TDEV as a function of Time Error

The ETSI noise model that was simulated outputs samples of the time error rather than fractional frequency error. Thus it is necessary to re-cast equation (4.3) in terms of x. By considering that from equation (3.6),

$$\overline{y(t)}=\frac{x_{i+1}-x_{i}}{\tau}$$

the above expression (4.3) for TDEV can be expressed in terms of the time error x(t) as

$$TDEV(\tau)=\sqrt{\frac{1}{6}\left<\left(\frac{1}{n(N-1)}\sum_{i=1}^{i=N-1}x_{i+2}-2x_{i+1}+x_i\right)^2\right\gt}.$$

assuming there are N+1 time error samples. By considering that the samples are spaced by integer multiples of $\tau_0$ any particular $\tau$ in the above equation can be chosen such that $\tau=n\tau_0$. This results in the equation

$$TDEV(\tau=n\tau_0)=\sqrt{\frac{1}{6}\left<\left(\frac{1}{n(N-n)}\sum_{i=1}^{i=N-n}x_{i+2n}-2x_{i+n}+x_i\right)^2\right\gt}.$$

To increase the confidence in the estimate and to maximise the data utilisation Wallis and Allan [23] substitute N-3n+1 for N in their expression for the Allan Variance and therefore limit n to values up to $\mathrm{int}(\frac{N}{3})$. Performing a similar procedure here yields

$$TDEV(\tau=n\tau_0)=\sqrt{\frac{1}{6n^2}\left<\left(\frac{1}{... ...}\sum_{i=1}^{i=N-3n+1}x_{i+2n}-2x_{i+n}+x_i\right)^2\right\gt}.$$

By taking the mean the above equation may be estimated as [1]  

$$TDEV(\tau)=\sqrt{\frac{1}{6n^{2}(N-3n+1)}\sum_{j=1}^{N-3n+1}... ...m_{i=j}^{n+j-1}\left(x_{i+2n}-2x_{i+n}+x_{i}\right)\right)^{2}}$$ (28)

where x_i is the i^th sample of the time error function; N is the total number of samples; $\tau=n\tau_0$ is the integration time; and $n=1,2,\ldots,$ $\mathrm{int}(\frac{N}{3})$ is the number of sampling intervals. It is this equation (4.4) that was used to calculate TDEV, as is discussed in section B.1.