 We are talking about binary passband signaling right and as a slightly more general case of binary signaling than the binary PSK we are actually looking at what we can actually call forery scheme although we are under the general title of binary scheme to the moment that is the quadrature phase shift scheme right. It is because we thought that each of the two quadrature carriers when modulated in a BBSK manner in a polar antipodal manner gives rise to the quadrature phase shift scheme that we talked about and this was a picture that I think I showed you at that time this constellation diagram corresponds to QPSK to recapitulate for you we have two quadrature carriers in QPSK each modulating each modulated by possibly one of the two bits that either we can think of these two carriers being independently modulated by two streams, two bit streams are on a block by block basis that is we take two bits at a time and use one of these points as the signal for representation of that those two bits. As far as the modulation is concerned therefore you can think of this being the cosine part of the carrier and this being the sine part of the carrier so you are doing an antipodal modulation on the cosine component as well as on the sine component on the quadrature component right. So depending on what bit sequence you have for example if two bits are 1 1 then we have cosine 2 pi f0 t and you also have sine 2 pi f0 t being simultaneously represented right that is this will be represented by this amplitude of this carrier and this amplitude of the sinusoidal carrier right and the resultant phasor that you will get is this point right. Similarly you have when you have 0 1 your quadrature the in phase carrier is minus cosine 2 pi f0 t and the quadrature carrier is plus sine 2 pi f0 t and the resultant of these two is this phasor this is the net signal that you end up sending and so on for each of the 4 possible phases that you can have right corresponding to the 4 bits that you can have. Now how you allocate these bits to the corresponding waveforms or the corresponding phases is somewhat arbitrary except that it is useful to follow a certain pattern for example if you might notice in this diagram I have got this point representing a 1 1 this point representing a 0 1 a 0 0 and 1 0 do you notice any pattern in this scheme of things in this representation of these 4 points on the constellation by these respective bit pairs the gray code transformation that is as you go from one point on the constellation to its neighboring point on either side of the neighborhood right you only change one bit in the representation that is I am representing this as a 1 1 and this as a 0 1 whereas this 1 has gone through a transition similarly as I go from here to here this represents a 1 0 can you guess a reason as to why this will be useful to do that any suggestion on that why it will be useful can you elaborate how yes that is a good answer but okay it is really very simple now when you are of course we are going to talk about it in detail when we talk about demodulation but roughly you can see that error in this case refers to the event that I might confuse or wrongly decide when I really transmitted this point on the constellation I might wrongly decide that the transmitted point was this or this or this these are the likelihoods when you are making your decision of the receiver in the presence of noise you will make one of these wrong decisions if ever you make a wrong decision right now the most probable wrong decisions will be those which correspond to adjacent symbols or adjacent points in the constellation because they are the ones which one can easily go to due to noise right so therefore when that happens we would like to make its impact as small as possible on the original binary bit pair right and therefore you want to make sure that your neighboring symbols do not differ from each other by more than one bit so if at all there is a simple error at the receiver it gets converted to only a single bit error rather multiple bit errors so that is the advantage of using gray coding representation or gray coding mapping on the constellation diagram so are these points clear now also to further recapitulate for you we had discussed certain disadvantages of the QPSK waveform right and the specific disadvantage was we could easily have an incoming bit pattern which takes us from succeeding bit pairs going from 1 1 to 0 0 or 0 1 to 0 1 0 right in which case we can expect 180 degree phase transitions to take place in the waveform right for example we had first pair as 1 1 and the next pair as 0 0 we are going from this phase to this phase which is 180 degree opposite to the previous phase or similarly I might go from here to here so it is quite possible that at various stages in the waveform you will have 180 degree phase shift transitions which are undesirable particularly when you finally band limit this waveform such 180 degree phase shift variations will cause the envelope of the signal to go through zeros so essentially you will get a non-uniform amplitude or a non-constant envelope waveform right after filtering if you have infinite bandwidth it will not create any problem if you have rectangular pulses but we know that finally we are not going to have rectangular pulses and therefore this 180 degree phase shift transitions really mean that you have to go through an envelope variation which goes through zeros right it becomes zeros in between the amplitude does not remain constant throughout after filtering. Now to