EEL5173 Fall 2009 Lecture _22

EEL5173 Fall 2009 Lecture _22 - vi. Co-prime LMFD via gcld...

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Theorem. Given a LMFD, Using elementary column operations, reduce the matrix H into a Row Hermite Form is co-prime. ), ( ) ( ) ( 1 s P s Q s G L L = ( ) ( ) L L L L P H Q H s G 1 1 1 ) ( = let [ ] L L P Q s H = ) ( Consequently, [ ] 0 ) ( L H s H H L is then a gcld of P L and Q L . p x q p x q p x p p x q p x p vi. Co-prime LMFD via gcld (continue)
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Ex. + + + + = = + + + + + = s s s s s s s s P Q s s s s s s s s s s G L L 2 1 2 2 2 1 2 2 2 2 2 ) 1 ( ) 2 ( 0 0 ) 2 ( ) 1 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 1 ( ) ( Find a gcld of P L and Q L and then a co-prime LMFD of G ( s ). [] + + + + + + + + = = s s s s s s s s s s s s s s s s P Q s H L L 0 ) 2 ( ) 1 ( ) 2 ( ) 1 ( 0 ) 2 ( 0 ) 1 ( 0 ) 2 ( ) 1 ( ) ( 2 2 2 2 2 2 2 2
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+ + + + + = + × + × + + + + = + + = + + + = × + + × + + + + + + + + + ) 2 ( ) 4 3 )( 3 ( ) 2 ( 0 4 0 ) 2 ( ) ( ) ( 0 ) 2 ( 0 ) ( 4 ) ( 0 ) 4 3 )( 3 ( 4 ) 2 ( ) 1 ( ) ( ) 2 ( 0 ) 2 ( 0 4 ) ( 0 ) ( ) 1 ( 0 ) 2 ( ) 1 ( ) 1 ( 4 4 ) 2 ( ) 1 ( 0 0 ) 2 ( ) 1 ( ) 2 ( ) 1 ( 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 s s s s s s s s s s s s s s w s s s s s w s w s s s s s s s s w where s s s s s w s s s s s s s s s s s s s s s s s s s s s
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() () + + + + + + + = + + + + + + + + + = + + + + × + + + + × + + + + + + + + + + + + + + + ) 2 ( ) 2 ( ) 1 ( ) 2
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EEL5173 Fall 2009 Lecture _22 - vi. Co-prime LMFD via gcld...

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