[Music] okay so in the last class whatever we were uh uh describing is uh we're trying to show that if we do a simplest form of modulation which we are calling as dsbsc that dsb part is clear probably SC is still not clear that will be clear after some discussion some amount of discussion but that particular modulation which is the simplest of amplitude modulation where just we multiply a signal empty with cos Omega CT so we have shown the frequency domain response of that so basically it'll be just centered around FC and that same criteria will be coming that this will will be MF minus FC this will be MF + FC and of course with half okay so it will be centered around FC so this is the frequency domain representation and the corresponding time domain representation we have understood from this particular thing that it's a it's a carrier whose amplitude is instantaneously varing with respect to the message signal which is a slow varing signal according to our assumption here also that assumption comes into picture because FC has to be much much bigger than b which is the highest frequency component that the message signal has so therefore that will be much slower varying signal and amplitude of that carrier will be slowly varying accordingly so that that means that Mt should be observable in the envelope of the signal that is being transmitted so it's a cosal signal only the envelope will be tressing that Mt okay so what is happening there is a possibility that Mt might be positive or Mt might be negative like over here so this is the cross over point right Beyond this Mt is positive just after this Mt will be negative so what will happen in the carrier so the carrier that cost is going up to this point it is positive immediately after this point it will be negative so in the frequenc means in the phase of the carrier what will happen from positive to negative immediately it jumps so there should be a 180 phase shift immediately right because the amplitude is just just getting means from positive to negative so therefore in the carrier phase there should be 180 phase shift so wherever there will be zero Crossing in this dsbsc in time domain if you just observe the signal you'll always see a 180 phase reversal whichever way whichever phase it cuts over there it'll just be a 180 phase reversal so every Point you'll be getting that that has a huge implication we'll we'll see that that makes the demodulation very uh costly we we we'll come across that and that's where the uh suppressed carrier and non-suppressed carrier comes into picture we will come to those things but we have from the time domain we're just trying to analyze whatever is happening okay so there is a we know as many times there will be this zero crossover that many 180 phase reversal will be happening if this was also having a pattern like this there also we would have observed similar things so it will be like this the envelope will be just like this and again if things were coming out like this there will be a 180 phase reversal immediately okay so as many zero crossover will be happening that many phase reversal will be happening so we have understood what's the modulation right what will be corresponding Dem modulation that means I will be getting receiving this Mt cos Omega CT signal from the air through my antenna now I have to get back my Mt okay so how do I get that very simple operation another multiplication will give me that if you just see Mt cos Omega CT I'm getting if I just multiply it with another cos Omega CT locally generated what will happen so this is if I just take half Mt this is 2 cos Square Omega CT cos Square Omega CT can be written as cos 2 Omega CT right right we can just write this immediately what we get we separate this two out I get half Mt + half Mt cos 2 Omega CT fine this is what I get now just see this signal now you'll appreciate why this FIA transform and all those things were so important why time domain and frequency domain we have to observe the signal let's try to do a FIA transform of this signal of this composite signal let's call this as 5T and then I wish to evaluate 5f what is 5f so I have half over here so that should be half Mt the FIA transform should be MF and this is multiplied by cos 2 Omega C so therefore there should be a frequency shifting property so I can write half and then this particular part should take me to cos f+ FC sorry not cos M f+ FC plus m f minus FC right is this fine okay so I'll get that particular thing but there should be a half also coming out of this one because uh this whenever I multiply by cost there should be another half so this must become 1x4 right sorry that there is 2 Omega C so it should be 2 FC okay fine so if I now just plot this frequency response how it will look like so this is suppose my m f was something like this so this is MF half MF so this is that half MF and at 2 FC there will be 1/4 of m f minus FC and at minus 2 FC there should be 1/4 of M f+ FC fine 2 FC yes fine now it's very simple you can see that just by putting a filter I can extract my signal back I'll be putting a low pass filter over here which has has a cut off frequency which is much lower than this 2 FC but bigger than this B okay if I just employ a filter like that I'll be getting my half MF back which has a for inverse transform