# Lecture 1 | The Fourier Transforms and its Applications

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

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