# APPLICATION NOTE 30 DFT or FFT? A Comparison of Fourier Transform Techniques

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2 Fourier Series The fudametal basis of the Fourier series is that a fuctio defied withi the iterval x ca be expressed as a coverget trigoometric series i the followig form; f(x) = a 0 + a 1 cosx + a 2 cos2x + a 3 cos3x + + b 1 six + b 2 si2x + b 3 si3x + where a 0, a 1, a 2, a 3 b 1, b 2, b 3 are real costats This ca be simplified to the followig; f(x) = a 0 + =1 (a cos x + b si x) (4) For the rage to ; a 0 = 1 2 a = 1 b = 1 f(x)dx f(x) cos x dx f(x) si x dx ( = 1, 2, 3 ) (6) ( = 1, 2, 3 ) (7) (5) A alterative way of writig the Fourier series above, which is of more relevace to digital sigal processig of complex electrical waveforms is derived from the followig relatioship; a cos x + b cos x = c si (x + α) (8) Applyig this to the Fourier series; f(x) = a 0 + c 1 si(x + α) + c 2 si(2x + α 2 ) + + c si (3x + α ) (9) a 0 is a costat c 1 = a b 1 2,. c = a 2 + b 2 are the magitudes of each coefficiet (10) α = arcta a b is the phase offset of each coefficiet (11) C represets the magitude of the th compoet, or th harmoic. Whe = 1, the harmoic is kow as the fudametal. α represets the phase agle of the th harmoic. It is at this poit that all of the mathematical tools required to decompose a periodic waveform have bee derived.

3 Mathematical Example, 2Vpk Square Wave I this example a 2Vpk (4Vpk-pk) square wave is decomposed ito its costituet harmoics. The fuctio for the above waveform is give by; Fig 1. 2, whe < x < 0 f(x) = 2, whe 0 < x < As the pure square wave fuctio is discotiuous, the itegratio will be performed i two discrete halves. Oe from - to 0 ad aother from 0 to. a 0 = 1 2 f(x)dx a 0 = 1 [ 0 2 dx dx] 0 (12) (13) a 0 = 1 {[ 2x] [2x] 0 } = 0 (14) a 0 equates to the mea value of the waveform (DC offset i electrical terms) over the complete fudametal period. 0 a = 1 [ 2 cos x dx + 2 cos x dx ] 0 (15) a = 1 2 si x 0 si x = 0 (16) Therefore, a 1, a 2, a 3 are all zero. 0 b = 1 [ 2 si x dx + 2 si x dx ] 0 (17) b = 1 cos x 2 0 cos x = 0 (18) whe is odd

4 b = (19) b = k = 4k as k = 2, b = 8 (20) (21) Therefore b 1 = 8, b 3 = 8 3, b 5 = 8 5 whe is eve b = = 0 (22) b 1, b 2, b 3 = 0 Therefore, the Fourier series for the 2Vpk square wave is as follows; f(x) = a 0 + =1 (a cos x + b si x) (23) f(x) = 0 + =1 (0 + b si x) (24) f(x) = 8 six + 8 si3x + 8 si5x + (25) 3 5 f(x) = 8 (six + 1 si3x + 1 si5x + ) (26) 3 5 The magitude ad phase of each harmoic ca be computed as follows; c = a 2 + b 2 are the magitudes of each coefficiet (27) α = arcta a b is the phase offset of each coefficiet (28) For the 3rd harmoic; c 3 = = 3 3 α = arcta 0 = 0 (30) 8 3 3rd Harmoic = 0.85V, 0 (29)

