CEMTool Tutorial. Fourier Analysis

Transcription

1 CEMTool Tutorial Fourier Aalysis Overview This tutorial is part of the CEMWARE series. Each tutorial i this series will teach you a specific topic of commo applicatios by explaiig theoretical cocepts ad providig practical examples. This tutorial is to demostrate the use of CEMTool for solvig electroics problems. I this tutorial, Fourier aalysis will be discussed. Topics covered are Fourier series expasio, Fourier trasform, discrete Fourier trasform, ad fast Fourier trasform. Some applicatios of Fourier aalysis, usig CEMTool, will also be discussed. Table of Cotets. Fourier series. Fourier trasforms 3. Discrete ad fast Fourier trasform. Fourier series If a fuctio g(t is periodic with period Tp, i.e., g ( t = g ( t ± T p ( ad i ay fiite iterval g(t has at most a fiite umber of discotiuities ad a fiite umber of maxima ad miima (Dirichlets coditios, ad i additio, ( T g t dt < ò ( the g(t ca be expressed with series of siusoids. That is, a g ( t = + å a cos( wt + b si ( wt (3 where = p w = (4 T p ad the Fourier coefficiets a ad b are determied by the followig equatios. a = g ( t ( w t dt = T t + T p ò cos,,... (5 t p

2 b = g ( t ( w t dt = T t + T p ò si,,... (6 t p Equatio (8.3 is called the trigoometric Fourier series. The term a i Equatio (3 is the dc compoet of the series ad is the average value of g(t over a period. The term ( w si ( w a cos t b t + is called the -th harmoic. The first harmoic is obtaied whe =. The latter is also called the fudametal with the fudametal frequecy of ωo. Whe =, we have the secod harmoic ad so o. a Equatio (3 ca be rewritte as g ( t = + å A cos( wt + Q (7 where A = a + b ad = æ - b ö Q = - ta ç è a ø (8,9 The total power i g(t is give by the Parseval s equatio: where A t + T p A P = g ( t dt A t dc T ò = å ( p æ a ö = ç è ø dc = The followig example shows the sythesis of a square wave usig Fourier series expasio. ( Example : Usig Fourier series expasio, a square wave with a period of ms, peak-to-peak value of volts ad average value of zero volt ca be expressed as where f 5 4 g ( t = - f t p å si é( p ù = ( - ë û ( 4 = Hz. if a(t is give as a ( t = si é( p ft ë - ù - û å (3 p = ( Write a CEMTool program to plot a(t from to 4 ms at itervals of.5 ms ad to show that a(t is a good approximatio of g(t. Solutio CEMTool script % fourier series expasio f = 5; c = 4/pi; dt = 5.e-5; tpts = (4.e-3/5.e-5 + ; for ( = ;<= ;++ { for (m = ;m<= tpts;m++ s(,m = (4/pi*(/(* - *si((* - **pi*f*dt*(m-;

3 } for (m = ;m<=tpts;m++ { a = s(:,m; a(m = sum(a; } f = a'; t =.:4.e-3:5.e-5; plot(t,f xlabel("time, s" ylabel("amplitude, V" title("fourier series expasio" Figure shows the plot of a(t Figure : Approximatio to square wave By usig the Euler s idetity, the cosie ad sie fuctios of Equatio (3 ca be replaced by 3

4 expoetial equivalets, yieldig the expressio T p / ( exp( w g t = å c j t (4 =- Tp / where ( exp( w p c = g t - j t dt ad = T ò / w T (5 - p p Tp Equatio (4 is termed the expoetial Fourier series expasio. The coefficiet c is related to the coefficiets a ad b of Equatios (5 ad (6 by the expressio - b c = a + b Ð - ta æ ç ö è a ø I additio, c relates to A ad φ of Equatios (8 ad (8.9 by the relatio A c = ÐQ (7 The plot of c versus frequecy is termed the discrete amplitude spectrum or the lie spectrum. It provides iformatio o the amplitude spectral compoets of g(t. A similar plot of c versus frequecy is called the discrete phase spectrum ad the latter gives iformatio o the phase compoets with respect to the frequecy of g(t. If a iput sigal x(t: x ( t c exp( jw t (6 = (8 passes through a system with trasfer fuctio Hw(, the the output of the system y(t is ( ( w exp( w y t H j t c j t = (9 The block diagram of the iput/output relatio is show i Figure. Figure : Iput/Output relatioship However, with a iput x(t cosistig of a liear combiatio of complex excitatios, ( exp( w x t = å c j t ( =- the respose at the output of the system is y ( t H ( jw c exp( jw t = å ( =- The followig two examples show how to use CEMTool to obtai the coefficiets of Fourier series expasio. 4

