) ( ) 1 ( ) ( ) N H z N N H z H z H z H z H z z H z------ +-- = = ±- ECE 700 WI 2009 1 The second condition makes H ( z ) “power-symmetric” and { H ( z ), H 1 ( z )} a “power complementary” pair. 2 2 ( ) 2 2 1 ( ) ( ) 1 ( ) ( ) 1 j j j j H e H e H e H e ϖ π ϖ ϖ ϖ- + = + = Power-symmetric leads to PR For the above choice of analysis and synthesis filters, ( ) 1 1 ( 1) 1 ( 1) 1 ( 1) 1 1 ( 1) det( ( )) ( ) ( ) ( ) ( ) ( )( ( )) ( )( ( )) [ ( ) ( ) ( ) ( )] m N N N...
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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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090202Lecture11 - ECE 700 ECE 700 WI 2009 Digital Signal...

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