# Relationship between the Real and Imaginary parts of a LTI causal system

## 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).

• never heard of it called KK, but it's easy to prove that for a causal LTI system that the real and imaginary parts of the frequency response are a Hilbert pair. i've always felt that the terms analytic signal and analytic function are too easily conflated. it is not easy to show that a minimum-phase system has a Hilbert transform relationship between the phase (in radians) and the natural log of the magnitude (in nepers). Jan 21 '16 at 7:41
• @robertbristow-johnson may I get some reference to your last statement in your comment? Jan 21 '16 at 10:16
• Google is your friend. ece-research.unm.edu/bsanthan/ece539/note7.pdf Wikipedia might also have something: en.wikipedia.org/wiki/Minimum_phase . looks like they have that K-K thing, which i need to look at. Jan 21 '16 at 19:54

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.