SIGNAL PROCESSING & SIMULATION NEWSLETTER

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1 SIGNAL PROCESSING & SIMULAION NEWSLEER Fourier aalysis made Easy Part Jea Baptiste Joseph, Baro de Fourier, While studyig heat coductio i materials, Baro Fourier (a title give to him by Napoleo) developed his ow famous Fourier series, approximately 2 years after Newto published the first book o calculus. It took Fourier aother twety years to develop the Fourier trasform which made the theory applicable to a variety of disciplies such as sigal processig where Fourier aalysis is ow a essetial tool. Fourier did little to develop the cocept further ad most of that work was doe by Euler, LaGrage, Laplace ad others. Fourier aalysis is ow also used i thermal aalysis, image processig, quatum mechaics ad physics. Why do we eed to do Fourier aalysis I commuicatios, we ca state the problem at had this way; we sed a iformatio-laced sigal over a medium. he medium ad the hardware corrupt this sigal. he receiver has to figure out from the received sigal which part of the corrupted received sigal is the iformatio sigal ad which part the extraeous oise ad distortio. he trasmitted sigals have well defied spectral cotet, so if the receiver ca do a spectral aalysis of the received sigal the it ca extract the iformatio. his is what Fourier aalysis allows us to do. Fourier aalysis ca look at a ukow sigal ad do a equivalet of a chemical aalysis, idetifyig the various frequecies ad their relative quatities i the sigal. Fourier oticed that you ca create some really complicated lookig waves by just summig up simple sie ad cosie waves. For example, the wave i Figure a is sum of the just three sie waves show i Figures b, c ad d of assorted frequecies ad amplitudes. Easy Fourier Aalysis Part Complextoreal.com

2 (a) - A complicated lookig wave (b) - Sie wave (c)- Sie wave 2 (d) - Sie wave 3 Figure - Sie waves Let s look at sigal a i three dimesios. With time progressig to the right we see the amplitude goig up ad dow erratically, we are lookig at the sigal i ime domai. From this agle, we see the sum of the three sie waves as show i Fig (b,c,d). Whe we look at the same sigal from the side alog the z-axis, what we see are the three sie waves of differet frequecies. We also see the amplitude but oly as a lie with its maximum excursio. his view of the sigal from this poit of view is called the Frequecy Domai. Aother ame for it is the Sigal Spectrum. Figure 3 - Lookig at sigals from two differet poits of view he cocept of spectrum came about from the realizatio that ay arbitrary wave is really a summatio of may differet frequecies. he spectrum of the composite wave f(t) of Fig () is composed of just three frequecies ad ca be draw as i Fig (3.). Easy Fourier Aalysis Part Complextoreal.com 2

3 his is called a oe-sided magitude spectrum. Oe-sided ot because aythig has bee left out of it, but because oly positive frequecies are represeted. (So what is a egative frequecy? Is there such a thig? We will discuss this i more detail i later. For ow, suffice it to say that a egative frequecy is simply a frequecy which is laggig i phase.) Figure 3. - he Frequecy Domai spectrum of wave i Figure Now let s look at the sigal i frequecy domai. hik of it as a recipe, with x- axis showig the igrediet ad the y-axis, how much of that igrediets. he x-axis for a sigal would show the differet frequecies i the sigal ad y-axis the amplitude of each of those frequecies. Let s expad o this cocept. V-8 juice for example has may differet igrediets such as celery juice, salt, water, spices, etc.. We ca remove most of these igrediets oe by oe ad the remaiig liquid would still taste essetially like V-8. What we ca ot remove ad have the item still retai its primary character is called the fudametal compoet. I V-8, that is tomato juice. Sigals carryig iformatio, similarly, have a fudametal frequecy alog with other lesser importat frequecies. A oisy sigal o the other has o sigle fudametal frequecy. It has a flat spectrum. All frequecies are preset i the sigal i the same quatities. So a spectrum does ot ecessarily have a fudametal compoet. he spectrum of such a sigal would be flat. Let s take the followig complicated lookig wave. Easy Fourier Aalysis Part Complextoreal.com 3

