Lecture 2 | The Fourier Transforms and its Applications

this presentation is delivered by the Stanford center for professional development alright so today I want to continue our study and begin a real serious mathematical study of the question of periodicity remember that we are essentially identifying the subject of Fourier series with the study with a mathematical study of periodicity and last time I went on at some length about the virtues of periodicity about the ubiquitous nature of periodic functions periodic phenomena in the physical world notes in the mathematical world and we made a distinction some perhaps a little bit artificial but but sometimes helpful between periodicity and time and periodicity in space the sort of two phenomena seem to be often come to you in different forms and it's sometimes useful in your help in your own head to sort of ask yourself which kind of periodicity are you looking at but in all cases actually periodicity is associated with the idea of symmetry that's a topic that will come up from time to time and if I don't mention it explicitly as with many other things in this course one of the things you should learn to sort of react to or think about yourself see what aspects of symmetry are coming up in the problem how does a particular problem fit into a more general context because as I've said before and we'll say it again one of the wonderful things about this subject is the way it all hangs together and how it can be applied in so many different ways all right if you understand the general framework and put yourself and orient yourself in a certain way using ideas and the techniques of the class you'll really find how remarkably applicable they can be okay so I says I said last time as we finished up last time when we were sort of still just crawling our way out of junior high a mathematical approach to periodicity is possible because there are very simple mathematical functions that exhibit periodically Hey namely the sine and the cosine but that's also the problem because periodic phenomena can be very general and very complicated and the sine and the cosine are so simple so how can you really expect to use the sine and the cosine to model very general periodic phenomena and that's really the question I want to address today so how can we use such simple functions sine of T and cosine of T to model complex periodic phenomena now first a general mark is how high should we aim here I mean how general can we expect this to be so how general all right that's really the fundamental question here and in answering that question led led both scientists and mathematicians very far from the original area that they were investigating ah let me say well let me say right now pretty general alright and we'll see exactly how general I'll try to make that more precise as we develop a little bit more of the terminology that really a plot that will apply and allow us to get more more careful statements but we're really aiming quite high here alright and we're really hoping to apply these ideas in quite general circumstances now not all phenomena are periodic alright and even in the case of periodic phenomena it may not be a realistic assumption I think it's important to realize here what the limits may be or how far the limits can be pushed so not all phenomena naturally although many are and many interesting ones are are periodic and even periodic phenomena in some sense you're making an assumption there that is not really physically realizable so even for periodic phenomena or at least functions where periodic in time even phenomena soon I think I'll start talking in terms of signals rather than phenomena but phenomena sounds a little grander at this point even phenomena that are periodic in time real phenomena that is died out eventually we only observe or at least we only observe something will refine my period of time where as as mathematical functions the sine of the cosine go on forever all right as a mathematical model sine cosine go on forever so how can they really be used to model something that dies out but a periodic function sine and cosine alright go on forever repeating over and over again alright so in what sense can you really use sines and cosines to model periodic phenomena when a real periodic phenomena when it really dies out well that'll take us a while to sort all that out let me just give one answer to this and one indication of how general these ideas really are and you have a homework problem that asks you actually to address this mathematically all right so if you have a fineness that you can still use still apply ideas of periodicity even if only as an approximation or a female even if only as an extra assumption so what I mean by that is as follows so suppose the phenomena looks like this suppose the signal looks like something like this let me write it over here so it dies out over a period of time so this is time and there's only a finite interval of time might be very large but there's only finite interval of time when the signal is nonzero I'm drawing just as a somehow generic signal here well that's not a periodic phenomena doesn't exhibit periodic behavior but if it dies out only after a finite outside of a finite interval then you can just repeat the pattern and make it periodic right force this you can force periodicity that is by repeating the pattern you can force extra symmetry you could force extra structure to the problem that's not there in the beginning so I mean by this very simple idea here's the original signal and I just repeat the pattern so this is the original signal these are may be sort of you know extra copies of it that I'm just inserting artificially and extend the function to be periodic and exist for all time you may only be interested in this part of it but for mathematical analysis if you make a periodic that'll apply to the whole thing all right this is sometimes called the periodization of a signal and it can be used and it is used to study signals which are non periodic to use even if you use methods of Fourier series in some cases other sort of free analysis the study signals which are not periodic so there's actually a homework problem on different sorts of periodization and it's it's a technique that comes up in various applications all right so see first problem set homework 1 now the point is that periodicity and the techniques for studying periodic phenomena are really pretty general all right even if you don't have a periodic phenomenon you can make it periodic and perhaps you can apply the techniques to study the periodized version of it and then restrict to study the special case of actually where you're interested in the function all right so it's pretty that's the point of this remark is that the phenomena that you're not restricting yourself so much by insisting that you're going to use sines and cosines or you're going to study periodic phenomena so the study that we're going to make here can be pretty general can apply really quite generally ok now let's do it or less let's get launched into the program so first of all let's fix a period in the discussion alright so we just just to just to specialize and fix ideas let's take let's consider periodic phenomena of a given fixed period and see what we can say about those see how we can model those matically so for the discussion let's fix the period for the discussion all right and there's a choice here a natural choice would be two pi because the sine of the cosine are naturally periodic of period two pi but I think for many formulas and for and for variety of reasons it's more convenient to fix the period to be one alright so we're going to look we're going to consider function signals which are periodic of period one so we'll use period one that is we consider signals I'll write things is a function of time but again it's not only periodicity in time that I'm considering all that I'm going to say can apply to any sort of periodic phenomena so we'll consider functions F of T satisfying f of T plus one is equal to f of T for all T all right and as the building blocks as the basic model functions we scale the sign in the cosine that is we don't just look at sine of T and cosine of T we look at sine of two pi T and cosine of two pi T so the model signals are sine of two pi T that has period 1 and cosine of 2 pi T that has grade 1 all right simple enough now one very important thing I want to comment before before we launch into particulars is that periodicity is a strong assumption and the analysis of periodicity has a lot of consequences if you know if you have a periodic function if you know it on an interval of length 1 and any interval of length 1 you know it everywhere because the pattern repeats if you just know a piece of the function you know it everywhere so if we know if we know when I say if we know if we analyze if whatever formulas we derive and so on this is an important Maxim so I'll put in quotes know to be a sort of generic infinite perfect godlike knowledge so if we know a periodic function say period 1 on an interval and not just a particular interval but any interval of length 1 then we know it everywhere all right these are all simple remarks okay these are remarks that you all have seen before in various context but again I want to lay them out because I want you to have them in your head and I want you be able to pull out the appropriate remark at the air mark at the appropriate time and you'd be amazed how far simple remarks can lead in the analysis of really quite complicated phenomena now how are we going to model how are we going to take such simple functions of sine and cosine individually and model a very complicated periodic phenomenon that is the sine of the cosine as endlessly fascinating as they maybe are just the sign in the cosine but the fact is they can be modified and combined to yield quite general results we can modify and combine sine of two pi T cosine of two pi T to model very general periodic phenomena of period one to model general periodic signals of again period one all right now here is the first big idea or here's a way of phrasing the first big idea when I talk about modifying and combining let's first talk about modifying and the the Magli maximum or the aphorism that goes on goes with what I have in mind is one period many frequencies there's a big idea one period many frequencies I think you can actually find this in the Dead Sea Scrolls okay what do I mean one period many frequencies well let me just take a simple example I mean for example G you have sine of two pi T and we know what the graph of that looks like I'll put the graphs over here it repeats once it has it's it's a period one it also has a frequency one that is completes one cycle in one second so if I think this is the time axis say although again although I'm thinking in terms of time it's a very general mathematical statement it repeats exactly once on the interval from 0 to 1 all right if I double the frequency and look at sine of 4 PI T all right then that completes two cycles so this is period one frequency one sine of 4 PI T is period 1/2 frequency one frequency two but period one half also means period one alright and the picture looks like this it goes up and down twice does it it does in one second alright 0 1 it repeats one side it goes through one cycle and a half a second goes through two cycles in one second says frequency - alright but it also has period one because if you consider this as the basic pattern that repeats and also repeats on an interval of length one okay it's true that repeats on an interval of length one half that the signal is already contained interval one one-half but the whole but the signal also has app it has a longer period all right it has a shorter period it also has a longer period and let me do one more if for instance I look at sine of 6 PI T very simple all my remarks are very simple this has period 1/3 this has frequency 3 but it also has period 1 you might think of this is I don't know the secondary period or somehow I don't know exactly the best way of saying it because really the best description is in terms of frequency not period and what does the picture look like well this time it has 3 cycles per second frequency 3 so that means it goes up and down three times in one second see if I can possibly do this one two three good enough one zero and one cycle is in what goes it goes up and down completely once in one third of a second then the next third of a second goes up and down again the next third of a second goes up and down a third time but it also has period one because if you consider this as the pattern that repeats that pattern repeats on an interval of length one although but in some sense the true repetition the true period is shorter than that now what about combining them so that's how you even modifies and you do the same thing with cosine that's how you modify the function one period many frequencies if we combine them together I actually have a picture of it here but I think I'll just try to sketch it fool that I am what about the combination and when I say combination I am thinking of a simple sum that is sine of two pi T plus sine of 6 PI T plus QV sine of 4 PI T plus sine of 6 PI T all right what does the graph of that look like well it looks like so and again I'm going to sketch this and then I'm going to make I want to make another comment about this in just a second but I actually have Mathematica plot this for me it looks something like this it goes this is plotted under interval interval of length 2 all right I've plotted it and it goes it's kind of nice it goes up and then in a little bit like this and then it goes up and down a little bit farther and up a little bit lifts and it goes down that's the sound that makes up down like that and then up excuse me that's not two it will be two and down and up and then there we go - and then it repeats here's one all right that's the sum now what is the period of the sum the period of the sum is one all right because although the things of the terms of higher-frequency are repeating more rapidly the sum can't go back to where it started until the slowest one gets caught up goes to where it's back to where it started all right the period of the sum is 1 1 period many frequencies there are three frequencies in the sum one two and three but added together there's only one period so in a complicated under this is this very important point and again it's I'm sure it's a point that you've seen before that's why I say one period many frequency and that's why for complicated periodic phenomena it's really better more revealing to talk in terms of the frequencies that might go into it rather than the period you're fixing the period you're fixing the period to have length 1 all right but you want but you might have a very complicated phenomena that complicated phenomena as it turns out is going to be built up out of sines and cosines of varying frequency as long as the sum has has period 1 then we're OK 1 period many frequencies that's the that's the aphorism that goes on with this that goes along with this now in fact what is that what we can what we can do more than just modify the frequency we can also modify the amplitudes separately and we can modify the phases of each one of those each one of those terms so to model complicated perhaps help a complicated will see a complicated signal of period one we can sum we can modify the amplitude the frequency and the phases of either sines or cosines but let me just take let me just tick with the signs of sine of two pi T and add up the results that is we can consider something like this something in the form say K going from 1 up to n and we can consider however many we want a sub K sine of two pi K times T plus that's a modifying the frequency plus V sub K modify allowing myself to modify the phase P sub K that's about the most general kind of sum that we can form out of just the signs or and I could do the same thing with cosines or I can combine the two and I'll say more about that just a second all right so again many frequencies one period all right the lowest the longest period in the sum is when K is equal to one period one the higher terms they're called harmonics because the connection with music and that's discussed in the notes because they model because simple sines and cosines model musical phenomena as a repeating pattern musical musical notes the higher harmonics the higher terms have higher frequencies have shorter periods but the sum has period one because the whole pattern can't repeat until the longest period repeats until the longest pattern is completed now I'm actually going to post on the website I'm going to give you a little MATLAB program that allows you to experiment with just these sorts of sums alright that is you choose n it forms sums exactly like this you can choose the A's the amplitudes you can choose the phases and it will plot what a sum looks like it's called so I have a MATLAB program with a graphical user interface I wrote this myself actually a couple years ago I was very pretty the first thing I read in MATLAB it was pretty clunky let me tell you but last year a student in the class in 261 took it upon herself to modify it which caused me to bump up her grade in the end and it's really quite a nice little little program all right it's not complicated but it'll show you how how complicated these sums can be all right so as a MATLAB program which I'll post on the website sign some actually it's called sign some - because sign someone was my own version which is now on the ash heap of history all right to plot these sums and it's really it's really quite I mean talk about fun you know I mean to see how complicated a pattern you can build up on a relatively simple building blocks like this it's really it's really pretty good so we may even we were actually trying to see if we can do a homework homework assignment on the base based on this based on the program there's a feature in the program that allows you to actually to play the sound that's associated with this that is if you consider these things as modeling if you consider the simple sinusoids as modeling a pure musical note then a combination models a combination of musical notes and if you put a little button play sound you'll get something that may sound good or may not sound good but it's interesting to try unfortunately the play sound feature doesn't work doesn't seem to work on the Mac it only seems to work on Windows and it requires Windows Media Player or something like that bill gates version of death for 0.2 or something I don't know anyway so we have we safety we have to see if we can fix it to work on the Mac but everything works on the Mac fine except for the playing that playing the note so I'll put that up on the web in a zip file and a zip file and you should pull around with a little bit okay because it is it does it'll give you a good sense of just how complicated these things can be all right now so how complicated can they be that's the question that I really want to address I mean this sum is already more complicated than just the sine and the cosine alone but it doesn't begin to exhaust the possibilities that we want to be able to deal with let me say to advance the discussion actually and to really get to where get to the point where I can ask the question in a reasonable way how general a periodic phenomena can we expect to model with sums like this let me say a little about the different forms that you can write this summoned because algebraically for algebraic reasons primarily algebraic reasons there are more or less convenient ways to write this sum okay and I think it's worth commenting about it just a little bit so different ways of writing the sum the sum this sort of sum K equals 1 to n of Asaph I use capital in here none man a sub K sine of 2 pi K T plus P sub K all right now you can lose the phase so to speak and bring in write it in terms of sines and cosines if you use the addition formula for the sine function that is the cite the formula for the sine of the sum of two angles so if you write sine of 2 pi K T + VK as the sine of 2 pi K T times the cosine of V K plus the cosine of 2 pi K T times the sine of C K just using the addition formula then that sum can be written in terms of sines and cosines you can write the sum in the form let me use four different coefficients say sum from k equals 1 to n of little a k times the cosine of 2 pi K T plus little B sub K times the sine of 2 pi K T alright and you know all the little a KS and the B KS are related to the capital A's and the phase just by working it out all right so Oh capital ace of K times this thing and then there's a there's there's a term coming from the face you haven't lost information about the phase in some sense it's still there but it's represented differently in terms of the coefficients how in the form of the sum alright this is a very common way of writing these sorts of trigonometric sums and matter fact I'd say it's more common in if you're looking in the applications you look in the textbooks it's more common to write the sum this form than it is to write it in this form but they're equivalent all right you can go back and forth between the two and you can also allow for a constant term you can shift the whole thing up and that's also usually done for purposes of generality right you can add a constant term and it's usually written in the form a 0 over 2 the reason why the a 0 over 2 is in there is because if you had another form of writing it let me just write out write out the rest of it a 0 over 2 plus again sum from N equals 4 k equals 1 to N to n of a sub K cosine of 2 pi K T plus B sub K sine of 2 pi K T alright and electrical engineers always call this the DC component I hate that alright but they always do all right whoo we learn to call this the DC component yeah I hate that not everybody I'm glad to see that's because you think of a periodic phenomena you think about alternate youth I don't know what you think about you think of alternating current or voltage somehow as a periodic part but then there's a direct part that doesn't alternate there's a DC part direct current part that doesn't alternate and that's that term the reason why I don't like calling this the DC component is because what if the problem had actually nothing to do with current you know you sometimes you sometimes trap by your language and the field again as I will say over and over again is so broad and so diverse that you don't want to trap yourself into thinking about it in only one way you know you may be modeling some very complicated phenomena and you say what does the DC component in somebody's going to look at you like what are you talking about you know all right now so that's a very common way of writing the form of the sum but by far the most convenient way algebraically and really in many ways conceptually is to use complex Exponential's to write the sum not the real sines and cosines by far and this is pretty much the last time I'm going to I'm going to use sines and cosines or pretty much the last time I'm going to write the expression like this so it's by far better and you'll have to be ought to convince you of this all right primarily algebraically but also also conceptually it's by far better to use to represent sine and cosine via complex Exponential's and write the sum that way all right so what do I mean by that let me just remind you of course that e to the 2 pi I and T by oles formula is cosine of 2 pi NT plus RK I guess I'm calling it K let me let me stick with the terminology there 2 pi 2 pi K T plus sine of 2 pi K plus I times the sine of 2 pi K T oh yes that's something else all right I'm going to announce a declaration of principle I is the square root of -1 in this class not J deal with it get over it all right for me for this class it's I you can use J if you want I will use I now because of Euler's formula Euler's famous formula you can express sines and cosines in terms of the complex exponential and its conjugate that is very simple formula the cosine is the real part and the sine is the imaginary part so cosine of 2 pi KT is then e to the 2 pi is the real part 2 pi K T plus e to the minus 2 pi KT over 2 and the sine of 2 pi K T likewise is the imaginary part that's e to the 2 pi KT minus e to the minus 2 pi K T divided by 2i there is an appendix in the notes on the algebra of complex numbers so if you're at all rusty on that you should review that right because you're going to be you're going to want to be able to manipulate complex numbers and I'm thinking primarily here in terms of working with complex conjugates with real parts and imaginary part you're going to want to be able to do that with confidence and gusto all right so if you're at all rusty and manipulating complex numbers and complex Exponential's look over the chapter all right matter of fact on the first problem set there are several problems that give you practice in exactly this and manipulating complex numbers all right they're complex Exponential's now because of this and I won't I won't write it out detail you can obviously then convert a sum which is now gone which looks in terms of sines and cosines in terms of complex Exponential's so you can convert a trigonometric sum as before to the form sum and I'll write it like this sum from k equals minus n to n so it includes the 0th term the constant term C sub K e to the 2 pi K T all right where now the C sub KS are complex numbers everything in sight is complex so alright the CK's are complex now they can be alright they can be expressed I won't do this and I was going to give you a homework problem in this but I decided not to you can do this just for fun you can see how the different coefficients are related so start with the expression in terms of sines and cosines make the substitution in terms of the complex exponential and see what happens of the coefficients alright you will find actually a very important symmetry property alright these are complex numbers but they're not just arbitrary complex numbers they satisfy symmetry property and that's and because it's because of the symmetry property that the total sum is real alright the symmetry property this comes up a lot and we'll see similar sorts of things reflected actually we talked about Fourier transforms that is C sub minus the sum goes from minus an up to n that they satisfy the property the C sub minus K is C sub K complex conjugates CK bar all right that that's a very important identity that's satisfied by the coefficients light for a real signal like that and it comes up often all right it's one of things you have to keep in mind alright that that's a cut that's a consequence of actually making the conversion that is starting with a formula in terms of sines and cosines and then getting the formula in terms of the complex numbers alright conversely conversely if you start with the sum of this form alright where the coefficients satisfy this symmetry property then the total sum will be real that's because you can group a positive term and a negative term and because of this relationship here you'll be adding a complex number plus its conjugate so you get a real result out of that alright so if the coefficients satisfy this then the signal is real conversely if the signal is real and you write it like that then the coefficients have to satisfy the symmetry property yeah jeez what is that line that line is indicates complex conjugate alright so for a general arts I have to say anymore first so for general complex number A plus bi the conjugate is a minus bi okay there are different notations for complex conjugates sometimes some people use differently some people use an asterisk a star some people even use a dagger alright but I think by far it's true I'm not making that up but this is the most common notation alright and that is notation that I will use ok now now now now we are ready at last to at least ask the question that's really at the heart of all of this how general can this be how general can this be I mean I'm in the form now algebraically well I'm in the form now where I can ask the question and as we'll see algebraically writing sums of this form is by far the easiest way to approach it so we can now ask a fundamental question why is there something rather than nothing let's kick that one around for a while the fundamental question so again F of T is a periodic function of period one alright can we write it f of T as that sort of sum K going from minus n to n of C sub K e to the 2 pi K T so again I'm assuming the signal is real here so the coefficients satisfy the symmetry relation just keep your eye on the ball here the fundamental question is this you have a general periodic function can you write it as a trigonometric sum can you express it in terms of sines and cosines can you express it in terms of the funding building blocks all right by the way linearity is playing a role here although again I haven't set it explicitly until now we're considering linear combinations of the basic building blocks we're considering a linear way of combining the basic trigonometric functions the basic periodic functions all right a linear way of doing that so that's the fundamental question and the answer I'll tell you next time but you don't think I'd make such a big deal out of it if the answer was no no so but there's a lot to do and answering this question answering this question led to a lot of it's very profound and far-reaching investigations all right now but I want to get started on it all right now let me give you a little clue yeah okay cordyline is named way no audio you was talking with one why am I suddenly starting with one well for one thing so the question is why does some go from minus n 2 N and why is the wise ins just go from 1 to N well for one thing if it went from 1 to N the co fit the signal wouldn't be real right remember there's this combination of the positive terms and the negative terms all right and the positive terms in the negative term because the symmetry relation the coefficient the positive and the negative terms combine to give you a real signal to give you a real part and it's a fact that if you start with a real signal in terms of sines and cosines and then use complex Exponential's to express it this way you will find that it's a symmetric sum it goes from minus n to n okay by the way I should I should have said something over here I suppose note one thing by the way that C sub 0 is equal to C sub C sub 0 is equal to C sub - zero zero being what it is C sub minus zero is equal to C sub zero is equal to c zero-bar what does it mean to say that a complex number is equal to its conjugate it's real all right so the one coefficient that you know for sure is real others may be real it may just work out that way but the one coefficient you know is real far sure is the zero coefficient all right so it must be real so c0 is real that's that's just a little aside c0 is real alright you have to be a little you have to cut me a little slack here like I said we all have to cut each other a little slack there's so many little bits you know to observe little pieces little comments and things like that I can't make all of them alright I hope I put all of them or most of them in the in the lecture notes alright in the notes do you see these things but as I say there's so many there's so many things along the way that you could point out that you can note that we just can't do it all because I want to keep my eye on the bigger picture alright but this is one thing that comes up often enough so they'll be you know they'll be instances where you have to read the read the notes carefully and try to make note of all those things that it's hard it's hard you know it's hard to know when you're going to need this little factor that little fact because there's so many little facts but you'll see when that when the whole thing when you if you keep the big picture in mind in many cases the details will take care of themselves really now where was I yes a secret of the universe all right here's here's a pretty big secret of the universe actually coming your way when when you try to apply mathematics works and when you try to apply mathematics to various problems you often have a question like this what if what how can something what I can something happen right is it possible to write something like this all right now a very good first approach and I'm serious about this if you're on when you're doing your own work and you're trying to look at a mathematical model of something you're saying can I do something like this often the first step is to suppose that you can and see what the consequences are alright then later on you can say alright then maybe I should try this because that seems to be what has to happen alright and then and then you and then you go backwards alright and mathematicians will never tell you this because they like to sort of cover their tracks they say well it obviously goes like this you know and we're obviously going to find this formula in that formula and isn't in life is going to work out so simply but what they don't show you is often that first step of saying suppose the problem is solved what has to happen all right so suppose you can do this we can write f of T equals the sum from k equals minus n to n of C sub K e to the 2 pi K T what has to happen all right now by that I mean if you can write this what are the coefficients if you can write any question like this then what I'm asking here is what are the mystery coefficients in terms of F coefficients C sub K in terms of F f is given to you alright so the unknowns in this expression are the coefficients and the question is can you solve for them alright suppose you can write it like that can you solve for the coefficients can we solve for the CK all right I'm gonna take a very naive approach all right I'm gonna isolate what do you do it like an algebraic equation start with an algebraic equation isolate the unknown so isolate like the I don't know M coefficient or something like that all right so isolate C M out of this that is it's a big old sum right so f of T is you know all these terms plus C sub m e to the 2 pi m t plus all the rest of the terms that is to say I can write C sub m e to the 2 pi M T is f of T minus all the terms that don't involve M so let me write it like this say sum over K different from M of C sub K e to the 2 pi I JT all right I've done anything you said algebraically manipulated the equation to bring the one mystery term or one fixed term on the other side all right all I did here was so f of T is this big sum one of those terms in the sum is C sub M e to the 2 pi M T I want to softly unknowns all right so softening unknowns 109 one unknown at a time so this is the M term in the sum bring that over the other side of the equation write C sub m e to the 2 pi M T is f of T minus all the terms that don't involve M