which is half Mt so I'll be getting this signal back okay so immediately you can see my demodulated circuit is almost ready what I need to do in Dem modulator whatever modulated signal Mt cos Omega CT I'll be getting I'll do another multiplication with cos Omega CT whatever I get I pass it through a low pass filter even the filter design is also known the cut of frequency must be suppose the cut of frequency f c cut off okay so that must be bigger than b we should not say much much bigger than bigger than b but that must be lesser than this 2fc at least as long as I'm doing that I'll be getting my signal half empty over here because this part will be rejected so very nice we we could realize see this is the importance of signal and from signal to system that is why probably we have devoted so much time in doing frequency uh response fora transform how to see a signal in frequency domain uh so those are the things just giving us tools enough tools to actually manipulate the signal and then understand what kind of systems I should put so that whatever I wish I will be able to achieve that so basically your modulator and demodulator is now ready for this dsbsc of course we still have not characterized what this multiplier circuit will be how do I achieve a multiplication okay but we have now understood that if I know how multiplication has to be done I need a multiplier circuit followed by a lowas filter that will make my uh demodulator I need in the modulator I need just a multiplier circuit nothing else okay so now let us try to see what are the difficulties over here the first difficulty that comes out is generating this one that's a big challenge because if you see very carefully the frequency and of course the phase also we are not writing the phase these two has to completely match over here then only that cost sare to Omega c will be coming out and then only the fua transform will give me very nicely empty later on we will prove that if there is a phase drift or frequency drift and if we multiply these two I have a chance of not getting anything over there okay so it's we have to be very cautious about generating this local coidal that has to be completely in sync with the carrier by which the signal has been originally modulated the problem is that nobody will give me that signal right because if the modulator and the modulator are sitting at the same place what's the point in doing communication because when we wish to do communication we wish to transmit it over a longer distance if I know that same carrier I can give in both places from the same circuit then probably my modulator and Dem modulator are already sitting in the same place or otherwise I have to separately again communicate the carrier as well right to a long distance so just to send my message over a carrier I have to again transmit my carrier along with it that's one difficulty okay the second difficulty is even if I try to send my carrier there is a possibility that this modulated signal and the carrier because they're going through the channel they might go through some frequency and phase drift it might happen due to dler effect due to other effects uh means uh that are there uh due to the channel effect mostly so there will be a drift in Phase as well as frequency of the carrier signal and that will be random as well as the modulating signal so if I wish to really means even if I have that carrier and I'm transmitting along with that there is a possibility that these two are going through different uh phase and frequency DFT and at the end they are not in sync in terms of frequency and phase again if I multiply I'll not get the proper representation so what I have to generally do that this particular signal that I'm getting already I know that it already has of course a contaminated coyal can I extract that carrier out of this if I can do that from there if I can generate the this cost Omega City then I am fine so that's called the carrier recovery there is a means there will be in this course only there will be a few classes devoted towards that that how do we do that carer recovery that's a big circuit again uh it must be locked with the incoming uh frequency and phase uh that's sted as uh phase lock loop we we'll see those circuitry but that's the part which is required otherwise your demodulation will not be good so that's the difficulty we are having having in this particular modulation scheme that we need to have another carrier which is completely in synchronism with the incoming carrier frequency and phase this is the one difficulty that we have that means the receiver design becomes little more complicated because we have to do this carrier recovery on top of this whole thing right so if I now ask when we are do uh designing a system is this desirable for let's say a bro broadcasting kind of uh system okay broadcasting means like the radio transmission uh we had those bharati and all other things uh where we used to uh just uh transmit something from a big antenna okay and that was broadcasted to everybody everybody was listening to uh same voice okay so this was broadcasted and everybody must have their own receiver and must detect it okay in that kind of thing I can actually make my transmitter little costly because that's common to everybody that cost will be shared