4 his wave is periodic with a period = sec. Figure 4 - Aother really complicated lookig wave he first thig we otice is that the wave is periodic. Fourier aalysis tells us that ay arbitrary wave such as the above that is periodic, ca be represeted by a sum of other simpler waves. Let s try summig a buch of sie waves to see what they look like. Figure 5a - his is a wave of frequecy Hz, amplitude = Figure 5b - his is a wave of frequecy 2 Hz, amplitude = Figure 5c - his is a wave of frequecy 3 Hz, amplitude = Easy Fourier Aalysis Part Complextoreal.com 4

5 Figure 5d - his is a wave of frequecy 4 Hz, amplitude = Each of the waves here have frequecies that are iteger multiples. I more scietific words, we say that they are harmoic to each other, similar to musical otes which are also called harmoic. What is a harmoic It is a frequecy that is iteger multiple of the other frequecy. Waves of frequecy 2 ad 4 Hz are harmoics to a wave of frequecy Hz sice they are both its iteger multiples. Frequecies 2.4 ad 3.6 Hz are harmoics to a wave of frequecy.2 Hz sice they are both iteger multiples of.2 Hz. Whe the multiple factor is eve, the harmoic is called a eve harmoic ad whe the factor is odd, it is called the odd harmoic. Frequecies 66,, 54 Hz are odd harmoics of frequecy 22 Hz, whereas 44, 88 ad 32 Hz are eve harmoics. We write the sum of N such harmoics as N f () t = si( ω t) () = Each wave has a frequecy that is iteger multiple of the startig frequecy ω, which is equal to 2 π () i this case sice f = Hz. Here is what a sum of four sie waves of equal amplitude, each startig with a phase of degrees at time looks like. f () t = si( ωt) + si( 2ωt) + si( 3ωt) + si( 4ωt) Figure 6 - his is the sum of all four of the above sie waves. Easy Fourier Aalysis Part Complextoreal.com 5

6 If we keep goig ad add a large umber of sie waves of equal amplitude, the summatio approaches a impulse fuctio as show below for N = 25. Sice we added together 3 sie waves of amplitude, the maximum amplitude is 25. Figure 7 - his is the sum of 25 sie waves. I the graph above, we allowed the amplitude of each harmoic to be oe. Goig to the ext level of abstractio, it is obvious that to represet a arbitrary wave, we eed to allow the amplitude of each compoet to vary. Otherwise, all we will get is the scaled versio of the sigal i Fig (7). So we modify equatio () by itroducig a coefficiet a to represet the amplitude of the th sie wave as follows: N f () t = a si( ω t) (2) = he coefficiet a allows us to vary the amplitude of each harmoic f (t) = si(ωt) to create a variety of waves. Here is what oe particular wave which is the sum of four sie waves of uequal amplitude looks like. Figure 8 - Sum of four sie waves of uequal amplitude But lookig at the origial wave, f(t) i Fig (4), we see that it starts at a o-zero value. No matter how may sie waves we add together, we ca ot replicate this wave because sie waves are always zero at time zero. But if we add some cosie waves to the sum i equatio (2) which do ot start at zero, we may be able to create the wave of Figure 2. Easy Fourier Aalysis Part Complextoreal.com 6

7 So let s add a buch of cosie waves of varyig amplitudes to our f(t) equatio. Figure 9a - A cosie wave of frequecy Hz, amplitude = Figure 9b - A cosie wave of frequecy 2 Hz, amplitude = Figure 9c - A cosie wave of frequecy 3 Hz, amplitude = Figure 9d - A cosie wave of frequecy 4 Hz, amplitude = Oce agai the sum of the cosie waves of equal amplitude looks like this. Easy Fourier Aalysis Part Complextoreal.com 7

8 Figure - Sum of four cosie waves of equal amplitude Figure - Sum of 3 cosie waves of equal amplitude A sum of 3 cosie waves looks like as i Fig (). It approaches a impulse fuctio just as the sum of sie waves did but this oe is a eve fuctio. Eve fuctio he fuctio that is symmetrical about the y-axis. Cosie wave is a eve fuctio. Odd fuctio he fuctio that is ot symmetrical about the y-axis. Sie wave is a odd fuctio. he sum of the cosies is a eve fuctio. Cotrast this with Fig (7), the sum of sies, which is a odd fuctio. hese characteristics, odd ad eve, are useful whe lookig at real ad imagiary compoets of sigals. Now let s allow the amplitude of each cosie wave to vary. Here is what oe particular sum of four cosies of uequal amplitudes looks like. Figure 2 - Sum of four cosie waves of uequal amplitude. Now let s modify equatio (2) to add the cosie waves. Easy Fourier Aalysis Part Complextoreal.com 8

9 N N (3) = = f ( t) = a si( ωt) + b cos( ωt) he coefficiets b allow us to vary the amplitude of each cosie wave. Puttig this equatio to work, we see i the followig figure the sum of four sie ad four cosie waves. Figure 3 - Sum of five sie ad cosie waves of uequal amplitudes We are very close to completig our equatio for arbitrary periodic waves. here is oly oe remaiig issue. Sums of sie ad cosies are always symmetrical about the x-axis so there is o possibility of represetig a wave with a dc offset. o do that we add a costat, a to the equatio. his costat moves the whole wave up (or dow) alog the y-axis offset. f ( t) = a + a si( ωt) + b cos( ωt) N N (4) = = he coefficiet a provides us with the eeded dc offset from zero. Now with this equatio we ca fully describe ay periodic wave, o matter how complicated lookig it is. All arbitrary but periodic waves are composed of just plai ad ordiary sies ad cosies ad ca de composed i its costituet frequecies.. Equatio (4) is called the Fourier Series equatio. he coefficiets a, a, ad b are called the Fourier Series Coefficiets. A equatio with may faces here are several differet ways to write the Fourier series. Oe commo represetatio is by liear frequecy istead of the radial frequecy. Replace ω by 2πf ad the write the equatio as f ( t) = a + a si( 2 πft) + b cos(2 πft) N N (5) = = Easy Fourier Aalysis Part Complextoreal.com 9

10 he Fourier series equatio allows us to represet ay wave, low or high frequecy, basebad or passbad, large badwidth or very small. N is the umber of harmoics used i the summatio. his is a variable ad we ca choose it to be aythig, but for complete represetatio, N is set to ifiity. his makes the equatio completely geeral ad we ca represet eve oise sigals this way. he harmoics themselves do ot have to be of iteger frequecies such, 2, 3 etc.. he startig frequecy ca be ay real or imagiary umber. However, the harmoics of the startig frequecy ARE its iteger multiples. f () t = f() t = f(t) the smallest frequecy is called the resolutio frequecy, determies how fiely we decompose the sigal. It ca be ay arbitrary umber, say for example From that poit o, the ext harmoic is 2 times this, ext oe 3 times ad so o., is the period of the first wave we pick, ad each f is a iteger multiple of the iverse of that period. We ca also start aywhere. We ca pick a small resolutio frequecy ad the start the aalysis with the th harmoic for example. Replace f by /, where is the period ad replace N by to write equatio (5) i a differet from. ( π / ) ( π / ) (6) f() t = a + a si 2 t + b cos 2 t = We ca also covert all sie waves ad make them cosie waves by addig a halfperiod phase shift. he cosie represetatio, used ofte i sigal processig is writte by addig a phase term to the equatio. si(2 π ft) = cos(2π ft + π / 2) o create the f(t) we would add two cosie waves of the same frequecy, except the oe of them would have a π /2 phase shift (that s a sie wave, really.) Now we have oly cosies. he ame of the coefficiet has bee chaged to c, to reduce cofusio betwee this term ad the terms a ad b. a ad C would be exactly the same as a. f() t = C + C cos( 2π f t+ φ ) = f() t = C + C cos( w t+ φ ) = 2π f () t = C + Ccos( t+ φ) = (6a) Easy Fourier Aalysis Part Complextoreal.com

