step to the absolute time is ($(\$ abstime - k \cdot \Delta t) \cdot \Delta f[k]$). By summing $\Sigma\Delta \varphi[k]$ and ($(\$ abstime - k \cdot \Delta t) \cdot \Delta f[k]$), the required $\varphi(t)$ is realized.

The proposed phase noise generation approach successfully overcomes the phase discontinuity issue identified in the existing Verilog-A models for oscillators [@MTJSTO_model3; @VCOmodel]. However, the proposed noise generation approach makes the proposed MTJ STO model less suitable for simulations which involve momentary variations in the operating conditions, such as the modulation of current or field. Nevertheless, at the early stage of evaluating STOs in various applications and designing STO-based building blocks towards applications, these simulations involving momentary variations in the operating conditions are not yet critical.

Simulation Results
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Transient simulations of the aforementioned analytical model implemented in Verilog-A, are carried out using the Cadence SpectreRF circuit simulator.

Simulation results of the MTJ STO model
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To validate the proposed $\varphi(t)$ function for generating the frequency fluctuation of the MTJ STO, transient simulations of the stand-alone (unloaded) MTJ STO model using the parameters from different MTJ STOs [@ref1]-[@muduli2010nonlinear] are performed. The time domain signals of the transient simulation using the parameters from [@ref1] and $I_\text{DC}$ 