# Evaluating Optical Clock Performance for GNSS Positioning
## 3 Clock Characterisation  
This section presents a brief description of clock characterisation including frequency stability measures and relevant noise behaviours, as these are essential in analysing clock frequency stability. As described in [3], all types of atomic clocks have a similar working principle that requires a source of oscillation to produce stable frequencies. Then, counting the cycles in relation to an initial time will determine the corresponding time. A time output is normally expressed as one pulse-per-second (1 PPS). Although atoms are isolated in a vacuum chamber, their natural frequencies can be perturbed by external environmental effects, normally from electric and magnetic fields and gravity, and therefore the causing errors in the measurement must be characterised in order to correct them. Frequency stability, sometimes referred to as frequency instability, is a critical characteristic of any atomic clock including optical clocks, which describes how consistent the clock frequencies are over a given time interval. Allan Variance (AVAR) is a standard measure of frequency stability of oscillators used to characterise the clock frequency stability rather than a standard variance due to the non-stationary noise behaviours [16]. AVAR is denoted as \(\sigma_{y}^{2}(\tau)\), where \(\tau\) is averaging time, also known as integration time or observation time. To measure the frequency stability of a clock, either a single or series of reference clocks is needed. In some cases, clocks were self-tested [49]. In the testing process, the output frequency of the clock under test is compared with reference clock frequency over various averaging times--measured by a reference clock, and the fractional frequency average \(y\) over that interval \(\tau\) is determined. Thus, AVAR is a function of averaging time, \(\tau\). The square root of AVAR is ADEV, also known as sigma-tau denoted as \(\sigma_{y}(\tau)\). ADEV is a unitless expression, but it can be read as follows. For example, an ADEV of \(\sigma_{y}(\tau)\) = 1 \(\times\) 10\({}^{-10}\) at averaging time \(\tau\) = 1 s means that there was a fractional frequency change over 1 s. In other words, if \(\sigma_{y}(\tau)\) = 1 \(\times\) 10\({}^{-10}\), the clock with a 10 MHz frequency standard at \(\tau\) = 0 s would differ by about 1 mHz after 1 s. Generally, clock ADEV degrades over long-term averaging time because of frequency drift and aging [16]. Several methods have been introduced for computing ADEV to improve the results statistically. For instance, the overlapping ADEV--adopted in this study--is the most widely used method as this can provide better statistical confidence by making use of more of the data within a dataset. In some studies, the Modified ADEV (MDEV) is used to distinguish the white and flicker frequency noises [50]. ADEV values are normally plotted in log-log graphs. Further information on the different methods and their calculation formulas can be found in [51] and references therein. The frequency stability of most clocks can be modelled through the combination of various power-law noises [52] such as White Phase Modulation (WPM), Flicker Phase Modulation (FPM), White Frequency Modulation (WFM), Flicker Frequency Modulation (FFM), and Random Walk Frequency Modulation (RWFM). AVAR cannot separate phase modulation noises; however, it can distinguish frequency noises such as white, flicker, and random walk. Figure 2 shows typical noise behaviours on ADEV vs. Averaging time plot where the black dash curve is ADEV of a notional clock, and the green, blue, and red lines are white, flicker, and random walk frequency noise contributions, respectively. Additionally, Table 1 provides the relationship between AVAR and power-law noises formulated in [51]. Conveniently, AVAR identifies the power-law noises differently and results in the estimation of their strengths. Figure 2: Typical dependency of clock stability against averaging time for a notional clock.