 So, this is lecture 36 and the last thing we saw was couple of ideas hopefully you remember the fractionally spaced equalizers and pass band equalizers. So, those while being similar in the ideal case while being similar to what is being achieved, similar to the previous constraints that previous structures that we saw like symbol rate equalization and even base band equalization. In practice there might be a lot of advantages to these two things because of various other reasons. So, these two things are used in several systems. So, now I am going to briefly talk about a complete PAM receiver. So, one version of a complete PAM receiver remember when I say receivers have if you actually see implementations of receivers are all various choices that you can make for different components and all possibilities exist. So, what I am going to draw is one version of it which is which one can say is a little bit canonical as in many things you will find similar structures repeated. So, let me see I mean it is a bit of a complicated picture. So, I will try to recreate it as clearly as possible. So, you have input coming in which is a band pass signal. So, maybe I will call it r of t or something. So, the first thing that is typically done is a usually a band pass filter. So, you have a certain band in which you are expecting a signal and since other bands might not carry all kinds of other signals you do not want to pick up any of those things. So, first thing that is typically always done is a band pass filter in the range of interest. So, after that what do you do after band pass filtering you will have to do the phase splitting and then move it to the base band multiplied by e power minus j. So, the phase splitting is basically is removing all the negative frequencies and keeping only the positive frequency it is a phase filter. So, you have to do one crazy filtering. So, that filter sometimes it is done in discrete time also. So, depending on the frequencies of your system depending on what is possible and not possible you might be able to even do that in discrete time. So, in general the band pass filter might have a sampling following it. So, this sampling will have to be maybe at a very high frequency if you want to do it properly, but maybe there are cases where you can play around with the sampling also. So, in general there might be sampling here, but that is optional. So, I will put a little bit of a dotted line around this. So, if you want to do the phase splitting in digital then you will have to do a sampling. So, but sometimes it is done in analog itself. So, you have a phase splitter. So, after that you have complex signals. So, till you have up to the phase splitter signal is real obviously and after that you have complex signals. What do you mean by complex signals? You have to pull two wires out. So, two wires matter to you not just one wire. It is two signals matter to you after that. So, the first thing you do is multiply this by e power minus j 2 pi fct. So, that is the first thing you do. So, you move it to baseband. Then after that you are going to start your filtering. So, you have to do match filtering in your baseband equivalent followed by symbol rate sampling ideally if you want to do that then followed by the whitener I mean precursor then a post-cursor. So, I will assume an MMSC DFE like structure. So, you have a precursor and a post-cursor. So, of course you can do it to be or something more complicated, but in general that is not very practical. If you have a lot of tabs that is not very practical. So, I will assume an MMSC DFE type equalizer and then typically one can also do it in the fractionally sampled domain. So, twice the symbol rate sampling you might want to do. So, here you have a sampler which is I will say k Ts kT by 2. So, twice the symbol rate you are sampling out of the from the signal and then this goes to the fractionally spaced equalizer. Remember the FSC is supposed to do both match filtering and the whitening. So, it is one filter and that is always nice to design. So, you have to adapt or do anything it is just one filter. The tabs are adopted in one way there is no further confusion. After the fractionally spaced equalizer you move to symbol rate. You sample at single rate so remember I am going to do so well. So, first thing I should be careful about is the FC. So, I may not have FC exactly at the receiver. So, I will say normally I have F1T and whatever delta FT that comes in I am going to take care of inside the equalizer in the passband equalizer. So, I am going to do both FSC as well as passband equalizers. So, I have to multiply here by some carrier and remember that delta F if you know it you can use it, but delta F has to be derived from the decision. So, I will not show you where it comes from I will show that later. Similarly, these sampling instances this one, this one and this one all have to come from some recovery something further down in the circuit. So, eventually we will connect it up but for now assume it is available. So, you multiply by a e power minus j delta FT. Then after that you have the addition from the post cursor this goes to the slicer this is your slicer. So, the slicer output on the one hand is your decision this is your s hat k but the output is also used for several things. So, one of the first things that is used for is it is used to compute the error for updating your equalizer. So, I am going to try and draw this as cleanly as possible, but it is a bit of a difficult picture to draw. So, the error slicer error is used first of all to update your equalizer, but it has to be again rotated for the delta F. In the pass band equalizer case the error once again is rotated before it can be given. So, to the precursor of course the post cursor works with whatever error is already there. So, remember that. So, what is the post cursor? So, it goes to the post cursor as it is for adapting the post cursor. The post cursor also works with the output of the slicer and gets it back into is it okay? So, make sure the picture is clear in your mind here I have a multiplication from the carrier then after that goes to the fractionally spaced equalizer for update. So, what? Now, the input and output of the slicer should also be available to two blocks, two blocks that I am going to call the first one as the carrier recovery block sits here and then you have a timing recovery block which sits here. So, let me see I think the carrier recovery block has moved a little bit to the right. So, the input and output of the slicer are going to go to the, it is not enough room is there. Anyway, so let me see let me take it a little bit twisted sorry about that for the timing recovery block and to the carrier recovery block. So, there might be other parts of your circuitry which are also connected to the carrier recovery and timing recovery which I am not going to show if I start showing all of that then it will go for a major toss. So, I will just say the slicer input and output are connected to the timing recovery carrier recovery I think it is pretty much enough but in some cases you might want in some special way other things to connect also it might connect you can use something else if you want for carrier recovery something else if you want for timing recovery in this case I am going to only do timing recovery and carrier recovery based on decisions around the slicer. So, that is why I am showing only those two inputs you might want other inputs also and if you want you can take them but I am not showing all that here. So, the carrier recovery gives you the carrier that you need to multiply here and the same guy is taken out here and multiplied with this oops okay so maybe I will move this and roll it okay. So, that is the one loop and the timing recovery like I said it produces three different timing pulses and they are used for the various samplings in your circuit okay. So, like I said the receiver involves a lot of messy calculations and to faithfully represent that this diagram is also equally messy okay. So, there is this version of this picture and well there is an exact same picture in Barry's book okay in John Barry's book. So, you are welcome to look at it if you want a better idea okay. So, so what okay so the various things to note first of all the first thing to note is there are a lot of loops okay. So, previously we never showed any loops in the receiver okay but you see there are a lot of loops one of the first loops is and all of the loops involve the slicer input and output the first loop is the equalizer update loop okay. So, there is a loop that way okay and the other loop is the timing recovery loop this is very critical it goes through a lot of things. The next loop is the carrier recovery loop okay. So, there are three different loops and all of them have to function properly within specified parameters to for the whole receiver to work if any of them fails the whole thing will go for a toss okay. So, that is where the the artistic input in the receiver comes in okay. So, how do you adjust all these loops carefully how do you pick the parameters what do you pick for each of these things because what goes inside carrier and timing recovery is usually something non-linear something heuristic so usually there will be these PLLs and all these things in order okay. So, once you have face lock loops and all that things get a little bit dicey so how you design them how nicely you do them is important but all of that can be done even digital you do not have to worry about analog PLLs but still a lot of careful things that need to be done okay. So, one thing I want to point out is the if you look at the carrier recovery loop right okay. So, if you look at the carrier recovery loop okay the output of the slicer goes output and input of the slicer go to the carrier recovery loop then it comes in feeds us a multiplication okay and then it goes back to the slicer okay. So, in the carrier recovery loop there is no filtering okay of course you can say there is the post cursor but the post cursor does not play any role in the carrier I mean does not play a significant role at least in the carrier recovery loop okay. So, within that loop there is no filtering okay. So, that happened because why did that happen that happened because the way we are doing the multiplication we were able to pull it out nicely okay. So, the other structure helped you in doing that if you were to do the multiplication on the inside around the slicer all kinds of crazy multiplications will happen and then maybe the post cursor will also play a role okay. So, right now the post cursor is