12 Getting Started With Fourier Analysis

Transcription

1 Commuicaios Egieerig MSc - Prelimiary Readig Geig Sared Wih Fourier Aalysis Fourier aalysis is cocered wih he represeaio of sigals i erms of he sums of sie, cosie or complex oscillaio waveforms. We ll sar wih he basic forms of he rigoomeric Fourier series, ad oulie he derivaio of possibly he mos useful equaio i sigal processig: he Fourier rasform.. Periodic Waveforms: he Cosie Series Firs, assume ha a periodic waveform x() ca be expressed i erms of he sum of a large umber of cosie waves all of which are periodic wih he same period. ha is, we ca express x() as: x( ) a cos o work ou he co-efficies a, we oe ha for ad m boh posiive iegers: / / / cos m cos cos ( m) cos ( m) / / / / m 0 / 0 / m 0 m (sice he iegraio over ay ieger umber of cycles of a cosie wave gives zero, bu whe = m, he secod iegraio is jus he iegral of he cosa cos(0) = ). his orhogoaliy propery of he fudameal ad harmoic cosie frequecies provides a simple mehod of calculaig he coefficies a, sice if we muliply he waveform x() by ay oe of hese cosies, ad iegrae over oe period, we ge: / / cos m x( ) cos m a cos / / / / m a cos m cos a as he oly erm i he summaio ha is o zero is he erm where = m. herefore, a m m / / cos m x( ) 007 Uiversiy of York Page 3/08/00

2 Commuicaios Egieerig MSc - Prelimiary Readig. Periodic Waveforms: he Sie Series However, o all period waveforms ca be expressed i erms of a sum of cosie waves. hose ha ca all share he propery of havig eve symmery: x( ) x( ). (he sum of ay umber of waveforms wih eve symmery also has eve symmery.) Aoher se of periodic waveforms have he propery of odd symmery: x( ) x( ). hese waveforms ca be represeed i erms of a ifiie umber of sie fucios, usig a similar echique o ha show above: ad sice for sie waveforms: x( ) b si / / / si m si cos ( m) cos ( m) / / / we ca derive: ad herefore: / m 0 m / / si m x( ) si m b si / / b m / / b cos m cos b m / / si m x( ) For ay waveform ha ca be described i erms of he sum of a sie fucios (i oher words ay odd symmeric waveform), his provides a simple echique for workig ou how much of which frequecy sie waves are required..3 Eve ad Odd Symmery Cosie waves have eve symmery x( ) x( ), ad herefore ay waveform ha ca be expressed i erms of cosie waves mus also have eve symmery. Sie waves have odd symmery, so ay waveform ha ca be expressed i erms of sie waves mus also have odd symmery. Wha abou waveforms ha have eiher form of symmery? We have see before ha ay arbirary waveform x(), ca be expressed i erms of he sum of a eve-symmeric ad a odd-symmeric waveform, e() ad o(), where: 007 Uiversiy of York Page 3/08/00

3 Commuicaios Egieerig MSc - Prelimiary Readig ( ) ( ) ( ) ( ) e( ) x x o( ) x x herefore, ay arbirary fucio ca be expressed i erms of he sum of a fucio wih eve symmery (which ca possibly be represeed i erms of a sum of cosie waveforms), ad a fucio wih odd symmery (which ca possibly be represeed i erms of a sum of sie waveforms)..4 he rigoomeric Fourier Series While all periodic waveforms ca be expressed i erms of he sum of a fucio wih odd symmery ad a waveform wih eve symmery, I will o aemp o prove here ha for all waveforms of ieres, he resula eve ad odd symmeric waves ca be expressed as he sum of a ifiie series of cosie ad sie waveforms. I fac, i geeral his is o rue, as ay waveform wih ay discoiuiies (i.e. sudde chages) i value or gradie cao be exacly represeed i his form. However, for a wide rage of waveforms of pracical ieres, he Fourier aalysis echique works well. here is oe fial erm ha mus be cosidered before ay waveform ca be represeed i erms of he Fourier series. Boh cosie ad sie waveforms have a average value of zero. If he waveform beig represeed i erms of cosies ad sies does o have a average value of zero, he is mea value (a 0 ) mus also be added o he series, where: a 0 / / x( ) his gives he geeral form of he rigoomeric Fourier Series as: x ( ) a 0 a cos b si where he coefficies a 0, a ad b ca be evaluaed usig he expressios above..5 he Complex Fourier Series Havig o work ou hree differe iegrals o deermie he coefficies a 0, a ad b ca be iresome. A aleraive complex form of he Fourier series exiss ha removes his requireme, ad allows all of he releva coefficies o be evaluaed usig jus oe iegraio. his form has he geeral equaio: x( ) c exp j Noice ha he coefficies c ow exed from mius ifiiy o plus ifiiy, raher ha sarig from oe. he egaive erms idicae erms wih egaive frequecies: o a cocep ha exiss i he real world a all, ad which migh eed a bi of explaaio..5. Posiive ad Negaive Frequecies O he Argad diagram, he complex umber Rexp(j) is show as follows: 007 Uiversiy of York Page 3 3/08/00

