There are so many definitions of the minimum phase transfer function, and these are two of them.
- The transfer function of the system which has no zeros or poles at right half plane.
- The transfer function which has the minimum phase angle range among the systems which has the same magnitude characteristic.
And these two sentences are describing the same thing. So I want to prove that these statements are equivalent. How to prove it?
PS : Suppose that the system is continuous.
Answer
To your second definition it should be added that you only consider causal transfer functions, because it is not difficult to find a smaller phase lag with a non-causal system:
A minimum-phase system is a causal and stable system with a phase lag that is smaller than the phase lag of any other causal and stable system with the same magnitude response.
Note that a real-valued zero in the left half-plane contributes a phase change of π/2 to the total phase as we move along the frequency axis from ω=0 to ω→∞:
0≤arg{jω+a}<π2,a>0,ω∈[0,∞)
A complex conjugate pair of zeros contribute a phase change of π.
On the other hand, a real-valued zero in the right half-plane contributes a phase change of −π/2 as ω moves from zero to infinity:
−π2<arg{a−jω}≤0,a>0,ω∈[0,∞)
Note that I've chosen the sign of the term s±a such that in both cases the phase is zero for ω=0, which is necessary for a fair comparison between the two cases.
Consequently, exchanging a zero in the left half-plane for a zero in the right half-plane (without changing the magnitude of the frequency response) will always result in an additional phase lag of π for ω→∞.
As an example, consider two first-order transfer functions:
H1(s)=s+2s+1andH2(s)=2−ss+1
The signs of H1(s) and H2(s) were chosen such that their phases are both zero for s=0 (and not ±π). The figure below shows the phase plots (arg{H1(jω)} in blue, and arg{H2(jω)} in green):
The pole contributes a phase change of −π/2 as ω moves from zero to infinity. The left half-plane zero of H1(s) contributes a phase change of π/2, resulting in a net phase change of zero, whereas the right half-plane zero of H2(s) contributes a phase change of −π/2, resulting in a total phase change of −π.
Another way to see the same thing is to note that any causal and stable transfer function can be written as the product of the minimum-phase transfer function with the same magnitude and a causal and stable allpass:
H(s)=Hm(s)Ha(s)
It can be shown that the phase of a causal and stable allpass is always non-positive for ω∈[0,∞), and, consequently, the phase lag of the minimum-phase system is always less than or equal to the phase lag of any other causal and stable system with the same magnitude response.
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