# Thread: How can these signal components be independent ???

1. ## How can these signal components be independent ???

Hi,
In wireless communication theory, the complex baseband representation of a signal passing through a multipath propagating channel is modeled as a complex random variable with real (the I component) and imaginary (the Q component) parts written as:

r_I(t) = sum_n(C_n * cos(phi_n))
r_Q(t) = sum_n(C_n * sin(phi_n))

C_n represents the random attenuation of path n, and has a not further specified distribution, with all C_n independent.

phi_n represents the phase of path n and is modeled as being a uniform random variable over [-pi, pi], all independent of each other and also independent of the C_n.

Then, as the sum goes to infinity, the central limit theorem is invoked to model the I and Q components as having the normal distribution, with zero mean. So far, so good.

The part I don't understand however, is that the I and Q components are always said to also be independent from each other, making them jointly Gaussian.
Since cos(phi_n) and sin(phi_n) are dependent, how can their sums over n (with identical values of C_n in both sums) be independent ?

I'd appreciate any help on this!

Paul.

P.S. the independence of the I and Q components is then used in the theory to model the envelope of the signal (i.e. its magnitude) as a Rayleigh distributed random variable, modeling the signal fading process in such multipath channels.

2. ## Re: How can these signal components be independent ???

Do you know how to convert Cartesian coordinates into polar coordinates?

3. ## Re: How can these signal components be independent ???

Can you? How does that help?

4. ## Re: How can these signal components be independent ???

diracsbracket: The "report" function is there to report spam, duplicate posts, or genuinely abusive posts. Using it to complain about the response of someone (Dason) who is a moderator himself and has simply asked you a question that would help him to help you is inappropriate.

5. ## The Following User Says Thank You to CowboyBear For This Useful Post:

Dason (04-28-2017)

6. ## Re: How can these signal components be independent ???

Originally Posted by CowboyBear
diracsbracket: The "report" function is there to report spam, duplicate posts, or genuinely abusive posts. Using it to complain about the response of someone (Dason) who is a moderator himself and has simply asked you a question that would help him to help you is inappropriate.
@CowboyBear. A moderator should know better than to post a reply that has NOTHING to do with the question asked, and which instead asks a question that makes me believe he must think I am an idiot or something. If you look at the problem I asked, you can see what you may call both cartesian and polar coordinates (I,Q = cartesian, C_n, phi_n = polar), so why does he asks if I know how to convert from cartesian to polar? That in itself is insulting enough to flag it as such. If this is how this forum works, then you may ban me ad vitam aeternam, and stante pede.

7. ## Re: How can these signal components be independent ???

Originally Posted by diracsbracket
@CowboyBear. A moderator should know better than to post a reply that has NOTHING to do with the question asked, and which instead asks a question that makes me believe he must think I am an idiot or something. If you look at the problem I asked, you can see what you may call both cartesian and polar coordinates (I,Q = cartesian, C_n, phi_n = polar), so why does he asks if I know how to convert from cartesian to polar? That in itself is insulting enough to flag it as such. If this is how this forum works, then you may ban me ad vitam aeternam, and stante pede.
It sounds like you need to chill out and stop perceiving insults that never happened. Act like an adult by restating your question and asking Dason to clarify how his question might be helpful for you (or just answer his question and see where it goes...).

8. ## The Following 2 Users Say Thank You to ondansetron For This Useful Post:

Dason (04-28-2017), GretaGarbo (04-28-2017)

9. ## Re: How can these signal components be independent ???

Originally Posted by ondansetron
It sounds like you need to chill out and stop perceiving insults that never happened. Act like an adult by restating your question and asking Dason to clarify how his question might be helpful for you (or just answer his question and see where it goes...).
OP this would have been a good response. Acting like an ass and reporting me for asking a simple question definitely doesn't make me want to help anymore.

By the way I get the feeling that you're probably not going to stick around but I do hope that you can learn something about how to behave yourself from this whole ordeal. If you do choose to behave better and stick around I'd be happy to help but you're not doing yourself any favors by acting like a child.

Also keep in mind that we deal with a wide variety of people. I have personally dealt with people that have been working on stuff similar to what you've submitted and didn't know/understand/remember much about coordinate transformations so really what was the harm in me asking? Why would you take a simple question and react like that?

10. ## The Following 3 Users Say Thank You to Dason For This Useful Post:

GretaGarbo (04-28-2017), ondansetron (04-28-2017), spunky (04-28-2017)

11. ## Re: How can these signal components be independent ???

Hey. You're all right. My bad! Cheers. @Dason: thank you for your question.

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