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.
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.
So, is the Gain-Phase relation a special case of KK relations accounting for stability too?
Causality in time domain implies analyticity in frequency domain, vice-versa. Is my statement true?
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 $H(\omega)=H_{Re}(\omega) +j H_{Im}(\omega)$.
How do these relationships extend to higher dimensions? (for images and videos).
Thank you for your patience.