{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

2.+Typical+Discrete-Time+Systems

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

This preview shows pages 1–7. Sign up to view the full content.

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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
. 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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern