[Music] okay so what we have done so far we have tried to discuss about Fourier series right and especially the origin of Fourier says means how furious is has come into picture so I'll just quickly recapitulate what we have achieved so far so given given a vector analogy we have said that any vector if you know the dimension or whatever vector space you have if you know the dimension and if you know means as many orthogonal vector can exist in that particular dimension suppose you have a two dimensional thing and you have two orthogonal vector so what we have said that as long as you know this two orthogonal vector you can represent any vector in that space in that two dimensional space with respect to these two vectors especially the linear combination of these two vector and these two orthogonal vector are actually termed as the basis vector right we have taken that same analogy in two signal so our target there was of course we have started targeting a periodic signal so what we have said given any periodic signal that can have any nature can we really find out some known signal a linear combination of them can give me back that same periodic signal so any signal possible in this world which are periodic it's some period let us say T zero okay so can we get set of basis signal or set of signals which are already known like this orthogonal signal so already known and can we represent and can you guarantee that any signal in that particular signal space can be represented as linear combination of all those known signals so what in that aspect we have defined few things I am NOT going into the details of those things so we have defined what we mean by orthogonal Tekin T in signal space okay so if two signals X 1 T and X 2 T if these two signal are they're both our periodic signal with a period t 0 let us say and then how do we actually prove that these two signals or how do you test that these two signals are orthogonal so this is something we have already evaluated ok we have evaluated two condition one is if both signals are real then we have evaluated one condition and then we have said if both signals are complex then also we have evaluated one condition so the ultimate condition which we have also proven is that X 1 T X 2 T over that period let us say minus T is 0 by 2 to plus t0 by 2 this must be 0 if both the signals are real and at the same time if signals are complex then you have to take X 1 T and a complex conjugate of the other signal X 2 T and you have to do the same integration over that same period and this must give me 0 if X 1 T and X 2 T are orthogonal to each other so what we have to now do we have a after that we have said we should have a quest of this known signal all the orthogonal signal in that signal space we must be able to find out as many are there ok so in that process we have defined two things one is trigonometric Fourier series that means all the signals must be decomposed into means series of sinusoidal or cosinusoidal so that's one another one was exponential Fourier series again all the signals must be decomposed as linear combination of exponential functions right remember if the signal targeted signal has a period t0 all those exponential signal or it is a complex exponential signal all the exponential signal also must have period of t0 so in that process trigonometric 1 we have already done exponential we have said that e to the power J 2 pi F 0 n T ok so that's the signal we are targeting and we are calling this as all those X I T or xn T so basically it is a as you can see it's exponential and you can put Oilers theorem and then you can get 1 cos plus J sine and then the period will definitely be 1 by F 0 right so f 0 is the frequency so for any value of n there will be a period which is defined by 1 by F 0 which is called as t 0 okay so this period will always be there so all these signals are periodic signal effectively with period this and remember these are all complex signal so we need to or through variety condition we need to actually prove it from that complex perspective or complex signal perspective that's the first thing ok and with this we have actually shown that these signals are orthogonal with respect to different values of n right so we take different values of n you can immediately show that it is orthogonal so whenever suppose we take two values m and n and then you put that orthogonality condition that we have just discussed you put that you will be saying that if m is not equal to n it will be always 0 ok so these signals are all for different values of n up n starting from 0 to infinity and minus infinity so for all these values of n they are all orthogonal to each other there is another aspect of it in vector also we have told that eventually what we were targeting any vector we're targeting to represent by another known vector right and in that process we are targeting the error of this representation so we have said that this is that error so if suppose this is my G and this is my X and if G has to be represented by X we are giving some linear coefficient C X and then this was the error and our target was to minimize the error same thing we have also done with respect to the signals and then we have evaluated all the values of C n that is required for different signals right so that is something we have already evaluated so it's the error that we are targeting to minimize right and eventually we have also said that in the vector space suppose in this one we are just trying to represent G which is a two dimensional vector by one vector so that is why there will be always a error we can minimize that but there will always be a error but if we have another vector which is orthogonal to this X let us say Y we can always completely represent G with respect to linear combination of x and y we know that already in a two dimensional space if we have two orthogonal vector we can always represent that so that means if you have a completeness of the basis definition whichever space you are defining if all the orthogonal vectors in that or all the orthogonal basis vector are already defined then you can always have a linear combination and you can make your error 0 same thing happens to the signal if I have a complete definition of basis set then I can always represent a signal with respect to linear combination of them and we also know how do we calculate the individual coefficients so this is just a complete recapitulation of what we have achieved so far okay so with that aspect we have talked about completeness and we are just saying without prove that for all values of M let's say all values of M and going from minus infinity touching 0 to plus infinity okay every integer value it takes if we take all this then this makes a complete mess is set this particularly to the power J 2 pi so we are just stating it without proof and then we can say any particular signal GT which has a period T 0 it might have any shape that can be always represented as a linear combination so with coefficient cn e to the power J 2 pi F 0 n D n going from minus infinity to plus infinity right so this is something we can always talk about where the C ends are evaluated just by that error minimization process and which was evaluated as 1 by T 0 integration whatever that GT is there e to the power minus J 2 pi F 0 n T DT and it should be integrated over that period let's say minus T 0 by 2 to plus T 0 so that's the Fourier series and corresponding coefficient we already know about right this is something we already know now let us try to see if this is the representation in this representation what we get we have already seen that in trigonometric representation we actually get different sinusoidal right so at different frequency this is also exponential representation so it must be a sinusoidal it is a complex sinusoidal so let us try to see what do we get so first let us try to evaluate what do we mean by this okay so see that coefficient evaluated at minus N and I am trying to take the complex conjugate of that so let us first put what is C minus n that should be whatever my signal is let us say my GT e to the power minus J 2 pi F 0 now here that coefficient should come at that coefficient it has to be evaluated so that particular coefficient minus n into T DT it must be in related from minus t0 by 2 to plus t0 by 2 and there is a complex conjugate of this so the whole thing has to be conjugated right so it should be minus n I forgotten that right because we are evaluating the coefficient at minus n so correspondingly the coefficient should be minus M and if this is minus n what happens this minus minus it becomes plus right and then we are taking complex conjugate so integration has nothing to do with it so it will come inside the integration GT is a real signal we have already talked about that that we are evaluating the Fourier series of a real periodic signal GT ok so GT is real so complex conjugate of that will be just GT and the complex conjugate of this one so what that will be it is already minus minus plus and you take the conjugate of that that should be again - so I should eventually get 1 by T 0 minus T 0 by 2 to plus T 0 by 2 GT it is the power again I get back - so there are 2 negation minus minus becomes plus and then you take complex conjugate so it again becomes minus we can identify this this is exactly CN right so what a very nice thing we have observed now what happens whenever we evaluate these coefficients C minus n is actually a complex conjugate of C n so always C minus n ne for any value of n they are always a complex conjugate of C n what does that means let's say my cm these are complex number so it must have a amplitude and a phase so I can represent that as CN and then there should be a phase which is let us say e to the power minus J theta n okay I can represent it this way because he N is a complex number so it must have a amplitude and must have a phase right so that's all we are doing we are just putting this representation as amplitude and phase okay so if this is the case what must be the complex conjugate of that the C minus n must have amplitude remains the same because it's complex conjugate so it must be CN and this will be just conjugate it so e to the power minus J theta in complex conjugate will be plus theta n so what is happening if because now every coefficient have two parameter if I can see it has an amplitude it has a phase so I can have eventually two plots one is with respect to this n I can plot the amplitude so what happens to the amplitude if at N+ I get some value at n- the amplitude should remain the same right because for CN also it's mod CN or C minus and also its mod CN right so the amplitude should remain the same therefore what do I get for different values of n the values might be different but whatever it is