 to talk about digital modulation schemes right and we have described the process of digital modulation as one of mapping a message sequence into a set of a sequence of waveforms right. This mapping essentially is carried out so that we can eventually communicate on a waveform channel right and we had made a review of what kind of constraints are present on a waveform channel namely power constraints and bandwidth constraints and how to design waveforms keeping these constraints in mind. Today we will basically last time we discussed the case of binary modulations and binary baseband modulations to be more specific and today we will extend our talk on baseband modulations to consider the case of emery situations that is emery digital modulations in the baseband situation. Now emery modulations, emery baseband modulations or baseband waveforms for emery modulations are what we are going to discuss today and as before we can do this discussion we can carry out this discussion in the context of either bandwidth constraint channels or in the context of power constraint channels. At the moment we are going to look at waveforms which in which there is no bandwidth constraint there may be a power constraint but there is no bandwidth constraint and for this kind of waveform channels primarily as I was telling you last time there are two basic kinds of emery alphabets that we can use right and these are orthogonal and simplex. Let me first take the case of emery orthogonal signal here we use a set of M waveforms so we use a set of this M is the same symbol that we use for the emery here a set of M waveforms let us call them S sub m t where small m takes the values let us say between 0 to M minus 1 having the following properties first they all have the same energy E sub p right they all have the same energy that is if you were to compute the energy of each of these signals this E sub p referring to the fact that this is the energy of the pulse p corresponding to the waveform S m t which is S sub m square d t more precisely we can write some amplitude A square and we can take the mod square if you are talking of complex waveforms right. So A square S m t whole square between 0 to infinity or minus infinity to infinity whatever you like to use. So they have the same energy and secondly they are orthogonal to each other in the same way that we discussed for the binary case. So the second property that they satisfy is that if you take any two of these waveforms let us say S m t and S m t conjugate multiply them integrate the result is 0, yes that is important for M not equal to N because for M equal to N this will reduce to the energy integral right. The value of this constant A that you have got here in this expression this can be chosen as per our convenience sometimes you choose A equal to 1 sometimes you can choose A equal to square root of E p so that this basic signal S m t is having unit energy right. So depending on the convenience we can choose A to be either 1 or square root of E p when you choose A equal to 1 that implies that energy in S m t is equal to E p when you choose A equal to square root of E p that means you are considering a normalized version of S m t whose energy itself is unity. So that is a matter of convenience whatever you like to choose you can choose. What about the value of M? For convenience of mapping sequences into waveforms and typically we are going to work with binary sequences right which we are going to map into waveforms it is preferred to choose M to be a power of 2 so M is equal to 2 to the power k which permits convenient mapping alright because we can take a sequence of k pulses k bits coming in and decide on depending on the sequence one of these M waveforms for mapping for transmission right. Now although I have talked about orthogonality in this sense usually and I have taken this time limits to be infinite usually each of these signals will be time limited right. So if for example we have a strictly time limited set of pulses then we can talk of in fact let us ignore that fact for the time being but what is more important is we are transmitting these waveforms from one set of bits to another set of bits, one set of k bits will be mapped onto this waveform then the next set then the next set and so on. There is another condition of orthogonality which if satisfied with the waveforms is really helpful at the receiver and that is S M T is not only orthogonal to S N T for M not equal to N but also to S N conjugate T minus L T and now it is for all values of M and N right including M equal to N that is we would prefer the signal set that we have selected to be orthogonal not only in this sense but also in this sense where we are looking at the correlation essentially you can think of this integral as some kind of a correlation between the two waveforms, the correlation between S M T and any translated signal from the set where the amount of translation is a multiple of the symbol duration the waveform duration L cannot be 0, L not equal to 0 right because that is already here that condition is already here for L equal to 0 we already, so when you put L equal to 0 then M and N have to be different that is why I have put that condition separately and this condition separately right for L equal to 0 we cannot allow M equal to N that is the only difference otherwise yes right they are the same. So this only says that the signal in the set is orthogonal to every other signal in the set with and without translation and the amount of translation