Discrete Fourier Transform

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1 Discrete Fourier Trasform V 3. November 5, 27 Christia Koll, christia.koll@tugraz.at, Josef Kulmer, kulmer@tugraz.at, Christia Stetco, stetco@studet.tugraz.at Sigal Processig ad Speech Commuicatio Laboratory, Iffeldgasse 6c/EG Abstract This ote provides a brief review of the Fourier trasform for the aalysis of discretetime sigals ad systems ad a descriptio of practical assigmets, which will be performed o the Raspberry Pi. Some tasks are to be performed with the help of MATLAB. It is assumed that the studet has eough theoretical backgroud to perform the practical part of the work. The review of some aspects of discrete-time sigals ad systems provided here is just a quick remider. Equipmet: PC with Netbeas 8.2 & MATLAB istalled Raspberry Pi 2 Model B Oscilloscope Agilet 54622D Sigal geerator Agilet 332A Cables Itroductio I sigal processig applicatios we ofte face the ecessity to combie ad separate sigals. Sometimes a oscilloscope will clearly show what happes. Whe you mix sigals, however, the oscilloscope is ot very useful i may situatios, sice the sum of sigals is ofte a messy thig to look at. The idea behid the Fourier trasform is pretty simple: oe ca thik of sigals as of a sum of siusoids with differet frequecies ad phase shifts. This approach ca be applied to a variety of sigals (see table ), icludig cotiuous-time, discrete-time, periodic, aperiodic, ad eve N-dimesioal sigals.

2 Discrete Fourier Trasform 2 Sigal type / Time domai Cotiuous-time Discrete-time Periodic sigals Fourier series Discrete Fourier trasform Aperiodic sigals Fourier trasform Discrete-time Fourier trasform Table : Fourier aalysis framework 2 Overview of Fourier trasform family 2. Fourier series (FS) Fourier series are used for periodic cotiuous-time (aalog) sigals x(t) with period T. There are several ways to write this trasform, but the most commo is the two-sided, complex expoetial represetatio: X[k] = T T /2 2πk j T x(t)e t dt, k =.... T /2 The time sigal x(t) ca be computed from the Fourier coefficiets X[k] by x(t) = k= X[k]e jkω t (complex form), or by x(t) = A 2 + A k cos(kω t+ϕ k ) (trigoometric form), k= usig A k = 2 X[k], ad ϕ k = arg(x[k]) = arcta ad ω = 2π/T the fudametal agular frequecy. ( ) Im{X[k]}, Re{X[k]} Accordig to these equatios we ca represet a periodic sigal x(t) by a series of coefficiets X[k], or A k ad ϕ k. I the trigoometric form A k ad ϕ k are the amplitude ad the phase of the harmoic sigal compoet with agular frequecy ω = kω. I the complex form both amplitude ad phase of the complex harmoic sigal compoet e jkω t with agular frequecy ω = kω are represeted by the complex coefficiet X[k]. The Fourier series coefficiet for k =, X[] resp. A represets the mea value of x(t), i.e., the DC compoet i the sigal (a harmoic sigal at frequecy ω = ). Questio: What is the phase arg(x[]) resp. ϕ of the DC compoet? A resp. X[] ad X[ ] represet harmoic compoets at the fudametal agular frequecy ω. Coefficiets for k > represet higher harmoics of the sigal x(t). Questios: What is the relatio betwee X[k] ad X[ k]? Why do we sum over k =... i the complex form of the Fourier series represetatio but oly over k =... i the trigoometric form? Recall that e jkωt = cos(kω t)+jsi(kω t).

3 Discrete Fourier Trasform Fourier trasform (FT) The FT is used for aperiodic cotiuous-time (aalog) sigals x(t). The most commo way to express this trasform is the two-sided, complex expoetial represetatio: X(ω) = x(t)e jωt dt. The iverse trasform to fid the time sigal x(t) agai, is x(t) = 2π X(ω)e jwt dω. Accordig to these equatios ay time sigal x(t) ca be represeted by ad perfectly re-costructed from the fuctio X(ω) i the spectral domai. X(ω) is called the Fourier spectrum of the sigal x(t). The Fourier spectrum is a cotiuous fuctio of agular frequecy ω [, ]. To calculate the exact FT, you eed to kow x(t) from t = to t = +. This is of course impossible for real-world applicatios. I most cases of practical iterest, X(ω) will ot be periodic (cf. discrete-time Fourier trasform, below). 2.3 Discrete-time Fourier trasform (DTFT) The DTFT is used for aperiodic, discrete-time or digital 2 sigals x[]. The most commo way to express this trasform is X(e jθ ) = = x[] e jθ. Here, θ is the ormalized agular frequecy, θ = 2πf/f s, with f s beig the samplig rate of x[]. Sice the term e jθ, i the trasform is periodic i θ, with period 2π, also the resultig spectrum X(e jθ ) is periodic (with period 2π, too). To idicate this periodicity, the argumet of the spectrum is commoly writte as e jθ (ot θ oly). Due to the periodicity, the Fourier spectrum of a discrete-time sigal has to be kow oly for a iterval of legth 2π, commoly the fudametal iterval θ [ π,π] is used to represet X(e jθ ). The discrete-time sigal x[] ca be recovered by x[] = 2π π π X(e jθ ) e jθ dθ. This expressio says that we ca use the fuctio X(e jθ ) (Fourier spectrum) to represet the discrete-time sigal x[]. The spectrum X(e jθ ) is a cotiuous fuctio i θ. The DTFT spectrum is always periodic with the period of 2π. 2 A digital sigal, as used i computers ad DSPs, is a quatized discrete-time sigal.

4 Discrete Fourier Trasform Discrete Fourier Trasform (DFT) The DFT is used for periodic, discrete-time or digital sigals x[]. The DFT for a sigal with period N is The iverse trasform (IDFT) is X[k] = x[] = N N = N k= 2πk j x[]e N. X[k]e j 2πk N. These equatios say that the coefficiets X[k] represet the periodic discrete-time sigal x[]. Notice, that oly N samples of the time sigal x[] are used to compute X[k] (for k =... ), ad also oly N of the coefficiets X[k] are used i the iverse trasform. A periodic discrete-time sigal has a periodic spectrum (just like ay other discrete time sigal), i.e., X[k +N] = X[k]. The X[k] terms are evely spaced samples (at θ = k 2π ) of the cotiuous spectrum N X (e jθ ) of the sigal x [] = x[] for < N; x [] = for <, N (Verify this usig the formula for the DTFT above!). Fast Fourier trasform (FFT) To represet the time sigal we eed to compute X[k] for N values of k, ad for each of these values we must perform N multiplicatios ad N additios, so computig the DFT requires N(2N )arithmeticoperatios, i.e., ithasacomputatioal complexity ofo(n 2 )(orderofn 2 ). However, specifically if N is a power of 2, may of the DFT calculatios are redudat. By carefully re-arragig the order of multiplicatios ad additios, the computatioal complexity ca be reduced to O(N log 2 (N)). The resultig algorithm is called fast Fourier trasform. The differece betwee O(N 2 ) ad O(N log 2 (N)) ca be substatial: If N = 24, ad log 2 (24) =, the computatioal complexity of the FFT is oly % of that of a DFT. 3 Widow fuctios As oted above the DFT ad IDFT are defied for periodic time sigals. We may, however, be iterested i a digital represetatio of the spectrum of o-periodic discrete-time sigals. For time-limited sigals, i.e., for sigals that differ from zero oly for < N, we foud that the DFT results i samples of the DTFT. For o-periodic, ifiite (or log) time sigals we have to restrict the calculatio of the DFT to a umber of N samples, meaig that we may oly use a umber of L N sigal samples. The procedure to pick oly a limited umber of samples from a possibly ifiitely log sigal is called widowig. If we bear i mid, that we ow calculate a local (short-term) spectral represetatio, ad that the IDFT will result i a periodic sigal, the DFTsuch ca be applied for all discrete-time sigals.

