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= [[ECE PhD Qualifying Exams|ECE Ph.D. Qualifying Exam]] in "Communication, Networks, Signal, and Image Processing" (CS)  =
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[[Category:ECE]]
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[[Category:QE]]
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[[Category:CNSIP]]
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[[Category:problem solving]]
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[[Category:random variables]]
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[[Category:probability]]
  
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= [[ECE-QE_CS1-2011|Question 1, August 2011]], Part 1 =
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<center>
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<font size= 4>
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[[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]]
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</font size>
  
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:[[ECE-QE_CS1-2011_solusion-1|Part 1]],[[ECE-QE CS1-2011 solusion-2|2]]]
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<font size= 4>
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Communication, Networking, Signal and Image Processing (CS)
  
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----
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Question 1: Probability and Random Processes
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</font size>
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}\text{1. } \left( \text{25 pts} \right) \text{ Let X, Y, and Z be three jointly distributed random variables with joint pdf} f_{XYZ}\left ( x,y,z \right )= \frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy} exp \left [ -\frac{1}{2}\left ( \frac{x-y}{z}\right )^{2} \right ] \cdot 1_{\left[0,\infty \right )}\left(y \right )\cdot1_{\left[1,2 \right]} \left ( z \right) </math></span></font>
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'''<math>\color{blue}\left( \text{a} \right) \text{ Find the joint probability density function } f_{YZ}(y,z).</math>'''<br>
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===== <math>\color{blue}\text{Solution 1:}</math>  =====
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<math> f_{YZ}\left (y,z \right )=\int_{-\infty}^{+\infty}f_{XYZ}\left(x,y,z \right )dx </math>&nbsp;
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math> =\frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy}\int_{-\infty}^{+\infty}exp\left[-\frac{1}{2}\left(\frac{x-y}{z} \right )^{2} \right ]dx\cdot 1_{[0,\infty)}
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\left(y \right )\cdot1_{\left [1,2 \right ]}\left(z \right )</math><br>
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<math>\text{But}\int_{-\infty}^{+\infty}exp\left[-\frac{1}{2}\left(\frac{x-y}{z} \right )^{2} \right ]dx \text{looks like the Gaussian pdf, so} </math>
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<math> =\frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy}
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\underset{\sqrt[]{2\pi}z}{\underbrace{\frac{7\sqrt[]{2\pi}z}{7\sqrt[]{2\pi}z}  \int_{-\infty}^{+\infty}exp\left[-\frac{1}{2}\left(\frac{x-y}{z} \right )^{2} \right ]dx}}\cdot 1_{[0,\infty)}
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\left(y \right )\cdot1_{\left [1,2 \right ]}\left(z \right )
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</math>
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<math>
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=\frac{3z^{2}}{7}e^{-zy}\cdot 1_{[0,\infty)}
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\left(y \right )\cdot1_{\left [1,2 \right ]}\left(z \right )
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</math>
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August 2011
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</center>
 
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<math>\color{blue}\text{Solution 2:}</math>
 
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here put sol.2
 
 
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==Question==
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'''Part 1. ''' 25 pts
  
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<math>\color{blue}\left( \text{b} \right) \text{Find}
 
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f_{x}\left( x|y,z\right )
 
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</math><br>
 
  
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<math>\color{blue}\text{Solution 1:}</math>  
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}\text{ Let } \mathbf{X}\text{, }\mathbf{Y}\text{, and } \mathbf{Z} \text{ be three jointly distributed random variables with joint pdf } f_{XYZ}\left ( x,y,z \right )= \frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy} exp \left [ -\frac{1}{2}\left ( \frac{x-y}{z}\right )^{2} \right ] \cdot 1_{\left[0,\infty \right )}\left(y \right )\cdot1_{\left[1,2 \right]} \left ( z \right) </math></span></font>  
  
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<font color="#ff0000"><span style="font-size: 17px;">'''<font face="serif"></font><math>
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'''<math>\color{blue}\left( \text{a} \right) \text{Find the joint probability density function } f_{YZ}(y,z).</math>'''<br>  
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= \frac{f_{XYZ}\left( x,y,z\right )}{f_{YZ}\left(y,z \right )}
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</math>'''</span></font><font color="#ff0000"><span style="font-size: 17px;">
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</span></font>  
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'''<font face="serif"><math>
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<math>\color{blue}\left( \text{b} \right) \text{Find }  
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= \frac{e^{-\frac{1}{2}\left(\frac{x-y}{z} \right )^{2}}}{\sqrt[]{2\pi}z}
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f_{x}\left( x|y,z\right ).
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</math>&nbsp;&nbsp;</font>'''
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</math><br>  
  
