[Music] okay so as we have discussed in the previous class that will be now concerned about a signal which is non periodic in nature most of the signals if you see we have defined periodic signals and the fundamental property of a Pury signal is that it must be defined from minus infinity minus infinity to infinity that means that periodic signal must repeat at some interval let us say T 0 which is the fundamental interval and it should go from minus in 2002 in otherwise that periodicity will not hold good okay so but most of our signals actually start at a particular instance whatever practical signal we can see suppose avoid Y signal generated by me it will start at a finer particular time instant that isn't a in that effort particular instance so they are finite in time they cannot start actually at plus infinity minus infinity and cannot go extend up to plus infinity so most of our real life signals are time bounded okay so therefore they have a finite energy because if you go from minus infinity to plus infinity you will probably means if you wish to evaluate power you'll see probably zero okay because the signals are not stretched up to minus infinity and plus it's easy so our signals are generally energy signal that means it is time bounded it starts at a particular time and ends at a particular time so we should have a strong analysis of those kind of signal otherwise our analysis is not true so whatever Fourier series gives us that is a very strong tool we can agree on that but still that particular thing is not good for real signal so we need to have good understanding about the real signal which are finite in time okay so let's try and take an example that's a very simple example signal to define as GT it starts at some value of P let's say a it ends at some value of TD so this is the time so if this is my present let's say zero times zero so this was my past at which time some value a which is negative in time it has started and it will end at B so that's the future of the city so this is the signal so it starts at a it it ends at B if this is a signal and I wish to still do similar analysis as Fourier has done in his Fourier series analysis right so I still want to see what are the frequency component it has how to define them can we get some spectrum out of heat all those things okay but the problem is whatever Fourier has done that was good for periodic signals so what we'll try to do or what Fourier has also done that for a periodic signal only we have some strong result so can we apply those result into this so for that we have to forcefully make the signal periodic so let us try to do that let's say beyond this we define some amount of time that's our period okay and the same signal is repeated at everything zero so basically what happens if this is T zero my signal was like this after another T zero there is another period of T zero where the same signal is repeated and they are also repeats and it gets stretched to minus infinity to plus infinity so basically from our signal which was defined from A to B we have created a period which actually takes into account this a and B inside that period and then we start repeating this signal whereas in the original signal there was nothing beyond a and B or beyond even this T zero but we have deliberately made this signal looks like a periodic signal very nice so this is a newly constructed signal from that original signal we call this as G t0 T so that means from this signal we have transformed into of periodic signals right the good part is by this by means of this clever construction we can always say if I take this limit T zero tends to infinity G t0 t that exactly goes to G T this is what we were targeting so we were actually trying to see a particular signal can that be represented as a special case of periodic signal okay so we have constructed that particular signal and then we are told okay now if we stretch this period to infinity okay or this t02 stretching from minus infinity to plus infinity then definitely these parts will not appear so they will disappear because the period gets stretched up to minus infinity plus infinity so that will exactly look like our GT so this is what we are now targeting so what you're targeting is deliberately we are making a periodic signal out of non periodic or a energy signal which is time bounded so this something you have done now for GT 0 T which is a speedy periodic signal I can do Fourier series analysis so for GT 0 T which is a periodic signal with period T 0 I can always do a Fourier series analysis for n minus infinity to plus infinity DN it before J n 2 pi F 0 T I can write this way where this F 0 is 1 by T 0 the period I have constructed the frequency fundamental frequency is 1 over that ok so this representation is true I can always do a representation like this and after that we have the value 8 DN so DN is something we already know it should be 1 by T 0 and integrate it from minus t0 by 2 to plus t0 by 2 over the period g t0 T which is the main signal and it will power minus J n 2 pi and 0 T DT so this is something we already know right very good okay now let's see what happens to our representation okay so in our representation let us try to do it this way let's define it just arbitrarily defining for our you will see why we are defining this so let us define a GF which is nothing but integration from minus infinity plus infinity GT original signal it is the power minus J 2 pi F 0 TPP is just defining this this is our by definition