5 Discrete Fourier Trasform 5 3. Rectagular widow The simplest way to restrict a ifiite time sigal to a legth of L samples is to multiply the sigal with a widow fuctio { < L, w[] = otherwise. This widow fuctio is called rectagular widow (for obvious reasos). Cosiderig that we multiply two sigals x[] ad w[] i the time domai, the correspodig operatio i the frequecy domai is a (circular) covolutio of the spectra X(e jθ ) ad W(e jθ ). E.g., the spectral lie X(e jθ ) = 2πδ 2π (θ θ ) of a harmoic sigal x[] = e jθ is trasfered to the spectrum of the widow fuctio shifted by θ after widowig: e jθ w[] 2πW(e j(θ θ ) ). To ivestigate the effects of widowig we thus should also look at the properties of the Fourier trasform of a widow fuctio. w[] -5 W(e jθ )/W(e j ).8.6 db π/L π -π/2 θ π/2 π Figure : Rectagular widow (L = 2) ad correspodig spectrum As depicted i fig. the rectagular widow results i a good frequecy resolutio due to the arrow mai-lobe. However, the amplitude measuremet capabilities are rather poor, due to the high amplitudes of the side-lobes. Except for self-widowig fuctios (impulses, bursts, etc.) the effects of spectral leakage (trasfer of spectral eergy due to the side-lobes of the widow fuctio Fourier trasform) are sigificat. The effect of itroducig spectral eergy at frequecies that are ot preset i the origial sigals spectrum by widowig ca also be explaied i the time domai: By widowig we itroduce discotiuities i the periodic sigal represeted by the DFT coefficiets i geeral. To reduce spectral leakage widow fuctios that decay i amplitude to-wards the begi ad ed of the widow are used. 3.2 Triagular (Bartlett) widow A simple widow fuctio that decays to-wards the eds is the triagular or Bartlett widow: (L )/2 < L, w[] = (L )/2 otherwise.

6 Discrete Fourier Trasform 6 A example of a triagular widow ad its spectrum is show i fig. 2. Notice the lower amplitude of the side-lobes i the spectrum as compared to that of the rectagular widow (differet scale!). For the triagular widow of (approximately) the same legth, however, we fid a cosiderably wider mai-lobe as for the rectagular widow. w[] - W(e jθ )/W(e j ).8.6 db π/(L-) π -π/2 θ π/2 π Figure 2: Triagular (Bartlett) widow (L = 2) ad correspodig spectrum 3.3 Geeralized cosie widows The Blackma, Hammig, Ha (aka. Haig), ad rectagular widows are all special cases of the geeralized cosie widow: { A Bcos(2π w[] = )+Ccos(4π ) < L, L L otherwise. where A, B, ad C are costats. The widow choice is usually task-depedet: the width of the the mai-lobe is iversely proportioal to height of the side-lobes. Thus, there is a trade-off betwee frequecy resolutio ad reductio of spectral leakage. The Hammig ad Ha widows (fig. 3) are the two-term geeralized cosie widows, give by the A =.54, B =.46 for Hammig ad A =.5, B =.5 for Ha widows (C = i both cases). The Blackma widow is a popular three-term widow, give by the A =.42, B =.5, C =.8. The flat-top widow that ca be selected for the FFT aalysis i the oscilloscope Agilet 54622D is a geeralized cosie widow with a sum over 5 terms, ad A =,B =.93,C =.29,D =.388, ad E = Kaiser widow The Kaiser widow is a approximatio to the prolate-spheroidal widow, for which the ratio of the mai-lobe eergy to the side-lobe eergy is maximized. For the Kaiser widow of the particular legth L, the parameter α cotrols the side-lobe height. For the give α, the sidelobe height is fixed with respect to the widow legth.

7 Discrete Fourier Trasform 7 w[] - W(e jθ )/W(e j ) db π/(L-) π -π/2 θ π/2 π w[] - W(e jθ )/W(e j ).8.6 db π/(L-) π -π/2 θ π/2 π Figure 3: Ha ad Hammig widows (L = 2) ad their spectra.8 w[] -2-4 W(e jθ )/W(e j ).6 db π -π/2 θ π/2 π Figure 4: Kaiser widows ad spectra for L = 2 ad α = 5 (circles ad solid lies), ad α = (squares ad dashed lies) w[] = I 2α L ( ) 2 L < L, I (α) otherwise. I ( ) is a modified Bessel fuctios of the first kid.

