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Copyright (C) EEE413 Digital Signal Processing 3 Example 1: 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
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Minimum-Phase Systems
Quote of the DayExperience 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, ©1999-2000 Prentice Hall Inc.
Copyright (C) EEE413 Digital Signal Processing 2
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
• A minimum-phase system has an additional useful property that the natural logarithm of the magnitude of the frequency response is related to the phase angle of the frequency response (measured in radians) by the Hilbert transform
• Minimum-phase and All-pass decomposition– Any rational system function can be decomposed as
zHzHzH apmin
Copyright (C) EEE413 Digital Signal Processing 3
Example 1: 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
1z2
11z31zH
1
11
1
1
11
1
1z3
11
z311
31z
z211133
1zz2
1113
z211
z31zH
zHzHz3
1131z
z211
z311
3zH apmin1
1
1
1
1
Copyright (C) EEE413 Digital Signal Processing 4
Example 2: Minimum-Phase System• Consider the following system
• One pole inside the unit circle: • Complex conjugate zero pair outside the unit circle
1
14/j14/j
2z3
11
ze231ze2
31zH
1
14/j14/j4/j4/j
1
14/j14/j
2
z311
ze32ze3
2e23e2
3
z311
ze231ze2
31zH
Copyright (C) EEE413 Digital Signal Processing 5
Example 2 Cont’d
14/j14/j
14/j14/j
1
14/j14/j
2ze3
21ze321
ze321ze3
21
z311
ze32ze3
249
zH
14/j14/j
14/j14/j
1
14/j14/j
2ze3
21ze321
ze32ze3
2
z311
ze321ze3
2149
zH
zHzHzH apmin2
Copyright (C) EEE413 Digital Signal Processing 6
Frequency-Response Compensation• In some applications a signal is distorted by an LTI system• Could filter with inverse filter to recover input signal
– Would work only with minimum-phase systems• Make use of minimum-phase all-pass decomposition
– Invert minimum phase part • Assume a distorting system Hd(z)• Decompose it into
• Define compensating system as
• Cascade of the distorting system and compensating system
zH1zHmin,d
c
zHzHzH ap,dmin,dd
zHzH1zHzHzHzHzG ap,d
min,dap,dmin,ddc
Copyright (C) EEE413 Digital Signal Processing 7
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
jap
jmin
jd eHargeHargeHarg
jap
jmin
jd eHgrdeHgrdeHgrd
Copyright (C) EEE413 Digital Signal Processing 8
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
0k
2min
n
0k
2 khkh
1kmin zz1zQzH
k1 zzzQzH
nqz1nqnh 1nqznqnh kkmin
Copyright (C) EEE413 Digital Signal Processing 9
Derivation Cont’d• Evaluate each sum
• And the difference is
• Since |qk|<1
n
0k
22kkk
2n
0k
2min
n
0k
22kkk
2n
0k
2
1kqzkq1kqzkq1kqzkqkh
kqzkq1kqzkq1kqz1kqkh
22k
n
0k
222k
n
0k
2min
n
0k
2 nq1z1kqkq1zkhkh
n
0k
2min
n
0k
2 khkh