we are saying that because this sir it should be on the F because this is our integration over T so that T will vanish it will be a finally a function of F we are defining that as a capital G F function of smallest right so this definition is correct no problem in that as long as this integration can be done so if we can if you say that this integration is possible we can always say there will be this will be a function of F because T goes away right so we can write that function as some capital G as some function okay so this destination is correct when this definition is correct what I can write about DN so let us see what is VN so now this DN if you carefully see this DN has 1 by p0 that is alright and in place of G F if I just write F 0 ok so if I just write F 0 then what will happen I will be getting this GT integration minus t0 by 2 to plus t0 by 2 and I will have e to the power minus J n 2 pi similar things are there just in place of F I have a F 0 right so basically what I can say this particular part if I put T 0 tends to infinity ok immediately this limit goes from minus infinity to plus infinity and as long as I say t 0 tends to infinity my G n t if you see the previous definition for GT 0 T if I put T 0 tends to infinity GT 0 T I can write as GT so I can write GT and replace that as GT 0 T ok so I can if I say that whenever I will put this particular limit I will be always getting this representation as we've G in place of F I will be putting F 0 so this is quite obvious as you can see what is G of 0 G of 0 will be just replace over here GF 0 so that should be GT e to the power J 2 pi F 0 T DT ok only problem is we have a GT over here so GT as long as T 0 is tending towards infinity which is this case GT and GTS 0 T are similar so I can GT I can replace by GT 0 T now if you go over here now this to limit T 0 tends to infinity means minus T 0 by 2 and plus T t0 by 2 goes to minus infinity and plus infinity so I can replace these two limit as if going to minus infinity to plus infinity so therefore DN exactly mimics these things ok so that is the beauty I could do by this definition and by our definition of GT and GTS 0 T or the relationship between these two ok so once I could do this I have got a very nice construction so what is that construction now I can see that what happens to my DN DN becomes 1 by T 0 which I have already written this is my DN and immediately I can represent my G t0 T which is the Fourier series expansion so which is summation n minus infinity plus infinity DN e to the power J n 2 pi n 0 T and in place of DN if I can replace this so it happens to be n minus infinity plus infinity 1 by T 0 GF 0 e to the power J n 2 pi F 0 e right this is what I get for this particular part ok so so far it's all good now let us try to see what that this particular thing means so what is happening in this case we have already assumed otherwise you could not have constructed this DN equal to this that T 0 tends to infinity so whenever T 0 goes to infinity I have a relationship F 0 equal to 1 by T 0 so what happens to F 0 F 0 tends to 0 okay so this 1 by T 0 or F 0 I can that becomes infinitesimally small okay so that becomes infinitesimally small instead of writing it as 1 by T 0 I can now write as del F where this so I am just replacing F 0 equal to del F which tends to 0 it's just for my convenience because we are used to see del F and this is G F 0 or sorry let's see there is things that we have missed here so it's nf0 which makes the f so it should be written as n f0 right so that that was a mistake I did okay so because it's if you see the construction GF is written as G te to the power minus J 2 pi F T now here in the construction we have e to the power minus J 2 pi T all of this and n f0 is there so therefore it must be n f0 right that's all right so this should be all replaced by n f0 so del F n f0 n f0 must be replaced as n del s right because f0 is replaced now by del F so GT 0 T may be constructed GT 0 T becomes summation n equals to minus infinity to plus infinity 1 by T 0 becomes del F G n f0 becomes del s into the power J n f0 becomes del s 2 pi T right so this is the new construction that we have got for GT 0 T and we know here T 0 tends to infinity and del F which is 1 by T 0 goes to 0 okay so this is something we already know now let us try to see what exactly is happening what is this part this was some function GS okay now whenever we start varying this n minus infinity plus infinity so what is happening whenever n is equal to 0 so it gives G 0 n equal to 1 it gives at del F whatever value of G n gs so this must be G on the del F so this is G 0 at n equal to 2 so this is a technique hwal to 1 n equal to 2 it gives me G 2 del s and so on and also so on in this direction so basically it's almost like there is a function G F which is a continuous function and whenever we are putting this different values of n we are actually sampling this function at del F interval now as long as this del F is becoming infinitesimally small so what is happening all these samples are coming closer and closer together okay and in the same method what is happening to this particular value so this is nothing but suppose I have a GF let's say this CF looks like this I take suppose n equal to zero I take this value and I take this box so what will be the area under this box that should be del F into G 0 or 0 into del F the next part will be again del F and what's this value this value is G 1 into del F into del s so it is almost like this