11 I complex represetatio, the Fourier equatio is writte as j t/ f() t = Ce π (7) = Complex otatio, first give by Euler, is most useful-albeit scary-lookig form. I ext part, we look at how it is derived ad used for sigal processig. All these differet represetatios of the Fourier Series (4), (5), (6), (6a) ad (7) are idetical ad mea exactly the same thig. How to compute the Fourier Coefficiets of a arbitrary wave I sigal processig, we are iterested i spectral compoets of a sigal. We wat to kow how may sies ad cosies make up our sigal ad what their amplitudes are. Alteratively, what we really wat are the Fourier coefficiets of our sigal. Oce we kow the Fourier coefficiets, we kow which frequecies are preset i the sigal ad i what quatities. his is similar to doig chemical aalysis o a compoud, figurig out what elemets are there ad what relative quatity. How do we compute the Fourier coefficiets? Computig a f () t = a + ( a siω t+ b cos ω t) = he costat a i the Fourier equatio above represets the dc offset. But before we compute it, let s take a look at oe particular property of the sie ad cosie waves. Both sie ad cosie wave are symmetrical about the x-axis. Whe you itegrate a sie or a cosie wave over oe period, you will always get zero. he areas above the x- axis cacels out the areas below it. his is always true over oe period as we ca see i the figure below. +Area Positive ad egative area cacel. +Area Positive ad egative area cacel. +Area -Area -Area Figure 4 - he area uder a sie or a cosie wave over oe period is always zero. Easy Fourier Aalysis Part Complextoreal.com

12 o o = = asi wt dt = asi wt+ bcos wt dt = he same is also true of the sum of sie ad cosies. Ay wave made up of sum of the sie ad cosie waves also has zero area over oe period. So we see that if we were to itegrate our sigal over oe period the area obtaied will have to come from coefficiet a oly. he harmoics ca make o cotributio ad they fall out f() t dt = a si cos odt+ a o wt+ b wt dt = (8) he secod term is zero i (8), sice it is just the itegral of a wave made up of sie ad cosies. Now we ca compute a by takig the itegral of our complicated lookig wave over oe period. he wave has o-zero area i oe period, which meas it has a DC offset. Figure 6 - Sigal to be aalyzed, looks like it has a dc offset sice there is more area above the x-axis tha below. All area comes from the a coefficiet. Figure 6a - he dc compoet Easy Fourier Aalysis Part Complextoreal.com 2

13 Area uder the wave whe shifted dow is zero. Figure 6b - Sigal without the dc compoet he area uder oe period of this wave is equal to f () tdt= adt o (9) Itegratig this very simple equatio we get, f() t dt = a () We ca ow write a very easy equatio for computig a a = f t dt () () Sice o harmoics cotribute to area, we see that a is equal to simply the area uder our complicated wave for oe period divided by, the itegral period. We ca compute this area i software ad if it is zero, the there is o dc offset. his is also the mea value of the sigal. A sigal with zero mea value has o dc offset. Computig a Now we employ a slightly differet trick from basic trigoometry to compute the coefficiets of the sie waves. Here is a sie wave of a arbitrary frequecy ω that has bee multiplied by itself. f ( t) = si ωt*si ωt Easy Fourier Aalysis Part Complextoreal.com 3

14 Figure 7 - he area uder a sie wave multiplied by itself is always o-zero. We otice that the resultig wave lies etirely above the x-axis ad has a et positive area. From itegral tables we ca compute the area as equal to ( )( ) a si ωt si mω t dt = a / 2 for = m (2) Where is the period of the fudametal harmoic. But ow let s multiply the sie wave by a arbitrary harmoic of itself to see what happes to the area. f ( t) = si ωt*si mωt Sie wave multiplied by aother of a differet harmoic Multiplyig oe sie wave by ay other causes the area uder the ew wave to become zero. Figure 8 - he area uder a sie wave multiplied by its ow harmoic is always zero. he area i oe period of a sie wave multiplied by its ow harmoic is zero. We coclude that whe we multiply a sigal by a particular harmoic, the oly cotributio comes from that particular harmoic. All others harmoics cotribute othig ad fall out. Easy Fourier Aalysis Part Complextoreal.com 4

15 ( ω ) ( ω ) a si t si m t dt = for m ( ω ) ( ω ) a si t si m t dt = a / 2 for = m (2) Now let s multiply a sie wave by a cosie wave to see what happes. f () t = si ωt*cosmωt Sie wave multiplied by a cosie wave for ay ad m Figure 9 - he area uder a cosie wave multiplied by a sie wave is always zero. It seems that the area uder the wave which is multiplicatio of a sie ad cosie wave is always zero whether the harmoics are the same or ot. Summarizig, by settig ω = ω ( ω ) ( ω ) a si t si t dt = for m m ( ω ) ( ω ) a si t si t dt = a / 2 for = m m ( ω ) ( ω ) a cos t si t dt = for all ad m m (3) Rules:. he area uder oe period of a sie or a cosie is zero. 2. he area uder oe period of a wave that is a product of two sie or cosie waves of o-harmoic frequecies is zero. Easy Fourier Aalysis Part Complextoreal.com 5

