Can complex envelope be writen in the form of quadrature components when it has symmetric spectrum? I am reading a chapter on VCO noise in &quot;Design of CMOS phase-locked loops from circuit level to architecture level by Behzad Razavi&quot;;I am confused when the upconverted noise is writen as $N_I\cos\omega t - N_Q\sin\omega t$, which means the baseband noise(complex envelope of the upconverted noise) have a form of $N_I+jN_Q$; I have this confusion because I think the complex envelope has symmetric spectrum which implies it is real signal so Nq should be 0; please help me understand this, thanks!
The noise is actually never upconverted, it was already there in the bandpass channel. The textbook is simply modeling the noise as a quadrature signal.
To recap:

The transmitted passband signal is usually assumed to be noise-free.
The passband signal is real, and it occupies frequencies from $-W_2$ to $-W_1$ and from $W_1$ to $W_2$.
The receiver adds noise to the signal, usually in the first few stages of the analog front-end. This noise is passband: it occupies the same spectrum as the received signal.
The passband noise goes through the quadrature receiver and its complex envelope is calculated. In the process, the spectrum from $-W_2$ to $-W_1$ is discarded, and the spectrum from $W_1$ to $W_2$ is shifted to baseband.
As a result, the spectrum of the complex envelope of the noise is not symmetric.
In conclusion, we can think of the passband noise as a quadrature signal.

Here's an illustration: