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# HW1d - Advanced Digital Signal Processing Homework1...

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Advanced Digital Signal Processing Homework1 1. Let 1 X and 2 X be independent random variables derived from a uniform distribution ( ) 1 , 0 U . a. Show that the random vector ( ) ( ) ( ) 2 1 2 1 2 sin , 2 cos X X X X π π is uniformly distributed over the unit disk { } 1 : , 2 2 2 1 2 1 + y y y y . b. By using a polar to rectangular transformation on ( ) 2 1 , X X , show that the following transformation is equivalent to Box-Muller transformation presented in Step 4 of the Box-Muller procedure. ( ) ( ) 2 1 2 2 1 1 2 sin ln 2 2 cos ln 2 X X Z X X Z π π = = c. Verify that the joint density of 1 Z and 2 Z is ( ) I N , 0 2 , i.e. bivariate Gaussian with zero mean and covariance equal to the ( ) 2 2 × identity matrix (Use Jacobian identity) d. How does the efficiency of this version of the Box-Muller procedure compare to the procedure presented in the class? What advantage, if any, is there in using the Box-Muller discussed in class? 2. Let 1 X and 2 X be independent exponential r.v.s with parameter 0 > α (Recall: the density of an exponential r.v. with parameter α is ( ) ( ) 0 , > = = x e x Exp x f x α α α ).

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HW1d - Advanced Digital Signal Processing Homework1...

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