offset that effect let us see what we can do let us return to this constellation diagram again in this constellation diagram I had represented this point as a let us say corresponding to an incoming bit pair of 1 1 and how did I generate this incoming bit pair of 1 1 I looked at two successive bits coming in right and then mapped those two successive bits on to this point. Now this is actually a slightly artificial way of doing things if you may notice because I have to wait for two bits look at what the two what the bit pair was and then carry out this mapping I could as well carry out a mapping as and when each bit is coming along and each bit is coming along at let us say specific interval t sub b right and you are doing this mapping as we have talked about at intervals of two t sub b right because you are going to wait for two bit intervals to look at the sequence or this sequence or this sequence and decide which of the constellation points needs to be transmitted. Now what I could do was I need not wait up to the second bit to come along and then decide as and when each bit comes along I let it decide now what the next phase will be right. So as if your choice on the constellation is decided by not the two bits by not looking at the complete pair of first and then deciding but it keeps on making a transition every t b seconds you are still looking at two bits you had you had a previous let us say you had a previous phase here somehow right now at the after t b seconds one of these bits is going to change one is going to remain constant because every t b seconds only one bit will be changing right let us say one of them changes and you come here it has to come here because only one of the bits is changing at a time. So let me first explain then you please ask your doubts that is a this corresponds to previous and this corresponds to new right now at the after t b seconds this will disappear right this will become previous and you will get one new bit. No okay depends on what the new bit is you will either go down or go up now okay right okay absolutely right yes because the new one because the previous one now is one you will come here right and I think I will elaborate further on this point the main point is yeah so either you will stay there or you will go on this side is it clear so either there will be no phase transition or there will be a phase transition of how much you have gone from here to here 90 degrees right similarly if you are here again this zero will move here right and therefore you should be either moving here if this is previous this is new this becomes one here right now depending on whether it is zero or one either will move up no we will not have 180 degrees so if you have zero one I can have 180 degrees no I thought 180 was out of the question that way okay let me return to this point although and then wait for next bit and then move to this one see to start with we have one zero and we are interpreting this to be the previous bit and this to be the new bit right yeah the previous one will disappear and a new one will come which could be either zero or one I think I am making some mistake in explaining it is actually should be quite obvious that you are only changing one bit right and therefore you can only go to one of the neighboring points because you are only changing one of the two bits. I think I will try to explain it in a different way first and then come back to this diagram maybe we will ignore this confusion for a time being yes I think I better finish this explanation otherwise this is going to create trouble I took it for granted that it will be very easy to explain now let us try waveform corresponding to this so let us say the point is that every T b seconds T sub b seconds you are going through it can we can we have some quiet place every T sub b seconds you are going to a bit transition right so let us say we are resolving it into basically what we are doing is we are extending each bit to really to be to an interval which is twice this I think this is the mistake which is which is that constellation diagram is not everything in the case of this particular scheme that I am talking about now we are really extending this to twice this interval right similarly the next one will come along and the transition is here I am sorry it should have been right this bit is extended up to here and the next bit goes up to here so this is 0 this represents this one here and this 0 is represented by this right let me take a fresh page you have an incoming bit stream corresponding to 1 then 0 let us say again 0 right and let us say then 1 and so on right what we are really doing is we are now looking the change is being made I think the reason why we are making we are getting confused there is because the change is being made only by 1 bit every T b seconds whereas the change is retained for a duration of 2 T seconds 2 T b seconds so we have these are your bit intervals what I really do is actually this is the point I should have first explained before I go to the phasor diagram we need to look at 2 bits in an interval of 2 T seconds but the way we are doing that looking is slightly different from what we are doing earlier since we are looking at 2 T seconds we really let this bit extend up to interval of so I am converting this 1 bit stream into 2 bit streams the I bit stream and the Q bit stream right you can think of this process as follows I am converting this into 2 quadrature bit streams one I am calling the I bit stream the other I am calling the Q bit