we are talking about is symbol duration or the waveform duration right of course it is better to call symbol duration as long as I am maintaining the limits to infinity here but it is obvious that if each of these waveforms is itself limited to T seconds then the second condition will be always satisfied without any problem right if not then we have to make sure that it is through this integral. So is this last point clear the third condition that I have talked about here will be always satisfied for a set of waveforms in which every waveform has a duration of T seconds because the moment you translate it by multiple of T seconds there will be no overlap between the original waveform and the translated waveform and therefore the product will be 0 and the integer will be automatically 0. So this is a very useful and important property just like this is at the receiver remember this condition is important so as to distinguish between different kinds of signals that you may like to transmit corresponding to different message sequences. This condition is important again from the point of view of inter-simple antifluorance and things like that that is signal transmitted in one interval does not interfere with that transmitted in the subsequent to right in some sense and the sense that you are looking here is that of correlation that it has no correlation with signal transmitted in another symbol duration okay. So the mapping let me although we have discussed the mapping sufficiently but just to complete the discussion the mapping that we are going to do is such that we will take the Mth symbol call it M we have M possible symbols capital M symbol 0 to M minus 1 M denotes the Mth such symbol this will be mapped on to the waveform SMT which in turn will be transmitted on to a waveform channel right and the way this will be done is that your bit stream that is coming in incoming bit stream because usually most sources will be binary in nature right will be broken up into K bit blocks right so broken up into K bit blocks or which you can call K bit bytes or K bit words or whatever you like to call them and the Lth word that is during the Lth time interval defines a specific number M sub L right which implies that you will have to transmit the waveform corresponding to the index M sub L right the symbol M sub L which will be S sub MLT and your overall transmitted waveform if we call that C of t will be the sum of these waveforms from symbol to symbol for all L right the sum of all these pulses is what you finally transmit of course these pulses are all mutually displaced with respect to each other by symbol relation remember I defined K a few minutes ago this K is related by M equal to 2 to the power K right so we take an incoming bit stream that is coming in coming along let us say we select K equal to 3 so we select the symbol value here, here and here look at this this will define a specific symbol by which you are going to denote this sequence right call it M sub L for example in this case you may denote use a symbol 0 to 7 right for a 3 bit block and then depending on what this value is you will choose a specific waveform for transmission and the final transmitted waveform will be a sum of all these pulses coming one after another right and of course this capital T here will correspond to the duration over which these 3 symbols have been accumulated or this K symbols have been accumulated because from one set of K symbols you have to go to the next set of K symbols right that is the way the mapping will be done so instead of each waveform carrying binary information it is carrying information about a block of bits rather than a single bit that is the essential difference between M-ary modulation, modulations and binary modulation right and there are advantages and disadvantages of doing things like that I am sure you can think of some of those yourself at the demodulator which we will discuss in detail separately but broadly what the demodulator will now have to do because you have talked about binary demodulators at least very crudely earlier that is very crudely talk about the M-ary demodulators it will have to somehow produce an estimate ML hat of ML from the received waveform which may be noisy so from the received noisy waveform if somehow we produce an estimate which is different from what was actually transmitted you have committed an error this indicates that you have committed a mistake that is you are now going to decode your bit sequence wrongly fine this is the basic concept of M-ary modulation clear and M-ary orthogonal modulation let me illustrate by means of a diagram okay that is something we will be talking about in detail when we come to demodulators right just like in the binary case we have to produce a decision that whether a 1 or a 0 was transmitted here that I am only indicating what the demodulator functionally has to do it has to produce functionally an estimate of M sub L right how it is to be done is something we will take up when we talk about demodulation in general. Let me give a few examples of orthogonal waveforms I have taken the value M equal to 4 for these examples for illustration here is a set of 4 waveforms which you may call S O T, S sub 1 T, S sub 2 T, S sub 3 T right which you may use for the situation that M equals 4 so essentially you see that over the simple duration between 0 to T I have essentially a sine wave of a different frequency going in and in the same way that we discussed the binary FSK you can think of this as a 4 array FSK case all that is needed is that each of these sine waves has a frequency which is an integer multiple of 1 by T 1 by 2 T right now I think at this point I like to go back to the discussion that we had this last time about orthogonal FSK kind of waveforms some of you have expressed the doubt that there will be DC standing on the wave on the transmitted waveform right for the examples that we selected it looked as if there will be DC on the wave DC on the channel well that is only that was only an example it was not to be taken that seriously that just showed an example of orthogonal set of binary waveforms right for example in this case you will see there is no DC because each waveform is balanced in terms of positive and negative cycles you could have selected even for those examples a waveform set like that right so that was just an example to indicate that those are the kinds of waveforms which can be called to be orthogonal right but if DC is a problem which you need to avoid you can choose these waveforms more carefully right so that was just to they will be there will be other considerations typically of course here we are talking about baseband if you are talking about passband even half a cycle of mismatch in terms of DC will not make that much difference and actually speaking orthogonal waveforms really are useful when they are used with large values of M not with small values of M and when you are using them with large values of M these considerations become very secondary okay so anyway I thought since that point came up for discussion last time which I could not satisfactorily address I should at least point this out to you at this point so this is one example of orthogonal signaling here is another example of orthogonal signaling very simple trivial kind of example so here is your S O T which is the waveform is to be regarded of duration T up to here so it starts here and ends here right again S 1 T starts here and ends here right S 2 T goes like that fine and S 3 T goes like that oops sorry this is 0 this is T so you can see that the waveforms are all containing a pulse over a different portion of time non-overlapping portions of time and for this reason you can possibly call this waveform a 4 array pulse position modulation right where the position of the pulse decides the nature of the waveform okay yet another example of orthogonal set of waveforms is a set of pulses like this right as you can see if you want to multiply them out and integrate the result they will be the integral will turn out to be 0 any pair of them right and this is called 4 array code shift key C standing for code okay so basically we have a different code for representing each waveform different binary code right so these examples are sufficient to show that one can construct a fairly large variety of orthogonal set of signals right now you want me to display that for a little longer all right please say so if there is any problem of that kind and at any stage okay one typically uses whenever one uses orthogonal set of signals one typically uses very large values of M right you in fact one can construct larger families of Amary orthogonal pulses in the same way it will not be surprising if you come across systems which use values as large as M equal to 32 or M equal to 64 okay they are in common use another point to note is and this point we discussed also in the context of binary orthogonal signaling loosely speaking motivated by the fact that FSK is a very important member of this family this family this class of signals are called collectively also loosely known by the name of Amary FSK so generally we may also refer to them as Amary FSK or Amary FSK type signals right or some sometimes simply refer to as MFSK right Amary FSK is briefly sometime denoted by MFSK let us talk about the energy budget here we may have so many different possible ways of constructing Amary signals we can choose different values of M and come up with a different modulation scheme right for the same situation now how do I compare all of them when I compare them in terms of energy obviously when I choose a different value of M the total energy that is being used to transmit that signal is representing different number of bits right because the moment I change the value of M the number of bits corresponding to that also changes so it is more useful therefore to talk about not just the total transmitted energy in the Amary case but the transmitted energy per bit right so if E p if remember each pulse here carries an energy E sub p right this energy E sub p is actually used for a symbol of length k in our binary to Amary mapping right so it is where k is so it is distributed over this energy is distributed over k bits where k is equal to log M to the base 2 so therefore the energy per bit I am sorry energy per bit will be how much E sub p upon k or E sub p upon log 2 M it is this energy which is important when you are comparing different modulation schemes or different Amary modulation schemes for that matter right so suppose you were to ask or you are interested in asking a question what happens when I go from M is equal to 2 to the power k to 2 to the power k plus 1 well you look at the performance and look at the corresponding energy per bit and then you can make a meaningful comparison okay so E b E sub b which is the energy consumed per bit will be equal to E sub p upon k or E sub p upon log M to the base 2 so this serves as a common basis of reference let us say for performance comparison of different modulations different Amary modulation schemes right different values of M so there is one important family of Amary signals which one can use and are very commonly used now the next family that I will consider we start with a different colour just to put some variety or simplex signals and the motivation one can derive is from the fact that perhaps you may feel that orthogonal signals may not be the best class of signals from some point of view intuitively you may feel like that at least from your binary experience you may feel like that because your binary experience that you conditionally have is that of on of king versus polar king right on of signalling is an example of orthogonal signalling that you are familiar with polar signalling is an example of anti-polar signalling that you are familiar with right and you have a reasonable appreciation even though we have not gone into detailed performance comparison so far because we have not looked at that we have not looked at even optimum demodulation at the moment so we cannot really talk about performance comparison but you have an intuitive appreciation of the fact that anti-polar signalling gives you better performance than polar signalling then on of our orthogonal signalling at least for the binary case right so a useful question to ask therefore in the Emory context is can we construct generalization of anti-polar class of signals which constructed from the binary is constructed in some way which have similar properties as that of anti-polar. Now what is the essential property that distinguishes anti-polar and orthogonal signals in the binary case in the context of correlation right now we are using correlation as a measure of similarity or dissimilarity of waveforms that we use right isn't it in the orthogonal case the similarity or dissimilarity is measured by finding out whether or not the correlation between various waveforms is 0 or not right in the anti-polar case what is the similarity or dissimilarity measure the correlation is negative in fact negative right we like to go from two signal sets which are not only zero which totally have zero correlation in fact they have less than zero correlation they have negative correlation right that is if it is p t then otherwise minus p t and in general in negative correlation between signals of a signal set in Emory schemes is a more desirable property than a zero correlation right that is something that will become clearer and clearer as we go along particularly for demodulation because that in the basically means that various signals in the set have a larger distance in some sense with respect to each other than in the case of orthogonal signals. So basically simplex signals are motivated from that kind of consideration okay so let us see how they are defined actually one can construct a family of so called simplex signals from any given family of orthogonal signals okay so any orthogonal family of impulse is let us say each of energy is p can be used to construct a simplex family right the way it is done is as follows from each signal SMT in the original orthogonal family I subtract the average value of all other pulses in fact of all the pulses including this call this let us say Q MT okay so from SMT for any value of M between 0 to capital M minus 1 I am subtracting the average value of the signal right average value of the signal set and generating a new waveform which I am calling Q sub MT so I get a new set of waveforms Q sub MT where M goes from 0 to M minus capital M minus 1 right this signals Q MT is formed the so called simplex family and I think the special case of binary becomes obvious that it will lead to when you choose M equal to 2 it will lead to anti-portal signals right I mean that is quite obvious for M equal to 2 it will lead to anti-portal signal for binary case it will lead to M is that obvious so this is you know the basic definition of anti-portal like basic definition of anti-portal is very simple what was the definition we talked about minus pd and plus pd minus pd and plus pd just precisely what you are going to get in the binary case of course we cannot talk about that concept in the amary case is it that is why we have to talk about a different concept which you are calling simplex but binary case becomes a special case of this all right isn't it suppose I start with the set 0 and pt which is the on-off set right that will lead to after this procedure minus pt by 2 and pt right so that will lead to a anti-portal sort of set so now let us talk about the properties of the simplex family first of all if you are to calculate the energy of each pulse that results that is if you are to do the exercise of calculating the integral of mod QMT or QMT square right you expect it to be the same no your infar is slight shock it will not be same I like you to do that it is a bit of very simple computation so please do that yourself I am not going to spend time on doing that it is very simple algebra and what you will find is it is 1 minus 1 upon M into e sub p so the energy carried by each signal in the simplex family is smaller than the corresponding pulse energy of the orthogonal family from which the simplex family has been constructed okay so that is the first thing that they have less energy than the orthogonal family right second important point to notice is that simplex pulses are not orthogonal right so simplex pulses are not orthogonal in fact they have the more desirable property of negative correlation right can you remove this