at minus n the value should be same right so we get a even symmetry in terms of the plot right whatever we get at positive half we get same thing at the negative half okay so that is why it is called even symmetric and for phase just the opposite whatever we get at positive will be getting just the negative of that right because you can see if it is minus theta n it should be plus theta if it is plus theta n it should be minus theta so the angle should be same but it is just negated right so we get odd symmetry for the phase plot so basically what we are getting now we have a signal GT right it's continuous in time what we can see now that that GT as long as it is periodic it can be decomposed into multiple complex sinusoidal is something we can immediately observe and those sinusoidal for different values of n it's actually representing different frequency component if n equal to 1 it's f 0 n equal to 2 it's 2 f 0 if n equal to minus 1 it's minus F 0 n equal to minus 2 it's minus 2 2 F 0 so it's actually representing different frequency component or we should say different sinusoidal with different frequency value okay so all we are doing is now because we have got if you see this we have got two equivalent representation of the same signal GT in time I can plot I can also plot GT or I can represent GT as a summation infinite sum of course as a summation of sinusoidal at different because this is just a complex sinusoidal which is known ok so it is just that complex sinusoidal all we need to know is at every frequency component what kind of amplitude it has and what kind of frequency it has so the information about this signal is completely carried over here because this is a known thing all these signals are known they are just representing different complex frequencies nothing else we just have to know the corresponding coefficient because as the coefficient change changes it will represent different different signal that's all we are targeting ok so therefore thanks to pop Fourier what we get is a separate representation of a signal now we do not see the signal in time domain whenever we have a signal we know that it has an equivalent Fourier series representation as long as it is periodic and immediately we can plot and effectively we have to give two plots one for amplitude one for phase and we also know that the amplitude plot should be symmetric or even symmetric so therefore positive of whatever it is it should be mirror imaged at the negative half and the phase plot should be odd symmetric that means whatever we get in the positive half that should be just negative of that in the negative half right so that should be the case so we should plot mod CN and theta so all we have to do is this and we know that for which point in this independent axis we have to plot these are just those n value or more precisely n into F zero values so those frequency those complex frequency component how much of that component is present in phase as well as in amplitude so how much of those components if I mix them all together my desired signal will be generated so this representation is called the frequency domain representation because we are actually decomposing our whole signal into different frequency components that's all all that we are doing is this on T there are two strikingly different [Music] representation when we do it in trigonometric series and when we do it in exponential C's but the function must have unique representation in frequency so let us now try to appreciate what exactly is the relationship between these two representation when we have started representing it in trigonometric series what we were getting we are also getting two plots actually because we had that sinusoidal as well as cosinusoidal at every frequency we were having both the things right barring zero zero was only having a DC value so it doesn't have it doesn't recognize whether it's cause of sine because it's not having any frequency component right it's DC so at that point it was just a zero but rest of the case it was always having a pair a N and B and corresponding a n is due to the basis function cos 2pi FC 2 pi F 0 n T and BN is due to sine okay so these two frequency component were present if you see that representation there was nothing called negative frequency okay which we are getting over here in the exponential representation so exponential representation have the spectrum starting from minus infinity to plus infinity so it has some component at the negative frequency as well as positive frequency whereas when we are representing it in trigonometric that also has and this also have two plots one is amplitude one is phase whereas whenever we are representing in trigonometric form it was also having to plot corresponding to sinusoidal and cosinusoidal but it does not contain any negative frequency right so this is something we have observed so we are always getting two plots and in this to plot what was coming it was just those mod CN at every value of n mod CN and theta n that's what you are getting okay and then we are getting this for the other case we are getting a n and B n now let us try to see if we can correlate these two things that's the first task and then we'll come back and try to explain what do we mean by this particular thing called negative frequency okay so first let us be concerned about this relationship so what happens we have seen that there is