5 Example : For the full-wave rectifier waveform show i Figure 3, the period is.333s ad the amplitude is 69.7 Volts. (a Write a CEMTool program to obtai the expoetial Fourier series coefficiets c for =,,,.., 9 (b Fid the dc value. (c Plot the amplitude ad phase spectrum. Figure 3:Full-wave rectifier waveform Solutio % geerate the full-wave rectifier waveform f = 6; iv = /f; ic = /(8*f; tum = 3*iv; t = :tum:ic; g = *sqrt(*si(*pi*f*t; g = abs(g; N = legth(g; % % obtai the expoetial Fourier series coefficiets 5

6 um = ; for( i = ;i<=um;i++ { for (m = ;m<=n;m++ cit(m = exp(-j**pi*(i-*m/n*g(m; c(i = sum(cit/n; } cmag = abs(c; cphase = agle(c; %prit dc value disp("dc value of g(t"; cmag( % plot the magitude ad phase spectrum f = (:um-*6; subplot(, stem(f(:5,cmag(:5 title("amplitude spectrum" xlabel("frequecy, Hz" subplot(, stem(f(:5,cphase(:5 title("phase spectrum" xlabel("frequecy, Hz" dc value of g(t as = Figure 4 shows the magitude ad phase spectra of Figure 3. 6

7 Figure 4: Magitude ad Phase spectra of a Full-wave rectificatio waveform Example 3: The periodic sigal show i Figure 5 ca be expressed as (i -t ( ( + = g ( t g t g t = - t < Show that its expoetial Fourier series expasio ca be expressed as ( g t = - (- ( e - e å exp( jp t ( =- ( + jp (ii Usig a CEMTool program, sythesize g(t usig terms, i.e., ( gˆ t = - (- ( e - e å =- ( + jp exp ( jp t 7

8 Figure 5: Periodic expoetial sigal Solutio (i g ( t c exp( jw t = å =- Tp / where = ( exp( - w c g t j t dt T c = exp -t exp - j t dt = ò ò T / ad w - p p ( ( p - thus g ( t = - (- ( e - e å =- (ii CEMTool script ( + jp exp - (- ( e - e ( jp t ( + jp p p = = = p, T p % sythesis of g(t usig expoetial Fourier series expasio dt =.5; tpts = 8./dt +; cst = exp( - exp(-; for ( = -;<=;++ { for( m = ;m<=tpts;m++ g(+,m = ((.5*cst*((-^/(+j**pi*(exp(j**pi*dt*(m-; } for( m = ;m<= tpts;m++ { g = g(:,m; g3(m = sum(g; 8

9 } g = g3'; t = -4:4.:.5; plot(t,g xlabel("time, s" ylabel("amplitude" title("approximatio of g(t" Figure 6 shows the approximatio of g(t Figure 6: A approximatio of g(t 9

10 . Fourier trasforms If g(t is a o-periodic determiistic sigal expressed as a fuctio of time t, the the Fourier trasform of g(t is give by the itegral expressio: ( = ( exp( - p ò (3 G f g t j ft dt - where j = - ad f deotes frequecy g(t ca be obtaied from the Fourier trasform G(f by the iverse Fourier trasform formular: ( ( exp( p g t = ò G f j ft df (4 - For a sigal g(t to be Fourier trasformable, it should satisfy the Dirichlet s. If g(t is cotiuous ad o-periodic, the G(f will be cotiuous ad periodic. However, if g(t is cotiuous ad periodic, the G(f will discrete ad o-periodic; that is g ( t = g ( t ± T p (5 where Tp = period æ ö G f = å cd f - (6 T ç p =- T è p ø c g t j f t dt T the the Fourier trasform of g(t is ( Tp / where = ( exp( - p ò T / (7 - p p Properties of Fourier trasform if g(t ad G(f are Fourier trasform pairs, ad they are expressed as g ( t G ( f the the Fourier trasform will have the followig properties: Liearity ( ( ( ( Û (8 ag t + bg t Û ag f + bg f (9 where a ad b are costats Time scalig ( g at Duality æ f ö Û G ç a è a ø G ( t Û g (- f (3 Time shiftig ( ( exp( p g t - t Û G f - j ft (3 Frequecy Shiftig ( j f t g ( t G ( f f exp C Û - C (33 Defiitio i the time domai (3

