090202Lecture11 - ECE 700 Digital Signal Processing/02/09...

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ECE 700 Digital Signal Processing ECE 700 WI 2009 Lecture 11 02/02/09 1
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2 Channel modulation matrices 0 1 0 0 0 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 - =   - - - - - dncurlybracketlefthorizcurlybracketexthorizcurlybracketexthorizcurlybracketextdncurlybracketmidhorizcurlybracketexthorizcurlybracketexthorizcurlybracketextdncurlybracketright dncurlybracketlefthorizcurlybracketexthorizcurlybracketexthorizcurlybracketextdncurlybracketmidhorizcurlybracketexthorizcurlybracketexthorizcurlybracketextdncurlybracketright ECE 700 WI 2009 ( ) ( ) [ ( )] ( ) ( ) ( ) ( ) is called the Analysis Modulation Matrix and ( ) is called the Sy m T m H z G z m m H z G z nthesis Modulation Matrix. 1 0 ( ) 0 1 ( ) 0 1 0 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
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PR FIR filterbank Suppose that we want all four filters H 0 ( z ), H 1 ( z ), G 0 ( z ) and G 1 ( z ) to be real-coefficient FIR filters. This is possible if ( ) ( ) 1 0 det( ( )) , in which case ( ) ( ) 2 . ( ) ( ) m k l k H z cz H z G z z H z G z c - - - = - = - - ECE 700 WI 2009 0 1 Choosing l = k , we get a real, causal FIR PR filterbank if 0 1 ( ) 0 1 1 0 ( ), ( ) :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.
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Orthogonal PR FIR filterbanks The FIR PR conditions allow us to choose suitable H 0 ( z ) and H 1 ( z ) to meet the PR conditions from the previous slide. For orthogonal PR FIR filterbanks, we make the choices: 0 1 1 0 0 0 0 ( 1) 1 1 0 ( ) 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 The second condition makes H 0 ( z ) “power-symmetric” and { H 0 ( z ), H 1 ( z )} a “power complementary” pair. 2 2 ( ) 0 0 2 2 0 1 ( ) ( ) 1 ( ) ( ) 1 j j j j H e H e H e H e ϖ π ϖ ϖ ϖ - + = + =
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Power-symmetric leads to PR For the above choice of analysis and synthesis filters, ( ) 0 1 0 1 ( 1) 1 ( 1) 1 0 0 0 0 ( 1) 1 1 0 0 0 0 ( 1) det( ( )) ( ) ( ) ( ) ( ) ( )( ( )) ( )( ( )) [ ( ) ( ) ( ) ( )] m N N N N H z H z H z H z H z H z z H z H z z H z z H z H z H z H z z - - - - - - - - - - - - = - - - = - - ± - = + - - = ECE 700 WI 2009 The job then is to design H 0 ( z ) that satisfies the power-symmetric condition.
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