8 Discrete Fourier Trasform Chebyshev widow (also Dolph-Chebyshev) The Chebyshev widow miimizes the mai-lobe width, give a particular side-lobe height. It is characterized by a equi-ripple behavior, that is, its side-lobes have the same height. Examples are give i fig. 5. w[] - W(e jθ )/W(e j ) db π -π/2 θ π/2 π Figure 5: Chebyshev widows ad their spectra (L = 2) desiged for 3dB (circles ad solid lies) ad 6dB (squares ad dashed lies) side-lobe suppressio 3.6 Widows usable o the oscilloscope.2 w[] Rect Flat Top Ha 2 W(e (jθ) )/W(e (j) ) Rect Flat Top Ha ormalized θ (x π rad/sample) Figure 6: Widows ad their spectra o the oscilloscope i the laboratory (Agilet 54622D). I the practical part, we will make use of the FFT capabilities of the Agilet 54622D oscilloscope. O this device, you ca select three differet widow fuctios: Rectagular, flattop ad Ha-widow. Figure 6 shows a compariso of those widows i time ad frequecy domai. Thik about applicatio scearios for each of these three widows! I Matlab, you ca also use witool to comfortably create ad compare differet widow fuctios.

9 Discrete Fourier Trasform 9 4 Applicatios of the FFT 4. Estimatio of the power spectral desity A very commo task i sigal processig is the estimatio of the power spectral desity (PSD) of a sigal x[] []. A first step would be to compute the ormalized squared magitude of the sigal s DFT: I[k] = LU X w[k] 2 where L is the legth of the widow, U is a ormalizatio costat that depeds o the chose widow ad X w [k] is the k-th coefficiet of the DFT of the widowed sigal x[]w[]. This estimate of the PSD is called the periodogram if a rectagular widow is used, otherwise it is called the modified periodogram. Aother method to estimate the PSD of x[] is to compute the average of R periodograms: I[k] = R I r [k], R r= k =,...,N This meas that the idividual I r [k] are periodograms of blocks of the sigal x[]. Depedig o the applicatio, these blocks ca have overlap or ot. The choice of the widow fuctio w[] of course also iflueces the PSD estimatio. If w[] is the rectagular widow the the method of periodogram averagig is called Bartlett s procedure. I case of the other widow shapes it is called Welch s procedure. Thik about advatages ad disadvatages of the two methods! 4.2 Fast covolutio Oe of the first applicatios of the FFT algorithm was to implemet covolutio by multiplicatio i the frequecy domai faster tha by evaluatig the covolutio i the time domai directly. The liear covolutio is defied as y[] = L h i= h[i]x[ i], where x[] is a legth-l x iput sigal, h[] is a legth-l h impulse respose of the filter, ad y[] is the output sigal. Examiatio of this equatio shows that the output sigal y[] has legth L y = (L x +L h ), ad computatioal complexity is of order O(L y 2 ). The way to compute afast covolutio isto calculatethe product Y[k] = X[k]H[k] offfts of the time sigals x[] ad h[] ad to use a iverse FFT to compute the output sigal y[]. For large L x ad/or L h we profit from the reduced computatioal complexity of the FFT of O(N log 2 (N)). However, remember that sice the DFT ad FFT represet periodic time sigals the direct applicatio of this approach would result i a cyclic covolutio, which is ofte udesirable. The solutio is to pad both sigals with appropriate umber of zeros before computig the FFTs.

