[Music] ok so so far what we have done is we have seen a vector analogy and try to put that into signal space okay so we are now targeting that a particular signal must be represented by another signal and then from from the vector analogy we also know that in the signal space if we can now we also now know what's the characterization of orthogonality of ah two signals so in a particular signal space if we can find out all the set of orthogonal signal then we know that exactly a particular signal can be represented in a linear combination of all those orthogonal signal this is something we have already understood and their linear coefficients also how to calculate that also we have characterized now ok so we know that exactly how to evaluate those optimal c which minimizes the whole error and if i have that complete definition that is called the completeness of ah the basis set so if i have complete definition of all the basis set that are possible in that particular signal space i will be able to represent a signal so this is what fourier means this was the background where fourier started means analyzing signal and this is how he started things so lets now our now target should be that how do we represent or how do we find out all the orthogonal signal in a particular signal space so initially fourier started with trigonometric signal or sinusoidal signal so he initially started with lets say we take this cos m two pi f zero t ok so he started with this and correspondingly i can also define a sine m two pi f zero t ok so these two signal we are targeting so whenever m is equal to one it is just a cosine ah cosine signal with frequency f zero where f zero is equals to one by t zero ok so t0 is the time period of that signal so whenever we are talking about cos signal or sign signal its a periodic signal we know that and the thing is that if the constituting signals are periodic then what it will represent must also be a periodic signal so what initially fourier device he told that ok i can represent signals but all those signals must be also periodic signal that i am targeting to represent that can be any signal it might be a triangular wave something like this ah let us say this kind of thing but it must be periodic with period let us say t 0 it must be a pulse and of course it gets stretched to minus infinity so it can be any periodic signal and he then started saying can we represent this in a particular fashion and then he started inquiring the basic signal that we know even the electromagnetic wave that propagates that propagates with this sinusoidal or sinusoidal so can we represent all these signals with respect to this cosinusoidal or sinusoidal but before doing that we we need to know the orthogonal signals in that signal space so what he started doing he started inquiring whether two co-sinusoidal lets say cos m two pi f zero t for different values of m ok so let us say one is cos m two pi f zero t and another is cos n two pi f zero t whether these two signals are orthogonal or not so if they need to be orthogonal what we have to prove it is a real signal cos is a real signal so all we need to prove is and it is a periodic signal so in within one period if the property is true it will be true for all other periods because it will be just mimicking the similar pattern so orthogonal means the integration must be zero right there multiplication and integration must be zero thats what we have understood g and x will be orthogonal f g in g t into x t d t over a particular time period must be zero so if within one period it is zero the other period also will be behaving similarly so entire period or entire signal duration it should be zero so this is what we wish to prove is it true so will probably do it from lets say minus t zero by two to plus t zero by two we can even do it from zero to t zero any period you can take even we can do it from t zero to t zero so over a single period we are trying to evaluate what will be happening if we do this integration because we know that that is the proof of orthogonality if this is zero for two different signal of course ah if it is same signal then we should not we should get the energy back right so if it if it is two different signal that means two different values of m and n or we are saying m not equal to n for this condition we are trying to see what will be the value ok so lets see let us try to evaluate this take out half it should be or i can do it over here only half two right now just put a trigonometric form two ah cosine into cosine so that should be cos a plus b plus cos a minus b so it should be cos m plus n two pi f zero t plus cos m minus n we are assuming that of course cos negative that will again become positive so here either m is bigger than m n or n is bigger than m it does not matter it will all be same there is a half outside now lets see the first integration zero to t zero cos ok so if m is not equal to n then both these numbers are nonzero ok so what is happening this is another cosine and whatever that f zero this m plus n or m minus n it will be either one or bigger than one so over that period t zero if i integrate a cosinusoidal which is see already two pi is there that means either it will just make the full cycle or it will make multiple full cycle and a cost signal if it completes a full cycle or multiples of full cycle always the integration what will happen sorry up to this always the integration will be zero right as long as it is a sinusoidal signal and we know that it is either completing a full cycle if the difference is lets say this addition of course it will not be it will be more than a full cycle it will be two three or some integer number so multiples of full cycle it is completing so if i integrate that should be zero definitely so this part will be zero even this part