hw03_sol

# hw03_sol - CS170 Spring 2010 HW3 Solutions 1(DPV 2.7 For n...

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Unformatted text preview: CS170 - Spring 2010 HW3 - Solutions 1. (DPV 2.7) For n 6 = 0 and ω = e 2 π n : n- 1 X i =0 ω i = 1- ω n 1- ω = 0 n- 1 Y i =0 ω i = ω P n- 1 i =0 i = ω n ( n- 1) 2 The latter is 1 if n is odd and- 1 if n is even. 2. (DPV 2.9) a) Use N = 4 and ω = e 2 πi/ 4 = i . The FFT of x + 1 is FFT(1 , 1 , , 0) = (2 ,i + 1 , ,- i + 1). The FFT of x 2 + 1 is FFT(1 , , 1 , 0) = (2 , , 2 , 0). Hence, the FFT of their product is (4 , , , 0), corresponding to the polynomial 1 + x + x 2 + x 3 . b) Use ω = i . The FFT of 2 x 2 + x +1 is FFT(1 , 1 , 2 , 0) = (4 ,- 1+ i, 2 ,- 1- i ). The FFT of 3 x +2 is FFT(2 , 3 , , 0) = (5 , 2 + 3 i,- 1 , 2- 3 i ). The FFT of their product is then (20 ,- 5- i,- 2 ,- 5 + i ). This corresponds to the polynomial 6 x 3 + 7 x 2 + 5 x + 2. 3. (DPV 2.30) (a) Observe that taking ω = 3 produces the following powers : ( ω,ω 2 ,ω 3 ,ω 4 ,ω 5 ,ω 6 ) = (3 , 2 , 6 , 4 , 5 , 1). Verify that ω + ω 2 + ω 3 + ω 4 + ω 5 + ω 6 = 1 + 2 + 3 + 4 + 5 + 6 = 21 = 0 ( mod 7) (b) The matrix M 6 (3) is the following: 1 1 1 1 1 1 1 3 2 6 4 5 1 2 4 1 2 4 1 6 1 6 1 6 1 4 2 1 4 2 1 5 4 6 2 3 Multiplying with the sequence (0 , 1 , 1 , 1 , 5 , 2) we get the vector (3 , 6 , 4 , 2 , 3 , 3). (c) The inverse matrix of M 6 (3) is easily seen to be the matrix 6 · 1 1 1 1 1 1 1 5 4 6 2 3 1 4 2 1 4 2 1 6 1 6 1 6 1 2 4 1 2 4 1 3 2 6 4 5 1 Verify that multiplying these two matrices mod 7 equals the identity. Also multiply this matrix with vector (3 , 6 , 4 , 2 , 3 , 3) to get the original sequence. (d) We first express the polynomials as vectors of dimension 6 over the integers mod 7: (1 , 1 , 1 , , , 0), and (- 1 , 2 , , 1 , , 0) = (6 , 2 , , 1 , , 0) respectively. We then apply the matrix M 6 (3) to both to get the transform of the two sequences. That produces (3 , 6 , , 1 , , 3) and (2 , 4 , 4 , 3 , 1 , 1) respectively....
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## This note was uploaded on 04/30/2010 for the course CS 170 taught by Professor Henzinger during the Spring '02 term at Berkeley.

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hw03_sol - CS170 Spring 2010 HW3 Solutions 1(DPV 2.7 For n...

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