Revision as of 16:31, 19 February 2019 by Wan82 (Talk | contribs)


ECE Ph.D. Qualifying Exam

Communication Signal (CS)

Question 1: Random Variable

August 2016 Problem 1


Solution

a)
$ F_x(\omega)= \begin{cases} 0 & \omega<0 \\ \dfrac{1}{2}hb-\dfrac{1}{2}hb(\dfrac{h-\omega}{h})^2=\dfrac{2\omega}{h}-\dfrac{w^2}{h^2} & 0\le\omega<h \\ 1 & \omega\ge h \end{cases} $

b)
$ f_x(\omega)=\dfrac{\partial F_x(\omega)}{\partial\omega} $
$ f_x(\omega)= \begin{cases} 0 & \omega<0 \\ \dfrac{-2}{h^2}\omega+\dfrac{2}{h} & 0\le\omega<h \\ 0 & \omega\ge h \end{cases} $

c)
$ X(\omega)\bar=\int_{-\infty}^{\infty} \omega f_x(\omega) dx =\int_{0}^{h} -\dfrac{2}{h^2}(\omega)^2 +\dfrac{2}{h}\omega d\omega =\dfrac{1}{3}h $

d)
$ P(x>\dfrac{h}{3})=\int_{\dfrac{h}{3}}^{+\infty} \omega f_x(\omega)dx = \int_{\dfrac{h}{3}}^{h} -\dfrac{2}{h^2}\omega+\dfrac{2}{h}d\omega =\dfrac{4}{9} $


Back to QE CS question 1, August 2016

Back to ECE Qualifying Exams (QE) page

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett