### Lecture 1 | The Fourier Transforms and its Applications

this presentation is delivered by the Stanford center for professional development we are on the air okay welcome one at all and as I said on the TV when you were walking in but just to make sure everybody knows this is e 261 the Fourier transform and it’s applications Fourier transforms at all Fourier and my name is Brad Osgood circulating around are two documents that give you information about the class there is a general description of the class course information how we’re going to proceed some basic bookkeeping items I’ll tell you a bit more about that in just a second and also a syllabus in a schedule and I’ll also say a bit more about that in just a second let me introduce our partners in crime in this course we have three courses fins thomas john thomas one stand up where’s Thomas there we go Rajiv Agarwal I smell that right very good Reggie going to stand up is Rajiv and Michael medias okay so far am gonna correct that okay that’s like a Metis everybody thank you all right so now and we will be setting up times for the review sessions and so on all right you know so that’ll be that will be forthcoming we have a web page for the course some of you may have already visited that but let me give you the and it’s the addresses on the one the sheet one of the sheets that’s being passed around but let me write that up now so you can be sure to visit it and register for the class because it is on the web page that you will find course handouts course information I will email people via the web page alright so you have to be registered I five to send an announcement to the class post an announcement and send out an email then that’ll be done through the web page and you have to be registered on the Paige in order to get those emails I won’t be doing it through access all right so it is at like many of the other classes HTTP slash slash however you do those where the colons go or is it here EE class Sanford et you you can find it very easily edu slash e 261 okay go there if you have not already and and register yourself for the class all right now let me say a little bit about the information that you have I want to say a little bit more about the mechanics I’ll talk more about the content in just a second the let me say a bit about the syllabus and schedule and the course reader the syllabus is as I said on the on the top an outline of what we’re going to be doing I hope a fairly accurate outline of what we’re going to be doing but it’s not a contract alright so there will be a natural ebb and flow of the course as things go along and when we get to particular material or what we cover in what order this is more or less I say accurate but it is not written in stone what you should use it for however is to plan your reading so I things will be much better for all of us if you read along with with the material as the syllabus has the schedule basically outlines all right because there’s there times what I’m going to want to want to skip around a little bit there are times and I’m going to derive things there are times when I’m not going to derive things and you’ll get much more out of the lectures our time together if you’ve read the material thoroughly before you come to class so that’s one thing I ask you to do we have two exams scheduled we have a midterm exam and a final exam I’m going to schedule the midterm exam midterm exam is is already actually on here at least tentatively sort of toward the end of October we’ll have it outside of class that is it’ll be a sit down regular exam but I want to do it for 90 minutes rather than 15 min 15 minutes is just too short at a time for a class like for material like this so it’ll be a 90 minute exam and we’ll schedule it several sessions outside of class this is the way I’ve usually done it hasn’t been any problem it’s worked out alright for everybody so we’ll have you know alternate times and so on and the final exam is scheduled by the char’s office do not come to me right before the final exam saying oh I scheduled a trip out of town I hope that’s not a problem right you know what the dates are ahead of time we’ll also have regular problem sets none of these things that I’m saying should be new to you you’ve been

through the drill many times the first problem the problem sets are going to be I had a starling innovation last time I taught the course where I handed out the problem sets on Monday and had them do the following Wednesday so you actually had you know like a week and a half to do the problem set so there was overlap between the two and people thought that was just a brilliant idea so we’re going to do that again this year except for the first problem set and I decided it was not such good policy to hand at the very first problem set on the very first day of class so I’ll hand that out on Wednesday and I’ll post that also or at least I’ll post it on that sure I’ll hand it out it’ll be available on Wednesday and it’ll be due the following Wednesday and again these sorts of things are pretty routine for you I’m sure even through many times it will be practice although again not necessarily every time without fail to have MATLAB problems on the homework one or two MATLAB problems on the homework so I’m going into the assumption that people have some experience with using MATLAB you don’t have to be terribly advanced and also access to using MATLAB so if you do not have experience using MATLAB and you do not have access to MATLAB get some experience and get some access won’t be hard okay now let me say us a little bit about some of the things this is the course reader for the course it’s available at the bookstore and also available on the course website all right doesn’t have the problems in it but it has the material that were going to be covering in class now this is a basically a stitched together set of lecture notes that I’ve been using for a number of years in the class and I sort of tinker with it every time I teach the class but because it is a stitched together set of lecture notes they’re the organization is sometimes a little bit odd like you have an appendix in the middle of the chapter and what that means is it was used to be an appendix to a set to a particular lecture that went on that particular day and it never got moved to anywhere else all right so the organization can be a little bit funny you can help on this all right that is if you find typos if you find errors if you find things that are less than clear in their in their wording if you want if you if you have some other ideas for you apples or other explanations please tell me I am working on this I have to say that the I’m hope because these these were written as a set of lecture notes these are meant to be a good and I hope helpful companion for the class that is they’re meant to be read and they’re meant to be used so you can help as generations of students in the past have helped try to refine these and turn them into something that’s really a good accompaniment to to the class as we go on okay one other thing that’s special this quarter is the class is as always taped and the lecture notes the lectures are going to be available to everybody but this time for the first time the lecture is going to be available to the world all right Stanford is decide on an experimental basis we’re sort of competing with MIT here I think to try to make some classes some of the materials for some classes available to the world all right so the Elektra notes are going to be everything’s going to be done through the website but instead of needing a Stanford ID to view the tape lectures I think anybody in the world can view these lectures was a little bit daunting I have to watch my language to try to dress well all right so we’ll see what we’ll see what goes with that I will however issue a warning I will not answer the world’s email all right I will answer email from the class but I will not answer and I think I speak for the TAS here the TAS neither will answer the world’s email on this all right how we’re going to keep the world out of our inboxes I’m not sure exactly whether this is going to be a problem or not but at any rate that’s what’s happening okay all right any questions about that any questions about the mechanics of the course or what your expectations should be what my expectations of you are okay all right now I always like to take an informal poll actually when we start this class that’s what it’s a number of times now and it’s always been a mixed crowd and I think that’s one of the things that’s attractive about this class so let me ask who are the E’s in this class who are an electrical engineering your undergraduate or graduate all right so that’s a pretty strong show of hands but let me also ask who are the non E’s in this class all right that’s also a pretty strong show of hands the EES are as is typical the majority of the students in the class but there’s also a pretty strong group of students in this class who are not elect engineers by training by desire by anything all right and they usually come from all over the place I was looking at the web I was looking at the classes before I got the class and I think there’s some people from chemistry somebody from chemistry anybody from chemists I thought there were somebody up see back there all right and other some people from Earth Sciences uh somebody from somebody is talking actually from Earth Sciences this morning somebody from Earth Science okay where else I think there was an Emmy couple of Emmys maybe yeah all right now that’s important to know I think the course is very rich in material all right rich in applications rich in content and it appeals to many people for many different reasons okay for the ease and who are taking the class you have probably seen a certain amount of this material I don’t want to

say most of the material but you probably seen a fair amount of this material scattered over many different classes but it’s been my experience that one of the advantages of this class for electrical engineering students either undergraduate or graduate students is to see it all in one piece all right to put it all in your head at one time at least once all right because the subject does have a great amount of coherence it really does hang together beautifully for all the different and varied applications there are core ideas and core methods of the class that it is very helpful to see all at once alright so if you have seen the material before that’s fine I mean that is I mean that you can you can draw on that and draw on your experience but don’t deny yourself the pleasure of trying to synthesize the ideas as we go along I mean there’s nothing so pleasurable as thinking about something you already know trying to think about it from a new perspective try think about it from a new point of view trying to try to fold it into some of the newer things you’ll be learning so I have I’ve heard this from electrical engineering students many times in the past that it’s a it’s a pleasure for them to see the material all together at once it may seem like a fair amount of review and in some cases it will be but not in all cases and even if it is a review they’re often slightly different twists or slightly new takes on things that you may not have seen before I may not have thought of quite in quite that way so so so that is my advice to the electoral engineering students for the students who have not seen this material before they’re coming out of from a field and maybe only heard you know secret tales of the Fourier transform and its uses well I hope you enjoy the ride because it’s going to be a hell of a ride a heck of a ride as we go along alright now for everyone I sort of feel like I have to issue I don’t know if I call this a warning or just sort of a statement a principal or whatever this is a very mathematical class this is one of the sort of Holy Trinity of classes in the Information Systems lab in electrical engineering the electril engineering is a very broad department and split up into a number of laboratories along research lines I am in the Information Systems lab which is sort of the mathematical part of the subject there’s a lot of signal processing coding Theory imaging and so on and this course has been for a number of years taught by faculty sort of thought of as a cornerstone in the signal processing although it has a lot of different applications to a lot of different areas the other courses in that Holy Trinity are 263 dynamically near dynamical systems and 270 a statistical signal processing who’s taken to say whose likes let me ask you so because this is also very common who’s taken to 63 in the class also a strong majority and who’s taken to 78 yeah ok so there’s a fair a little little bit less but still number of people we will actually see not so much with 270 oh well actually with both classes with 263 in 278 you’ll actually see some overlap that I also hope you find interesting the language will be slightly different the perspective will be slightly different but you see this material in this class melding over into the other classes and vice-versa and again I think it’s something that you can really draw on and I hope you enjoy all right so it is those those classes and the perspective that we take the faculty your teaching those classes is a pretty mathematical one but it’s not a class in theorems and proofs you can breathe a heavy sigh of relief now all right I can do that but I won’t all right I will derive things I’ll derive a lot of a number of formulas I’ll derive it and I’ll go through those derivations or I’ll hope that you go through the derivations in the book when I hope and I think that they will be helpful all right and when in some case that is there’s an important technique or there’s an important idea that you’ll see not only in the tick Euler instance but over all that you’ll see the same sort of derivation the same sort of ideas be applied not only for one formula but for other sorts of formulas and also in some cases to my mind as twisted as that may be I sometimes think of the derivation of a formula almost as identical with a formula I mean to use the formula effectively almost as to know the derivation because it’s to know where it applies and to know how it applies and where to expect to use it all right so that’s why I will go through those things for the purpose of teaching a certain amount of technique and for the purposes of sort of having those techniques really at your fingertips so that you can apply them again in a situation that may not be quite identical with with what we did but will be similar enough so that the simp so that the ideas may apply in this situation that’s that’s very important we will also do plenty of different sorts of applications but again because the field the subject is so varied and because the clientele because the students in the class are also varied will try to take applications from different areas will have applications from electrical engineering but will also have applications from physics and from other areas i i’ve also done in the past and will see if i get to this some applications from Earth Sciences for example and we’ll just see how they go so we all have to cut each other a little bit of slack and if an application or particular area is not exactly to your liking well chances are it might be to somebody’s liking to your right or left so you say cut everybody

should cut each other a little slack and just enjoy the ride I should also say that many of the more specialized applications are found in more specialized courses all right so we will touch on a lot of things and I will use the words that are used in a lot of different courses and a lot of different subjects but we won’t always do see an application to its bitter end so to speak or we won’t do every pot we certainly won’t do every possible application because there are just so many of them so you will find you will not run out of ways of using the Fourier transform and Fourier analysis techniques in any classes here they go it goes on and on and on but we’ll only be able to see a certain amount of a certain amount of that all right and actually that leads to a very important point release of the start of the class that is where do we start all right that is this subject which is so rich and so diverse forces you forces me forces all of us to make hard choices in some ways about where what we’re going to cover where we’re going to start what direction we’re going to go and all the different choices are defensible you will find books out there that take very different taps toward the subject they take different starting points they have different emphases they go off in different directions and you can make a good argument for any one of those choices but you have to make a choice so for us we are going to choose I have chosen not we me I have chosen to start the class with a brief discussion of Fourier series and go from there to the Fourier transform all right whereas it is also very common choice to forget about Fourier series and maybe pick them up a little bit along Angier or pick them pick them up a little bit on the edges or assuming that everybody seen Fourier series then go right into the fray transform I don’t want to do that because I think that the subject of Fourier series is interesting enough in it we’re not going to do very much with it but it’s interesting enough in itself again it’s something you may have seen in different context but it provides a natural transition to the study of the Fourier transform and it is historically actually the way the subject developed okay so that’s how we’re going to that’s how we’re going to do things will start with Fourier series and use them as a transition to Fourier transform now first of all what is this concerned with overall I it may be a little bit too strong a statement but for our purposes I want to identify the idea of Fourier series as almost identified with the study of periodic phenomena alright so for us it’s identified most strongly with a mathematical analysis of periodic phenomena now it certainly shouldn’t be necessary for me to justify periodic phenomena as an important class of phenomena you have been studying these things for your entire life pretty much ever since the first physics course you ever took where they do the harmonic oscillator and then the second physics course you took where they did the harmonic oscillator and then the third physics course you took rhythm they did the harmonic oscillator you have been studying periodic phenomena alright so that shouldn’t be a controversial choice Fourier series goes much beyond that but it is first and foremost for us associated with a study of periodic phenomena the Fourier transform in although again it doesn’t maybe doesn’t do it’s just justice completely is can be viewed as a limiting case of Fourier series it has to do with a study of the mathematical analysis on phenomena so if you want to contrast Fourier series and Fourier transforms then that’s not a bad rough-and-ready way of doing it doesn’t it say it doesn’t capture everything but it captures something so Fourier transform as a limiting case and in a meeting that I’ll make more precise later is limiting case of Fourier series Fourier series of free series techniques is identified with or has to do with is concerned with how about that for weaseling way out of it is concerned with the analysis of non periodic phenomena so again it doesn’t say everything but it says something and one of the things that I hope you get out of this course especially for those of you who have had some of this material before are these sort of broad categorizations that help you sort organize your knowledge all right it’s a very rich subject you’ve got to organize it somehow otherwise you’ll get lost in the details all right you want to have certain markers along the way that tell

you how to think about it how to organize it what what what a particular formula what cat it what general category it fits under okay now it’s interesting is that the ideas are sometimes similar and sometimes quite different and sometimes it’s the situation is simpler for periodic phenomena sometimes the situation is more complicated for periodic phenomena so it’s not as though there’s sort of a one-to-one correspondence of ideas but that’s one of the things that we’ll see and one of the reasons why I’m starting with Fourier series is to see how the ideas carry over from one to the other see where they work and see where they don’t work alright some ideas carry easily back and forth between the two some phenomena some ideas some techniques some don’t and it’s interesting to know when they do and when they don’t sometimes the things are similar and sometimes they’re not now in both cases there are really to kind of inverse problems there’s a question of analysis and there’s the question of synthesis two words that you’ve used before but it’s worthwhile reminding what they mean in this context the analysis part of Fourier analysis is has to do with breaking a signal or a function I’ll use the term signal and function pretty much interchangeably alright I’m a mathematician by training so I tend to think in terms of functions but electrical engineers tend to think in terms of signals and they mean the same thing all right so analysis has to do with taking a signal or a function and breaking it up into its constituent parts and you hope the constituent parts are simpler somehow then the complicated signal that as it comes to you so you want to break up a signal into simpler constituent parts I mean if you don’t talk in just in terms of signals here or you don’t use exactly that language that’s the meaning of the word analysis I think close enough whereas synthesis has to do with reassembling a signal or reassembling a function from its constituent parts a signal from its constituent parts kind of stitch one alright and the two things go together all right you don’t want one without the other you don’t want to you don’t want to break something up into its constituent parts and then just let it sit there all these little parts sitting on the table with nothing to do you want to be able to take those parts maybe modify those parts maybe see which parts are more important than other parts and then you want to put them back together to get that to get either the original signal or a new signal and the process of doing those things are the two aspects of Fourier analysis I use I use the word analysis they’re sort of in a more generic sense now the other thing to realize about both of these procedures analysis and synthesis is that they are accomplished by linear operations series and integrals are always involved here both analysis and synthesis free analysis analysis and synthesis are accomplished by linear operations this is one of the reasons why the subject is so I don’t know powerful because there is such a body of knowledge on and such a deep and advanced understanding of linear operations linearity will make this a little bit more explicit as I go as we go on further but I wanted to point it out now because I won’t always point it out all right because when I say linear operations when I’m thinking of here integrals in series all right eg ie integrals and series both of which are linear operations the integral of a sum is the sum of the integrals the integral of a of a constant times a function is a constant interval the function and so and similarly with sums alright because of this one often says or one often thinks that Fourier analysis is part of the study of linear systems alright in engineering there’s there’s a there’s their courses called linear systems and so on and sometimes Fourier analysis is thought to be a part of that because the operations involved in it are linear I don’t think of it that way I mean I think it’s somehow important enough on its own not to think of it necessarily as subsumed in a larger subject but nevertheless the fact that the operations are linear does put it in a certain context in some in some ways in some cases more general context

that turns out to be important for many ideas alright so often so you see you often hear that Fourier analysis Fourier analysis is a part of the subject of linear systems the study of linear systems so I don’t think that really does complete justice to Fourier analysis because of because of the particular special things that are involved in it but nevertheless you will you’ll hear that okay now let’s get launched alright let’s start with with the actual subject of Fourier series and the analysis of periodic phenomenon a periodic phenomena and Fourier series as I said it certainly shouldn’t be necessary for me to sell the importance of periodic phenomena as something worth studying you see it everywhere all right the study of periodic phenomena is for us the mathematics and engineering or mathematics and science and engineering of regularly repeating phenomena that’s what’s always involved there’s some pattern that repeats and it repeats regularly right so it’s the mathematics and engineering so this is an engineering course I’ll put that before science or maybe I won’t even mention science mathematics and engineering of regularly repeating patterns I’m relieving a couple of terms here I’m leave all these terms somewhat vague what does it mean to be regular what does it mean to repeating what is a pattern in the first place but you know what you know what I mean you know it when you see it and the fact you can mathematically analyze it is what makes the subject so useful now I think although again it’s not ironclad trouble is this subject is so rich that every time I make a statement I feel like I have to qualify it well it’s often true but it’s not completely true and sometimes it’s not really true at all but most of the time it’s true that it’s helpful but not always helpful but most of the time helpful occasionally helpful to classify periodicity as either periodicity in time or periodicity in space all right you often see periodic phenomena as one type or the other type although they can overlap so you often periodic phenomena often are either periodicity in time a pattern repeats in time over and over again you wait long enough and happens again so for example harmonic motion so eg harmonic motion a pendulum I think bobbing on a string G harmonic motion or periodicity in base periodicity in space the city in space alright now what I mean he is there is often a physical quantity that you are measuring that is living on some object in space one dimension two dimensions whatever that has a certain amount of symmetry alright and the periodicity of the phone on is a consequence of the symmetry of the object so it’s often the cow giving example just a second so here you have say some some physical quantity physical not always but often you know physical quantity distributed over a region with symmetry the region itself repeats all right the region itself as a repeating pattern all right so the periodicity of the phenomenon the periodicity of the physical quantity that you’re measuring is a consequence of the fact that it’s distributed on on over some region that itself has some symmetry so the periodicity arises from the symmetry for periodicity here of the

object of the of the physical quantity that you’re measuring arises because the periodicity of the are the symmetry of the object where tributed where it lives I’ll give you an example there from the symmetry matter of fact I’ll give you the example the example that really started the subject and we’ll study this is the distribution of heat on a circular ring so eg the distribution of heat on a circular ring alright so the object the the physical quantity that you’re interested in is the temperature but it’s a temperature associated with a certain region and the region is a ring all right the ring has circular symmetry it’s around okay so you’re measuring the temperature at points on the ring and that’s periodic because if you go once around you’re at the same place so the temperature is periodic as a function of the spatial variable that describes where you are on the ring time is not involved here position is involved all right it’s periodic in space not periodic in time periodic in a spatial variable that gives you the position and the periodicity arises because the object itself is symmetric because the object repeats that’s why this sort of example is why one often sees and this actually turns out to be very far-reaching and quite deep that free analysis is often associated with questions of symmetry in a sort of most mathematical form you often find for a series developed in and in this context and Fourier transform is developed in the context of symmetry so you often see so you see Fourier analysis let me just say free analysis analysis is often associated with problems or just not off with with analysis of questions that have to do with that have some sort of relying symmetry so let me say often associated with problems with symmetry just leave it very general this is the very first of all that for the problem of distribution of heat on a ring we’re going to solve that problem that was the problem that Fourier himself considered alright they introduced some of the methods into the into the whole subject let’s launch everything all right so again it’s not periodicity in time its periodicity in space and for those of you who have had or may have courses in this that the mathematical framework for this very general way of looking for a analysis is group theory because the theory of groups in mathematics is a way of mathematize the ADEA of symmetry and then one extends the ideas for elseís into to take into account of groups that is to say to take into account the symmetry of certain problems that you’re saying and it really stays very quite it’s quite far-reaching we’re not going to do it we’ll actually have a few occasions to to go to go into this but but with a light touch all right I’m just telling you I’m just giving you some indication of where the subject goes all right now what are the mathematical descriptors of periodicity well nothing I’ve said so far I’m sure it is new to you at all you just have to trust me that at some point before you know it some things I say to you will be new I hope but one of the mathematical descriptions of periodicity again that in the two different categories say the numbers the quantities that you associate with either either a phenomena that’s periodic and timer function or a phenomenon that’s periodic in space for periodic and time for periodicity in time you often use the frequency all right frequency is the word that you hear most often associated with a phenomena that is periodic in time you use frequency the number of repetitions the number of cycles in a second say if a pattern is repeating whatever the pattern is again if I leave that term sort of undefined or sort of vague it’s the number of repetitions of the pattern in one second or over time all right that’s the most common descriptor

mathematical descriptor of a phenomenon is periodic that’s periodic in time for a function for a phenomenon is periodic in space you actually use the period that’s the only word that’s really in use in general for the particulate well one thing a time so for periodicity in space you use the period all right that is sort of the physical measurement of how long the long the pattern is before repeats somehow all right the measurement of how whether its length or some other quantity measurement of how let me just say how big the pattern is that repeats they’re not the same all right they have a different feel they rise off from from different sorts of problems that’s probably too strong a statement but I think I think it’s fair to say that mathematicians tend to think in terms of mostly in periodic they tend to think in terms of the period of a function or the period is the description of periodic behavior whereas engineers and scientists tend to think of systems evolving in time so they tend to think in terms of frequency they tend to think of how often a pattern repeats over a certain period of time all right that’s like everything else is that statement has to be qualified but I get tired of qualifying every statement so I’ll just leave it at that now of course the two phenomena are not completely separate or not always completely separate they come together periodicity and time and periodicity in space come together in for example wave motion all right that is traveling disturbance a travelling periodic disturbance so the two notions of periodicity come together two notions here periodicity and time periodicity in space come together in EEG wave motion understood very generally here as a periodic as a regularly repeating pattern that changes in time that moves because more jumps up a little bit I think of their skipping so a regular a moving a subset regularly moving disturbance you know a group of freshmen through the quad you know just they’re everywhere mostly regular mostly moving all right now there again the two descriptors come in the frequency and the wavelength so again you have frequency and wavelength you have frequency nu and wavelength usually associated usually denoted by this is for periodicity in space and for periodicity and time frequency nu for periodicity in time that’s the number of times and repeats in one second this is cycles per second the number of times that the pattern repeats in one second so for example you fix yourself at a fix your position in spate both time and space are involved so you fix yourself at a point in space and the phenomenon washes over you like a water wave all right and you count the number of times you’re hit by the wave in a second and that’s the frequency that’s the number of times that the phenomenon comes to you for periodic for periodicity and time the function the phenomenon comes to you for periodicity in space you come to the phenomenon so to speak all right so I fixed myself at a point in time the wave washes over me at a certain characteristic frequency over and over again regularly repeating it comes to me new times per second the wavelength you fix the time and allow the platen and see what the phenomena looks like to distribute it over space so for periodicity in space fix the time and see how the phenomena is distribute

to see the pattern distributed over space distributed my writing is getting worse distributed then the length of one of those a complete to speak is the period or the wavelength length is a term that’s associated with the periodicity in space for a traveling traveling phenomena for a wavelet wave for wave motion so the length the length of the disturbance I say one complete disturbance if I can say that one complete pattern is the wavelength now like I say ever since you were a kid you’ve studied these things and especially don’t know the number by lambda but I bring it up here because of the one important relationship between frequency and wavelength which we are going to see in a myriad of forms throughout the quarter that is there’s a relate in the case of wave motion there is a relationship between the frequency in the wave length determined by the velocity and there could be two different phenomena all right periodicity in time and periodicity in space may not have anything to do with each other but if you have a wave traveling if you have a regularly repeating pattern over time then they do have something with to do with each other and they’re governed by the formula distance equals rate times time which is the only formula that governs motion all right so there’s a relationship between frequency and wavelength that is distance equals rate times time I love writing this in a graduate course because it’s the up the equation in calculus actually in all of calculus I think this is pretty much the only equation used in very clever ways but the only equation and in our case if the rate is the velocity of the wave then this translate V is the velocity the rate of the wave of the motion and the equation becomes as I’m sure you know many times lambda that’s the distance that this this the the wave travels in one cycle it traveling it’s traveling at a speed V if it goes nu cycles in one second then it goes one cycle in 1 over nu seconds let me say that it going to make sure I got that right if it goes nu cycles in one second if it just passed you nu times in one second then in 1 over nu seconds it rushes past you once rushing past you once means you’ve gone through one wavelength so distance equals rate times time the time it takes to go one wave length is 1 over nu seconds so I have lambda equals V times 1 over nu or lambda nu equals V again a formula européenne many times now why did I say this if you’ve seen it many times because I never have the confidence that I can talk my way through that formula for one thing so I always have to do it secondly it exhibits a reciprocal relationship to quantities all right there’s a reciprocal relationship you can see it more clearly over here where the constant of proportionality or inverse proportionality is the velocity all right lambda is proportional to the reciprocal of the frequency or the restore the frequency is proportional to the reciprocal of the wavelength at any rate or the or expressed this way lambda times nu is equal to V so there’s a reciprocal relationship between the frequency and the wavelength all right this is the first instance when you talk about periodicity of such reciprocal relationships we are going to see this everywhere all right it’s one of the characteristics of the subject hard to state as a general principle but but they’re plain to see that in the prop in in in the analysis and the synthesis of signals using methods from Fourier series or Fourier analysis there will be a reciprocal relationship between the two between the quantities involved all right I’m sorry for being so general and but you’ll see this play out in case after case after case and it is something you should be

attuned to all right all right so you may never have thought about this in these types of simple enough formula you’ve used millions of times all right you may not have thought about it somehow in those terms but I’m asking you to think about stuff use you once saw in very simple context and how those simple ideas sort of cast shadow into much more involved situations all right the reciprocal relationship between as well as we’ll learn to call it the reciprocal relationship between the two domains of Fourier analysis the time domain in the frequency domain or the tie or the store the time domain and the spatial domain or the spatial domain in the frequency domain and so on is something that we will see constantly alright and I will point that out but if I don’t point it out you should point out to yourself you should be attuned to it because you will see it and it’s one of those things that helps you organize your understanding of the material because sometimes when you’re called upon to apply these ideas in some context that you haven’t quite seen you have to ask yourself it’s at least the good starting place is to ask yourself questions like well should I expect a reciprocal relationship here you might lead you to guess what the formulas should be or guess what the relationship should be so you say well somehow I want to use for a analysis to do this problem so I’m sure I should be looking for some sort of reciprocal relationship the quantities that I’m interested in somehow should be related in some kind of reciprocal way and what that might mean might be more or less involved depending on the particular kind of problem but you’ll see it trust me you’ll see it okay right now we’re almost done for today why does mathematics come into this in the first place I mean periodicity is evidently sort of a very physical type property why is it allow any kind of mathematical description well it does because there are very simple maybe not so simple mathematical functions that exhibit periodic behavior and so can be used to model periodic phenomena so math comes in because there are simple mathematical functions that model that are periodic that repeat and so can be used to model periodic phenomena I am speaking of course of our friends the sine and cosine now you may think again we’ve only talked about elementary things in very elementary contexts but you know I have a PhD in this subject and I get excited talking about sines and cosines I mean you know and it’s not just creeping old age I mean I think there you know there’s a lot there’s a lot to reflect on here and sometimes the miraculous nature of these things cosine of I’ll use I’ll use T is the variable cosine of T and sine of T our periodic of period two pi that is cosine of T plus two pi is equal to cosine of T for all values of T and sine of two pi + t + 2 pi is equal to sine of T why dead silence because the sine and cosine are item don’t tell me I want to do it because this I’ll do it over here because the sine and the cosine are associated with periodicity in space because the sine of the cosine are associated with an object that regular repeats the simplest object that the regularly repeats does circle you didn’t meet sine and cosine that way first you met sine and cosine in terms of ratios of psiy lengths of sides in triangles that’s fine but that’s an incomplete definition the real way of understorey way but the but them but the more sophisticated way the ultimately more far-reaching way of understanding sine and cosine is as associated with the unit circle where the cosine of t is the x-coordinate and the sine of t is the y-coordinate and T is Radian measure I’m not going to go through this in too much detail but the point is that the sine of the cosine are each associated with the phenomenon of periodicity in space they are periodic because if you go once around the circle that is to say T goes from T to T plus 2 pi you’re back where you started from all right that’s why it’s periodicity in space all right that’s the definition of sine and cosine that exhibits their their periodic phenomena not the definition in terms of right triangles

it’s not the definition it’s not that the definition in terms of right triangles is wrong it just doesn’t go far enough it’s incomplete all right it doesn’t reveal that fundamental link between the trigonometric functions and periodicity and it is fundamental if not for that mathematics could not be brought to bear on the study of periodic phenomena and furthermore this clear and will quit in just a second that is not just 2pi but any multiple of 2pi positive or negative I can go clockwise or I can go counterclockwise I can say the cosine of t plus 2pi n is the same thing as cosine of 2t and the sine of 2pi t plus 2pi n is the sine of T for n any integer n 0 plus or minus 1 plus or minus 2 and so on and so on the interpretation is that when n is positive I’m going count and it is just an interpretation is just a convention when n is positive I’m going counter clockwise around the circle when n is negative I’m going clockwise around the circle but it’s only when you make the connection between periodicity and space and the sign of the cosine that you see this fundamental property all right now all right I think we made it out of junior high today that’s that was my goal all right what is what is most amazing and what and what was what we’ll see you next time is that such simple functions can be used to model the most complex periodic behavior all right the simple from such simple things some simple acorns mighty oaks grow or whatever you excuse me whatever whatever stuff you learn out there that the simple these simple functions that associated with such a simple phenomena can be used to model the most complex really the most complex periodic phenomena and that is the fundamental discovery of Fourier series all right and is the basis of Fourier analysis and we will pick that up next time thank you very much see you then

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