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# Minimum Phase and Allpass Systems Minimum Phase Systems A system function is said to be a minimum phase system if all of its poles and zeros are within the unit circle PDF document - DocSlides

cheryl-pisano | 2014-12-11 | General

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Minimum Phase and Allpass Systems Minimum Phase Systems A system function ) is said to be a minimum phase system if all of its poles and zeros are within the unit circle. Consider a causal and stable LTI system with a diﬀerence equation representation of the form: =0 ]= =0 the system function ) of this LTI system takes the form )= =0 =0 In pole-zero format this system function is given by: )= =1 (1 =1 (1 max The minimum phase condition on the system function can then be translated into the pole zero parameters as: max 1) max The implication of this statement is that inverse system with system function ) also has it poles and zeros within the unit circle, i.e., ) is also a minimum phase system: )= =1 (1 =1 (1 max This means that the minimum-phase system is a system where both the system and its inverse are causal and stable.

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Allpass systems An allpass system is a system whose frequency response magnitude is constant for all frequencies, i.e., j c, π, The general form of a implementable allpass system is: ap )= =1 =1 )( (1 )(1 +peciﬁcally let us look at the causal and stable LTI system with system function ) given by: )= az re j re j The corresponding frequency response for this system is: j )= j re j j re j j The magnitude response of this system is given by: j re j j re j j =1 Consequently this system is an all-pass system of ﬁrst order. The phase response of this system is given by: A-.( j )) = 2 tan sin( cos( , π, The group delay of this system obtained after taking the derivative of the ex- pression above is given by: grd( j )) = 10 cos( , π, 1rom the expression above we can see that the allpass system is a positive group delay system, i.e., grd( j )) ,r< It is easily seen that this implies that the continuous phase is negative over [2 , ], i.e., Arg( j )) , [2 ,

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System Function Factorization Consider a stable and causal LTI system with system function ). +ince the system is a stable system the poles of the system function are required to be inside the unit circle (3C). The zeroes are however, free to wander outside. Let us assume that there are zeroes of ) that are outside the 3C. Then the system function ) can be rewritten as: )= =1 +peciﬁcally the system function ) is a minimum phase system function because it has all its poles and zeroes inside the 3C. -ewriting this expression to include the allpass factor we obtain: )= =1 (1 min ap =1 This implies that system function ) can then be factorized into two parts as: )= min ap where min ) is deﬁned as the minimum phase component of ) and ap is deﬁned as the allpass part or component of ). Implications of the Factorization In terms of the magnitude response j this factorization implies that: j min j In other words the magnitude of the frequency response of the system and the magnitude of the frequency response of the minimum phase part are identical. In terms of the phase response this factorization implies that: Arg( j )) = Arg( min j )) 0 Arg( ap j )) This means that the minimum phase system is also a minimum phase lag system. In terms of group delay response this factorization implies: grd( j )) = grd( min j )) 0 grd( ap j )) 3sing the positive group delay property of the allpass part we can infer that the minimum phase system is also a minimum group delay system.

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Factorization Algorithm The algorithm for obtaining the components is : 1. Assign the poles and the zeroes of ) inside the 3C to the minimum phase part min ). 2. -eﬂect the zeroes of ) that are outside the 3C to their conjugate reciprocal locations. 4. The allpass part will be composed of the zeroes that lie outside the 3C and the poles at the complex reciprocal locations needed to compensate for the reﬂection of the zeroes to the complex reciprocal locations. Magnitude Square Factorization Consider a system function ) that is positive, real and analytic on the 3C. +ince the system function ) is positive, real and analytic on the 3C, the corresponding frequency response can be expressed in the form: j )= j , π, K> The corresponding quantity system function in the Z-domain is: )= KH ,z ≤| This mean that the poles and zeroes of ) come in conjugate reciprocal pairs. We can then split the poles and zeroes into two groups, one group inside the 3C and the other group outside the 3C as described by: )= min =1 =1 =1 =1 max Then the system function ) can be factorized in the form: )= KH min max ,K> where min ) is the minimum phase part or component of ) and max )= min is the maximum phase part of ). The corresponding in the time domain is: ]= Kh min min