not playing a role because a multiplication happens in a outside of it completely okay. So, it turns out that is nice for carrier recovery right. So, you do not want when you have loops like that you do not want to put filters and play around with the carrier in any other way okay. So, subtle things like that become crucial in a receiver design and you do not want to put too many things in your loops how do you reduce it etc etc okay. So, all those things I will not go into any more detail because not much time on this course but we will see a few things few of these blocks closely and look at some maybe alternative ways of implementing them and make some comments okay. Any questions just by looking at this picture yes it could have. So, in this version I am just doing purely decision directed carrier recovery okay. So, remember C1 after the fsc everything is symbol rate okay symbol rate twice symbol rate okay well fsc works a twice symbol rate everything is symbol rate. So, when I say something is multiplying the carrier do not imagine it is some sine wave or something which is multiplying it is only a complex number every symbol gets multiplied by a complex number okay. So, it is just one rotation e power minus j delta f k t you are trying to estimate that but it is only one complex number on the unit circle okay. So, how you estimate that is can change symbol to symbol and you can completely control that with decision directed itself. For instance if slicer input if you notice it is at some point is at some point right the previous slicer input would have been at some other point the slicer output would have been something. So, based on that you figure out approximately where the rotation is then make do that some such thing I am talking about when I draw something like this. But you could for instance taken the the band pass signal okay from the band pass signal you might be able to recover the carrier there are ways of doing that okay. So, if you do that then it is a different kind of carrier recovery problem okay. So, maybe we will talk about that later because right now I am just doing purely decision directed carrier recovery and decision directed timing recovery nothing nothing more than that okay. So, the pros cursor and the fsc are adapted according to the LMS algorithm if there is a training phase then you will have another block which holds the training data symbols which will be used for the error computation for the adaptation period and then you have decision directed period when the actual slicer output is used okay. So, in the training phase you can even for the carrier recovery timing recovery you can use the training symbols which you know exactly you do not have to use the slicer output get confused with that. Anything else any other question you know fine okay all right. So, let me make a few more comments now. So, okay so what I am going to try to do is to draw the spectrum and maybe even the signal only the spectrum I think only the spectrum at different points as we go along in this receiver okay. So, when you are actually building this stuff that is what will help you okay. So, when you are trying to debug when something is not working all the theory you learned will not really help you have to look at actually the signal that you see before the band pass filter after the band pass filter after before the phase split after the phase split do you do you see what you expect to see okay. So, you should know what to expect first and then you should know how to see it okay. If you do know both then you can quickly debug what is wrong in your receiver otherwise you can never do it okay. You will only keep staring at the final data and say I am having 50 percent error rate 75 percent error rate some crazy numbers like that does the 75 percent bit error rate make sense all right. So, just flip everything you will get only a 25 percent error rate okay. So, 50 percent error rate 50 percent error rate is the only thing you can start you have to know what to expect at each point in your receiver and actually see it see it in a real system or assimilated in a MATLAB based system and then see what what you expect and then debug what is going wrong okay. So, maybe we will see that more elaborately later but for now I just want to show the spectrum at various points and make a few comments about alternative ways of smartly implementing these things okay. So, the first thing is that the input okay before the band pass filter what is the spectrum like okay. So, it is very easy to see that. So, you have a center frequency and the spectrum will be something around that okay. So, it may not be symmetric about FC right. So, that is your spectrum at the input okay. So, now of course there are two choices like I said you can sample at Nyquist rate this whole thing and then run a digital phase splitter if you want okay or you can do analog phase splitting. So, you do a phase splitting after the phase splitter what will be the spectrum. So, only the positive part right the negative part is gone you only have the positive part okay and then you do a multiplication by e power minus j 2 pi f 1 k t okay in the digital case and in the analog case it is 2 pi f t f 1 t okay. So, usually so let us assume it is digital or analog one of those things and then you multiply okay but there FC and f 1 need not be exact okay. So, maybe this is f 1 this is f c okay. So, there will be a small offset between the two of them okay and so you might want to multiply okay. So, you do a preliminary demodulation and you get your thing down to this is f 1 and now there will be a bit of a weird thing about that okay. So, this is what you get after a multiplication by e power minus j 2 pi f 1 t you can call this the preliminary demodulation that is the rest of the demodulation happen inside your equalizer okay. So, you are doing pass band equalizer inside the equalizer you are doing the rest of the demodulation in a decision directed way okay. So, that is the way of thinking about it okay. So, there is another way of getting from here to here without doing multiplication by e power minus j 2 pi f 1 t can anybody suggest smart way of getting there have you come across this something called band pass sampling have you heard about it okay. So, suppose you pick of a sampling frequency which divides f 1 okay. Suppose you pick a sampling frequency which divides f 1 okay and it is such that okay some f s I pick so that this is okay. So, I am getting totally confused here. So, let me just erase that and draw it properly okay I will write the f 1 and f c on top f c f 1 I pick a sampling frequency such that f 1 by f s is a integer okay. So, it divides f 1 and then it is long enough so that f s by 2 is going to include my entire band of interest. So, now if I sample at f s what will happen the aliasing will automatically bring it to base band and then I do a low pass filter to get rid of whatever else that might have been picked up okay. So, it is a very simple way of making sure you get this signal down to base band instead of multiplying by e power minus u. So, this is a common trick it is very nicely done but and if I f s and if I f c is not an integral multiple of f s or something you will always get this delta f you might be able to avoid the delta f by carefully designing it even otherwise you can do it okay. So, all this aliasing is quite fancy but what is actually happening here you have a base band signal it gets you are imagining that it is multiplied by 2 pi f 1 t and you are putting t equals what n by f s and f 1 by f s is an integer. So, what happens to the e power term it is completely drops out it is only becomes 1 okay. So, basically the envelope of the complex base band signal is going to be the complex pass band signal is going to be a base band signal and there are points in that complex pass band signal where your base band signal gets exactly reproduced. So, you simply sample there and you get your base band signal okay. So, there is no problem there okay and it is enough if you sample at the base band okay. So, that is the idea here. So, that is one more way of getting to doing the preliminary demodulation okay. So, you pick your clever sampling frequency so that you get you need a little bit of a gap also because after that you want to filter it down in case something aliased filters in okay. So, you want to filter it down and you want a gap so you have to pick the f s smartly so that it works okay and it is possible since these numbers are always there it is possible okay. So, that is one more way of doing it it is a smart way of doing things it happens. So, now you have complex base band okay and then you can bring in all the theory you want okay. So, you do you want to do symbol so you want to do white and match filtering followed by Viterbi decoding for optimal detection but nobody is going to do that in practice because once the number of tabs becomes 6 or 7 or 8 you cannot do Viterbi okay. So, the best structure next is the MMS EDFE and that is always done and in practice you want to do also fractionally spaced equalization and adjust for the delta f inside your equalizer by doing career decision directed career recovery okay. So, that is the way in which the whole thing works okay. So, let me just quickly summarize all those things that happened here. So, instead of doing the optimal thing so we pick the MMS EDFE with a fractionally spaced implementation okay a fractionally spaced MMS EDFE that is what we are doing. So, it is got two filters the first filter is a precursor but the precursor does what? Both match filtering and the equalization precursor and then the post precursor which works at symbol rate okay this is twice symbol rate this is this works at symbol rate okay and you want to do that all right and the next next crucial role is played is played by the slicer error okay it is going to give you all the loops that you want the first loop it gives you is equalizer adapting loop okay you have to set up a proper LMS or some such algorithm carefully and in the fractionally spaced equalizer case every second output only can be adapted for the precursor post cursor you adapt adapted single signal rate another loop it drives is the career recovery loop remember this is a symbol rate career recovery loop okay. So, you are only doing you are just finding different complex numbers different rotations different values on the unit circle to rotate by okay. So, that is what you are finding that is that is the way I drew it you could also have like he was pointing out could also take a band pass signal a version of the band pass signal and feed