also will be zero because its either it will be one full cycle because m not equal to n so m minus n at least it will be one and it it might be even bigger than that so whatever it is it is either a full cycle or multiples of full cycle if i integrate again this will be zero so the overall thing will be zero if m not equal to n so therefore we know that all the co sinusoidal signal with all values of m going from one to infinity are all orthogonal to each other thats a very important findings so what he could find in that signal space in the vector space if you remember when we had this page two dimensional we had only two orthogonal vector now we actually have infinite orthogonal vector right any value of m you put so m starting from integer value of course starting from one and then going up to infinity any value m or n you put as long as m not equal to n all of them are actually orthogonal to each other and these are actually called the harmonics ok so you have a co sinusoidal you have twice frequency equal sinusoidal thrice frequency co sinusoidal and so on all those frequencies or all those harmonics first second third fourth all those harmonics are included and they are actually orthogonal to each other thats one finding the second part is if i have a dc value which is just 1 okay so between that 0 to t 0 whatever period i have talked about so it just takes value 1. ok if this is the case will this be orthogonal to all those cos of course because what will happen this has to be multiplied with those cos and again will be integrated so if i just do that integration should be one into cos m two pi f zero t d t and integrated over zero to t zero right so now again its a co sinusoidal with m taking value 1 to infinity so therefore this is actually because the 2 pi is already there and f 0 is 1 by t 0. so basically this is either a full period or some multiples of full periods so it will all always be again integrated to zero right so even this particular signal where its just taking a constant dc value that is also orthogonal to all those cosinusoidal so the dc value which for a single period it takes same value so if i just consider all the periods it will just take a dc value so the dc value even is orthogonal to all the co sinusoidal signal right so this is what we have already demonstrated now the next part is is this also orthogonal to of course the dc as well as all those cos values will now again show that if i multiply with some cos lets say n two pi f zero t for any value of m and n now we do not have a restriction because even if m and n are equal these two are two different signals okay so they if i just do this integration again you will see with the same logic it will again become some cos or sign and then within again some either full period or multiples of full period the integration has to be done and it will all be zero so therefore what we have done is we could identify now few mutually orthogonal signal and i should not say few its infinite okay so all the cos m two pi f zero t for all values of m positive integer all the sine two pi m f zero t for all value of m positive integer and dc value these are all all these infinite number of signals are all mutually orthogonal to each other the other part which i am not covering over here which is little bit more involved that if i take this whole set that is a complete set of basis like that two dimensional vector we are saying x and y that makes it complete because only with those two vectors i can represent any vectors in this signal space for periodic signal we can actually prove which will be little bit more involved so we are just giving the ah outcome that if i just take consider these basis set this actually makes a complete basis set because these are all the basis set that can be defined in this particular representation ok so if i consider all of them i have actually i could actually finish representing all of them ok and any periodic signal therefore should be means we should be able to represent any periodic signal with respect to these basis functions therefore ok so what should happen then i know that now my any periodic signal gt i should be able to represent them linear combination with the linear combination of all these basis functions right so therefore i can write first is dc so a 0 into dc term which is 1 plus sum a n into cos 2 pi n f 0 t where f 0 is actually 1 by t 0 and this t 0 is the period of this signal ok and i have to take because all bases should be taken so i have to take a summation over all possible cos this thing or all possible values of n so n one to infinity plus the sinusoidal is there so b n sine two pi lets say n f 0 t summation that is the famous fourier series representation of any signal so because we have proven that these are the basis set which are mutually orthogonal to each other and because we have also without proof we have stated that these are all that i can get in that signal space so this is actually a complete definition of basis set so therefore without error i can be means i should be able to represent g t as a linear combination of all this basis set where the coefficient of linear combinations are this a zero and a n and b n ok and how do we evaluate this values let us say a zero how should we evaluate that this is something we have already seen the evaluation of a 0 should be with those dot product right so all we have to do is this g t with respect to our whatever signal is so therefore one d t must be integrated over a period zero to t zero and this one should be also multiplied with itself so one into one d t from zero to t zero this has to be done ok so as long as we are doing this we will be able to represent the whole thing right so a zero is like this similarly a n must be whatever that g t is we multiply that with cos two pi n