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----
 
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==Solution 3==
 
==Solution 3==
<math>Y=AX=\left(\begin{array}{cc}
+
<math>Y=\left(\begin{array}{c}Y_1 \\ Y_2\end{array} \right)=AX=\left(\begin{array}{cc}
 
a & b\\
 
a & b\\
 
c & d
 
c & d
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Thus, <math>Y_1,Y_2</math> are Gaussian distributed random variables. If they are uncorrelated, then they are also independent.
 
Thus, <math>Y_1,Y_2</math> are Gaussian distributed random variables. If they are uncorrelated, then they are also independent.
  
<math>r = \frac{COV(Y_1,Y_2)}{\sigma1\sigma2}</math>
+
<math>r = \frac{COV(Y_1,Y_2)}{\sigma1\sigma2} = 0</math>
 +
<math>r = E(Y_1Y_2) - E(Y_1)E(Y_2) = 0</math>
  
  

Revision as of 23:03, 22 February 2017


ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2013



Part 2

Let $ X_1,X_2,... $ be a sequence of jointly Gaussian random variables with covariance

$ Cov(X_i,X_j) = \left\{ \begin{array}{ll} {\sigma}^2, & i=j\\ \rho{\sigma}^2, & |i-j|=1\\ 0, & otherwise \end{array} \right. $

Suppose we take 2 consecutive samples from this sequence to form a vector $ X $, which is then linearly transformed to form a 2-dimensional random vector $ Y=AX $. Find a matrix $ A $ so that the components of $ Y $ are independent random variables You must justify your answer.


Solution 1

Suppose

$ A=\left(\begin{array}{cc} a & b\\ c & d \end{array} \right) $.

Then the new 2-D random vector can be expressed as

$ Y=\left(\begin{array}{c}Y_1 \\ Y_2\end{array} \right)=A\left(\begin{array}{c}X_i \\ X_j\end{array} \right)=\left(\begin{array}{c}aX_i+bX_j \\ cX_i+dX_j\end{array} \right) $


Therefore,

$ \begin{array}{l}Cov(Y_1,Y_2)=E[(aX_i+bX_j-E(aX_i+bX_j))(cX_i+dX_j-E(cX_i+dX_j))] \\ =E[(aX_i+bX_j-aE(X_i)-bE(X_j))(cX_i+dX_j-cE(X_i)-dE(X_j))] \\ =E[acX_i^2+adX_iX_j-acX_iE(X_i)-adX_iE(X_j)+bcX_iX_j+bdX_j^2-bcX_jE(X_i)\\ -bdX_jE(X_j)-acX_iE(X_i)-adX_jE(X_i)+acE(X_i)^2+adE(X_i)E(X_j)\\ -bcX_iE(X_j)-bdX_jE(X_j)+bcE(X_i)E(X_j)+bdE(X_i)^2]\\ =E(ac(X_i-E(X_i))^2+(ad+bc)(X_i-E(X_i)(X_j-E(X_j))+bd(X_j-E(X_j))^2]\\ =(ac)Cov(X_i,X_i)+(ad+bc)Cov(X-i,X_j)+(bd)Cov(X_j,X_j)\\ =ac\sigma^2+(ad+bc)\rho\sigma^2+bd\sigma^2 \end{array} $

Let the above formula equal to 0 and $ a=b=d=1 $, we get $ c=-1 $.

Therefore, a solution is

$ A=\left(\begin{array}{cc} 1 & 1\\ -1 & 1 \end{array} \right) $.



Solution 2

Assume

$ Y=\left(\begin{array}{c}Y_i \\ Y_j\end{array} \right)=A\left(\begin{array}{c}X_i \\ X_j\end{array} \right)=\left(\begin{array}{c}a_{11}X_i+a_{12}X_j \\ a_{21}X_i+a_{22}X_j\end{array} \right) $.

Then

$ \begin{array}{l}E(Y_iY_j)=E[(a_{11}X_i+a_{12}X_j)(a_{21}X_i+a_{22}X_j)]\\ =a_{11}a_{21}\sigma^2+a_{12}a_{22}\sigma^2+(a_{11}a_{21}+a_{22}a_{11})E(X_iX_j) \end{array} $

For $ |i-j|\geq1 $, $ E(X_i,X_j)=0 $. Therefore, $ a_{11}a_{21}+a_{12}a_{22}=0 $.

One solution can be

$ A=\left(\begin{array}{cc} 1 & -1\\ 1 & 1 \end{array} \right) $.


Critique on Solution 2:

1. $ E(Y_iY_j)=0 $ is not the condition for the two random variables to be independent.

2. "For $ |i-j|\geq1 $, $ E(X_i,X_j)=0 $" is not supported by the given conditions.


Solution 3

$ Y=\left(\begin{array}{c}Y_1 \\ Y_2\end{array} \right)=AX=\left(\begin{array}{cc} a & b\\ c & d \end{array} \right)\left(\begin{array}{c}X_i \\ X_j\end{array} \right)=\left(\begin{array}{c}aX_i+bX_j \\ cX_i+dX_j\end{array} \right) $

We know that the sum of two independent Gaussian distributed random variables is still Gaussian distributed.

Thus, $ Y_1,Y_2 $ are Gaussian distributed random variables. If they are uncorrelated, then they are also independent.

$ r = \frac{COV(Y_1,Y_2)}{\sigma1\sigma2} = 0 $ $ r = E(Y_1Y_2) - E(Y_1)E(Y_2) = 0 $




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