that is a stronger we can say that they contain a or they have a stronger property stronger in the sense that they are more desirable in this kind of applications of negative correlation for example if you choose two different values of M M and M prime let us say and they are not equal the correlation is given by Q M t Q M Q sub M prime t of course prime should come here and conjugate should come here integral between minus infinity to infinity will be equal to minus e sub q upon M or minus can I write it in terms of e sub p what will it be okay I think just leave it like that just check whether this is correct or not I am having a small doubt about it for the binary case it is okay because because that will give rise to 2 right e q by 2 that is correct okay in any case please verify this now why we call it a stronger property is I already mentioned let me put it in writing over here we study later this we will study this point later but I will just like to mention this property here that you will find that the simplex set gives the same error probability as the orthogonal set from which it has been derived right so what is the advantage advantages it is giving the same performance with a smaller energy right yes point here is that the energy transmitted might be less but fact that we are constructing it from the orthogonal signal that much energy is being used would not transmitting that energy what is important is how the energy is used on the channel right how much energy is actually how much power is a pump onto the channel right the construction is a very trivial process as a transmitter what I am trying to say is that some this encoding seen SMT minus summation of all this by that encoding we are bringing down the energy level but of each pass yes but the energy that we are giving to the system channel included is still the same because we are giving SMT that is being first generation mechanism is a local mechanism of your circuits of the transmitter right I mean you may do the whole exercise at a very low power level what is actually important is how much energy finally pumped into the channel right so in fact this construction mechanism is only artificial in that sense that does not define how much energy is actually being put on the channel that will be finally decided by the power amplifier you have right okay the point that I wanted to mention which I have mentioned to you simplex set gives the same error probability as the orthogonal set with smaller energy so it is more energy efficient right and something that is more energy efficient is more useful at least in a power constrained channel right that is the meaning of power constraint that we are short of energy and we like to make very efficient use of energy that we might have at our disposal like in satellite communication and now this is something that I will just mention to you as an interesting thought and maybe some of you can take that up for your own personal research there is a very long standing conjecture although nobody has so far been able to prove this or disprove it for that matter which is a conjecture because it is neither has been proved nor a counter example found so far of all the amary pulse alphabets of given energy is sub p the there is no others alphabets which can give you smaller probability of error than the simplex set okay let me mention it of all the amary pulse alphabets of energy is sub p or is sub q whatever none has smaller p sub e than simplex of course you have to specify the conditions on the normal kind of channels we do this analysis for which is the additive white Gaussian noise channel that is when you encounter white Gaussian noise on the channel then simplex signal set amary simplex signal set is the best and you already know that for the binary case right we have discussed that for the binary case this is the corresponding result for the amary case but it is an interesting point that I have made here for you that maybe some of you can try to prove or disprove it okay so what we have done so far is the amary orthogonal family and the amary simplex family which can be constructed or expressed in terms of the amary orthogonal family. Now there is one more family of amary signals if one can derive from the orthogonal family and is simply called the bi-orthogonal family it is very simple you take any set of you start with any set of m by 2 orthogonal pulses and we can now construct a set of m so called bi-ortho normal pulses which is simply the m by 2 orthonormal pulses that we started with you include those in the set along with this is I am just saying union with the negative of each of these pulses okay so for example if you start with s sub 0 t to s m by 2 minus 1 t right then just add the negative of each of these to the set you have a family of m signals which are called bi-orthogonal okay you can think of an n dimensional space right in which each orthogonal axis of the space orthogonal basis function of the space is used to represent one different signal and then you are also taking the signals corresponding to negatives of each of these basis functions right that is a bi-orthogonal family for example in the signal space representation which we will be taking up separately later suppose these are these represent two orthogonal sets orthogonal signals in a binary orthogonal set for example this may represent a signal p 1 t or let us say s o t and this represents s 1 t right they are just being used as basis functions for an abstract space of this kind then