all the way up to the cut-off. For cut-off frequencies of 100 MHz or 1 GHz, this means that the virtual time step $\Delta t$ should be set to lower than 1 ns or 100 ps respectively. Failing to set a short enough $\Delta t$ will result in a too narrow white band in the frequency noise spectrum, resulting in a decreased spectral linewidth. In this work, $\Delta t$ = 100 ps is employed. The length of $\Delta f [~]$ is the ratio between the simulation time $t_{sim}$ and the virtual time step $\Delta t$.

The second step in implementing $\varphi(t)$ is to create another vector $\Sigma\Delta \varphi[~]$, which is used to store the phase deviation accumulated from the reference time ($t$=0) to each virtual time step. The relationship between $\Delta f [~]$ and $\Sigma\Delta \varphi[~]$ is illustrated in Fig. 2. In the first period between $t$=0 and the first virtual time step ($t$=1), $\Sigma\Delta \varphi[j=0]$ is 0. During the period between $t$=$j$ and $t$=$j$+1 ($j$&gt;0), the total accumulated phase deviation caused by the frequency fluctuation(s) in the past period(s) (from $\Delta f [0]$ to $\Delta f [j-1]$) is stored in $\Sigma\Delta \varphi[j]$ . A large $\Delta f [j]$ leads 