Computing Techniques ME 2016
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This 0 page Class Notes was uploaded by Chloe Reilly on Monday November 2, 2015. The Class Notes belongs to ME 2016 at Georgia Institute of Technology - Main Campus taught by Aldo Ferri in Fall. Since its upload, it has received 13 views. For similar materials see /class/234238/me-2016-georgia-institute-of-technology-main-campus in Mechanical Engineering at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
ME2016 Dr Ferri DFT Example Consider the function ft 0507500st 03sintcos2t 04sin2t080033t From this expression it is eVident that a0 10 a1 075 b1 03 a 1 b2 04 a3 08 The function is created using the following Matlab code and is plotted below using N256 points over a time period of 211 seconds N256 delt2piN TNdelt tOdelt N ldelt aOlO alO75 bl O3 a308 a2l b2 4 faO2alcostblsinta2cos2tb2sin2ta3cos3t plottf xlabel39t39 ylabel39f39 In this case T 211 thus mo ZnT l rads The rst 11 DFT values are calculated using the following Matlab commands for k111 Fk O for r1N angle k 12piNr1 Fk Fk frexp iang1e end end Note that the index k goes from 1 to 11 which corresponds to the subscript k in for example fk and Fk going from 0 to 10 In tabular form the rst 11 values of F are given by k kmo RealFk ImagFk 0 00 1280 0 1 10 960 384 2 20 1280 512 3 30 1024 0 4 40 0 0 5 50 0 0 6 60 0 0 7 70 0 0 8 80 0 0 9 90 0 0 10 100 0 0 These 11 complex values Fk are identical to those that would have been obtained using the Matlab tm command gtgt Ff f t f The magnitude of F is plotted vs the frequency kmo in rads below 140 1207 447 DFT O Based on exact freq amps 100 Q 80 607 4o 7 DFT Frequency rads Also shown in the gure are the values predicted using the following correspondence between the DFT and the Fourier series coefficients In general the following relations hold a0 2F0 N and 2 2 a RealF b Ima F kl23 k N k k N g k a0 The constant term is given by a0 ZFON 2128256 10 a1 The factor multiplying cost cosl wot is 2rea1F1N 296256 075 b1 The factor multiplying sint sinl mo t is 2imagF1N 2384256 03 a2 The factor multiplying cos2t cos2 mo t is 2rea1F2N 2128256 10 b2 The factor multiplying sin2t sin2 mo t is 2imagF2N 2512256 04 a3 The factor multiplying cos3t cos3 mo t is 2rea1F3N 21024256 08 Note that absFgFg Nell2 For kgt0 absFk 1Re alFk 2 1m agFk 2 1 a b The commands used to generate the above plot are W0 2piT w O10WO AmpO Na0 Ampl Nsqrta102b1022 Amp2 Nsqrta2A2b2A22 Amp3 Na32 amps AmpO Ampl Amp2 Amp3 p10twabsF39 39O3amps39o39 xlabel 39Frequency rads39 y1abe139DFT39 1egend39DFT3939Based on exact freq amps39 The last 11 values of Fk as computed using F fft f are given by k kwo RealFk ImagFk 245 2450 0 0 246 2460 0 0 247 2470 0 0 248 2480 0 0 249 2490 0 0 250 2500 0 0 251 2510 0 0 252 2520 0 0 253 2530 1024 0 254 2540 1280 51 2 255 2550 960 384 It may be seen that FNk conj Fk for kgt0 Aperiodic Signal O en we use FFT s to analyze systems that don t repeat For example we can have a signal that is zero after some transient event is over Here we consider a unit pulse of duration 516 sec followed by zero values for t gt 516 sec the signal is said to be padded with zeros beyond 516 sec Using a sampling interval of 116 sec and N 16 samples the signal is as shown below Although the true signal is zero for t gt 1 sec the FFT has no way to know that and it assumes that it repeats with period 1 sec The signal shown plotted below can be generated using the commands tO1161516 f1 1 1 1 1 O O O O O O O O O O O 1 The DFT of the signal can be calculated using the Matlab command gtgt Ff f t f The magnitude of the DFT of f is shown below Note that the magnitude plot is symmetric about the 9th point since N2 8 which corresponds to the 9Lh element in the vector F Magnitude of F N U a N u I 0 10 20 30 40 50 60 70 80 90 100 Frequency rads
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