stream right the in each of this bit streams basically I have half the bit rate right I take alternate bits of each let us say the even bits go into the I bit stream and the odd bits can go into the Q bit stream right so the even let us say the first one the 0th one it is 1 is extended over an interval of 2 T B seconds and this is how it is represented the next even bit is this 0 right and this is how it goes right and now let us see what happens to the Q bit stream to start with you have a 0 let us say before that suppose this was 1 I am not sure that so at this point you have a 0 and this 0 will persist up to this point and the next one is a 1 so it will persist up to this point right and now actually this is the important thing what you will see is that the transitions in the 2 bit streams are really offset with respect to each other by a bit interval right they are not occurring together this was a mistake we are making when we are looking at the constellation diagram without regard to the physical processing that we are carrying out right when you regard them as occurring together basically that is what QPSK is you just look at these 2 bits together and then decide a point in the constellation right now what we will do is we look at what is happening at each of these points right so as you can see either in a particular interval of TV seconds whichever you might take the previous one of the bits will remain constant and the one of them is going to a transition right no matter which interval of TV seconds you take right so you can only go from the present bit pair value to one of the neighboring bit pair values is it clear now and therefore you can only move from any point in this constellation to one of the neighboring point in the constellation right not clear still suppose you have one right this one of this ice streams is going to remain at one now this one may change or may not change depending on what you had here right but no matter how it happens whether it changes or not only one of the bits one of the bits is 2 bits is changing right if only one of the 2 bits is changing and you have a gray code mapping you can only go to a neighboring point because of gray code mapping right is it clear because you had a gray code mapping and only one of the 2 bits is changing after TV seconds so every TV seconds you are moving from any point in the constellation to one of the neighboring points that is the whole point right so I think the explanation that we are giving earlier was faulty because it was not taking into account this particular mechanism by which we were creating the phase transitions is it clear so therefore this kind of quadrature phase shifting it is still quadrature because we will still have at any time 4 possible phases right but you are only going to go through phase transitions of 90 degrees by a mechanism of offsetting of the bit streams in the INQ streams the bit patterns in the INQ streams are offset with respect to each other by a bit interval or by half the symbol interval if you call 2 T serve be a symbol interval then you are offsetting these 2 bit streams by half the symbol interval if you can address your question this way please I think others will also benefit whatever doubts you might have we have the punker speak out is it clear anybody is it fine is it really clear you can speak out if it is not clear I can explain again it does not really matter if you slope process a little bit because it is a fairly important point to understand okay what is really important is to realize that Q bit stream is based on this I mean this 0 here corresponds to this 0 here right the next 0 here will come after sorry this 0 here is here right this one here comes here so every bit is therefore being extended to twice its duration because you are looking at a new bit every TB seconds but the processing is done on a symbol basis the symbol is 2 bits that is right you can think of the gray code basically remember each of the quadrature phase quadrature carriers is being is modulating is being modulated by you can think of as the INQ bit streams right so it is either being modulated by I stream which is the which is modulating the cosine carrier and the Q stream is modulating the sinusoidal carrier right so depending on where you are on this and where you are on this you will be on one of these four, so we are transmitting the signal every 2 TB seconds but no we are transmitting the signal continuously but now the transitions will occur every TB seconds in the QPSK the transitions will occur in every 2 TB seconds so what we have really done is we have traded off the points of transition which were earlier occurring at a slower rate right by the magnitude of transitions which but the transition themselves are now occurring more frequently every TB seconds in this manner right earlier we are looking at these two things together as a bit pattern 1 0 and transmitting this phase then we are waiting for 2 TB seconds looking at the next bit pattern 0 1 then transmitting this bit phase right now I will go through many more transitions I will go through a transition here another transition here and so on and so forth. In the earlier case the error detection would have been not there if we were talking to the same thing. No the error detection there is no error detection even now the pre-coding advantage will still persist. See if we have talked with 2 bits at a time then there will be 2 packets coded at one bit. Not as one bit as one symbol. One symbol. Right. Yes. So then the concept of