Positive and negative frequencies: how to prove they are independent? This seems to be quite a basic question but is there some detailed proof why interference power (i.e., power spectral density) coming from the positive and negative frequencies are independent? I am considering two real baseband OFDM signals which are interfering with each other and where $2N-f=-f$ is relation between positive and negative frequencies while $2N$ is total number of frequencies (tones). Assume that two signals are not sync and that is the source of interference leakage coming from the negative spectrum. To simplify, I want to prove why is the received PSD equal to : $PSD_{rec.,k}=A^2_{pos.,k}+A^2_{neg.,k}$ instead of $PSD_{rec.,k}=A^2_{pos.,k}+2A_{pos.,k}A_{neg.,k}+A^2_{neg.,k}$ where $A^2$ is squared amplitude (i.e.,DFT coefficient) of tone $k$. Could anyone recommend me some literature or give me a short answer on this question?
This is a bit waffly, but gives the general idea.

The usual reason for this is because of expectations used when we talk about Power Spectral Densities:
$$
S_{xx}(\omega) = \int_{-\infty}^{+\infty} \gamma(\tau) e^{-i\omega \tau} d\tau
= \int_{-\infty}^{+\infty} E\left[ x(t) x(t+\tau) \right] e^{-i\omega \tau} d\tau
$$

So what you have to look at is how the $E[\cdot]$ works on the cross-products between the positive and negative frequencies. Assuming $x$ is just:
$$
x(t) = \sum_{k=1}^{N_+} a_k e^{j\omega_{+,k} t} +  \sum_{k=1}^{N_-} b_k e^{j\omega_{-,k}t} 
$$
then each of the cross terms will look like:
$$
\lim_{T\rightarrow \infty} \frac{1}{T} \int_{-T}^{+T}e^{j\omega_+t}e^{-j\omega_-t}dt = \delta(\omega_+ - \omega_-)
$$
which will be zero if $\omega_+ \not= \omega_-$. That makes all the cross-terms zero, so you're just left with the $x_+(t)x_+(t+\tau)$ and $x_-(t)x_-(t+\tau)$ terms, i.e. your $A^2_{pos,k}$ and $A^2_{neg,k}$ terms.