TDEV Generated by the Top Branch of the Noise Model

 In this section the TDEV approximation (4.9)  

 $$TDEV(\tau)\approx\sqrt{\frac{1}{2.5\tau}S_x\left(\frac{0.3}{\tau}\right)}$$

will be used to predict the noise types generated by the top branch of the model. The power spectral density of the noise generated is given by (5.7)

$$S_{out_{f}}(f)=\frac{\sigma^2}{2B_n} K_f^2 \frac{f_0}{f}\frac{B^2}{f^2 + B^2}$$

Substituting the above equation into (4.9) one obtains an expression for the time deviation of the noise produced by the top branch, TDEV_f in seconds

$$TDEV_f(\tau)\approx\sigma K_f B\sqrt{\frac{f_0}{5B_n(0.3)}}\sqrt{\frac{\tau^2}{0.3^2+B^2\tau^2}}[s]$$

Given that from section 2.4, $f_0\approx 5\cdot 10^{-6}Hz, B_n=5Hz$,

$$\sqrt{\frac{f_0}{5B_n(0.3)}}\approx \frac{10^{-3}}{\sqrt{1.5}}\approx 10^{-3}$$

Hence

$$TDEV_f(\tau)\approx\sigma\cdot K_f\cdot B \cdot 10^{-3}\sqrt{\frac{\tau^2}{0.3^2+B^2\tau^2}}[s]$$

The factor $\sigma=10^{-9}s$ also defined in section 2.4 may be removed to give an expression in nanoseconds  

$$\mathrm{TDEV}_f(\tau)\approx K_f\cdot B \cdot 10^{-3}\sqrt{\frac{\tau^2}{0.3^2+B^2\tau^2}}[ns]$$ (49)