it to your career recovery to help it okay it is possible I will describe that briefly you will see why it is possible see BPSK what are you doing for instance in BPSK modulation you are sending either cos 2 pi fct or minus cos 2 pi fct. So, if you square it what will happen you will get cos a frequency a pure frequency at 2 fc right and then you divide by 2 dividing by 2 is quite okay. So, you do that you get fc that is one more way of doing career recovery there is various ways of doing career recovery. So, if you do that then there will be other components here you can do it in various ways okay this career recovery loop and then you have a timing recovery loop okay. So, it produces three different timing pulses okay one at symbol rate one at twice symbol rate and one which which if necessary if you are doing Nyquist rate but also that will also be some multiple of this okay. So, you never want to produce clocks which are not multiples of each of them right. So, usually you never do that in boards okay. So, this is what this is what it drives okay. So, so those are the various components of a receiver and for all you see receiver boards they might be floating around you might be able to get it I think a couple of places where you suggest if you are interested one is the telematics lab downstairs but they have a satellite receiver board okay. So, you can go there and you can see all these components and figure out what chips they are using what is the what is the coolest PLL chip out there for doing timing recovery etc etc etc. So, these things keep changing with all these chip makers keep changing their chips once in a while. So, what chips to use and all you can see and also another place is IWL Intel Wireless Lab where I think I think there is some there are some wireless receiver boards might be able to see some extra components okay. So, you can see that if you want. All right. So, what we are going to do next I know I had a big list for you but what we are going to do next is a bit of slowing down okay. So, the first thing I want to do is I want to go back and look at simple BPSK and draw basically signal diagrams okay. So, what does the signal look like at different points for simple BPSK okay. So, I think people who do the lab already know this because they have actually seen the signals but anyway I think everybody should see them because you should know how the signal actually looks right. I never did that carefully enough before. So, we should just I think I am going to do that for BPSK and maybe for QPSK and then with with different types of pulse shapes initially with rectangular pulse shape and then with raised cosine pulse shapes. So, just to get to see what the signals look like then you can play around with it okay. So, this is mostly review just to make sure you do not forget the very basics which are very important okay. So, the first thing I am going to review is BPSK with rectangular pulse shape and ideal channel this is the easiest okay. So, once again how does this look like you have a symbol sequence which is plus 1s and minus 1s okay and you have a rectangular pulse shape which is 1 between 0 and t, t is the symbol rate and you get a signal out which you are going to up convert by multiplying by e power j 2 pi f c t and then what should you do take the real part and that gives you the x2 of t. The reason why I am drawing like this is if you want to do QPSK what will you do? You will have one more line which takes the imaginary part okay. So, well think about it okay so roughly something like that you would do okay you have x2 of t then this goes through a channel which I am saying is ideal okay this is an ideal channel so it is basically delta of t okay. So, do not worry about the channel too much you get a received vector to which I am going to add noise at the receiver of course anything you do will add noise and then you get the actual received vector which you have to deal with yes it is okay yeah it is true the real part and imaginary part will come before I know the receiver you would do something else x1 of t is going to be the signal here okay so what I want you to do is I am going to say s of k is something like this it is at 0 it is plus 1 at t it is minus 1 at 2t it is minus 1 and 3t it is plus 1 okay so I want you to draw what x1 of t will be what x2 of t will be what r1 of t will be and what r2 of t will be roughly okay this is a transmitter okay x1 of t is the easiest it is going to be 1 between 0 and t minus 1 from t and 3t am I right then 1 again from 3t to 4t okay that is x1 of t what do you do for x2 of t so it is basically x1 of t multiplied by what cos 2 pi fct so you need some information about the relationship between fc and t so I am going to say let us say for a simple case four cycles of the carrier will show up in each symbol okay so suppose you assume that plot the plot x2 of t so you can take that 4 to be any other number also of course take 3 if you like but in practice it will be much larger you know you expect it to be very very large x2 of t okay so I am taking cos 2 pi fc and I said four cycles maybe three cycles we will see one two okay so I am able to fit in only three cycles very nicely so I will take three cycles okay so that is how the first thing will look what will happen next there will be a