Question 19.14: Find the PDF of τk for a sinusoid with amplitude A and angul......
Find the PDF of τ_{k} for a sinusoid with amplitude A and angular frequency \omega _{0} in additive white gaussian noise, n(t), with variance \sigma _{n}^{2} .
The sinusoid may be described as:
v(t) = A\sin (\omega _{0}t) + n(t) (19.77)
As in Example 19.9, assuming the noise n is much smaller than the amplitude of oscillation, \sigma _{n}~«~A , the absolute jitter is linearly related to n _{k} by the slope of the sinusoid around its zero crossings.
\tau _{k} \cong \frac{n_{k}}{A\omega _{0}}
Since n _{k} is gaussian-distributed, absolute jitter is also gaussian-distributed with the following PDF:
f_{\tau }(t) = \frac{1}{\sigma _{\tau }\sqrt{2\pi }}exp\left\lgroup\frac{t^{2}}{2\sigma _{\tau }^{2}} \right\rgroup (19.78)
Where \sigma _{\tau } is given by (19.57).
\sigma ^{2}_{\tau } = \frac{\sigma ^{2}_{n}}{A^{2} \omega _{0}^{2}} (19.57)