The time series data in this example are obtained from sampling a function describing the free decay of a torsion oscillator for time t > t o

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1 The Excel FFT Fucti v2 P T Debevec July 5, 28 The discrete Furier trasfrm may be used t idetify peridic structures i time series data Suppse that a physical prcess is represeted by the fucti f time, ht ( ) The fucti is sampled at times, t = Δ t where =,,2,, Frm these measuremets, h, cmplex amplitudes, H, are determied which satisfy the equatis H = 2π i = h e The sampled fucti the has the discrete Furier expasi 2π i h = He This equati ca be cast i familiar frm with ( ) ( ) = 2π = 2π T T = ωδ t= ωt h iω t He = = The right-had side is the discrete aalgue t the cmplex frm f the Furier expasi where the cmplex cefficiets, i t ht () = ce ω =, are give by c iωt c = ht () e dt T T The Excel data aalysis pacage has a Furier aalysis rutie which calculates the cmplex cefficiets, H, frm the time series data, h The rutie requires that the umber f samples i the time series data be a pwer f 2, ie = 2 m The example i this te uses = 248 The Excel fucti is t well dcumeted, but it is straightfrward t use This te describes the Excel wrsheet, Furier_examplexls, which is i the Physics 4 web site uder Tutrials ad Lectures, Experimet

2 The time series data i this example are btaied frm samplig a fucti describig the free decay f a trsi scillatr fr time t > t, () ( ) ( π ( )) at t θ t = Ae si 2 f t t The fucti is calculated i time steps f 2 s, which crrespds t samplig rate f 5 Hz The time series data are shw i the Fig belw free decay damped scillatr raw cuts time (s) Fig Plt f time series data decay f trsi scillatr The data ccupy cells B3 t B25 i the data wrsheet f the wrb Clic Tls i the Excel meu bar, ad select Data Aalysis I Data Aalysis select Furier Aalysis, ad a simple dialg bx appears The dialg bx is shw belw i Fig 2 Fig 2 Furier Aalysis dialg bx

3 Fr Iput Rage eter $B$3:$B$25, the lcati f the time series data, ad fr utput rage eter a cveiet place the wrsheet, fr example, $J$3:$J$25 After selectig the OK butt, Excel returs i clum J the cmplex cefficiets, 248 f them A prti f clum J is shw i Fig 3 belw Fig 3 Prti f Excel wrsheet shwig FFT utput Excel isists i displayig 5 digits fr bth the real ad imagiary part f the cefficiet, ad thus the utput ccupies csiderable space Examiati f the clum f cmplex umbers shws that the first etry (i cell J3) ad the 25 th etry (i cell J27) are real These cells represet the cefficiets fr the zer frequecy ad the fldig frequecy, f c Sice the data were sampled at 5 Hz, the fldig frequecy is ½ 5 Hz = 25 Hz Cells J3 thrugh J27 represet the cmplex cefficiets fr 2+ frequecies frm Hz t 25 Hz The frequecy icremet is the Δ f = 2 f c It is cveiet t mae a clum f frequecies ext t the clum f cefficiets This clum icludes cells I3 t I27 The Furier Aalysis rutie perates ly the time series data The crrespdece betwee frequecy ad cmplex cefficiet must be calculated idepedetly The 23 etries frm cell J28 t cell J25 are idetical t the 23 etries frm cell J26 t J4 These etries ctai additial ifrmati, ad they culd have bee mitted frm the display The Furier Aalysis f time series data yields the cmplex cefficiets fr ly 2+ frequecies The pwer i each frequecy bi is prprtial t the square f the mdulus f the cmplex cefficiet The cstat f prprtiality adpted i this te has the prperty that the mea squared amplitude f the time series data is equal t the ttal pwer i the frequecy dmai as shw belw Excel has a umber f fuctis fr cmplex argumets Multiplicati f tw cmplex umbers is accmplished with the fucti IMPRODUCT(z, z 2 ) Multiplicati f the cefficiets i clum J by prduces the rmalized cefficiets i clum The abslute value f a cmplex umber is accmplished with the fucti IMABS(z) Applyig the IMABS fucti t the rmalized cefficiets i clum ad multiplyig by a factr f 2 prduces the magitude f the Furier cefficiets i clum P The magitude f the Furier cefficiets as a fucti f frequecy is shw i Fig 4 belw The maximum value ccurs i the frequecy bi at 4883 Hz

4 magitude f Furier amplitude magitude f Furier amplitude versus frequecy frequecy (Hz) Fig 4 Magitude f rmalized Furier cefficiet versus frequecy Fially, the pwer at each frequecy (except zer frequecy) is the square f the magitude The pwer is calculated i clum Q ad shw i Fig 5 belw The pwer at zer frequecy is the square f the magitude divided by 2 This factr f 2 als appears i -discrete Furier aalysis This pwer distributi is characteristic f expetial decay E+5 Furier pwer versus frequecy E+3 Furier pwer E+ E- E frequecy (Hz) Fig 5 Furier pwer versus frequecy

5 The rmalizati adpted i this te has a cveiet prperty Defie the mea squared amplitude f the time series data as θ = mea squred amplitude = ( ) 2 t The square f the amplitude is calculated i clum C The mea squared amplitude is calculated i cell F9 Fr these data it has the value f A pure sie wave with a amplitude f 2, fr which a itegral umber f perids ccur i the ttal sampled time, has a mea squared amplitude f uity Thus the mea squared amplitude has the same rle as the rms value f a peridic fucti The ttal Furier pwer, the sum f the etries i clum Q, is calculated i cell G9 Fr these data it als has the value f , because f the rmalizati If ly the relative pwer i differet frequecy bis is eeded, the the rmalizati is t required The relative pwer is give by the square f the mdulus f the Excel utput directly