16 3. he area uder oe period of a wave that is a product of two sie or cosie waves of same harmoic frequecy is o-zero ad ot equal to a /2, where is the period of the resolutio frequecy we have chose. 4. he area uder oe period of a wave that is a product of a sie wave ad a cosie wave of ay frequecies (differet or equal) is equal to zero. Recall that i vector represetatio, sie ad cosies are orthogoal to each other. So all harmoics are by defiitio orthogoal to each other. A very satisfyig iterpretatio of the above rules is that sie ad cosie waves ca act as filterig sigals. I essece they act as arrow-bad filters ad take out all frequecies except the oe of iterest. his forms the basic cocept of a filter. Now let s use this iformatio. Successively multiply the Fourier equatio by a sie wave of a particular harmoic ad itegrate over oe period as i equatio below. o ( ) = si ( ) + ( ω ) ( ωt) d + cos( ω ) si ( ω ) f() t si wt dt a wt dt a si t si t b t t dt We kow that the itegral of the first ad the third term is zero sice the first term is the itegral of a sie wave multiplied by a costat (Rule ) ad the third is a sie wave multiplied by a cosie wave (Rule 3). his simplifies our equatio cosiderably. he itegral of the secod term is a a si ( ωt) si ( ω t) dt = (3) 2 From this we write the equatio to obtai a, which are the coefficiets of each of the sie waves as follows 2 a = f()si t ( ω t) dt (4) he a is the computed by takig the sigal over oe period, successively multiplyig it with a sie wave of times the startig fudametal frequecy ad the itegratig. his gives the coefficiet for that particular harmoic. Imagie we have a sigal that cosists of just oe frequecy, we thik it is aroud 5 Hz (ad is a sie wave from). We begi by multiplyig this sigal by a sie wave of frequecy.2 ad each of its harmoics which are.4,.6,.8,.. ad so o. Actually Easy Fourier Aalysis Part Complextoreal.com 6

17 sice we kow it is i the rage of 5 Hz, we ca dispese with the lower harmoics say up to 4 ad start with 4.2 ad go to 5.8 Hz. Here is all the math we do.. Multiple the wave with a sie wave of frequecies 4.2 ad itegrate the result. Most likely the result will be zero. 2. Go to ext harmoic, which 4.4. his is 22 d harmoic of the resolutio frequecy.2 Hz. 3. Repeat step ad 2 ad cotiue util harmoic frequecy is equal to 5.8 Hz. he results will show that the itegrals of all harmoics frequecies are zero, except for the 25 th harmoic, the itegral of which will be equal to a 2 25 = = a = a 25 Oe period itegral 2.5 Where = /f = /.2 = 5 sec. he coefficiet ca ow be calculated which gives the amplitude of the wave. (We already kow its frequecy, which is 5 Hz, sice the itegral is o-zero for that compoet.). Computig coefficiet of cosies, b Now istead of multiplyig by a sie wave we multiply by a cosie wave. he process is exactly the same as above f t o t d t d ( ω ) = a cos( ωt) dt+ a si ( ωt) cos( ω ) t + ( ω ) ( ω ) ()cos t b cos t cos t dt Now terms ad 2 become zero. (First term is zero from rule, the secod term due to rule 3.) he third terms is equal to b b cos( ωt) cos( ω t) dt = (4) 2 ad the equatio ca be writte as Easy Fourier Aalysis Part Complextoreal.com 7

18 2 b = f()cos t ( ωt) dt (5) So the process of fidig the coefficiets is multiplyig our sigal with successively larger frequecies of a fudametal wave ad itegratig the results. his is easy to do i software. he results obtaied successively are the coefficiet for each frequecy of the harmoic wave. We do the same thig for sies ad cosie coefficiets. Followig this process, we compute the coefficiets of the followig wave Figure 2 - he sigal to be aalyzed Without goig through the math, we will give the aswers i two vectors, first is the coefficiets of the sie ad secod the cosie waves ad the dcoffset. a = [ ] b = [ ] a =.32 From this we ca write the equatio of the above wave as f() t = si(2π) t +.3si(4 π) t +.7si(6 π) t +.3si(8 π) t +.3si( π) t+.3si(2 π) t+.2si(4 π) t cos(2π) t+.2cos(4 π) t+.7cos(6 π) t+.5cos(8 π) t+ +.2cos( π) t+.2cos(2 π) t+.cos(4π) t +... he coefficiets are the amplitudes of each of the harmoics. he resolutio frequecy is Hz ad the harmoics are iteger multiples of this frequecy. Now we kow exactly what the compoets of the received wave are. If the trasmitted wave cosisted oly of oe of these frequecies, the, we ca filter this wave ad get back the trasmitted sigal. Easy Fourier Aalysis Part Complextoreal.com 8

19 Summary he Fourier series is give by f ( t) = a + a si( ω t) + b cos( ω t) where ω = 2πf = = he coefficiets of the Fourier series are give by a = f t dt () 2 a = f()si t ( ω t ) dt 2 b = f()cos t ( ω t) dt where ω is the fudametal frequecy ad is related to by ω = 2π f = 2π Coefficiets become the spectrum Now that we have the coefficiets, we ca plot the magitude spectrum of the sigal. Easy Fourier Aalysis Part Complextoreal.com 9

20 Magitude Frequecy Sie Cosie Figure 2 - he Fourier series coefficiets for each harmoic You may ow say that this spectrum is i terms of sies ad cosies, ad this is ot the way we see it i books. he spectrum ought to give just oe umber for each frequecy. We ca compute that oe umber by kowig that most sigal are represeted i complex otatio where sie ad cosie waves are related i quadrature. he total power show o the y axis of the spectrum is the power i both the sie ad cosie waves i the real ad imagiary compoets of the same frequecy. We ca compute the magitude by from the root sum square of the sie ad cosie coefficiets for each harmoic icludig the dc offset of the zero frequecy value. Magitude = a + b 2 2 Plot the modified spectrum Magitude Frequecy Magitude Easy Fourier Aalysis Part Complextoreal.com 2

21 Figure 22 - A traditioal lookig spectrum created from the Fourier coefficiets Voila! Although this is ot a real sigal, we see that it ow looks like a traditioal spectrum. he largest compoet is at frequecy = 3. he y-axis ca easily be coverted to db. I complex represetatio, the phase of the sigal is defied by φ = b a ta ( / ) For every frequecy, we ca also compute ad plot the phase. Phase plays a very importat role i sigal processig ad particularly i complex represetatio ad shows useful iformatio about the sigal. Oe thig you may ot have oticed durig this computatio of the coefficiets is that they will be differet depedig o what you pick as the resolutio frequecy. We will get differet aswers depedig o the choice we make for this umber. I essece depedig o the resolutio, the sigal eergy leaks from oe frequecy to the ext so we get differet aswers, but the overall picture remais the same. he issues of leakage will discussed later. We also stated that the wave has to be periodic. But for real sigals we ca ever tell where the period is. Radom sigals do ot have discerible periods. I fact, a real sigal may ot be periodic at all. I this case, the theory allows us to exted the period to ifiity so we just pick ay represetative sectio of our sigal or eve the whole sigal ad call it he Period. Mathematically this assumptio works out just fie for real sigals. Figure 23 - We call the sigal periodic, eve though we do t kow what lies at each ed. Figure 24 - Our sigal repeated to make it mathematically periodic, but eds do ot coect ad have discotiuity Easy Fourier Aalysis Part Complextoreal.com 2

22 he part of the sigal that we pick as represetig the real period is oly a sample of the whole ad ot really the actual period. he ed sectio of the chose sectio will most likely ot match as they would for a real periodic sigal. he error itroduced ito our aalysis due to this ed mismatch is called aliasig. Widowig fuctios are used to artificially shape the eds so that they are zero at the eds ad so the chose sigal portio is made artificially periodic. his itroduces errors i the aalysis which have to be dealt with by other techiques. Next the complex represetatio. Copyright 998, All rights reserved C. Lagto Revised 22 I ca be reached at mtcastle@earthlik.et Other tutorials at Easy Fourier Aalysis Part Complextoreal.com 22