lect6 - MINIMUM PHASE SYSTEMS A causal rational system is...

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MINIMUM PHASE SYSTEMS A causal, rational system is stable if all its poles are inside the unit circle. Such a system is called called minimum phase if, in addition, all its zeros are inside the unit circle. Equivalently it has no zeros outside the unit circle, including no zeros at . We now show that any causal, stable, real rational system, ) ( z H , can be factored as ) ( ) ( ) ( z H z H z H a m = (6.1) where ) ( z H m is minimum phase and ) ( z H a is all-pass. Before starting the proof, some discussion is in order. Suppose that ) ( 1 z H and ) ( 2 z H are two causal, stable, real rational systems such that ) ( ) ( ) ( 2 1 z H z H z H a = . (6.2) This relation can be thought of in the following way. The all-pass, ) ( z H a , is factored out of ) ( 1 z H , leaving the quotient ) ( 2 z H . Note that ) ( ) ( ) ( ) ( 2 2 1 w w w w j j a j j e H e H e H e H = = so that ) ( 1 z H and ) ( 2 z H have same magnitude response. With regard to the phase response, let ) ( ), ( ), ( 2 1 w j w j w j a and denote the respective phase response functions. Then we have ) ( ) ( ) ( 2 1 w j w j w j a + =

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so that [ ] ) ( ) ( ) ( 2 1 w j w j w j a - + - = - . Recalling that
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lect6 - MINIMUM PHASE SYSTEMS A causal rational system is...

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