discontinuity right there will be a sharp discontinuity here it will go down to minus 1 there will be a 180 degree phase shift okay so once again one two three but there will be no phase discontinuity at 2t because the same symbol is going through one three okay I am drawing it with different frequencies there's no frequency modulation going on here okay so it's the same frequency nothing changes okay then once again there will be a abrupt phase shift of 180 here okay one two three then it'll come down to zero and stop this is what you expect x2 of t to be okay suppose I give you only x2 of t how do you identify the symbols okay just by looking at it yeah so you have to look at places where phase change happened okay so you also know given you know t you know exactly where to look suppose you don't know capital t suppose you don't know zero okay suppose you don't know zero when will you not know zero you don't know right at the receiver you're receiving whole bunch of things you don't know where zero is okay and you won't even know capital t suppose you don't know both of those can you still identify where what the bits are and where they occurred roughly okay suppose roughly if you have to do how will you find t for instance okay so assuming there are enough transitions in your data which will happen if your data is random you can look for points of where there is 180 degree phase shift and then roughly see what the time difference is between 180 degree phase shifts and then do what look at the gcd or something rough gcd of those numbers to get your capital t a rough estimate of capital t so then based on that you can figure out whether or not a phase change happened and and did not happen the only thing you can't figure out is the first phase you won't know whether it was plus or minus right I conveniently picked the zero phase here but you don't know what phase it started right you can never know that so only thing you can't fix is the initial bit after that everything else you can fix okay so this this is very important in practice when you're debugging your receiver right you should know if your input signal is okay or not okay I should check if the 180 degree transitions happen okay if it is qpsk what will happen let me see who gives me the answer qpsk what will be this these signals shifted by 90 degrees yeah the phase change will be multiples of 90 degrees it could be zero could be 90 degrees it could be 180 degrees it could be 270 degrees so but still there will be phase changes so you just look for places where the phase is changing and doing that okay but all those things are obviously not optimal receiver practices right that's just for debugging your receiver circuit what's the optimal receiver circuit first thing is who was a band pass filtering followed by phase phase splitting followed by a preliminary demodulator and then a match filter those that's the way that's the way you do actual reception okay so you can just look at it and kind of eyeball the base base band band pass signal and figure out if it's okay or not that's okay to do but don't use that as your receiver principle okay so it's just an approximate thing we are doing this okay so let me show you how the receiver would look and then try to plot well the other things I have not plotted are r1 of t what will r1 of t look like if you say channel is ideal it will roughly look similar okay so in most cases if you're using very little bandwidth like we are doing here right for this for this assumption we'll use very little bandwidth right channel will be roughly flat most most channels will be flat and you'll not see any much distortion here and if your noise is small enough r2 of t will also look roughly like this so this is a good way of debugging if your transmitter is working correctly or not just look at this signal how will you capture such a signal in a scope can you capture it is it possible what will happen what do you expect okay so think about it might be possible okay anyway so that's those are questions to uh what are you about so let's see so you have r2 of t coming into your receiver you do a band pass filter that's fine followed by well you have to do a phase splitter also right so band pass filter and a phase splitter okay so I'll write ps here okay phase splitter and then you do a preliminary demodulation e power minus j so I want to be a little bit accurate here so I'll write f1t 2 pi f1t plus theta minus theta okay so minus theta is just for convenience plus or minus theta some theta okay okay the reason why I'm writing that is I don't know if my the frequency of demodulation is equal to fc and you and you can never be sure about the phase right it's very tough to lock all these things very correctly so initially when I do it there might be a phase problem also you might be able to lock it either here or later on in the circuit that's okay but you may not be able to lock exactly okay so then you do what here you have to take real part again okay so here you might take the imaginary part and feed it to your q channel kind of thing is that okay or maybe some some such thing you'll have to do to get the q channel okay so maybe maybe you demodulate differently for the q channel as well okay so you take real part at least in bpsk you simply take real part right there's no problem simply