having a chain from only from one point to other doesn't remain. So you can see that if you can go only half a sheet of 19 not 180 between that. That is the point. It is not the question of error correction it is the fact that earlier in our earlier scheme of things we could have 180 degree phase shifts or transitions. In our new scheme of things we could only have 90 degree phase shift or phase transitions but the transitions will now occur every bit interval basis rather than every pair of bit interval basis right. This scheme of QPS case called offset QPS in which the incoming bit stream is first split into 2 streams I and Q and then each of them modulates one of the 2 quadrature carriers and then they are added together right. So that is offset quadrature phase. The 2 pulses in the 2 stream 2 bit patterns are offset with respect to each other by an interval of 1 bit interval or half the symbol interval right. So I hope this is now sufficiently clear even though I have had to rather experience rather painfully it was actually a very simple thing. Let's therefore talk about OQPSK mathematically now. If you have physically understood what is happening here let's go through a mathematical characterization of the offset QPSK. So basically what we have seen is that offset QPSK delays the quadrature waveform by half a pulse width okay and where I am defining the pulse width as that corresponding to a pair of pulses a pair of bits and then result is that the phase changes can you see this colour alright? That is not writing very smoothly let me try this I think I will try green okay I will change that colour no problem. Phase changes this you can see clearly only by 90 degrees at every bit interval so mathematically we can say something like this if you want to write my modulated waveform you can think of this as actually a sum of 2 modulated waveforms in which the I stream modulates that is the in phase carrier and the Q stream modulates the quadrature phase carrier right that is one we are looking at it or the even bits modulate one of the carrier and the odd bits represent another carrier and the even and odd bits have pulse shapes associated with each other which are essentially offset with respect to each other by half the bit interval half the pulse width or half the simple interval not bit interval half the bit interval. So you can think of this even bits being represented by A sub 2 L and associated pulse shape being S t occurring at 2 L T not 2 L T because you are having a new pulse every T seconds okay let me write you are absolutely right but this T here this T is 2 T sub okay plus J times I am taking the odd bits here which are going to represent by A 2 L plus 1 into a pulse shape S t minus L T minus T by 2 okay so this pulse is offset by this pulse T by 2 where T by 2 is equal to T sub B right so this is one way of representing mathematically the modulation process of offset QPSK right. Sir in the previous phase you said that the phase changes by 90 degrees at every bit interval. It may change I am sorry if I convey that impression it is wrong it may change in fact whether the phase changes or not will depend on the bit pattern that is coming along right but the magnitude by which it may change is at the most 90 degrees right where A sub L is going to be either plus A or minus A depending on whether the Lth data bit is 1 or it is 0 right and T is here the duration of a pair of bits I have already given that up over there duration of a pair of bits alternatively as one of you wanted to write it in terms of T B you can do that T sub B and then this will become minus 2 L T sub B and so on. Sure remember that the duration of each of these STs is how much this pulse shape has a duration of 2 T B equal to T right but a new pulse is coming and interfering with the previous pulse every T B seconds right and deciding on the new phase right we are going from the previous phase to a new phase every T B seconds right. So therefore is it alright we can remove this now we can also alternatively write this modulation process as follows I can write C T is equal to incidentally these are all baseband expressions I am writing as far as the passband expression is concerned you know how to go from a complex baseband representation to the corresponding passband representation you multiply this with e to the power j omega ct and take the real part of that right that will give you this modulating the cosine component and this modulating the sine component that is standard. So I am only writing the baseband complex envelope representations yeah coming back to this alternative representation we can also write this as B L S T minus L T right where if you were to see you can think of this as a complex number right this B L will be now what no this is the point very good that is the point I have expected you to respond with B L will be purely here for even bits or even L yes right and the value will be equal to A sub L I will be purely imaginary for odd L and the value will be again let us say j times A sub L right basically that is it that same i qubit stream concepts right at even times you are only changing the real part of the carrier right changing that to either plus amplitude to minus amplitude or vice versa right at odd bit intervals when the Q stream becomes active you are changing the quadrature carrier that is a physical picture behind this statement. So any questions about the OQPSK representation so one thing you will therefore appreciate is that unlike QPSK where I just had to specify a constellation