okay and then write that as that's almost isolating C sub M but not quite because it's got a complex exponential in front of it so multiply both sides this board is not so great multiply both sides by e to the minus 2 pi M T so C sub M is e to the minus 2 pi M T times f of T minus the sum over all K different from M of C sub K e to the minus 2 pi M T times e to the 2 pi K T get with me nothing on my sleeve all right all right now that's brilliant I have isolated one unknown in terms of all the other unknowns all right so I don't know if one can say that we use it we have really made progress here so we need another idea this is as far as algebra can take you all right algebra says you want to solve for the unknown fine isolate you know what are your eighth grade teach you tell you put the one unknown on one side of the question put everything else on the other side of the equation hope and pray alright so we put the one unknown on one side of the equation everything else is on the other side of the equation hope and pray now the desperate mathematician at this point looking for something to do let me actually take this take this out one more algebraic step let me just combine those two Exponential's there and write this as C sub M is e to the minus 2 pi i mt minus the sum over all terms different from M of C sub K e to the 2 pi I 2 pi i k- m times t I'm just combining the two complex Exponential's there all right great what if F of T picky picky all right F of T - all right now good so now they say that we need another idea and the desperate mathematician at this moment will think of one or two things one of two things the other differentiate or integrate I mean what's beyond algebra calculus what's in calculus derivatives and integrals all right so here's a clue derivatives won't work but integrals will alright we need another idea and set a good idea it's an inspired idea it's inspired it because it works and I'll show you why so I'm going to integrate both sides from 0 to 1 over one period all right all I have to worry about here is one period everything's periodic a period one so I integrate over one degree from zero to one what if I do what do I get well certainly if I integrate 0 to 1 of C sub M DT that just gives me C sub M all right so what about the rest of it so I get I get C sub M is equal to the integral from 0 to 1 e to the minus 2 pi i MT F of T DT minus the interval the sum is the sum of the integral we write this sum from k all the different all the terms k differ from M of and this end a constant comes out C sub K the integral from 0 to 1 e to the Monty to the 2 pi K minus M K minus M T DT that's a T there C ouch all right I've integrated both sides from 0 to 1 all right now watch this alright watch I can integrate that complex exponential that's a simple function I can integrate that just like I integrated in calculus all right the integral from 0 to 1 e to the 2 pi K minus M T DT now K is different from M all right if K is equal damn I'm just eat of the 0 here I just get 1 again so the K is different from it so the integral of this is 1 over 2 pi I trust me integrating this is the same as integrating an ordinary function 2 pi I as you did in calculus 1 over 2 pi K minus M to the 2 pi K minus M T evaluated from T going from 0 to 1 straightforward integration straightforward integration which is equal to we are almost done we're almost there 1 over 2 pi K minus M that makes sense right because K is different from M so that's not a problem e to the 2 pi K minus M times 1 so it's either 2 pi K minus M minus e to the 0 all right but e to the 2 pi K minus M that's e to the 2 pi I times an integer that's like sine of 2 pi times an integer cosine of 2 pi times an integer that's 1 and e to the 0 is also better known as 1 so this is better known as 1 minus 1 which is better known as 0 nothing all this crap integrates to 0 excuse me all right incredible what is the upshot it all goes away what is the upshot the upshot is that C sub M what's left what's left is C sub M is the integral from 0 1 that all that all the terms in the sum here are gone are gone there they integrate it out to 0 what remains is the interval from 0 to 1 of e to the minus 2 pi M T f of T DT all right now in principle this is known because you start out by assuming new F suppose I know F what has to happen all right well nice to me suppose I've been given F and I write f as this sum what has to happen here is the answer all right here's the answer let me summarize we have solved for the unknowns alright so given F of T periodic of period 1 suppose we can write f of T as the sum K equals minus n to n of ck e to the 2 pi K T what has to happen what has to happen is the coefficients have to be given by this formula then you must have I'll just I'll just write C sub K instead of C sub M C said the caithe coefficient is the integral from 0 to 1 of e to the minus 2 pi K T F of T DT that's what has to happen all right so this an important first step in applying mathematics to any given problem whether it's a mathematical part more non mathematical problem suppose the problem is solved what has to happen if the problem is solved means suppose you have this representation then the coefficients have to be given by this formula all right so next time I'm going to turn this around saying suppose we give these coefficients by the formula do we have something like that and in what sense and that will lead us to great things all right so more on that next time

Channel: Stanford

Views: 368,895

Rating: 4.7135944 out of 5

Keywords: Electrical, engineering, computers, math, physics, geometry, algebra, technology, functions, linear, operations, Fourier, transformations, periodicity, symmetry, sin, cosin, phenomenon, signals, periods

Id: 1rqJl7Rs6ps

Length: 52min 56sec (3176 seconds)

Published: Thu Jul 03 2008