4 Commuicaios Egieerig MSc - Prelimiary Readig posiive frequecy R exp j egaive frequecy R exp j If he agle is made a fucio of ime, he a complex oscillaio resuls: he complex umber moves i circles aroud he origi. he rae of chage of phase is kow as he agular frequecy (measured i radias per secod), ad sice here are a oal of radias i a circle, he period of he oscillaio /. Ay complex umber circlig aroud he origi i a ai-clockwise direcio (so ha he phase is always icreasig) is said o have a posiive frequecy. Similarly, ay complex umber circlig aroud he origi i a clockwise direcio (so ha he phase is always decreasig) is said o have a egaive frequecy. A real oscillaio of a real quaiy i he real world ca be represeed as he sum of wo complex oscillaios: oe of posiive frequecy, ad oe of egaive frequecy, i such as way ha he wo imagiary compoes of he frequecies cacel ou. For example, a cosie wave is represeed as: ad a sie wave as: cos si exp exp j exp j j exp j ad ay sigle-frequecy wave of ampliude R ad iiial phase as: j j j exp ( ) exp ( ) Rcos R R exp( j ) R exp( j ) exp j exp j Noe ha i all cases he coefficie of he egaive frequecy erm is he complex cojugae of he coefficie of he posiive frequecy erm. his is rue for all real waveforms..5. Orhogoaliy of Complex Oscillaios o deermie he coefficies c of he complex Fourier series, we firs oe ha: 007 Uiversiy of York Page 4 3/08/00

5 Commuicaios Egieerig MSc - Prelimiary Readig / / exp j m exp j exp j ( m) / / m 0 m so ha, / / exp j m x( ) exp j m c exp j / / c m / c exp j m exp j / ad hece: / cm exp j m x( ) / his equaio is he mos usually quoed form of he complex Fourier series. I shows how ay periodic waveform wih a period ca be represeed i erms of a ifiie umber of complex oscillaios wih frequecies give by m/. he erm wih coefficie c 0 ad frequecy zero is he jus he DC erm: hece c 0 = a Power i he Complex Fourier Series he mea power i a periodic waveform is jus he eergy i oe period divided by he period. Sice he oal power i he Fourier represeaio of he waveform mus be equal o he oal eergy i he sigal, he we have: / / P x ( ) c exp j / / Usig he orhogoaliy propery of he complex expoeials, he oly erms ha remai i he righ-had side afer muliplyig ou he summaio are: / / ( ) P x c We ca herefore defie a power desiy he amou of power associaed wih a rage of frequecies. Cosider a rage of frequecies equal o he frequecy differece bewee wo harmoics, /. Well, o quie ay periodic waveform, i sill has o have o discoiuiies i value or gradie. 007 Uiversiy of York Page 5 3/08/00

6 Commuicaios Egieerig MSc - Prelimiary Readig ( Hz ) freq (Hz) he amou of power i his frequecy rage is he power i he compoe wih ha frequecy: which is jus c. So we ca say ha he power desiy aroud a frequecy / Hz is: P ( W / Hz) c sice he power i his rage of frequecies is: d 007 Uiversiy of York Page 6 3/08/00 Power( Hz) P ( W / Hz) Freq.Rage c c.6 he Fourier rasform d he Fourier rasform provides a echique for applyig he ideas of Fourier aalysis o operiodic waveforms. he idea is o le he period ed o ifiiy, ad he o argue ha here is o differece bewee a o-periodic waveform ad a periodic waveform wih a ifiie period. I his case, he harmoics become closer ad closer ogeher, so ha i he limi, here is some eergy a all possible frequecies, ad he specrum ceases o be a se of coiuous lies a well-defied harmoic frequecies, ad sars o be a coiuous fucio, X(). We defie his fucio so ha he ampliude of X() a a paricular frequecy is equal o he produc of he period ad he ampliude of he frequecy compoe a his frequecy. I oher words: / X ( ) lim c lim x( )exp j m / ad for a compoe a a agular frequecy, his gives: / X ( ) lim c lim x( )exp j / x( )exp j he iverse Fourier rasform ca be derived i a similar way, by cosiderig ha: x( ) lim c exp j