f zero t integration should be zero to t zero and zero to t zero it should be cos square right cos square two pi n f zero t d t that's the fourier coefficient ah if you go back and try to see whatever you have learned in fourier series this is actually the fourier coefficient this is something we have already proven this is the optimal a n for representing that signal okay so similarly b n also will be we we can get a formula of b n 0 to t 0 it can be 0 to t 0 or minus t 0 by 2 2 plus t zero whatever it is it should be that single period ok so the result will be same so again g t this time it should be sine two pi n f zero t d t divided by ok so in your fourier transform you might have seen that this happens to be if you do this integration its just one or d t integrated zero to t zero so that should be one by t zero so a zero becomes one by t zero that is the whole formula right g t d t over the period ok this if you do integration it should be t zero by two if you just do that integration there will be a half coming out and two going in and then if you do that integration you will just find one ok so this should be two by t zero so therefore this thing goes away you get two by t zero same thing happens over here so this becomes two by t zero so that is the famous ah fourier series and the fourier coefficient calculation which we could directly now prove without any ambiguity ok this is very clear to everybody so once we have done this there is another representation probably whenever we ah so this was all fine we know that any signal now we can actually represent with respect to corresponding sinusoidal so what is actually happening all the sinusoidal harmonics are now coming into picture so any sinusoidal means any periodic signal you want to represent its nothing but a infinite summation of different harmonics nothing else any signal can completely be represented if you can evaluate through this the coefficients appropriate coefficients will be knowing exactly how linearly you have to combine those harmonics to get this particular signal so that is a very nice representation you actually now know that which sinusoidal with what strength you have to take and you have to add them to get this particular signal in a way it is also giving us a strong tool you get a signal you immediately know what are the constituent sinusoidal with what kind of strength should be present in that particular signal ok so whenever we started start talking about what are the constituents sinusoidal that means we are now talking in terms of what frequency components are actually there in this particular signal this is where a signal gets represented in the frequency domain okay so if we just give one example probably we just represent this particular signal so the signal is something like this let us say it is omega t it is defined from minus pi by 2 to plus pi by 2 its like this and then again this is pi this is three three pi by two it starts again this is two pi and so on so if i represent a signal like this which is a periodic signal and then if i the way we have learnt to evaluate the fourier coefficients if we just calculate that we can see all the fourier coefficients and we can then start representing this signal with respect to frequency component so right now i wont give the example fully because we still have not means ah equipped ourselves fully with the frequency component but what we will do will take this example and come back after little bit of more analysis and come back and draw actually the frequency spectrum will see what are the harmonics which are present and how they are present and accordingly we can actually draw the spectrum of the signal another representation not in time domain but in frequency domain so basically will be saying the representation of that equivalent representation will have like this so f 0 to f 0 3 f 0 what are the components it has so it will just give another representation in frequency domain so here it is just f 0 to f 0 that means frequency is increasing and what are the components of those frequencies which are present that will be able to characterize so this is how fourier series actually gives us a frequency representation or frequency domain representation of a signal its nothing its just it says for a periodic signal what are the whenever we say frequency component we are actually saying what are the basis components that has that it has and basis components means its sinusoidal so all those frequency component whenever we talk about whenever we plot the frequency domain we are actually saying at that point there is a sinusoidal corresponding sinusoidal whichever frequency component we take we say at that point there is a sinusoidal with that strength whatever strength will be putting so frequency domain representation is nothing but every frequency component we pick actually there is a equivalent sinusoidal that constitu means that actually makes the signal so you take of that strength sinusoidal you add all of them you actually get the signal back so therefore there is a equivalent representation we should say in time you see also in frequency domain you start seeing the similar representation of that signal its just thanks to fourier he could actually give that extra representation of signal which you will see that which helps us in many way for the processing of the signal ok so before going into that and before characterizing that lets try to see another representation which is also a fourier representation we have already seen a representation with respect to real signal so right now whatever basis set we have constructed those are all co sinusoidal sinusoidal or dc value those are all real signal now what will do we take a signal which is a real signal probably gt but will try to now represent it with respect to some