small small boxes I am actually taking the area under that okay and I am adding all this one ID once I do summation I'm actually adding all these things right so as long as my del F tends to zero what is happening this summation almost becomes the integration because integration also gives me area under a particular function so if I just now think that my overall function is this particular thing that is gf into e to the power J 2 pi F T where F is replaced by n del F so basically what is happening this particular function I am adding with or I am multiplying with del F and I am summing it up whenever we put F as discrete values so it is nothing but this summation becomes the integration and my G t0 T just happens to be our integration and because it goes from minus infinity plus infinity whenever we put n equal to minus infinity what happens this n into F 0 goes to minus infinity so if my variable is f that goes to minus infinity and whenever I put n equal to plus infinity it goes to plus infinity this del F becomes DF and I have this gf it will be power J 2 pi this is replaced by F T so basically n del F is almost becoming a continuous variable F this is very true we have already stated that if del F is small then all the values I am taking almost becoming continuous in time or sort in frequency so basically I was taking initially I had this function which is gf into this I was sampling that function and I was trying to calculate the area under it because I was multiplying by del F and taking that box and I was adding all those things so this is nothing but integration that we have already understood and what is also happening I am making this del F smaller and smaller so those samples are coming closer together and getting very closely packed and actually those samples are almost indicating now a continuous variable on F so whenever I write n del F that goes to F okay whether f varies that becomes the integration variable so immediately I get this and because my T 0 tends to infinity already for this construction so this can be written as G t because GTI and GTD rho t becomes equivalent as long as 0 goes to infinity that was our definition of the signal so we made this a periodic signal to be periodic with period t 0 and then we have told that this p 0 if we stretch to infinity it becomes a similar a periodic signal this is something we have already stated so as long as T 0 goes to infinity we get this very relationship so what has happened now we have a two relationship that we have said one was our original statement that gf must be related as this this was by definition GT it is the power minus J 2 pi FC DT and now from the construction we have got another relationship where we say GT is nothing but minus infinity to plus infinity gf in the power plus J 2 pi s t DF so this is the famous Fourier transform and Fourier inverse transform theorem you can immediately identify okay so if I have a signal GT I can always get a corresponding Fourier transform which is gf and if I have a signal G means if I have Fourier transform I can always do a inverse transform to get my signal back so this is the Fourier transform and inverse transform in this whole process the most important part is this gf which is intermediately constructed so what is this gf if you carefully see the construction this gf okay is actually almost similar to as you have seen it is almost similar to our DN we have derived this art here so basically we have already stated that DN must be 1 by T 0 into G and LF what was our DN DN was for every frequency component the corresponding amplitude and phase DN was giving me that now G if as long as del F tends to 0 this becomes a continuous variable ok so this end LS as long as del F is very small this almost becomes a continuous variable of F so this is nothing but our T 0 into DN or I can write this DN equal to this so basically what is happening the corresponding DN can be calculated if I know that gf whatever that GF is and I multiply that with that very insa infinitesimally small frequency component del F so what is happening suppose I have a particular let's say of Fourier transform representation gf okay so each one of this for a particular value of n are actually representing suppose this is my gf right so that sample value if I multiply that with del F so this area okay or let's say this area so GF x that del F this area as if giving me this TN at that frequency what is the overall coefficient of that frequency component so what is eventually happening as I put this del F tends to zero very small and small I am actually getting all frequency component so it is almost becoming continuous earlier in my periodic signal I was only getting those harmonics of zero to f0 and all those things now for a periodic signal what we have observed that because for that appearing signal we have represented it with respect to a periodic signal so if we faithfully represent back the periodic signal t0 has to be infinity once t0 is going towards infinity this del F is becoming very small so that means it is now almost those frequency components are getting closely packed so we are getting almost all the components all the frequency component in the continuous domain right and everywhere if I multiply by this del F that gf whatever I have if I multiply by this del F I get the strength of the frequency component similar like this pier right so what might happen suppose in a hypothetical experiment I keep on because I have to go means stretch my t0 towards