a already we have explored that there is a relationship between C plus N and C minus n so what we will try to do is we try to pair up these two things so we have a representation of GT as summation CN e to the power J 2 pi F 0 and T n going from minus infinity plus infinity okay so let us separate out the c0 term so at C 0 n is 0 so this becomes e to the power 1 right so that's alright and then we start pairing each of each of those N and minus n right so here will do a pair of C 1 and C minus 1 so on let us try to do it for C n e to the power J 2 pi F 0 n T plus C minus n e to the power definitely it will be minus n so minus J two pi f 0n t let us try to see what this term together gives me see if I can evaluate all these terms I am almost coming to a conclusion that it is almost like this c0 is nothing but a zero and this is actually almost like an cos and plus BN sine if I can finally represent them in that format then I can give a relationship between this CN and a and B n right so let us try to see now all we know is CN is a complex conjugate of C minus n therefore I can immediately put our representation so I can say that suppose CN is represented as mod CN e to the power J theta n then C minus n must be represented as mod CN e to the power minus J theta n right this is something I can always do as long as I know this theta N and mod CN and that I should be knowing because if I know the CN I can immediately calculate its modulus and its phase right so this is something I can always do now let us try to see if I put this replace this over here what do I get so for that in it Tom I can get mod CN e to the power J theta n e to the power J 2 pi F 0 NT plus mod CN e to the power minus J theta n e to the power minus J 2 pi F 0 NT okay so this is the NF time I am getting which is nothing but mod CN gets common and I get e to the power J theta n plus 2 pi F 0 n t plus the negative of that right e to the power minus J same thing so if I just say it say e to the power J some theta or let us say Phi plus e to the power minus J Phi put euler's theorem what do you get 2 into cos of that cause of inside whatever Phi is there so I can write this easily as 2 into cos theta n plus 2 pi F 0 NT right what I have eventually got is a single cosinusoidal term right I have got a single cosinusoidal term nothing else is something I am getting okay similarly for the trigonometric series for every n ed value I will be getting one a n cos 2 pi F 0 NT plus BN cos 2 pi F 0 and sorry sine right I will be getting this now this one what I can do a n I can write as some R cos Phi and BN I can write as R sine Phi this is something I can always write write because immediately I can see I can calculate R R should be what root over N square plus BN Square and Phi should be so I just divide these two tan inverse B n divided by n so as long as I know this relationship I will be always able to put this representation put this over here so I get R cos Phi cos 2 pi F 0 and T plus R sine Phi sine 2 pi F 0 nt why I'm doing this I just want this representation again right can I can I see a single cost representation it's cos a cos B plus sine a sine B so that must be cos a minus B right so I can write as cos 2pi f 0n t minus Phi fine so now I can see I have got two series one was exponential series another one was trigonometric series from both the series I could actually get I was trying to see the NH term and I could see that by unifying this cosinusoidal and sinusoidal and by unifying the CN and C minus n I could get similar representation only thing is that those coefficients has to be now matched right because I will be getting exactly similar term for every n so therefore the coefficient because it's a unique signal so it must have unique representation unique addition of sinusoidal okay so therefore coefficient must match so therefore R must be 2 of mod CN right and this Phi must be minus of this theta n right so this is what we are getting now Phi is tan inverse be n by n that must be minus theta n so this is one relationship I get and R is basically root over a n square plus BN square that must be 2 mod CN so basically what is happening I have got a trigonometric series I have got a exponential series exponential series whatever amplitudes for plot I get I immediately can get a relationship between the corresponding trigonometric series okay so if these two are equal the representation should be equal and what has happened fundamentally as you can see that because of this complex conjugate see finally when I added those two terms CN and corresponding to C minus n I have actually got a real signal so that means my signal was real people might be thinking that when I do for exponential Fourier series I am representing them in terms of complex signals so what is happening to my real signal complex signals are now creating real signal but that is not the case the C NS are arranged in such a way that if you take two NF terms they will those two complex signal will actually give you back the real signal okay so eventually it's nothing but both the representation tells you that it is a particular representation where the for a particular frequency there is a amplitude and there is a phase not nothing else both the representation gives you similar value okay so with this I will end this class a next class what we'll try to do we'll try to evaluate what do you mean by this negative frequency