complex signal ok so first of all will try to define a family of complex signal which are mutually orthogonal to each other then we will say what is the again the technique is same will then say what is the overall complete basis set what is the definition of complete basis set once we get that we can get a representation again so that particular thing is represented as exponential complex signal so it is like e to the power j 2 pi f zero n t so this is my test x t ok so any x t which is taking this value where n now can take integer but it can be positive negative up to plus infinity minus infinity okay so n can now take any value from minus infinity to plus infinity but its integer ok now let us first try to test whether this infinite number of signals we have got are they orthogonal to each other or not so that is something we have to prove the signal is a complex signal because we have this complex part already so first we have to prove that this is orthogonal or not so now we have to employ the complex part or complex equivalent orthogonality criteria so that means suppose we take a x t or x one t which is e to the power j two pi f zero n t and we take another x two t which is e to the power j two pi f zero m t ok so now all we have to prove is if m not equal to n then should we get the orthogonality criteria equal to zero so what is the orthogonality criteria that is actually x one t into x two t star so therefore it should be e to the power j two pi f zero n t into the complex conjugate of this one so that should be e to the power minus complex conjugate of exponential is just minus of that e to the power minus j two pi f zero m t d t and if you carefully see this is a complex signal but this is also a periodic signal because this signal can be represented as a complex co-sinusoidal and sinusoidal so it should be cos this particular factor 2 pi f 0 n t plus j sine 2 pi f 0 n t so therefore again the fundamental frequency is related to this f zero which is equal to one by t zero so the integration needs to be just done over that fundamental time period we do not have to do that whole integration because we know that in one integration if you can prove this just gets repeated so same technique will be employing so 0 to t 0 we can integrate the same signal okay so now this is let us try to evaluate this so it should be 0 to t 0 e to the power j 2 pi f 0 m n minus m t d t right now this integration how much it should be it should be just e to the power j 2 pi f 0 n minus sorry n minus m t divided by the coefficient j 2 pi f 0 n minus m and then we have to put value 0 and t 0. now see if we just put t 0 t 0 is 1 by f 0 right so this gets cancelled n is not equal to m right so this must be this cannot be 0 right so this must be some positive integer okay therefore it is just e to the power j two pi into some integer it is for j two pi into integer right that we can represent as cos and sine now cos two pi integer plus j sine two pi integer cos two pi integer that should be always having value zero right so what do we get if i just represent this so it should be so if i just represent this it should be sorry cos two pi that should be one right so we get one for t 0 and then for sign it is 0 so j part is cancelled minus i have to put this limit so again i put 0 it should be 1 and for sine it will be 0 so 1 minus 1 it gets cancelled so this becomes 0 right so whenever m not equal to n it is always 0 if m equal to n this becomes already 1 right because it becomes e to the power 0. so we know that this particular things are all orthogonal as long as m is not equal to n that means it is not of course with himself it cannot be orthogonal so with all other signals they are mutually orthogonal to each other so this is something we have now proven any value of m and n you take as long as m is not equal to n we could from the basic principle of orthogonality of the complex signal we could prove that they are all mutually orthogonal to each other and again we are saying that we without proof we are stating that these are all the signal that are required to represent the entire complete basis set ok so as long as we are taking that to be true we know that this x t can be represented with respect to these things so now it is very simple all we have to do is this g t must be a summation of some coefficient let us say that is c n e to the power j two pi f zero n t and n goes from minus infinity to plus infinity thats famous fourier series representation where the c n must be evaluated with the same criteria of complex so that must be integrated over that period g t into complex conjugate of x that is where the complex conjugate c comes into picture and that is why it becomes e to the power minus j two pi f zero n t d t divided by the this into complex conjugate of that so they will actually cancel each other it will be just one d t zero to t zero so that becomes just one by t zero so this becomes one by t zero zero to t zero g t e to the power minus j two pi f 0 n t d t this is the reason why in the inverse 1 or the coefficient 1 whenever you calculate you have to put e to the power minus j and whenever you are actually representing g t you get it is about plus j and you will see later on will from this series will go to fourier transform and that is why fourier transform always gets e to the power plus j and inverse transform always get or vice versa ok so this things happens due to that reason so now what we have seen that we can represent any signal again with respect to another basis set which are complex in the next what will try to do is we will try to get a means relationship between these two representation what the exponential fourier series tells us and what the trigonometrical trigonometric fourier series tells us and what are the relationship between these two things