7 Commuicaios Egieerig MSc - Prelimiary Readig ad sice he summaio is ow over a ifiie umber of erms separaed i frequecy by a amou df = / Hz, his ca be expressed as: x( ) lim c exp j X ( ) exp j df X ( ) exp j d X ( ) exp j d his is he mos-ofe quoed form of he iverse Fourier rasform, a leas i egieerig..6. Eergy i he Fourier rasform Sice he Fourier rasform cosiders sigals ha las for a ifiie ime, i makes more sese o alk abou he eergy i he sigals, raher ha he power. he oal eergy i a sigal ca be expressed i he ime domai as : Eergy x ( ) I he frequecy domai, we mus cosider he derivaio of he Fourier rasform agai. he power i a small rage of frequecies / Hz was c, ad he eergy is jus he power muliplied by he ime. herefore, iegraig he power over all frequecies gives aoher expressio for he eergy i he sigal: X ( ) Power lim c lim lim X ( ) df Eergy Power X ( ) df ( ) X d Noe ha his provides a physical meaig for X(), i is he eergy specral desiy i Joules per Hz. Seig hese wo expressios for eergy equal o each oher provides a useful expressio kow as Parseval s heorem: A leas his is rue whe he sigal is real. If he sigal is complex, i mus be wrie * Eergy x( ) x( ) x ( ) 007 Uiversiy of York Page 7 3/08/00

8 Commuicaios Egieerig MSc - Prelimiary Readig x( ) X ( ) d.7 he Fourier rasform of Sampled Sigals I may cases i sigal processig, we are ieresed i a sampled versio of a coiuous sigal: a sigal ha has a defied value oly for a series of regularly spaced imes. his is wha resuls from he process of aalogue-o-digial coversio. For example, cosider he coiuous fucio y() beig digiised, ad represeed by he series of discree samples y, where he sample values y are he values of y() a imes. We ca wrie he sampled versio of he sigal as: y ( ) y ( ) s ha is, a series of impulses spaced a ime apar (so he samplig frequecy, he rae of akig samples, is jus /). We wrie i his way so ha he area uder he curve of he sampled versio of he sigal is fiie his meas we ca ake he Fourier rasform. o deermie he Fourier rasform of y s (), we jus apply he formula: Y ( ) y ( ) exp( j) s y ( )exp( j) y exp( j ) Ad his is very ieresig cosider he value of Y s (+/): Y ( / ) y exp j( / ) s y j exp exp Y ( ) s y exp j sice exp(-) = for all ieger values of. I oher words, Y s () is a periodic sigal. I repeas exacly every / rad/s, or / Hz. A compariso bewee he origial specrum of y(), ad he specrum of he sampled versio y s () is show below for wo cases of ieres (oe solid lie, oe doed). 007 Uiversiy of York Page 8 3/08/00

9 Commuicaios Egieerig MSc - Prelimiary Readig Specrum of y() Specrum of y s () -/ Hz 0 / Hz Noice ha i he case of he doed lie, he periodic (sampled) specrums overlap. his meas i is impossible o filer ou he effecs of he samplig a ay furher processig sage: eergy a wo differe frequecies i he origial coiuous waveform y() appears a he same frequecy i he sampled versio of he waveform y s (). his is he pheomeo kow as aliasig..7. he Nyquis Samplig heorem he Nyquis samplig heorem gives he miimum samplig rae required o avoid aliasig. As ca be see from he diagram above, he requireme is ha he samplig frequecy / mus be a leas wice he maximum frequecy of ay eergy i he coiuous waveform. Sice i mos cases real coiuous sigals have eergy ha exeds o very high frequecies, ai-aliasig filers are placed before aalogue o digial coverors o remove hese frequecies before he samplig process..8 A Useful Resul from he Fourier Series Oe key resul ha is used i a few impora derivaios i commuicaios heory is: or is corollary i he frequecy domai: exp j exp j so i s worhwhile highlighig he derivaios of hese resuls. Here, I ll jus derive he firs oe: he secod oe ca be obaied from he firs oe by jus replacig wih, ad wih /. (he equaio does mid wheher is a ime, or a frequecy, or ayhig else: i s sill rue.) his equaio comes from cosiderig he Fourier series of a se of dela fucios repeaig wih period of. We ca wrie: 007 Uiversiy of York Page 9 3/08/00