diagram that was a complete specification of QPSK right in the case of OQPSK just giving the constellation diagram is not enough right I know these are the four phases but I no longer can just say that that how these four phases are going to be invoked there is no immediate way by which I can associate with the incoming bit stream the phase sequence that will finally come out of the modulator right in the OQPSK case right. So yes please. Sir in this upper one we are adding the two at C at a particular instant was some real class imaginary. Yes. Here it is either real or imaginary how can you equate the two? Sir some of those. This is another way of looking at what is being done right that is all there are two ways to explain that again let me go back to these diagrams right every bit in every even bit intervals let us say this is an even bit right nothing is going to happen to I mean the new pulse that is coming along is this one right the previous one stays this expression does not say that the previous one has disappeared the one which was active at the previous bit interval is still there in the sum and to that we are adding a new pulse right corresponding to this. So this previous one this one is continuing but to that we are adding a purely real component here you are adding a purely imaginary component in the next bit interval while the previous one is still there right. So at any point the number is still complex it has to be because you are going to have to what? Sir how did you say that BL is purely real? Sir you are written down there BL is purely real and purely real. That is true. It is true isn't it? It is the STs which are overlapping right every TV seconds there is a new ST coming in while the previous ST is still active right so at any stage therefore you are getting contributions from two of these STs right the previous one and the present one think about it a bit carefully this is something that you can easily understand right okay. Now the point that I was making was that because of the time offset OQPSK not completely characterized by a signal constellation you require something more than that for its characterization not completely specified or characterized by a constellation diagram. You need to specify rules which need to be imposed to constrain the phase transitions right because certain rules are being followed right and that is how you are going from one point in the constellation to the any other point in the constellation and these transitions are now governed by certain rules in the QPSK there were no rules at random you could go from one point in the constellation to another point in the constellation but now we can we have to follow certain rules that and this is best explained by viewing the modulation process of OQPSK as if you are working with a finite state machine with which you are familiar with your in your sequential logic and in the digital modulation context it is conventional to represent this finite state machine action by means of a diagram called the trellis diagram. So I am going to introduce to you what is a trellis diagram the concept of a trellis diagram this is a trellis diagram for OQPSK modulation. Now first what is a trellis diagram? Basically the trellis diagram consists of a set of nodes which represent the different possible states in which your machine can be right our constellation diagram has 4 points so we can be in one of these 4 states corresponding to 4 points in the constellation and you know the states are 1 plus j 1 minus j this 1 plus j 1 minus j refers to your constellation diagram right you can think of this point as 1 plus j this has 1 minus j this is minus 1 plus j and this is minus 1 minus j right. So these are the 4 possible states so you start with having these nodes corresponding to these 4 possible states. Now these nodes are replicated every bit interval right along time so you have a set of 4 nodes at this time a set of 4 nodes again after 1 time interval and so on and so forth right. Now so therefore at each time you can be in one of these 4 nodes right and as you step through the trellis you are really going through a particular bit sequence and therefore a particular phase particular phase pattern on the constellation diagram. Now let us say you are you are not let us discuss the OQPSK trellis diagram let us say you are in state 1 plus j right now where can you go let us say these are the even bit times and these are the odd bit times at even bit times what is going to happen you are only going to change the imaginary components because your real component is going to stay the same is it clear at even bit times the real component is going to stay the same come back to this diagram right. So what is really changing is depends on what you are calling even or odd right. It is a matter of definition it really does not matter okay it is a matter of definition it does not really matter I have taken it to be that way right. If we take this to be these alternate intervals to be even and odd it does not matter how you start right. So here I have only allowed a change into the change of the imaginary components so 1 plus j can at best become 1 minus j or it can stay as 1 plus j right. So depending on let us say whether incoming bit stream is incoming new bit value is 1 or 0 so 1 I have indicated but the transition that will take place these connections show the transitions right in from the first set of nodes to the next set of nodes and so on. So as a new bit is 1 you go to 1 plus j if it is 0 you can go to 1 minus