a bi-orthogonal family will be simply obtained by using this along with this and using this along with this right so what you will notice is that the dimensionality of the space will not increase we will be working in the same space m by 2 dimensional space but the number of signals that we are going to use is larger right right. Now some other points of interest in the context of emery orthogonal and other emery schemes that we have discussed so far we have talked about energy calculations and how we will like to compare the energies whenever required we initially mentioned that these are signals that can be used when bandwidth is not a limitation right I like you to appreciate that fact better here. Let us talk about what happens to bandwidth as m increases in this class of signals if you look at typical examples that I have given you you will get a feel for that right for example look at the Fourier FSK example that I gave you and so on as I increase the value of m you will have to put more and more such signals and it may appear intuitively that necessarily the bandwidth has to go up right increase in the value of m that is putting using larger and larger chunks of k bits to map into waveforms is necessarily associated with increase in bandwidth so all the emery schemes we have discussed so far have this problem that is bandwidth literally grows with m and there is no attempt on our part to constrain the bandwidth right all we are interested in is that the orthogonality condition must be satisfied or the simplex condition must be satisfied and so on we are not even explicitly or consciously try to do anything about the bandwidth right yes we will have to see with the bandwidth efficiencies but the overall bandwidth is going to go up right as the value of m is going to increase we have to still see how bandwidth grows with respect to increase in number of bits that we are simultaneously carrying out no it will depend on the kind of signals that we use you cannot make a very general statement about that all we can say is it will grow in some sense but the bit rate is only increasing log 2 m times whereas the bandwidth will be increasing much more than much faster than that right because you are having a set of m signals necessarily your bandwidth is going to go perhaps at least linearly it is not more right so therefore that increase in number of bits that you are representing with these m set of m waveforms is not necessarily an offsetting factor as far as bandwidth is concerned okay so there is no attempt to constrain bandwidth and increasing m is associated with more bandwidth therefore the question arises what should we do in bandwidth constrain channels when we want to go for emery schemes what approach to take where we cannot allow any increase in bandwidth at all no matter what is the value of m okay now what kind of approach comes to your mind we briefly talked about it last time we are going to necessarily have to work with band limited pulses right and typically we will decide on a band limited pulse shape or perhaps a set of band limited pulse shapes if we so desire usually it is convenient to 0 on to a single pulse shape PT right and then construct the emery waveform around that pulse shape and then the options that you have are very few in number okay you it is going to be now very difficult to construct orthonormal or simplex or other kinds of families once you put this bandwidth constraint right the most commonly used option is in fact what is called emery ASK or emery PAM okay so what a signal space representation of the same would be a chosen a pulse shape I think I have figure for that somewhere you might have chosen a pulse shape and let us say this line represents that particular pulse shape this is the basis function in this one dimensional space which is represented by PT so this point here represents the signal PT then in the binary case we know what the corresponding thing is you can use minus PT you can construct a 4 level signaling by using different 4 different amplitudes right or an 8 level signaling by using 8 different amplitudes but the basic pulse shape remains PT only you are using that pulse shape with different amplitudes okay so these are the kind of constellations or signal space diagrams that you work with when you are working with a bandwidth constraint emery signaling an example here of a waveform if for a waveform or responding to emery ASK where are just for the sake of illustration chosen a triangular pulse with 4 possible different amplitudes right this is one amplitude this is another a third and a fourth these are the 4 amplitudes that are coming up and you have associated with each of these amplitudes a binary sequence of length 2 right so when the 2 successive bits are 1 1 maybe you transmit this amplitude 0 1 this amplitude so on okay so that is an example of emery ASK of course this is not a good example in the sense that this triangular pulse is not going to be strictly band limited also right typically we are going to use pulses which satisfy the Nyquist criteria right namely the sync pulses and that family I think we will stop here and we will next time do or consider passband modulations for both binary as well as emery cases they are quite different in philosophy and style okay the most of the material again will be available in any of the books though not in this form the book that I am following at the moment will not be easily available to you so it is called Blahut digital communications by Blahut yes I could