Theoretical Analysis of the TDEV behaviour of WGN shaped by filter A

In this section the TDEV of behaviour of noise at the output behaviour will be predicted in order to ensure that the simulation for filter A is behaving correctly. This is achieved by using the estimator formula for TDEV (4.9) printed again below:

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

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Consider a White Gaussian noise source input into Filter A as depicted in figure 8.21. Since from equation (5.6)

$$\frac{1}{2\eta}=\frac{\sigma^2}{B_n}$$

and substituting the transfer function for filter A, namely (2.2) from page

$$H_{A}(s)\approx \frac{ \sqrt{2 \pi f_0}}{s^{\frac{1}{2}}} \q... ...here} f_{0}=\frac{\alpha_{1}}{2 \pi \sqrt[4]{7}}\approx 5\mu Hz$$

into (5.2)

S_out(f)=|H(f)|²S_in(f),

the TDEV at the output of filter A is simply

$$S_{A}(f)=\frac{\sigma^2}{2B_n}\frac{f_0}{f}.$$

Substituting the above and the values B_n=5Hz and $f_0\approx5\mu Hz$ into (4.9)

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

results in

$$\mathrm{TDEV}_A(\tau)\approx\sigma\sqrt{\frac{5}{2\tau}\frac{f_0\tau}{10\cdot 0.3}}[s]$$

Upon rearranging and removing $\sigma$ to obtain a value in nanoseconds the approximation

$$\mathrm{TDEV}_A(\tau)\approx \sqrt{\frac{f_0}{1.2}}\approx \sqrt{f_0}[ns]$$ (58)

is obtained.