Basic Definitions

The essential function of an oscillator is to produce a desired nominal frequency $\nu_{nom}$. In order to characterise how well a particular oscillator performs this function, it is necessary to first define certain useful physical measures commonly used in frequency metrology and briefly discuss some of the issues involved in measuring the frequency stability of an oscillator.

Consider a clock that produces an ideal timing signal of nominal frequency $\nu_{nom}$ and with phase

$$\Phi(t)=2 \pi \nu_{nom}(t)$$

A non-ideal clock whose phase has some added random noise $\varphi(t)$ will produce a signal with phase  

$$\Phi(t)=2 \pi \nu_{nom}(t)+\varphi(t)$$ (5)

The time error x(t) resulting from phase noise $\varphi(t)$ is given by  

$$x(t)=\frac{\varphi(t)}{2 \pi \nu_{nom}}$$ (6)

It is important to realise that any frequency measurement must necessarily utilise two oscillators. This is because it is only possible to measure the frequency difference of a clock under test with respect to another, usually higher quality oscillator. It is therefore useful to define the dimensionless quantity  

$$y(t)=\frac{\nu(t)-\nu_{nom}}{\nu_{nom}}$$ (7)

as the fractional frequency deviation of an oscillator with respect to a reference oscillator with nominal frequency $\nu_{nom}$.