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Prelude

I am writing an elaborate text on the relationship between the real and imaginary parts of a LTI causal system and how stability, causality and analyticity imposes various constraints on its magnitude response and the phase response. I am trying to explain three topics

  • Kramers Kronig Relationship
  • Bode's Gain-Phase Relationship
  • Hilbert Transform relationship

And finally, explain the connection between the three and their practical implications. This, I feel, is a much-needed text for people like me entering into this field. I have a list of questions, which I feel must come under the same context - please be patient in answering and kind enough to point out any mistakes.

Questions

The text, Kramers–Kronig Bode and the meaning of zero by John Bechhoefe, provides a short derivation of the gain-phase relation.

  1. What is the motivation behind taking the logarithm of the response function in deriving the gain-phase relation?

    Kramers Kronig (KK) relation requires only a causal system (analytic in the upper half of the complex $\omega$-plane) and does not talk about stability.

    The Bode's Gain Phase relation requires a minimum phase system with no zeroes in the upper half of the complex $\omega$-plane and for non-minimum phase it becomes an inequality. In other words, Bode's Gain-Phase relationship necessitates that the response function must obey KK relations.

  2. So, is the Gain-Phase relation a special case of KK relations accounting for stability too?

  3. Causality in time domain implies analyticity in frequency domain, vice-versa. Is my statement true?

  4. How are Hilbert transform relationship and KK relationship related? By Hilbert Transform relationship I mean the following (for a discrete case): $$ H_{Im}(\omega)=-\frac{1}{2\pi}\int_{-\pi}^\pi H_{Re}(\lambda)\cot(\frac{\omega-\lambda}{2})d\lambda. $$ For a given response function $H(\omega)=H_{Re}(\omega) +j H_{Im}(\omega)$.

  5. How do these relationships extend to higher dimensions? (for images and videos).

Thank you for your patience.

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  1. I believe the motivation is to make the math easier. Without the logarithm, you have $G(\omega)=|G(\omega)|e^{j\angle G(\omega)}$. By taking the logarithm, you have $\ln G(\omega)=\ln |G(\omega)|+j\angle G(\omega)$; by removing the exponential function and turning it into a sum, the relationship between $|G(\omega)|$ and $\angle G(\omega)$ becomes much easier to handle.

    KK only requires causality and linearity. Bode is more strict because it also requires no zeros in the upper half of the plane (but make sure to look at note 16 in the paper).

  2. As far as I can see, the value of the real part of the zeros is irrelevant; only their existance matters. So, stability is not really involved in this analysis at all.

  3. The implication (causal -> analytic half plane) is true. I believe the "vice versa" (iff) is only true for stable systems, but I'm not 100% sure.

I don't know the answers to your questions 4 and 5.

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Selected answers for the ones I know

(1.) The Bode gain-phase relation gives the relationship for one possible realization of the system. The minimum phase constraint "nails it down", so to speak and determines the phase response if the magnitude response is known. However, it is not the only physically realizable system that has said magnitude response, just the one with minimum phase. In other words, the magnitude response does not uniquely identify the system. This book is an excellent reference that will probably answer all 5 questions you have.

(4.) As far as I understand, confirmed by this, The Kramers-Kronig relations are simply stating that the real and imaginary parts of the frequency response function form a Hilbert transform pair.

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