10 Commuicaios Egieerig MSc - Prelimiary Readig ad for ay Fourier series: where here: hece: f ( ) m m f ( ) c exp j / / / / c f ( )exp j m ( )exp j m exp j ad he fial sep is i oicig ha i does maer if you add up from mius ifiiy o plus ifiiy, or add up from plus ifiiy o mius ifiiy, you ge he same aswer, so we ca replace wih m o he righ had side, ad ha gives: m m m exp j m exp jm m 007 Uiversiy of York Page 0 3/08/00

11 Commuicaios Egieerig MSc - Prelimiary Readig.9 Examples of he Fourier rasforms he Fourier rasform has some simple properies ha are worh kowig. hey ca all be derived from he form of he Fourier rasform iegral, ad doig so is good pracice i usig hese equaios. Some of he mos commo ad useful oes are coaied i he followig wo ables; I ll pu hem here for referece purposes. (We do expec you o kow all of hese, jus o udersad wha hey mea, ad kow where o fid hem.).9. Fourier rasform Properies. rasform Operaio Sigal Specrum. Iverse rasform f f ( ) e j j F e d F 3. Complex Cojugae f F (Real Sigals) f f F F 4. Symmery f Real & Eve f Real & Odd F F 5. Ierchage F f 6. Ampliude Scalig Af AF 7. Superposiio Af Bf AF BF 8. Delay or ime Shif f j F e 9. Level Shif A f A F 0. Frequecy-Shif or f e c F c raslaio. ime Scalig f a Real & Eve Imagiary & Odd F a a. ime Reversal f F 3. Sigal Differeiaio d f j F j 4. Crosscorrelaio 5. Auocorrelaio f f F F (Aperiodic) f f F F F 6. Sigal Covoluio 7. Specrum Covoluio F F f f f f F F f f 007 Uiversiy of York Page 3/08/00

12 Commuicaios Egieerig MSc - Prelimiary Readig.9. Fourier rasforms of Some Commo Sigals Fucio. Recagular Pulse f F rec. Sic Pulse sic 3. Raised Cosie Pulse cos for 4. riagular Pulse for 0 for 5. Double Pulse for 0 for 0 0 for sic rec sic sic j si 4 6. Ui impulse 4 7. Cosa 8. Sig Fucio sg for 0 for 0 9. Ui Sep u j 0. Ui Ramp u. Modulus j j. Expoeial pulse ue a j a 3. Smoohed Expoeial Pulse ue a j a 4. Refleced Expoeial Pulse I 5. Refleced Expoeial Pulse II a e sge a a a j a 007 Uiversiy of York Page 3/08/00

13 Commuicaios Egieerig MSc - Prelimiary Readig 6. Log Modulus log e 7. Gaussia Pulse e e 8. Cosie Wave cos 9. Sie Wave si j 0. Cisoidal Sigal e j. Damped Cosie u e. Damped Sie 3. Carrier Pulse u e a a a j cos a j si a j rec cos sic sic 4. Sawooh Pulse 5. Refleced Smoohed Expoeial Pulse 6. Limier Respose sg for 0 for e a sic j a a for sg for sic j.0 Problems ) Express he co-efficies of he complex Fourier Series c i erms of he co-efficies of he rigoomeric Fourier Series a ad b. ) Wha is he rigoomeric Fourier Series of a square wave, where he value is oe for he firs half of he period, ad zero for he secod half? 3) By cosiderig he Fourier rasform ad Parseval s heorem, evaluae: 4 a sic a 4) Prove ha if f() has he Fourier rasform F(), he he Fourier rasform of f( ) is F()exp( j). d 007 Uiversiy of York Page 3 3/08/00

14 Commuicaios Egieerig MSc - Prelimiary Readig 5) Prove ha if f() has he Fourier rasform F(), he he Fourier rasform of (jf(). df is 6) Derive he resul for he Fourier rasform of he recagular pulse give i able above. 007 Uiversiy of York Page 4 3/08/00