infinity I keep on reducing my sorry increasing by t0 so earlier suppose this was my representation okay so this was my del s now what I do I actually increase my p0 so immediately what will happen if I increase my t0 so immediately this will suppose I make t0 double so if I make p0 double immediately what will happen this DN value because t0 is doubled DN value will be reduced by half okay so what happens the N value will be reduced by half but the samples will be now more so basically instead of this it will be all reduced by half but the samples will be closely packed double close the pack because del F is now becoming half of the previous this one so there will be double closely packed and the sample values which is represented as DN which is the integration or we should say which is the area multiplied by del F G NF n del F ok so that area is represented by these values so they will be becoming 1/2 because I have increased my T 0 to double corresponding by a0 has become half so basically sample will be there will be double samples now more closely packed but this strength will be reduced and if I keep on doing this experiment what will happen I will still get this samples or their values as long as T 0 goes to infinity their relate means individual values will go to 0 so basically what I am getting I am getting at every point in continuous frequency domain I have some value but the corresponding value is 0 because T 0 goes to infinity ok but the relative values are still being constructed by this year so however small you make them they have a relative nature so if this is the strongest one and this is little weaker they will still remain the same their individual value will tends to 0 ok but their relative nature which is captured by this GF will still be the same and that characterizes the corresponding signals the relative nature but individual values at a particular frequency is no longer existent is a very typical understand there is a typical understanding behind this so probably we need to have some clarification one very simple example which is often used in probability theory probably will have better understanding so suppose I have a P voted rod and I give a random force to this rock and this rod can freely rotate around its axis and depending on that random force it will go and stop somewhere okay let us say where it stops so it says it stops over here that angle is called theta now this theta becomes the random variable okay now if I just ask you that can you tell me the probability that it will stop at angle 30 degree now how many possible angles are there suppose the force that I will be exerting are almost means the way it is being exerted that whichever angle it will stop they are all equally likely so it has an equal probability of stopping of stopping to any angle right so if that is the case every angle is equal probable so they are equally favorable so now I am targeting a particular angle 30 degree okay so how many angles in between are there there are in finite number of angles because 30.1 is the valid angle thirty point zero one is a valid annual zero zero one is validated so there are infinite number of angles so the favorable case for my that thirty degree is only one that it should stop at exactly thirty degrees thirty point zero zero zero zero up to infinity okay but there are infinite number of possible cases so my probability that frequency definition of probability will give me just one upon infinite let zero so stopping at a particular angle is zero okay so it doesn't have any probability of stopping at a particular angle but if I now specify a angle range let us say between zero to ninety degrees it will stop because things are equally likely now let us try to see it has four quadrant so it has almost 360 degree out of that I want to cover this 90 degree right so that must be my probability which is one by four so whenever now I am specifying a range I have a valid probability value whereas if I specify a single value I do not have anything no probability values are recorded same thing happens to this gf whenever we are talking about gf I plot a GF individual value I do not get anything but if I specify a range from frequency f1 to f2 it has some value because it will be the integration over that and I will get some value any function if DN is the way DN is defined if you see DN is the length at a particular frequency okay so if I say from this frequency to that frequency in Fourier series term I will be just summing all DN okay so from that particular frequency to another frequency I will be summing all this DN whereas in this particular representation if I have to sum DN what I have to do the summation becomes integration we have already seen that DN was nothing but that del F into so its area actually del F into G whatever that n del F right so basically and then summing over that n value whichever in value we are targeting so f1 to f2 will have some n value to some n value if I am doing this it's just area and we are summing that it just becomes in a continuous variable that becomes the integration from f1 to f2 I am integrating it so basically what is happening whenever we are saying that an individual value that does not exist because it was Del F tends to zero this area tends to zero so I do not get a value but if I start integrating from a particular frequency range to another frequency range I get some value okay so that's the beauty of it in the next class we will again what's the implication of this thing