MINIMUM PHASE SYSTEMS: phase analysis and example

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
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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)
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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
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5. re
im
re
im
re
im
re
im
re
im
re
im
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6. PROPERTIES OF MINIMUM-PHASE
SYSTEMS
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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
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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
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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
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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].
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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 

































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12. THANK YOU
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