11 ( dg t dt ( Û jp fg f (34 Itegratio i the time domai t - ( G g ( t dt Û G ( f + d f jp f ( ò (35 Multiplicatio i the time domai ( ( Û ( ( - g t g t G l G f l dl - covolutio i the time domai ò (36 ( t ( -t t Û ( ( ò g g t d G f G f (37-3. Discrete ad fast Fourier trasfrom Fourier series liks a cotiuous time sigal ito the discrete-frequecy domai. The periodicity of the time-domai sigal forces the spectrum to be discrete. The discrete Fourier trasform of a discrete-time sigal g[] is give as N - G[ k] = å g [ ] exp( - j p k / N k =,,..., N - (38 = The iverse discrete Fourier trasform, g[] is where N - g [ ] = å G[ k] exp( j p k / N =,,..., N - (39 k = N is the umber of time sequece values of g[]. It is also the total umber frequecy sequece values i G[k]. T is the time iterval betwee two cosecutive samples of the iput sequece g[]. F is the frequecy iterval betwee two cosecutive samples of the output sequece G[k]. N, T, ad F are related by the expressio NT = (4 F NT is also equal to the record legth. The time iterval, T, betwee samples should be chose such that the Shao s samplig theorem is satisfied. This meas that T should be less tha the reciprocal of f H, where f H is the highest sigificat frequecy compoet i the cotiuous time sigal g(t from which the sequece g[] was obtaied. Several fast DFT algorithms require N to be a iteger power of. A discrete-time fuctio will have a periodic spectrum. I DFT, both the time fuctio ad frequecy fuctios are periodic. Because of the periodicity of DFT, it is commo to regard poits from = through = N/ as positive, ad poits from = N/ through = N - as egative frequecies. I additio, sice both the time ad frequecy sequeces are periodic, DFT values at poits = N/ through = N - are equal to the DFT values at poits = N/ through =.

12 I geeral, if the time-sequece is real-valued, the the DFT will have real compoets which are eve ad imagiary compoets that are odd. Similarly, for a imagiary valued time sequece, the DFT values will have a odd real compoet ad a eve imagiary compoet. If we defie the weightig fuctio W N as Equatios (38 ad (39 ca be re-expressed as G k N - [ ] [ ] = k N jp N - jp FT N = = (4 W e e = å g W (4 N - [ ] [ ] g = å G k W (43 k = -k N The Fast Fourier Trasform, FFT, is a efficiet method for computig the discrete Fourier trasform. FFT reduces the umber of computatios eeded for computig DFT. For example, if a sequece has N poits, ad N is a itegral power of, the DFT requires N operatios, N N whereas FFT requires log ( N complex multiplicatio, log ( N complex additios ad N log ( N subtractios. For N = 4, the computatioal reductio from DFT to FFT is more tha to. The FFT ca be used to (a obtai the power spectrum of a sigal, (b do digital filterig, ad (c obtai the correlatio betwee two sigals. CEMTool fuctio fft The CEMTool fuctio for performig Fast Fourier Trasforms is fft(x where x is the vector to be trasformed. fft(x,n is also CEMTool commad that ca be used to obtai N-poit fft. The vector x is trucated or zeros are added to N, if ecessary. The CEMTool fuctios for performig iverse fft is ifft x(. The followig three examples illustrate usage of CEMTool fuctio fft. Example 4: Give the sequece x[] = (,,. (a Calculate the DFT of x[]. (b Use the fft algorithm to fid DFT of x[]. (c Compare the results of (a ad (b. Solutio N - (a From Equatio (4 G[ k] [ ] From equatio (4 W 3 = = å g W = k N

