Classification of Noise Types present in a Clock

 The internal phase noise typically affecting oscillators obey the power-law model . Such a model can be expressed as follows in terms of the Power Spectral Density of the fractional frequency deviation at Fourier frequency f from the nominal frequency nu_nom:  

$$S_{y}(f)=\left \{ \begin{array} {ll} \sum_{\alpha=n_{1}}^{n_... ...lpha}& 0\leq f \leq f_{h}\ 0&f\gt f_{h}\ \end{array} \right\}$$ (11)

where f_h is an upper cut-off frequency. Since it can be shown [16] that  

$$S_n(\omega)=\frac{1}{\left\vert\omega\right\vert^2}S_{\dot{n}(\omega)}$$ (12)

and since from equation (3.5)

$$y(t)=\dot{x}(t)$$

therefore

$$S_y(\omega)=\frac{1}{\left\vert\omega\right\vert^2}S_x(\omega)$$

i.e.

$$S_y(f)=\frac{1}{(2\pi f)^2}S_x(f)$$ (13)

Therefore the power-law model is also applicable to the power spectral density of the time error and one may write

$$S_x(f)=\frac{h_{\alpha}f^{\beta}}{(2\pi)^2}$$ (14)

where $\beta=\alpha-2$.

The most common noise sources found in oscillators are itemised in table 3.1.

 

$$\beta$$ Noise Type Power Spectral Density

$$\beta=0$$ White Phase Modulation S_x(f)= constant

$$S_x(f)\propto\frac{1}{f} \beta=-1$$ Flicker Phase Modulation

$$S_x(f)\propto\frac{1}{f^2} \beta=-2$$ White Frequency Modulation

$$S_x(f)\propto\frac{1}{f^3} \beta=-3$$ Flicker Frequency Modulation

$$S_x(f)\propto\frac{1}{f^4} \beta=-4$$ Random Walk Frequency Modulation