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----
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<math>\color{blue}\left( \text{c} \right) \text{Find }
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f_{Z}\left( z\right ).
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</math><br>
  
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<math>\color{blue}\text{Solution 2:}</math><br>  
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<math>\color{blue}\left( \text{d} \right) \text{Find }
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f_{Y}\left(y|z \right ).
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sol2 here
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</math><br>  
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----
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<math>\color{blue}\left( \text{c} \right) \text{Find}  
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<math>\color{blue}\left( \text{e} \right) \text{Find }  
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f_{Z}\left( z\right )
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f_{XY}\left(x,y|z \right ).
 
</math><br>  
 
</math><br>  
  
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<math>\color{blue}\text{Solution 1:}</math>
 
  
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<font color="#ff0000"><span style="font-size: 17px;">'''<font face="serif"></font><math>
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:'''Click [[ECE-QE_CS1-2011_solusion-1|here]] to view student [[ECE-QE_CS1-2011_solusion-1|answers and discussions]]'''
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=\int_{0}^{+\infty}{f_{YZ}\left(y,z \right )dy}
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----
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</math>'''</span></font><font color="#ff0000"><span style="font-size: 17px;">
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'''Part 2.''' 25 pts
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</span></font>
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'''<font face="serif"><math>
 
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=\frac{3z^{2}}{7}\cdot1_{\left[1,2 \right ]}(z)
 
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</math>&nbsp;&nbsp;</font>'''
 
  
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}  \text{Show that if a continuous-time Gaussian random process } \mathbf{X}(t) \text{ is wide-sense stationary, it is also strict-sense stationary.}
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</math></span></font>
  
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<math>\color{blue}\text{Solution 2:}</math><br>
 
  
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sol2 here
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:'''Click [[ECE-QE_CS1-2011_solusion-2|here]] to view student [[ECE-QE_CS1-2011_solusion-2|answers and discussions]]'''
 
----
 
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'''Part 3.''' 25 pts
  
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<math>\color{blue}\left( \text{d} \right) \text{Find}
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Show that the sum of two jointly distributed Gaussian random variables that are not necessarily statistically independent is a Gaussian random variable.
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f_{Y}\left(y|z \right )
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</math><br>
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<math>\color{blue}\text{Solution 1:}</math>
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<font color="#ff0000"><span style="font-size: 17px;">'''<font face="serif"></font><math>
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=\frac{f_{YZ}\left(y,z \right )}{f_{Z}(z)}</math>'''</span></font><font color="#ff0000"><span style="font-size: 17px;">
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</span></font>
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'''<font face="serif"><math>
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=e^{-zy}z\cdot1_{\left[0,\infty \right )}(y)
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</math>&nbsp;&nbsp;</font>'''
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:'''Click [[ECE-QE_CS1-2011_solusion-3|here]] to view student [[ECE-QE_CS1-2011_solusion-3|answers and discussions]]'''
 
----
 
----
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'''Part 4.''' 25 pts
  
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<math>\color{blue}\text{Solution 2:}</math><br>
 
  
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sol2 here
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Assume that <math>\mathbf{X}(t)</math> is a zero-mean continuous-time Gaussian white noise process with autocorrelation function
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<math>\color{blue}\left( \text{e} \right) \text{Find}
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f_{XY}\left(x,y|z \right )
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</math><br>
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<math>\color{blue}\text{Solution 1:}</math>  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <math>R_{\mathbf{XX}}(t_1,t_2)=\delta(t_1-t_2).
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</math>
  
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<font color="#ff0000"><span style="font-size: 17px;">'''<font face="serif"></font><math>
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Let <math>\mathbf{Y}(t)</math> be a new random process ontained by passing <math>\mathbf{X}(t)</math> through a linear time-invariant system with impulse response <math>h(t)</math> whose Fourier transform <math>H(\omega)</math> has the ideal low-pass characteristic
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=\frac{f_{XYZ}\left(x,y,z \right )}{f_{Z}(z)}
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</math>'''</span></font><font color="#ff0000"><span style="font-size: 17px;">
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</span></font>  
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'''<font face="serif"><math>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>H(\omega) =
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=\frac{e^{-zy}}{\sqrt[]{2\pi}}e^{-\frac{1}{2}\left(\frac{x-y}{z} \right )^{2}}\cdot1_{\left[0,\infty \right )}(y)
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\begin{cases}  
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</math>&nbsp;&nbsp;</font>'''
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1, & \mbox{if } |\omega|\leq\Omega,\\
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0, & \mbox{elsewhere,}
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\end{cases}
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</math>
  
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where <math>\Omega>0</math>.
  
