Following the answer in this post: Phase noise and frequency noise relationship, I can calculate the phase noise spectrum density with the unit from $\frac{dB_{c}}{Hz}$ to $\frac{rad}{\sqrt{Hz}}$. However, when I want to convert it to frequency noise spectrum density, I am confused about the frequency I should use to multiply the phase noise spectrum density: $\Delta f = f_{m} \phi_{p}$.
Is $f_{m}$ the carrier frequency or the frequency offset from the carrier frequency?
$f_m$ is the offset from carrier frequency. When we translate the signal from one frequency to another (another carrier frequency), we do not change the frequency or phase modulation characteristics. In fact, we can translate to 0 (DC) as long as we represent the signal with its complex real and imaginary components as the baseband equivalent signal. The point is, it doesn’t matter what the carrier frequency is, regardless of that frequency the phase and amplitude variation vs time is unchanged.
