Minimum-Phase Systems
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Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck,
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Minimum Phase Systems-Digital Signal Processing-Lecture Slides, Slides of Digital Signal Processing
This lecture is part of lecture series delivered by Dr Muhammad Fasih Uddin Butt for Digital Signal Processing course at COMSATS Institute of Information Technology. Its main points are: Minimum-phase, Systems, Causal, Stable, Magnitude, Unit, Circle, Frequency, Response, Compensation
Typology: Slides
2011/2012
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Minimum-Phase Systems
Quote of the Day
Experience is the name everyone gives to their
mistakes.
Oscar Wilde
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck,
Minimum-Phase System
- A system with all poles and zeros inside the unit circle
- Both the system function and the inverse is causal and stable
- Name “minimum-phase” comes from the property of the phase
- Not obvious to see with the given definition
- Will look into it
- Given a magnitude square system function that is minimum
phase
- The original system is uniquely determined
- Minimum-phase and All-pass decomposition
- Any rational system function can be decomposed as
H z Hmin zH (^) ap z
Example 2: Minimum-Phase System
- Consider the following system
- One pole inside the unit circle:
- Complex conjugate zero pair outside the unit circle
1
j / 4 1 j / 4 1
2
z 3
e z 2
e z 1 2
H z
1
j / 4 j / 4 j / 4 1 j / 4 1
1
j / 4 1 j / 4 1
2
z 3
e z 3
e z 3
e 2
e 2
z 3
e z 2
e z 1 2
H z
Example 2 Cont’d
j / 4 1 j / 4 1
j / 4 1 j / 4 1
1
j / 4 1 j / 4 1
2
e z 3
e z 1 3
e z 3
e z 1 3
z 3
e z 3
e z 3
H z
j / 4 1 j / 4 1
j / 4 1 j / 4 1
1
j / 4 1 j / 4 1
2
e z 3
e z 1 3
e z 3
e z 3
z 3
e z 3
e z 1 3
H z
H 2 z Hmin zH (^) ap z
Properties of Minimum-Phase Systems
- Minimum Phase-Lag Property
- Continuous phase of a non-minimum-phase system
- All-pass systems have negative phase between 0 and
- So any non-minimum phase system will have a more negative
phase compared to the minimum-phase system
- The negative of the phase is called the phase-lag function
- The name minimum-phase comes from minimum phase-lag
- Minimum Group-Delay Property
- Group-delay of all-pass systems is positive
- Any non-minimum-phase system will always have greater group
delay
j ap
j min
j
argHd e argH e argH e
j ap
j min
j
grdHd e grdH e grdH e
Properties of Minimum-Phase System
- Minimum Energy-Delay Property
- Minimum-phase system concentrates energy in the early part
- Consider a minimum-phase system Hmin(z)
- Any H(z) that has the same magnitude response as Hmin(z)
- has the same poles as Hmin(z)
- any number of zeros of Hmin(z) are flipped outside the unit-circle
- Decompose one of the zeros of Hmin(z)
- Write H(z) that has the same magnitude response as
- We can write these in time domain
^
n
k 0
2 min
n
k 0
2
hk h k
1
Hmin z Q z 1 zkz
k
1
Hz Qz z z
hmin n q n zkqn 1 h n qn 1 zkq n