Consider a causal and stable LTI system with a di64256erence equation representation of the form 0 0 the system function of this LTI system takes the form 0 0 In polezero format this system function is given by 1 1 1 1 max The minimum phase cond ID: 22138

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Page 1

Minimum Phase and Allpass Systems Minimum Phase Systems A system function ) is said to be a minimum phase system if all of its poles and zeros are within the unit circle. Consider a causal and stable LTI system with a diﬀerence equation representation of the form: =0 ]= =0 the system function ) of this LTI system takes the form )= =0 =0 In pole-zero format this system function is given by: )= =1 (1 =1 (1 max The minimum phase condition on the system function can then be translated into the pole zero parameters as: max 1) max The implication of this statement is that inverse system with system function ) also has it poles and zeros within the unit circle, i.e., ) is also a minimum phase system: )= =1 (1 =1 (1 max This means that the minimum-phase system is a system where both the system and its inverse are causal and stable.

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Allpass systems An allpass system is a system whose frequency response magnitude is constant for all frequencies, i.e., j c, π, The general form of a implementable allpass system is: ap )= =1 =1 )( (1 )(1 +peciﬁcally let us look at the causal and stable LTI system with system function ) given by: )= az re j re j The corresponding frequency response for this system is: j )= j re j j re j j The magnitude response of this system is given by: j re j j re j j =1 Consequently this system is an all-pass system of ﬁrst order. The phase response of this system is given by: A-.( j )) = 2 tan sin( cos( , π, The group delay of this system obtained after taking the derivative of the ex- pression above is given by: grd( j )) = 10 cos( , π, 1rom the expression above we can see that the allpass system is a positive group delay system, i.e., grd( j )) ,r< It is easily seen that this implies that the continuous phase is negative over [2 , ], i.e., Arg( j )) , [2 ,

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System Function Factorization Consider a stable and causal LTI system with system function ). +ince the system is a stable system the poles of the system function are required to be inside the unit circle (3C). The zeroes are however, free to wander outside. Let us assume that there are zeroes of ) that are outside the 3C. Then the system function ) can be rewritten as: )= =1 +peciﬁcally the system function ) is a minimum phase system function because it has all its poles and zeroes inside the 3C. -ewriting this expression to include the allpass factor we obtain: )= =1 (1 min ap =1 This implies that system function ) can then be factorized into two parts as: )= min ap where min ) is deﬁned as the minimum phase component of ) and ap is deﬁned as the allpass part or component of ). Implications of the Factorization In terms of the magnitude response j this factorization implies that: j min j In other words the magnitude of the frequency response of the system and the magnitude of the frequency response of the minimum phase part are identical. In terms of the phase response this factorization implies that: Arg( j )) = Arg( min j )) 0 Arg( ap j )) This means that the minimum phase system is also a minimum phase lag system. In terms of group delay response this factorization implies: grd( j )) = grd( min j )) 0 grd( ap j )) 3sing the positive group delay property of the allpass part we can infer that the minimum phase system is also a minimum group delay system.

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Factorization Algorithm The algorithm for obtaining the components is : 1. Assign the poles and the zeroes of ) inside the 3C to the minimum phase part min ). 2. -eﬂect the zeroes of ) that are outside the 3C to their conjugate reciprocal locations. 4. The allpass part will be composed of the zeroes that lie outside the 3C and the poles at the complex reciprocal locations needed to compensate for the reﬂection of the zeroes to the complex reciprocal locations. Magnitude Square Factorization Consider a system function ) that is positive, real and analytic on the 3C. +ince the system function ) is positive, real and analytic on the 3C, the corresponding frequency response can be expressed in the form: j )= j , π, K> The corresponding quantity system function in the Z-domain is: )= KH ,z ≤| This mean that the poles and zeroes of ) come in conjugate reciprocal pairs. We can then split the poles and zeroes into two groups, one group inside the 3C and the other group outside the 3C as described by: )= min =1 =1 =1 =1 max Then the system function ) can be factorized in the form: )= KH min max ,K> where min ) is the minimum phase part or component of ) and max )= min is the maximum phase part of ). The corresponding in the time domain is: ]= Kh min min

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