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# hwB27 - x and y have DFT’s X = £ 1 0 1-1 T and Y = £ 3...

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ENEE 241 02* HOMEWORK ASSIGNMENT 27 Due Mon 04/14 (i) (7 pts.) Determine the circular convolution s = x ~ y of x = £ 5 - 2 0 - 1 / T and y = £ 3 1 1 2 / T Using the FFT function in MATLAB, verify that the DFTs X , Y and S satisfy X ƒ Y = S . (ii) (7 pts.) If 1 3 - 2 1 4 4 1 3 - 2 1 1 4 1 3 - 2 - 2 1 4 1 3 3 - 2 1 4 1 x 0 x 1 x 2 x 3 x 4 = 4 0 - 1 7 4 , explain how the vector x can be obtained using DFTs (as opposed to a conventional solution of a 5 × 5 system of equations). Implement this solution in MATLAB to obtain x . (iii) (6 pts.) Let x and y be vectors of length N . Suppose that x is a sum of Fourier sinusoids of even-indexed frequencies ( k = 0 , 2 , . . . ), while x is a sum of Fourier sinusoids of odd-indexed frequencies ( k = 1 , 3 , . . . ). Without further information, can you determine the circular convolution vector s = x ~ y ? If so, what is the value of s ? Explain. Solved Examples S 27.1

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Unformatted text preview: x and y have DFT’s X = £ 1 0 1-1 / T and Y = £ 3 5 8-4 / T (i) Is either x or y real-valued? (ii) Does either x or y have circular conjugate symmetry? (iii) Without inverting X or Y , determine the DFT of the signal s (1) deﬁned by s (1) [ n ] = x [ n ] y [ n ] , n = 0 , 1 , 2 , 3 (iv) Without inverting X or Y , determine the DFT of the signal s (2) deﬁned by s (2) = x ~ y S 27.2 ( P 3.21 in textbook). The time-domain signals x = £ 2 0 1 3 / T and y = £ 1-1 2-4 / T have DFT’s X and Y given by X = £ X X 1 X 2 X 3 / T and Y = £ Y Y 1 Y 2 Y 3 / T Determine the time-domain signal s whose DFT is given by S = £ X Y 2 X 1 Y 3 X 2 Y X 3 Y 1 / T...
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hwB27 - x and y have DFT’s X = £ 1 0 1-1 T and Y = £ 3...

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