take real part you get your you get your signal okay so so so so I'm going to call this guy as r3 of t okay after this what do I do I have to do match filtering right now I have the complex baseband signal I have to do match filtering okay so in this case the match filtering is what rectangular pulse is what is match filtering same okay so you have to basically integrate and dumb right so that's what you have to do that's what you do so you do integration from 0 to t and then you do a sample okay so once again there's a kt plus tau because there'll be some delay because of all these things you won't you will never know where it is so that's your filter like I said r2 of t is roughly the same as x2 of t there'll be some noise so I want you to try and plot r3 of t okay so that's where the trick comes assume f1 is close to fc but not necessarily equal to fc and theta is not necessarily zero it could be something else and try to compute what r3 of t will be and then try to plot it okay so first try to write an expression for r3 of t in terms of say x1 of t okay try to compute an expression for r3 of t in terms of x1 of t which is your transmitted complex baseband signal well it's actually real baseband signal but complex baseband signal try to compute r3 of t like that okay everything going well r3 of t should agree with x1 of t do you see that right it should agree exactly with x1 of t that's your that's your complex baseband signal you've recovered your complex baseband signal now before the up converter go back and see that before the up converter you had x1 of t so x1 of t should have come back as exactly r3 of t ideally okay but assuming a delta f and a theta try to derive an expression for r3 of t in terms of x1 of t roughly okay and then of course there'll be a plus noise we don't care about the noise just the signal component of it what does it work out zoom channel is ideal okay so we're quickly running out of time so what does it work out to x1 of t times cos 2 pi 2 pi delta f t plus theta plus some noise okay so I'll call it simply n prime of t I don't care about it okay that's some noise okay so this is your signal you expected x1 of t you're getting cos 2 pi delta f t plus theta okay if delta f is 0 theta is 0 or you know theta roughly so it's it's exactly x1 of t there's no problem but if you don't know theta and if you don't know delta it can be a different signal okay so one way to just try and get a feel for it is to try and plot x1 of t or r3 of t assuming a different delta f a delta f not equal to 0 okay so nominally you expect x1 of t to be like something like this right roughly you expect this for x1 of t okay what will happen when you multiply by cos 2 pi f ct delta f delta f t plus theta okay so it depends on what delta f is suppose I say delta f is really small okay it's going to be very small so in fact even this is a very small frequency right the symbol rate is very very small so it might actually show up significantly in this even if delta f is small okay so you might get something like a cosine picture which is like this but what will happen there there will be a 180 degree phase shift so you it won't continue like this it'll continue in this way okay it'll go off like this and then maybe come down here and then once again there'll be a 180 degree phase shift depending on am I right I'm making a mistake it'll again go up positive no no it'll be minus right it'll come here and then do what oops what happened what is this okay anyway I'm not able to figure out how to erase it okay so it should it should what should go up or come down come down right go up you sure it'll go up okay so because it's coming this way so this should go up again and then maybe I don't know where it'll go maybe it doesn't go all the way up okay so it looks it'll look roughly like this so so it'll look a bit too crazy okay so it's nothing to do with what you expected before okay so of course if delta f is 0 theta 0 then you would have got the nice flat thing so the next stage in checking your receiver is to see what after your after the complex baseband signal does it seem to be what you expected okay if you if you expect a delta f then such crazy shapes will show up okay so it's difficult to make sense of them otherwise if you've never seen it before you'll look at it and say looks like something else is happening this is dipping that is dipping you might come up with all kinds of philosophies but ultimately delta f can cause things like this delta f is really really small also right so it will show up in a strange way stuff to understand okay so hopefully you also see theta is important even if delta f is 0 if theta is close to say pi by 2 then everything is gone so you're not you're not the face of your carrier is also crucially important okay so that's why these detectors are called coherent detectors you have to know the frequency as well as the face it should be matched properly okay so it's not the face is not that serious a problem okay the reason is equalizers can handle the face okay that's why face is not a serious problem but you still it's good to have a matched face okay all right so we'll stop here yes if you look at the imaginary part okay yes due to the theta something else will come from there to the area those things we'll see next thing will be QPS