5 Harmoic Aalysis It is widely kow that may of the waveforms withi the electroics idustry ca be represeted by simple mathematical expressios. By utilisig the Fourier series, the magitude ad phase of the harmoics ca be derived. Some waveforms do ot fit ito this category ad aalysis of such waveforms is performed through umerical methods. Harmoic Aalysis is the process of resolvig a periodic, o-siusoidal waveform ito a series of siusoids of icreasig order of frequecy. Followig o from the mathematical expressios derived earlier i this applicatio ote, a umerical method is ow preseted which itroduces the fudametals behid the approach used withi harmoic aalyzers. Ultimately, Fourier coefficiets a 0, a ad b will eed to be determied - see equatios (5), (6) ad (7), this will require itegratio. Numerically, the itegral fuctios (5), (6) ad (7) ca be described as mea values, as follows; a 0 = 1 f(x)dx = f(x)dx (5) a 0 = mea value of f(x) (31) a = 1 f(x) cos x dx = 1 2 f(x) cos x dx 0 ( = 1, 2, 3 ) (6) a = twice mea value of f(x) cos x (32) b = 1 f(x) si x dx ( = 1, 2, 3 ) (7) b = twice mea value of f(x) si x (33) This itegratio is performed withi the sigal processig elemets of a harmoic aalyzer (traditioally a DSP) with the trapezoidal rule. As most waveforms withi the realm of electroics are periodic, the period of the waveform ca be referred to as w (widow), which ca be divided up ito s (sample poits) of equal parts. Where d(time delta) is the time iterval i betwee the sample poits. A complex waveform is ow used as a example (fig.2) to illustrate how the waveform is sampled ad umerical Fourier aalysis methods are explaied. Fig 2.

6 Fig 3. = total umber of samples, i th sample, d = time betwee samples a 0 = Mea Value = a = 1 v[i] cos 2hi i=1 b = 1 v[i] si 2hi i=1 area legth of base = v[i] i=1 = 1 v[i] i=1 (34) (35) (36) I electrical terms, a 0 is the DC offset, a is the quadrature compoet of the th harmoic, ad b is the iphase compoet of the th harmoic. Simple trigoometry ca be applied to derive the magitude ad phase of the harmoics.

8 If a FFT harmoic spectrum aalysis is performed, sigificat harmoic leakage is evidet as a result of the o itegral cycle acquisitio. Fig 5. Various complex waveforms are ow aalyzed ad results from FFT ad DFT algorithms are compared. If the data acquisitio widow ecompasses a iteger umber of cycles, the harmoic leakage is sigificatly reduced. Although this is urealistic i a dyamic, real time measuremet eviromet this is performed for clarity of the mathematics i fig 6 ad fig 7. Fig 6. Fig 7. FFT Aalysis with ideal widow size

9 If the frequecy detectio or sample rate limitatio (ad subsequet widow size) drifts by oly 3~4Hz, the errors itroduced ito the FFT are quite sigificat. The graphs below illustrate this effect; Fig 8. Fig 9. Harmoic 1 (Fudametal) is sigificatly lower tha the 100Vpk origial compoet, there is also sigificat leakage ito other harmoics which i reality have o spectral voltage compoets. This is caused by the fact that the "sie ad cosie" multiplicatio of the FFT is ot performed at the correct harmoic frequecies, the fudametal compoet is i fact a 29.3Hz sie ad cosie multiplicatio. The FFT frequecies are derived from the sample rate ad the widow size. FFT Frequecy = fs sa = sample frequecy umber 1,2,3 1024, fs = sample rate, sa = umber of samples The FFT frequecy deotes the actual sie cosie calculatio, if the sample frequecy is ot ifiitely adjustable the FFT frequecies will be close but ot completely matchig the actual harmoic frequecies.

10 Error i FFT calculatio Harmoic Frequecy Actual (Vpk) Calculated with FFT Error % Fudametal 30.3Hz Hz Hz Hz Hz Hz Table 1. The further the fudametal frequecies drift from the FFT frequecy, the greater the leakage ad greater the error i the measured harmoics. To Illustrate this effect, a fudametal frequecy of 31.3Hz (equatig to a drift of 2Hz) results i the followig spectral aalysis ad subsequet errors. Fig 10. Harmoic Frequecy Actual (Vpk) Calculated with FFT Error % Fudametal 30.3Hz Hz Hz Hz Hz Hz Table 2. As previously metioed, it is impractical to achieve a exact iteger umber of waveform periods to fit withi the data acquisitio widow whe usig the FFT i a dyamic eviromet. It is therefore ot desirable to utilise the FFT Fourier aalysis techique i moder day power aalysis. There are widowig methods available to the egieer which will reduce the spectral leakage, these algorithms will improve values at some frequecies but also worse values at other frequecies. Ultimately, oe of the widowig algorithms available whe applied to a FFT will match the accuracy of the DFT.