j right similarly if you are initially in 1 minus j if it is 0 you stay there if it is 1 you change to 1 plus j right and so on. So basically you will see that in this interval you will have these are the various possible paths through which you can go depending on where you initially are. If you are in this node you can either go here or here if you are on this node you can either go here or here, if you are here you can either go here or here, here this way or that way right. These are the only ways in which you can progress from time this time to the next time in next bit time. At the next bit time which you can regard as odd bit times right you will only be changing the real part of the component right. So you can go from 1 plus j to minus 1 plus j or stay there right. This 1 may change to either plus 1 remain plus 1 or it may change to minus 1. Similarly 1 minus j can stay at 1 minus j or become minus 1 minus j right. So these are the possible transitions you can have at this time. Again you can progress like this. So the sequence of states through which you go will of course depend on your initial state and the specific bit pattern that comes along later. So as a particular bit pattern comes along you will be following through a particular path in this trellis right. So as you trace a path through the trellis you are specifying a specific bit pattern. So a complete specification of a modulation scheme like OQPSK requires you to specify it by means of a trellis diagram like this right. It is not only the fact that you have 4 phases that is 4 points in the constellation space but also the mechanism by which this phase transitions occur is best depicted in this kind of a diagram. You will notice we will see later that this kind of a diagram is useful in depicting many kinds of modulation schemes which have memory in it right where the next phase depends on what the previous phases were. There is a particular phase trajectory that you follow and therefore there is a particular path through which you have to trace the trellis. So it is the concept of a trellis diagram clear and what it can do for us. So basically you are specifying a sequence of complex numbers right. Suppose you are following this path specifying a sequence of complex numbers and what is the sequence of complex numbers specifying for us come on. The phase the successive phases of the carrier that will be transmitted right. So you start from suppose you start from here and let us say you go here and then you go here but you know that these phases now you cannot possibly go from here to let us say here. This is the constraint imposed by our modulation scheme right. So if you are here then we can either go here or here that is all. If you are here we can either go here or here right. So anyway once we are given a pattern we can follow a specific path and predict the sequence of complex numbers which will be finally used to modulate your carrier. So that is the concept of a trellis diagram. Incidentally even if you are not able to draw this here it is available in books and maybe I will zerox some portion of this and give you this material. If you are able to draw it it is fine. Let me tell you that there is going to be this is not the last we have heard of OQPSK. A small variation of OQPSK can give rise to further improvement in properties. What is the improvement in properties we have got in OQPSK as compared to QPSK? Let us quickly just capture on to this point. The improvement is in the waveform that will result. The phase transitions will be of smaller magnitude plus minus 90 degrees rather than plus minus 180 degrees right. Now still we will still have the trouble when we filter this waveform because we are still having phase transitions right and therefore we are going to go through variations in envelope magnitude. It is not going to stay constant after you band limit this waveform right because when you do band limitation sharp phase transitions whether they are of plus minus 180 degree kind or plus minus 90 degree kind are not going to be admissible. They are going to be smoothened out and therefore amplitude variations are going to come in. And remember amplitude variations are not good if they are nonlinearities in your system which are typically the case in channels like the satellite communication channels. There are going to be a lot of nonlinearities you have to work with. For example the TWT amplifiers travelling rate to amplifiers that you use at the satellite end are essentially nonlinear devices. If you operate at a particular amplitude you work fine but if you allow amplitude variation to take place you no longer show that your gain will be going to be a linear function of the amplitude right. So it is best to keep the gain and I mean you have to work at a small neighbourhood in the operating region. Amplitude variations not permissible. Also these amplitude variations are not good from the point of view of transmitter efficiency. We talked about that last time. So there is still need for looking for other variations of OQPSK which will not have any amplitude variations whatsoever. Fortunately it is possible to do so and we will discuss one such scheme next time. It is called minimum shift keying and we will think of this in two different ways. One in which it is regarded as an extension of OQPSK and in the other in which it is regarded as special case of FSK. So it is a very interesting modulation scheme. There are two different ways of looking at it. I think I will stop here and we will meet tomorrow.