10 Discrete Fourier Trasform 5 Practical part The practical part of the work cosists of several tasks that should be performed usig the measuremet equipmet available ad the Raspberry Pi. Detailed explaatio of how ad what tasks should be accomplished is foud below. Experimet : Fourier aalysis of periodic sigals The first task of the laboratory will be accomplished without usig the RPi.. Switch o sigal geerator ad oscilloscope. Set the sigal geerator to produce a square waveform with amplitude of V ad a frequecy of 5 Hz. 2. Kowig the amplitude ad the period of the oscillatios, compute (aalytically) the coefficiets A k of the Fourier series of the iput sigal for k O the oscilloscope push Math FFT Settigs. Set Source to the chael that correspods to the output of the sigal geerator. The frequecy resolutio ca be adjusted by selectig the time-base of the iput sigal. Set a appropriate value. (Try to reach SPAN = khz ad Ceter = 5 khz.) Select More FFT ad set Widow =FLAT TOP. Set up other optios ( Scale ad Offset) to coveiet values. 4. Measure the amplitudes of the first 8 harmoic compoets of the correspodig Fourier represetatio. Determie the error betwee the theoretical ad the measured values. What is show o the scope, which iformatio is lost? Note: The FFT is plotted i the db domai. db correspods to V RMS amplitude. Produce ad fill i a table like the followig: Harmoic umber Frequecy (Hz) Theoretical amplitude A t (V) Measured amplitude A m (V) Error E = A t A m (V) 5. Chage the sigal type o the sigal geerator to a sawtooth sigal. What differece ca you observe (you do ot eed to measure the harmoics agai)? Now icrease the frequecy to 5 Hz. What do you observe?

11 Discrete Fourier Trasform Experimet 2: Time- ad frequecy-domai properties of differet widows I this task, MATLAB is used to demostrate basic properties of widowig fuctios.. Start MATLAB ad load the script file fftdemo.m. Set a breakpoit i the first lie of the program, ru it, ad discuss it usig the sigle-step debug-mode. Chage the frequecy from.2f s to (6/52)f s ad discuss the results. I lie 2, other widow fuctios ca be implemeted ad discussed, if there is time left. 2. Apply a sie-wave to the oscilloscope, view its FFT spectrum ad adjust the parameters of the oscilloscope i order to have the same ratio of the sigal- ad the FFT-samplig frequecies as i the previous task. Use the rectagular widow! Discuss the result ad compare it to the MATLAB-figures. To further ivestigate what happes i the oscilloscope, go step-by-step through the MATLAB script fft hp scope.m. 3. Set the waveform of the sigal geerator to sie, the amplitude to.44 V(correspodig to dbv RMS o the scope), ad the frequecy to khz. O the oscilloscope, set the time-base to msec div to get a frequecy-resolutio of Hz. Select the rectagular widow. Chage the frequecy betwee khz ad.2 khz i steps of Hz ad measure the amplitudes. Plot the measured amplitudes vs. frequecy ad discuss the result! Repeat for the other widows ad idetify which widow provides the most accurate measuremet of the amplitude. Note: To make your measuremets more accurate, chage the FFT scalig factor to db div ad also make use of cursors ad Quick Meas capabilities 4. Geerate a AM modulated sie-wave usig the sigal geerator(press Shift + AM to activate ad de-activate AM). Set the carrier frequecy ( Freq ) to khz, the amplitude to Vpp, ad the waveform to sie. For the modulatig sigal set the frequecy to Hz ( Shift + Freq ), modulatio depth to % ( Shift + Level ), ad the waveform to sie (is default). Discuss how the spectrum of the resultig sigal should look like! 5. O the oscilloscope, activate the iput sigal ad set the time-base of the iput sigal to 2 msec. As the result you will see the modulated siusoid with two periods of the div modulated waveform. Set the trigger to the exteral sigal, which will cause a stable picture. (Set Mode to Normal, if you do t obtai the stable picture.) 6. Press Math FFT Settigs. Make sure that Source is set to the correct chael (the output of the sigal geerator). Set Spa= 2 khz ad Ceter= khz. Press More FFT ad select Widow = Rectagular. What ca be see? What is the frequecyresolutio of the FFT used i the oscilloscope? (Cosider the widow-duratio, which correspods to the visible portio of the time-domai sigal.) 7. Usig the cursors, mark the frequecy of the carrier ad the frequecy of the upper sidebad at. khz. Ca you distiguish the carrier ad the sidebad sigals? How ca the frequecy-resolutio be ehaced? Which frequecy-resolutio is required to distiguish the sigals, without takig spectral leakage ito accout? Explai your aswer.