among all the users whereas there are multiple users who are trying to willing to receive this signal their receiver must not be very costly okay because if the receiver is costly that cost directly will come on the user so in a broadcasting system generally my target should be that the receiver is little bit simplified and the transmitter probably is little more complicated okay why I'm saying all these things this will give me another design Direction where I'll probably take out this difficulty of getting this carrier recovery circuit into it and then multiplying it so this entire stuff I'll take out and I'll employ another modulation scheme which will be just a simplified modified version of this where the receiver will be be uh will be becoming very simple but the transmitter will be a little more complicated we'll show you what kind of complicacy we'll be having in the transmitter probably transmitter will be little less efficient okay but that's pretty obvious whenever we have one to one communication probably this is better because then I cannot make the transmitter very costly because it's one to one communication again if I increase the transmitter cost that will come to the user so there I need to have a balance that transmitter receiver must be almost similarly equivalent complex so there will probably we can employ this particular technique so that is why that short range radio communication people have used SSB uh sorry dsb double side band uh suppress area this particular modulation technique that we are discussing about so whenever we have one to one that short range uh radio communication which is not not broadcasting in nature there we can employ this kind of technique okay now let us try to see that we have talked about this demodulation now let us first for dsbsc okay let us try to see what are the different kind of modulator that we can generate so we have talked about that multiplier let's say there are if you go uh into Market you'll see that there are multiplier chips with differential amp ifier and all those things it's it's a little complicated okay so we can always employ a direct modulation by that technique but there are other techniques and which we'll be discussing other very simplified techniques to do modulation we'll discuss those things the first one is called the nonlinear modulator so in that case we'll be using a nonlinear device okay what do we mean by nonlinear device that if I give a input XT the output will just not be a linear scaled factor of this okay what will happen it will produce some Square term as well it's just a nonlinear device so output will be means showing nonlinearity with respect to the input so suppose my XT is this and output is YT and if I have YT following this relationship a XT plus b x² T that's the simplest nonlinearity we can get that's the quadratic one okay so we can also have other nonlinearity or other higher order like cubic so it will be having some another constant C into X Cube or so on higher polinomial also but we can easily get this quadratic nonlinearity uh and realize this by some devices which we all know like transistor or diode okay so they if you see their characteristics function that has a non linearity because the characteristics function generally goes like this right and if you bias it in certain region you'll see that it will follow quadratic nature okay so this kind of nature so if you give input output will be just uh in a quadratic form with some A and B that will depend on the diode characteristics okay but if we have a nonlinear device which is let's say a diode properly biased so that we get a quadratic uh nonlinearity into it and that's this device and now if we C connect this in this fashion so my two input if you know are Mt and cos Omega CT I need to produce a multiplication term so what I do is something like this I have two adders so these are just uh you can put them as opamp Adder okay just add the signal so this goes over here this comes over here this is actually uh this is a Adder actually sorry and this is a subtractor so it's plus and minus I get x1t over here x2t over here and then I pass it through a nonlinear device of this nature okay again I pass it through a nonlinear device of this nature so I get y1 T and y2t after passing it through this I pass it through another adder or I should call it subtractor and then whatever I get I pass it through a band pass filter centered around plusus Omega C or FC okay and the output I'll be getting will be proving that it is actually 4 B Mt cos Omega CT if I adjust my B to to be 1x4 then it is actually Mt cos Omega C whichever is our Target okay so how this works it's very simple you just if you just do uh those algebraic manipulation so if I have this X1 and x2t after nonlinear device so suppose this is JT what is JT JT is y1t minus y 2 T whereas y1t is actually a x1t + b X1 t² minus y 2 T is a X2 T minus B X2 t² right or I can write as a X1 T minus x2t + b T X1 2 T minus X2 2 T right now what is x1t x1t is Mt plus cos Omega T and what is x2t that is Mt minus cos Omega T CT okay so X1 - x2t will be just Mt will remain right so that should be 2 Mt and this is a + b square - A- be whole Square because X1 is if this is or let's say I should not say a let's say C and D so C + D whole Square minus cus d square that should be four right C into D so therefore it should be 4 into b c into D is Mt into cos Omega CT so Mt into cos Omega CT right now what we are doing now the frequency domain term will come into picture so this Mt if I take it into frequency domain that should be MF and this if I take into frequency domain that should be M plus MF sorry m f plus FC and MF minus FC if I put my band pass filter around FC then it this term should pass through and this will be not going through it so therefore at the end only this term Will Survive which is this right so we can see we can actually devise a multiplier circuit by two nonlinear device and three Adder okay adders are very easy to uh device just take a opamp and you can you can make a Adder right so three app opamp and two nonlinear device properly biased let's say diode properly biased so that we get this quadratic relationship immediately and a band pass filter again B pass filter can be designed uh using opamp and some active filtering okay so that's now you can see that's the multiplication circuit so this is one way of doing multiplication there is another way that's called the switching modulation so we just trying to show you which are the devices that can be employed to do this modulation right so for switching modulation what we'll be doing we know that a particular transistor or simos circuit can work as a switch so in the gate of a particular uh this switching transistor if we just put a signal like this which is having let's say plus 5 volt for half the duration and it's being zero for half the duration so what will happen whenever this is being put in the gate it will be on so it will pass the signal so if I just put uh in the emitter to collector if I just put my message signal so whenever at the gate it is getting plus 5 volt it will be on then that signal will be passing through it and whenever it is off that will not pass through it so if I just uh put a resistor across that if I take the voltage I'll see that it's get switched so basically if my signal is something like this and if I just switch it through this so whenever this part is on the signal will follow rest of the part it will be zero again the signal will follow rest of the part it will be zero so basically what is happening my message signal is getting switched through this pulse okay so almost what we are doing message signal multiplied by this pulse okay so if I represent this as WT and this is Mt I can actually connect this WT in the gate of that switch or transistor and uh in the emitter to collector I can put a resistor across which I'll be taking the voltage and I across the bias this one and uh emitter I can put my empty signal okay so then the the output of the resistor will be this modulated signal right or whether it's modulated or not I don't know it's a switched signal right now basically that output will be just multiplication of these two as I can see if this is I I put this as one then immediately it will be just a multiplication okay so I get my output which we say 5 T is just Mt into Omega T but the Omega T if you carefully see that's a periodic signal so I can do a Fier series analysis this is where you can see all those techniques that we have used will actually be used over here so Omega T I can just expand it in Fier cies so you can uh just do it it's uh represent this one as this which we have already done probably something like this okay so it's if this is overall D this is sorry this is this is T and this is minus t by2 this is T by2 and this is minus t by2 and with this period period it gets repeated okay so this one if you just do for year series analysis you you can represent it as this half so that means the DC part will be half that coefficient next coefficient will be 2 by pi into cos Omega CT the next coefficient 2 Omega CT will not be there it's the 3 omega CT part which will come 1X 3 cos 3 omega CT plus so it gets alternative plus and minus so 1X 5 cos 5 Omega CT and so on okay so basically it has all the odd frequency harmonics and the coefficients will be alternatively plus minus and the corresponding coefficient becomes 2 pi into 1 by whatever the frequency uh whatever the harmonics okay so this is what Omega T is therefore my 5 T will be immediately I can see I multiply this so half Mt + 2 by piun Mt cos Omega C T minus 2 by piun into 1 by 3 Mt cos 3 omega CT and so on okay again do a FIA transform of this because f f I wish to see so if I do a FIA transform I can see there will be some part at base band next Mt cos Omega CT so that should be around FC C next will be around 3 fct now if I just employ a band pass filter around FC properly designed then I'll be just getting this signal all other THS will be neglected so immediately I get my modulator because this is Mt cos Omega CT okay so if I just so basically what I have to do I have to put a switch which has two input one is that WT and the other one is that Mt after that whatever I get that is this ft must be passed through a band pass filter centered around plusus Omega C whatever I get that is actually 2 by pi Mt cos Omega CT right so that's another way of doing multiplication this is called the switching modulator so in the next uh class probably we'll be discussing more about the relative advantage and disadvantage of all this circuitry