20091ee114_1_hw7_sol

20091ee114_1_hw7_sol - EE114 Winter 2009 Problem Set 7...

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EE114, Winter 2009 Problem Set 7 Solution ________________________________________________________________________ Problem Set #7 Solution 1) a) Confirm that the four basis vectors of the N = 4 1-D DFT form an orthonormal set, that is, a a k l k l = = 1 0 , , , , otherwise where, * denotes conjugation. Confirm that the above relationship holds for the four possible combinations of k and l = 0 or 1 (e.g. k,l = 0,0; 0,1; 1,0; and 1,1). Here, the dot in a a k l means dot product (or inner product). The basis vectors are . 1 1 2 1 , 1 1 1 1 2 1 , 1 1 2 1 , 1 1 1 1 2 1 3 2 1 0 - - = - - = - - = = j j a a j j a a Now we show they are orthonormal. [ ] ( 29 1 4 4 1 ) 1 1 1 1 ( 4 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 4 1 0 0 = = + + + = + + + = = = a a [ ] ( 29 ( 29 0 1 1 4 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 4 1 1 0 = + - - = + - + - + = - - = -
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This note was uploaded on 05/10/2010 for the course EE EE114 taught by Professor Vanderschaar during the Spring '09 term at UCLA.

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20091ee114_1_hw7_sol - EE114 Winter 2009 Problem Set 7...

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