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2.+Typical+Discrete-Time+Systems

2.+Typical+Discrete-Time+Systems - 2 Typical Discrete-Time...

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2. Typical Discrete-Time Systems 2.1. All-Pass Systems (5.5) 2.2. Minimum-Phase Systems (5.6) 2.3. Generalized Linear-Phase Systems (5.7)
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2.1. All-Pass Systems An all-pass system is defined as a system which has a constant amplitude response. That is, |H( )|=A, (2.1) where H( ) is the frequency response of the system, and A is a constant. Now consider a typical all-pass system. Assume that a stable system has the system function . az 1 a z ) z ( H 1 * 1 (2.2) Note that the zero and the pole of H(z) are conjugate reciprocal (that is, they have reciprocal amplitudes and the same phase). Then, it can be shown that this system is an all-pass system. Proof. Letting z=e j in (2.2), we obtain
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. ae 1 ) ae 1 ( e ae 1 a e ) e ( H j * j j j * j j (2.3) From (2.3), we obtain |H(e j )|=1. (2.4) Thus, this system is an all-pass system. Now assume that the above all-pass system is causal. Then, it can be shown that this system must have a positive group delay. Proof. Letting z=e j in (2.2), we obtain . ae 1 a e ) e ( H j * j j (2.5) Since |H(e j )|=1, (2.5) can be written as . ae 1 a e )] e ( H j exp[ j * j j (2.6)
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Differentiating both the sides of (2.6) with respect to , we obtain . )] e ( H j exp[ ) ae 1 ( e ) | a | 1 ( )] e ( H [ grd j 2 j j 2 j (2.7) Substituting (2.6) into (2.7), we obtain . | ae 1 | | a | 1 )] e ( H [ grd 2 j 2 j (2.8) Since the system is causal and stable, |a|<1. Thus, grd[H(e j )] must be positive. Now consider a stable system with the system function . z a 1 a z A ) z ( H M 1 m 1 m * m 1 (2.9) Evidently, this system is an all-pass system. Moreover, if causal, this system must have a positive group delay.
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2.2. Minimum-Phase Systems 2.2.1. Definition of Minimum-Phase Systems A system is a minimum-phase system if it has a rational system function, is causal and stable, and has a causal, stable inverse. Besides a causal, stable inverse, a minimum-phase system may have other inverses. If a minimum-phase system has system function H(z), then H(z) has the following properties: (1) All the poles of H(z) are inside the unit circle centered about the origin. (2) All the zeros of H(z) are inside the unit circle centered about the origin. (3) The denominator and the numerator of H(z) have equal orders of z.
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2.2.2. All-Pass and Minimum-Phase Decomposition Assume that a system can become a minimum-phase system if its factors of form (1 az 1 ), |a|>1, are all removed. Then, this system can be decomposed into the cascade of a causal all-pass system with a unit amplitude response and a minimum-phase system.
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