![MINIMUM PHASE ⇒ MINIMUM
ENERGY DELAY
Given a causal system with impulse response h[n], we can define the partial
energy of the impulse response as
n
E[n] = Σ |h[k]|2
k=0
A minimum phase system with impulse response hmin[n] has the property
n n
Emin[n] = Σ|hmin[k]|2 ≥ Σ |h[k]|2 = E[n]
k=0 k=0
for all n ≥0 and all h[n] with the same magnitude response ashmin[n].
10](https://image.slidesharecdn.com/dspala-210510115943/85/MINIMUM-PHASE-SYSTEMS-10-320.jpg)
MINIMUM PHASE SYSTEMS
- 1. MINIMUM PHASE SYSTEMS PREPARED BY:- SHIVANGI SINGH DIGITAL SIGNAL PROCESSING ( 2171003) E.C DEPT.
- 2. INTRODUCTION A system with all poles and zeros inside the unit circle is called minimum phase system. Both the system function and the inverse is causal and stable Minimum phase systems are important because they have a stable inverse G(z) = 1/H(z). We can convert between min/max/mixed-phase systems by cascading all pass filters. When we say a system is “minimum phase”, we mean that it has the least phase delay (or least phase lag) among all systems with the same magnitude response 2
- 3. Minimum phase systems with real-valued impulse responses have ∠H(ej0) = 0. H1(z) = 6 + z−1 − z−2 = 6(1 + z−1/2) (1 − z−1/3) H2(z) = 1 − z−1 − 6z−2 = (1 + 2z−1) (1 − 3z−1) H3(z) = 2 − 5z−1 − 3z−2 = 2(1 + z−1/2) (1 − 3z−1) H4(z) = 3 + 5z−1 − 2z−2 = (1 + 2z−1) (1 − z−1/3) 3
- 4. PHASE ANALYSIS Phase response of a rational H(z) can be written as To have the least phase delay for a causal system, we must have 4
- 7. 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 j ap j min j d e H arg e H arg e H arg 7
- 8. 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 d e H grd e H grd e H grd 8
- 9. MINIMUM ENERGY-DELAY PROPERTY Minimum-phase system concentrates energy in the early part. n 0 k 2 min n 0 k 2 k h k h 9
- 10. MINIMUM PHASE ⇒ MINIMUM ENERGY DELAY Given a causal system with impulse response h[n], we can define the partial energy of the impulse response as n E[n] = Σ |h[k]|2 k=0 A minimum phase system with impulse response hmin[n] has the property n n Emin[n] = Σ|hmin[k]|2 ≥ Σ |h[k]|2 = E[n] k=0 k=0 for all n ≥0 and all h[n] with the same magnitude response ashmin[n]. 10
- 11. EXAMPLE : MINIMUM-PHASE SYSTEM Consider the following system One pole inside the unit circle: -Make part of minimum-phase system One zero outside the unit circle: -Add an all-pass system to reflect this zero inside the unit circle 1 1 1 z 2 1 1 z 3 1 z H 1 1 1 1 1 1 1 1 1 z 3 1 1 z 3 1 1 3 1 z z 2 1 1 1 3 3 1 z z 2 1 1 1 3 z 2 1 1 z 3 1 z H z H z H z 3 1 1 3 1 z z 2 1 1 z 3 1 1 3 z H ap min 1 1 1 1 1 11
- 12. THANK YOU 12