## 4 Impact Analysis of Phase Noise in Time Domain  
According to the power law spectral model of phase noise, the purpose of simulating the phase noise sequence in the time domain is to generate various power law spectral component noises, the most critical of which is the generation of 1/f noise components. Time-domain simulation methods for power law spectral noise can be divided into three categories [20]: methods based on autoregressive moving average (ARMA) models, models based on discrete fractional integration and models based on wavelet analysis. The noise generated by the method based on ARMA is stationary and has no scale invariance, and this generation method is not easy to operate in practical use. Due to the inherent time-frequency analysis characteristics of wavelets, the noise sequence generated by wavelets is non-stationary and to scale. Invariant phase noise sequences generated by discrete fractional integration methods are also ideally non-stationary and scale invariant, and are easy to manipulate in practice. Based on the above analysis, considering that the discrete fractional integration method has mature mathematical models available in the MATLAB system (essentially an extension of the IIIR filter), it is proposed to use the fractional integration-based method to generate phase noise sequences. Generating the components \(1/f^{\alpha}\) using the discrete fractional integration method is based on the Brownian motion model with one-dimensional discrete random walk characteristics, and its transfer function can be approximately expressed as [21; 22]  
\[H(z)=\frac{1}{\left(1-z^{-1}\right)^{\alpha}} \tag{14}\]  
Assuming that the sampling step size of the simulation is \(\Delta t\), according to the signal processing theory, the white noise sequence with variance \(Q_{d}\) passes through the discrete system, and the power spectral density of the continuous noise corresponding to the output sequence can be obtained as  
\[S(f)=\frac{Q_{d}\Delta t}{\left(2\sin(\pi f\Delta t)\right)^{\alpha}} \tag{15}\]  
Figure 7: Tracking error introduced by thermal noise and phase noise of third-order PLL. In order to facilitate the simulation of discrete systems, (14) is expanded according to the series, and we get  
\[H(z)=\frac{1}{1-\frac{a}{2}z^{-1}-\frac{a(1-a)}{2}z^{-2}-\ldots} \tag{16}\]  
This transfer function is equivalent to the IIR filter given by  
\[x_{n}=-a_{1}x_{n-1}-a_{2}x_{n-2}-a_{3}x_{n-3}-\ldots+w_{n} \tag{17}\]  
The filter coefficients are expressed as  
\[ \begin{array}{l}a_{0}=1\\ a_{k}=(k-1-\alpha)\frac{a}{k}\end{array}  \tag{18}\]  
In order to facilitate modeling in MATLAB, the series of infinite IIR needs to be truncated. After many calculations, the frequency decade \(N\) at which the noise sequence generated by the truncated IIR filter exhibits the desired power law spectral characteristics has the following approximate relationship with the series \(n\) of the IIR filter:  
\[N=\log_{10}(n)+1 \tag{19}\]  
The following simulation example is implemented for a certain type of OCXO crystal oscillator (5 MHz). The phase noise power spectral density curve of the crystal oscillator is basically white noise above 1 KHz, as shown in Figure 8 below. According to [PERSON]'s theorem, combined with the simulation efficiency, the sampling frequency of the phase noise time-domain sequence can be set as 100 KHz. Typical values in Figure 8 are extracted into Formula (7) and the synthetic time-domain phase noise sequence of each component corresponding to the power law spectrum model is simulated according to the fractional-order integration algorithm, as in Figure 9. Figure 10 is the time-domain sequence corresponding to each power law spectral component. Figure 11 is the phase noise single-side single-power spectrum model calculated by Figure 9, and the analysis result is highly consistent with the measured result in Figure 8.  
Figure 8: Phase noise measurement of 5 MHz OCXO. Figure 10: Phase noise time-domain sequence curve of 5 MHz OCXO.