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<math>\color{blue}\text{Solution 2:}</math><br>
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a) Find the mean of <math>\mathbf{Y}(t)</math>.
  
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sol2 here
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b) Find the autocorrelation function of <math>\mathbf{Y}(t)</math>.
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"Communication, Networks, Signal, and Image Processing" (CS)- Question 1, August 2011
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c) Find the joint pdf of <math>\mathbf{Y}(t_1)</math> and <math>\mathbf{Y}(t_2)</math> for any two arbitrary sample time <math>t_1</math> and <math>t_2</math>.
  
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Go to
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d) What is the minimum time difference <math>t_1-t_2</math> such that <math>\mathbf{Y}(t_1)</math> and <math>\mathbf{Y}(t_2)</math> are statistically independent?
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*Part 1: [[ECE-QE_CS1-2011_solusion-1|solutions and discussions]]
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*Part 2: [[ECE-QE CS1-2011 solusion-2|solutions and discussions]]
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:'''Click [[ECE-QE_CS1-2011_solusion-4|here]] to view student [[ECE-QE_CS1-2011_solusion-4|answers and discussions]]'''
 
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[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]]
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[[ECE PhD Qualifying Exams|Back to ECE Qualifying Exams (QE) page]]
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[[Category:ECE]] [[Category:QE]] [[Category:Automatic_Control]] [[Category:Problem_solving]]
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Latest revision as of 15:40, 30 March 2015


ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2011

Question

Part 1. 25 pts


 $ \color{blue}\text{ Let } \mathbf{X}\text{, }\mathbf{Y}\text{, and } \mathbf{Z} \text{ be three jointly distributed random variables with joint pdf } f_{XYZ}\left ( x,y,z \right )= \frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy} exp \left [ -\frac{1}{2}\left ( \frac{x-y}{z}\right )^{2} \right ] \cdot 1_{\left[0,\infty \right )}\left(y \right )\cdot1_{\left[1,2 \right]} \left ( z \right) $

$ \color{blue}\left( \text{a} \right) \text{Find the joint probability density function } f_{YZ}(y,z). $

$ \color{blue}\left( \text{b} \right) \text{Find } f_{x}\left( x|y,z\right ). $

$ \color{blue}\left( \text{c} \right) \text{Find } f_{Z}\left( z\right ). $

$ \color{blue}\left( \text{d} \right) \text{Find } f_{Y}\left(y|z \right ). $

$ \color{blue}\left( \text{e} \right) \text{Find } f_{XY}\left(x,y|z \right ). $


Click here to view student answers and discussions

Part 2. 25 pts


 $ \color{blue} \text{Show that if a continuous-time Gaussian random process } \mathbf{X}(t) \text{ is wide-sense stationary, it is also strict-sense stationary.} $


Click here to view student answers and discussions

Part 3. 25 pts

Show that the sum of two jointly distributed Gaussian random variables that are not necessarily statistically independent is a Gaussian random variable.

Click here to view student answers and discussions

Part 4. 25 pts


Assume that $ \mathbf{X}(t) $ is a zero-mean continuous-time Gaussian white noise process with autocorrelation function

                $ R_{\mathbf{XX}}(t_1,t_2)=\delta(t_1-t_2). $

Let $ \mathbf{Y}(t) $ be a new random process ontained by passing $ \mathbf{X}(t) $ through a linear time-invariant system with impulse response $ h(t) $ whose Fourier transform $ H(\omega) $ has the ideal low-pass characteristic

               $ H(\omega) = \begin{cases} 1, & \mbox{if } |\omega|\leq\Omega,\\ 0, & \mbox{elsewhere,} \end{cases} $

where $ \Omega>0 $.

a) Find the mean of $ \mathbf{Y}(t) $.

b) Find the autocorrelation function of $ \mathbf{Y}(t) $.

c) Find the joint pdf of $ \mathbf{Y}(t_1) $ and $ \mathbf{Y}(t_2) $ for any two arbitrary sample time $ t_1 $ and $ t_2 $.

d) What is the minimum time difference $ t_1-t_2 $ such that $ \mathbf{Y}(t_1) $ and $ \mathbf{Y}(t_2) $ are statistically independent?

Click here to view student answers and discussions

Back to ECE Qualifying Exams (QE) page

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