12 Discrete Fourier Trasform 2 8. Aswer the followig questios: How are the samplig period T s, samplig frequecy f s, the time-duratio of the FFT-widow T w, ad the frequecy-spacig f (frequecyresolutio) betwee FFT-lies related to oe aother? Use the parameter N to specify the umber of FFT-poits used. 9. Chage the time-base to icrease the spectral-resolutio. Deote for which resolutio the sidebads become visible first. Experimet 3: PSD estimatio Implemetatio of the periodogram ad Welch s method o the RPi.. Set up the sigal geerator to produce a 5 Hz sawtooth oscillatio with 5 mv amplitude. O the oscilloscope press Math ad eable FFT fuctio followig the kow procedure. 2. I Netbeas, ope the project Uit3 which ca be foud o the webpage. Go to the file welch.c ad look for the costat FFTSIZE, which sets the FFT size used by the program. 3. The FFT computatio is doe with the library fftw 3 which is a optimized FFT computatio library for gcc ad g++. Note that we compute the DFT of a purely real-valued sigal, i.e., we make use of the Hermitia property of the DFT. The output of the DFT computatio is therefore oly the positive half of the spectrum. This is beeficial i reducig computatioal complexity. 4. At the begiig of the source file welch.c you will see the defiitios of two importat parameters: #defie AVERAGING is used to specify the umber of periodograms to average ad #defie ADDNOISE is used to add a certai amout of oise (simulated withi the program) to the iput sigal to test for the robustess of the estimator. Make sure that AVERAGING is (such that o averagig is performed) ad that the ADDNOISElie is commeted out. If ecessary, rebuild the project. 5. The followig outputs are stored i.dat files for further processig with MATLAB. (i) iput file - cotais the iput data (ii) widowed iput file - cotais the widowed iput data (iii) DFT output file - cotais the N-poit FFT (iv) periodogram file - cotais the computed periodogram 6. To visualize the results, we will use MATLAB. Therefore, we eed to iclude the.dat files ito the MATLAB workspace. Beforehad, we eed to trasfer the ecessary files to our local machie by the followig commad: scp rp:./etbeas/remote/rp2/dsplabxx-liux-x86 64/home/user/DSP Lab WS78/Uit3 /dist/debug/gnu-liux/*.dat /home/user where XX is the umber of the local machie. 3 for further details take a look at

13 Discrete Fourier Trasform 3 7. I MATLAB, make sure that you achieve the correct scalig of the frequecy axis. How ca you covert the frequecy bis k ito frequecies give i Hz? 8. Explore the power spectral desity. Restart the program to view several periodogram estimates. Capture oe output for your report. Note which harmoics ca be idetified. 9. Add some oise by icludig the lie #defie ADDNOISE ad rebuildig the program. Agai ote which harmoics ca be idetified. Now icrease the FFT-size to 256 ad 52. Describe the effect of the loger FFT o the PSD estimate!. Now use periodogram averagig by settig #defie AVERAGING 6 ad with the FFT-size agai reduced to 28. Rebuild ad ru the program. Note agai which harmoics ca be idetified. Also try the loger FFT sizes with averagig eabled ad also higher umbers for the umber of periodograms to average. What are the advatages ad disadvatages of the Welch method?. For your report, capture some of the outputs for the PSD estimatio that you fid to be explaatory. Refereces [] Oppeheim, A. V., Schafer, R. W., ad Buck, J. R., Discrete-Time Sigal Processig, 2d editio, Pretice-Hall, Ic., 999.

Signal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform

Signal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform Sigal Processig i Mechatroics Summer semester, 1 Lecture 3, Covolutio, Fourier Series ad Fourier rasform Dr. Zhu K.P. AIS, UM 1 1. Covolutio Covolutio Descriptio of LI Systems he mai premise is that the