| Line 23: | Line 23: | ||
Such that<br> | Such that<br> | ||
<math>Y(t)=c_1X(t)-c_2X(t-\tau)</math> <math>\sim N((c_1-c_2)\mu_x,(c_1^2+c_2^2)\sigma_x^2-2c_1c_2R_{xx}(\tau))</math><br> | <math>Y(t)=c_1X(t)-c_2X(t-\tau)</math> <math>\sim N((c_1-c_2)\mu_x,(c_1^2+c_2^2)\sigma_x^2-2c_1c_2R_{xx}(\tau))</math><br> | ||
| − | <math>\Rightarrow </math> | + | <math>\Rightarrow |
| + | P(Y(t)<=\gamma) | ||
| + | =\int_{-\infty}^{r} \dfrac{1}{\sqrt{2\pi\sigma^2}}e^{-\dfrac{1}{2\sigma^2}(x-\mu)^2}dx | ||
| + | =\dfrac{1}{\sigma}\int_{-\infty}^{r} \dfrac{1}{\sqrt{2\pi}}e^{-\dfrac{(\dfrac{x-\mu}{\sigma})^2}{2}}dx \\ | ||
| + | =</math> | ||
---- | ---- | ||
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Revision as of 08:15, 19 February 2019
Communication Signal (CS)
Question 1: Random Variable
August 2016 Problem 4
Solution
Since $ X(t) $ is a wide sense Gaussian Process $ \Rightarrow X(t) $ is SSS.
$ Y(t) $ is a combination of two Gaussian distribution.
$ R_{x(t)x(t+\tau)}=R_{xx}(\tau) $
Such that
$ Y(t)=c_1X(t)-c_2X(t-\tau) $ $ \sim N((c_1-c_2)\mu_x,(c_1^2+c_2^2)\sigma_x^2-2c_1c_2R_{xx}(\tau)) $
$ \Rightarrow P(Y(t)<=\gamma) =\int_{-\infty}^{r} \dfrac{1}{\sqrt{2\pi\sigma^2}}e^{-\dfrac{1}{2\sigma^2}(x-\mu)^2}dx =\dfrac{1}{\sigma}\int_{-\infty}^{r} \dfrac{1}{\sqrt{2\pi}}e^{-\dfrac{(\dfrac{x-\mu}{\sigma})^2}{2}}dx \\ = $
