Timeline for Why does causality imply that the system function is analytic?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 13, 2023 at 13:04 | vote | accept | David Cian | ||
| Oct 9, 2023 at 11:53 | answer | added | V.V.T | timeline score: 2 | |
| Oct 9, 2023 at 2:16 | history | became hot network question | |||
| Oct 8, 2023 at 20:57 | history | edited | Matt L. | CC BY-SA 4.0 | edited title |
| Oct 8, 2023 at 19:39 | comment | added | Matt L. | @robertbristow-johnson: I agree that the word "clearly" may have been an exaggeration. However, note the similarity between Cauchy's formula and the definition of the Hilbert transform. That's where it all comes from. | |
| Oct 8, 2023 at 19:36 | answer | added | Matt L. | timeline score: 6 | |
| Oct 8, 2023 at 19:11 | comment | added | robert bristow-johnson | I'm thinking that if $$ h(t) = 0 \qquad \forall t<0 \text{ and } h(t) \in \mathbb{R} $$ (what we mean by "causal impulse response") then $$ \Im m\{H(f)\} = -\mathscr{H} \Big\{\Re e\{ H(f)\} \Big\} \qquad \forall f \in \mathbb{R} $$ or that $H(f)$ is an "analytic signal". | |
| Oct 8, 2023 at 19:06 | comment | added | robert bristow-johnson | How is $$ \frac{\mathrm{d}\Im m \{f(z)\}}{\mathrm{d}\Re e \{z\}} = \frac{\mathrm{d}\Re e \{f(z)\}}{\mathrm{d}\Im m \{z\}} $$ clearly related directly to the Hilbert Transform? | |
| Oct 8, 2023 at 18:48 | comment | added | Matt L. | @robertbristow-johnson: The OP means analytic in the complex analysis sense of the word, which clearly has to do with the Hilbert transform. | |
| Oct 8, 2023 at 18:41 | comment | added | robert bristow-johnson | You should be explicit with what you mean by "system function". (Might you mean "frequency response"?) And by what you mean by "analytical". (Might you mean something that has to do with the Hilbert Transform?) | |
| Oct 8, 2023 at 18:26 | history | edited | Matt L. | edited tags | |
| Oct 8, 2023 at 18:12 | history | asked | David Cian | CC BY-SA 4.0 |