11 DFT (Discrete Fourier Trasform) The Discrete Fourier trasform is able to resolve its samplig widow to ay iteger umber of samples, this equates to; widow resolutio = 1 fs I the time domai, this equates to a time resolutio of 100uS for a 10ks/s sample rate. If a 50Hz waveform is cosidered, its time period is 20ms. A 100us time resolutio eables the widow to perfectly ecompass both a 50Hz ad 50.25Hz, it is logical to coclude that a higher sample rate will result i greater frequecy resolutio as well as higher badwidth, if multiple cycles are used this resolutio is further improved. Newtos4th power aalyzers feature sample rates of 1Ms/s ad above, coupled with very accurate aalogue performace, this provides extremely sesitive ad accurate harmoic aalysis with fast update rates. Newtos4th Frequecy resolutio o a 50Hz waveform N4L PPA5500 widow resolutio(cycle by cycle) = 1 2Ms/s = = 500S 1 N4L PPA5500 cycle by cycle frequecy resolutio o 50Hz waveform = = = 0.001Hz The same waveform illustrated i fig.10 was used ad a DFT was performed, for a appropriate compariso the same 10ks/s sample rate was used. Whe utilisig a DFT waveform, it is possible to sychroise the data acquisitio widow to the fudametal time period with a resolutio of 1 sample poit. Oce this is performed, the DFT is able to calculate the harmoic compoets of the waveform very accurately. I this example the fudametal time period is calculated as follows; Fudametal Time Period = 1 = 1 = secods f fud 31.3 The umber of samples required to sychroise the data acquisitio widow to the fudametal time period is calculated as follows; No of samples required = fud time period sample time iterval = = The closest iteger umber of samples calculated is used as the data widow acquisitio size, N4L power aalyzers compute the fudametal frequecy "real-time" with a proprietary frequecy detectio algorithm withi a dedicated DSP. It should be apparet that for the DFT to be successful, frequecy detectio is of paramout importace, as all N4L power aalyzers utilise sample rates i excess of 1Ms/s, it is clear that the widow ad subsequet frequecy resolutio will be very accurate.

12 Fig 11. Fig 12. Figure 11 ad Figure 12 illustrate the effectiveess of the DFT eve whe utilisig low sample rates ad a sigle cycle. Higher sample rates ad multiple cycle aalysis will achieve greater accuracies through better widow sychroisatio (providig frequecy detectio is accurate) as well as greater averagig. Harmoic Frequecy Actual (Vpk) Calculated with FFT Error % Table 3. The calculated error achieved with the DFT is sigificatly better tha the equivalet FFT, further beefits of the DFT iclude the flexibility of the data acquisitio widow providig cycle by cycle aalysis of ay waveform without the restrictios of 2 FFT widowig.

13 Referece Measuremets A N4L PPA5500 Precisio Power Aalyzer, which utilises a DFT Harmoic Aalysis algorithm was bech tested agaist the Fluke 6105A power stadard. The tests were carried out withi the UKAS ISO17025 Test Laboratory (Lab o. 7949) based at N4L Headquarters i the UK, results from the tests are show below; Fluke 6015A Calibratio of PPA5530 Power Aalyzer - No Adjustmet Harmoic Frequecy Applied (Vpk) Measured Error Ucertaity % 0.01% <10mV NA NA <10mV NA NA <10mV NA NA % 0.03% <10mV NA NA The errors above represet the total error of the measuremet istrumet, icludig the error of the aalogue iput chaels. This calibratio procedure demostrates the power of the DFT whe combied with extremely liear aalogue iput desig. Summary Whilst the FFT has its place withi the electrical egieerig field, for high accuracy power aalysis the FFT is ot the optimum solutio. A high ed power aalyzer will eed to meet the demads of moder idustry i which harmoic distortio of varyig fudametal frequecies is required to be measured. It is the resposibility of the power aalyzer maufacturer to itegrate sufficietly powerful processors ad iovative digital sigal processig techiques i order to meet the computatioal demads of the DFT. All N4L power aalyzers have bee developed with this approach i mid ad years of experiece fie tuig both aalogue hardware performace ad DFT sigal processig algorithms have resulted i a extremely accurate solutio uder a wide rage of iput waveforms. For more iformatio about ay of the Newtos4th Power Aalyzers, visit Author : Sales ad Applicatios Egieerig, Newtos4th Ltd, UK Refereces [1] J. Bird, Higher Egieerig Mathematics - Fifth Editio, Elsevier, 2006

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