hmwk6_sol.pdf - ECE 513 DIGITAL SIGNAL PROCESSING Cranos M Williams Assigned October 9 2017 HOMEWORK 6 DUE SOLUTION 1 FFT Decimation in Time(20 Pts

hmwk6_sol.pdf - ECE 513 DIGITAL SIGNAL PROCESSING Cranos M...

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ECE 513 - DIGITAL SIGNAL PROCESSING Cranos M. Williams Assigned: October 9, 2017 HOMEWORK 6, DUE OCTOBER 16, 2017 SOLUTION 1.FFT - Decimation in Time: (20 Pts)Assume that we are given a signalx(n) such thatx(n) ={4,-3,2,-1,-5,3,-1,0}forn= 0, . . . ,7.(a) Using only 2-point FFTs, calculate the 8-point FFT of the sequencex(n) usingthe Decimation in Time Approach. Plot in Matlab the magnitude of the resultingspectrum fork= 0,1, . . . , 7. Solution: x 00 ( n ) = [ x (1) , x (5)] x 01 ( n ) = [ x (2) , x (6)] x 10 ( n ) = [ x (3) , x (7)] x 11 ( n ) = [ x (4) , x (8)] X 00 ( k ) = x (1) + x (5) e - j 2 πk 2 X 01 ( k ) = x (2) + x (6) e - j 2 πk 2 X 10 ( k ) = x (3) + x (7) e - j 2 πk 2 X 11 ( k ) = x (4) + x (8) e - j 2 πk 2 X 0 ( k ) = X 00 ( k ) + e - j 2 πk 4 X 01 ( k ) X 1 ( k ) = X 10 ( k ) + e - j 2 πk 4 X 11 ( k ) X ( k ) = X 0 ( k ) + e - j 2 πk 8 X 1 ( k ) (1) 1
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ECE 513 - Homework 6 2
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ECE 513 - Homework 6 3 figure(3)stem(abs(abs(Xk2) - magX)) ;title(’Absolute error between (a) and (b)’)Figure 1 gives a plot of Magnitude of spectrum forX(k) usingDecimation intime. 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 14 16 18 k |X 2 (k)| Magnitude plot for spectrum using Decimation in Time Figure 1: Plot of Spectrum for X ( k ) using Decimation in Time Pb1-(a).
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