## 2 Theory Introduction  
### Phase Noise  
In general, the mathematical model of the oscillator output signal can be expressed as the following:  
\[V_{o}(t)=A\sin(2\pi f_{o}t+\phi(t)) \tag{1}\]  
The amplitude noise caused by the amplitude fluctuation is omitted. \(\phi(t)\) in the phase term represents all phase and frequency deviations from the nominal frequency \(f_{o}\) and phase \(2\pi f_{o}t\), which can be called phase perturbations. There are two reasons for introducing \(\phi(t)\): one is the initial offset of the oscillator frequency and the drift of the frequency with time caused by causal elements such as ambient temperature changes and device aging, which mainly affect the medium and long-term frequency stability of the oscillator output signal. The other is the random phase fluctuation caused by random factors such as thermal noise and flicker noise inside the device, which mainly affect the short-term frequency stability of the oscillator. Phase noise is the random fluctuation of the oscillator phase caused by this random factor. This random fluctuation causes the signal spectrum to no longer be an infinite impulse function, as sidebands appear at the center frequency, causing spectrum spread, as shown in Figure 1 below.  
To characterize phase noise, the following two basic concepts are introduced: instantaneous relative frequency offset \(y(t)\) and instantaneous relative phase offset: \(x(t)\)  
\[y(t)=\frac{1}{2\pi f_{o}}\frac{\mathrm{d}\phi(t)}{\mathrm{d}t} \tag{2}\]  
\[x(t)=\frac{\phi(t)}{2\pi f_{o}} \tag{3}\]  
Characterization of phase noise can be performed in the time and frequency domains, respectively. The commonly used characterization method in the time domain of phase noise is in the form of the Allan variance [17]:  
\[\sigma^{2}(\tau)=\frac{1}{2}<(y_{k+1}-y_{k})^{2}>=\frac{1}{2\tau^{2}}<(x_{k+2} -2x_{k+1}+x_{k})^{2}> \tag{4}\]  
\(\tau\) is the sampling interval of the Allan variance (such as 1 s, 10 s \(\ldots\) 1000 s, etc.). \(y_{k}\) and \(x_{k}\) represent the discrete sequences obtained by sampling the instantaneous relative frequency offset and phase offset at intervals of \(\tau\), respectively. The method of characterizing phase noise in the frequency domain is mainly the method of power spectral density. The power spectral density of phase noise is generally expressed as \(S_{\phi}(f)\). Phase noise has a strictly one-way mathematical relationship to the Allan variance. Among the methods for quantifying phase noise, the single-sideband power spectral density \(L(f)\) is the most commonly used indicator for the characterization of phase noise in the frequency domain, and it is also the result given by a general phase noise analyzer. \(L(f)\) is defined as the ratio of the single-sideband noise power to the carrier signal power of the oscillator output signal within a 1 Hz bandwidth off-center carrier frequency \(\Delta f(Hz)\), as in Figure 2. When in decibel form, the unit of \(L(f)\) is \(\mathrm{d}\mathrm{b}\mathrm{c}/\mathrm{Hz}\). When the amplitude of \(\phi(t)\) is much less than 1 rad, \(L(f)\) and \(S_{\phi}(f)\) have the following relationship, in which the unit does not include dB.  
\[L(\Delta f)=10\mathrm{log}_{10}\bigg{[}\frac{P_{SSB}(f_{0}+\Delta f,1 Hz)}{P_{S }}\bigg{]} \tag{5}\]  
\[L(f)=10\mathrm{log}_{10}\bigg{(}\frac{1}{2}(S_{\phi}(f)\bigg{)} \tag{6}\]  
Figure 1: Spectral spreading due to phase noise. (The left is the ideal spectrum, the right is the actual output spectrum).  
Figure 2: Definition of phase noise.