Timeline for Bandpass Stationary Stochastic Process
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 4, 2022 at 13:15 | history | edited | Vito | CC BY-SA 4.0 |
Added sentence
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| Nov 4, 2022 at 13:07 | comment | added | Dan Boschen | @Vito Right, right right--- Very good, thank you! Can you make that clearer in your first sentence, I think it's a great point. | |
| Nov 4, 2022 at 12:58 | comment | added | Vito | @DanBoschen Actually the other way around. Since $\phi_{xx}=\phi_{yy}$ must be true to ensure stationarity of the passband process, this constraint is somehow eating up one degree of freedom. | |
| Nov 4, 2022 at 12:31 | comment | added | Dan Boschen | @Vito is your opening sentence to mean that the flaw in my thought process is that I am (incorrectly) seeing the quadrature components as two independent degrees of freedom when they will not be if $\phi_{xx} \neq \phi_{yy}$ (but are when $\phi_{xx}= \phi_{yy}$)? | |
| Nov 4, 2022 at 12:29 | comment | added | Dan Boschen | @MattL. Thank you, this is likely the gap in my intuition-- my thinking w/o basis if I and Q are stationary, then the sum of the two must be, but I can start to sense the idea of a time varying effect when not equal. I need to wrap my head around that and then see if that resolves my core question (since that means there is a cross dependence). I'm going to read your links. This is helpful, thanks | |
| Nov 4, 2022 at 11:01 | comment | added | Matt L. | @Vito: I should've said "power spectrum"; the autocorrelation is indeed time dependent, but the power spectrum is not defined, at least not in its standard form as a (purely) frequency dependent function. | |
| Nov 4, 2022 at 7:47 | comment | added | Vito | Why not? You can, but it would be a time-dependent autocorrelation $\phi_{nn}(t,\tau)$. | |
| Nov 4, 2022 at 7:31 | comment | added | Matt L. | @DanBoschen: That's the point, the bandpass process will not be stationary unless the autocorrelations of the real and imaginary components are identical. So in that case you couldn't even define the autocorrelation of the bandpass process. | |
| Nov 4, 2022 at 7:22 | comment | added | Vito | No, we can't, because that would violate 4-1-44. I agree that it may be not intuitive, but the math is clear. | |
| Nov 4, 2022 at 1:56 | comment | added | Dan Boschen | Also I don't see how that is a consequence of stationarity alone: Can't we have a stationary process on the real component, and a different but stationary random process on the imaginary component with both autocorrelations not equal (as a simple example completely different power levels)-- the resulting complex process will also be stationary, right? | |
| Nov 4, 2022 at 1:52 | comment | added | Dan Boschen | Thanks for this response. I get that, my question is specific to this and regardless of that point -- I can without that restriction follow the math from his equations 4-1-42 to 4-1-44 and reach a similar result as long as the real and imaginary components are independent. (There is likely a very simple math error here on my part, but I don't yet see it...If I get more time I will add those equations specifically to my post with the derivation proceeding as such so it's even clearer) | |
| Nov 3, 2022 at 21:25 | history | answered | Vito | CC BY-SA 4.0 |