13 jp - 3 W3 = e = j.866 j4p - 3 W3 = e = j W = W = W 3 3 = W Usig Equatio (4, we have [ ] å [ ] G = g W = + + = 4 = 3 [ ] = å [ ] = [ ] + [ ] + [ ] G g W g W g W g W = ( ( = j j.866 = j.866 [ ] = å [ ] = [ ] + [ ] + [ ] G g W g W g W g W = ( ( = j j.866 = j.866 (b The CEMTool program for performig the DFT of x[] is % x = [ ]; xfft = fft(x The results are xfft = i i (c It ca be see that the aswers obtaied from parts (a ad (b are idetical. -t Example 5: Sigal g(t is give as g ( t = 4e cos ép ( tù u ( t ë û (a Fid the Fourier trasform of g(t, i.e, G(f (b Fid the DFT of g(t whe the samplig iterval is.5s with N=. (c Fid the DFT of g(t whe the samplig iterval is.s with N=5. (d Compare the results obtaied from part a,b, ad c. Solutio é ê ë -t jpt - jpt (a g(t ca be expressed as g ( t = 4e e + e u ( t Usig the frequecy shiftig property of the Fourier trasform, we get ( G f = + + jp f - + j f + ( p ( (b,c The CEMTool program for computig the DFT of g(t is % DFT of g(t % Sample, Samplig iterval of.5 s ù ú û 3

14 ts =.5; % samplig iterval fs = /ts; % Samplig frequecy = ; % Total Samples m = :; % Number of bis sit = ts*(m - ; % Samplig istats freq = (m - *fs/; % frequecies gb = (4*exp(-*sit.*cos(*pi**sit; gb_abs = abs(fft(gb; subplot( plot(freq, gb_abs title("dft of g(t,.5s Samplig iterval" xlabel("frequecy (Hz" % Sample, Samplig iterval of. s ts =.; % samplig iterval fs = /ts; % Samplig frequecy = 5; % Total Samples m = :; % Number of bis sit = ts*(m - ; % Samplig istats freq = (m - *fs/; % frequecies gc = (4*exp(-*sit.*cos(*pi**sit; gc_abs = abs(fft(gc; subplot( plot(freq, gc_abs title("dft of g(t,.s Samplig iterval" xlabel("frequecy (Hz" The plot is show i Figure 7. 4

15 Figure 7: DFT of g(t (d From Figure 7, it ca be see that with the sample iterval of.5s, there was o aliasig ad spectrum of G[k] i part (b is almost the same as that of G(f of part (a. With the samplig iterval beig. s (less tha the Nyquist rate, there is aliasig ad the spectrum of G[k] is differet from that of G(f. Example 6: Give a oisy sigal g ( t si ( p f t.5( t = + where f Hz =, (t is a ormally distributed white oise. The duratio of g(t is.5 secods. Use CEMTool fuctio rad to geerate the oise sigal. Use CEMTool to obtai the power spectral desity of g(t. Solutio A represetative program that ca be used to plot the oisy sigal ad obtai the power spectral 5

16 desity is % power spectral estimatio of oisy sigal t =.:.5:.; f =; % geerate the sie portio of sigal x = si(*pi*f*t; % geerate a ormally distributed white oise =.5*rad(size(t; % geerate the oisy sigal y = x+; subplot(, plot(t(:5,y(:5, title("nosiy time domai sigal" % power spectral estimatio is doe yfft = fft(y,56; le = legth(yfft; pyy = yfft.*coj(yfft/le; f = (5./56*(:7; subplot(, plot(f,pyy(:8, title("power spectral desity", xlabel("frequecy i Hz" The plot of the oisy sigal ad its spectrum is show i Figure 8. The amplitude of the oise ad the siusoidal sigal ca be chaged to observe their effects o the spectrum. 6

17 Figure 8: Noisy Sigal ad its spectrum Refereces. CEMTool 6. User s Guide. Joh O. Attia, Electroics ad Circuit aalysis usig MATLAB, CRC Press, first editio