Lecture 18

[Music] welcome to the class and what we are going to see today is basically extension of hard decoding now we've discussed hard dis decoding where in where it was used it was used in the demodulation block and what was happening there you were basically thresholding the received symbol to the nearest constellation Point okay now coding I briefly said that we could use something as soft decoding and it could give us better performance than hard decoding so even in the hard decoding I just said that it is basically like we it works according to the nearest distance rule okay I'll show you that how this nearest distance rule is arrived okay and then build on top of it to basically give you how soft decision is done okay right so that will complete our understanding of this Dem modation block okay first part will be like devoted to understanding this this hard decoding we'll develop the framework okay for how this nearest distance rule is derived and then we'll use this framework to develop the soft decoding okay so if you understand this well extending it to this is very easy okay so this lecture is sufficient for understanding this okay no other notes are required and so if you look at it basically what are we doing in that transmitter and the receiver part okay what happens is if you look at in this complete schematic what have I done I have abstracted the transmitter and the receiver part when I say this abstraction is I've combined all these blocks okay and then taken the input and then given the equivalent output okay so what is this basically what is this transmitter what are different blocks of transmitter are in principle doing okay they are taking a message okay right some information bit which is transfer block which needs to be transmitted and finally what is happening here is it is modulated mapping into some symbols correct so that's what I'm writing over here some message is being mapped into some symbols okay they can be qpsk 16 quam 64 quam they can as well be bpsk symbols also so in this study we will restrict ourself to like bpsk but this is applicable for all allation levels okay right so you see this for corresponding to each message we're mapping it to a particular symbol okay let's say so this is whatever processing you do right okay so what is happening at the end of it you're taking certain bits mapping it to some symbols then what happens after that is you have R = to S + n this is our awgn channel on which the symbols are being transmitted okay noise gets added now this receiver has to basically decide based on the receive signal what message M hat was transmitted right okay and that's the job of the receiver correct and finally it has to give you the whatever way it process it has to give you this message back correct okay so that is the abstraction of the receiver so before we and develop this receiver let's understand this the model of thermal noise how does this work okay what is how is this model okay so basically caused by random motion of electrons in electronic components and there are like millions of these random electrons and you don't know how they are moving okay so now they're all like they move in random Direction and then what does the central limit theorem tell us you add all this like random like numbers okay right and what it gives you is that those sum of those random quantities okay can be modeled as lotion because there are like millions of these random quantities which are being added okay correct so this thermal noise is modeled as goian due to centrer limit theorem okay so that's the reason that we we call it we use this El White goian Noise Channel okay and it is of the order of 10us 15 wats in a mobile phone and why does this become relevant that is the first question we need to ask and why does what happens is these mobile phones they are designed to operate at a receive power of 10us 14 wat as low as that okay and just to give you an example that your mobile phone when it transmits it transmits at a power of 250 M okay by the time it reaches the base station it signal strength would have deteriorated that much okay so depending on the distance and then you look at it the signal power is becoming close to the noise power at this signal power that is only 10 times stronger than the noise power so now the noise is becoming relevant right okay so you have to consider this that's where this AWG Channel comes into picture and what is our design basically to design what is our objective objective is to design this optimal receiver for this communication system okay and what do we mean by Optimal receiver okay the optimal receiver will basically maximize probability of correct decision okay and uh it makes sense because we want to maximize the probability of correct decision okay whatever this m hat is being detected over here its probability of being detected correctly should be maximized okay that's the job of the receiver now that's the basic framework wherein like the receiver needs to be developed okay now as I said earlier that we will restrict ourself to this to Binary shift keying in this understanding okay and then but this is applicable for any quam 16qam or any other higher order modulations so what we are saying here is that the system transmits either message m0 or M1 okay so binary and then it trans MIT this message m0 with some a Priory probability P of m0 and M1 with probability P of M1 what is that mean that it is not likely equally likely that there'll be like the probability of transmitting m0 is different from probability of transmitting M1 okay and these probabilities are basically known already known to both transmitter and receiver okay what is not known is whether m0 is trans or M1 is transmitted and this noise is statistically independent of this transmit message okay right because this is imposed onto Us by the nature and nature doesn't like bias whether is not biased whether m0 is transmitted or M1 is transmitted okay and this noise okay because this has goshan PDF is goshan okay what will it's pdfb this is this is the PDF okay and this if you look at it what is what what is this PDF telling us that what is the power of this or the variance of this goian it is Sigma square and the mean of this is basically zero okay right and so this is the PDF of the rice okay and it has a goian PDF so this is bpsk where m is either m0 or M1 now what is this the decision rule which is going to maximize the probability of correct decision okay or minimize the probability of error what is it called maximum a posterior probability okay and what is this okay look at this carefully what are we decare what how are we like defining this rule we are saying that declare this estimate of the message as m0 if look at this P of m0 given you have received R is equal to row okay what is this this is the a posterior probability of m0 being transmitted okay why are we calling this as a posterior probability a posterior is basically the probability of after the event has happened okay and how are we if you keep this in mind how are we calculating this probability probability of m0 of message m0 given this event has happened right okay that's what is the a posterior probability okay so and how are we maximizing it we are basically like calculating two different probabilities probability of M given this signal is received okay and we are calculating this probability of M1 given this uh signal is received so we'll calculate these two numbers okay and we will pick we will say that m0 is was transmitted if this probability is greater than this posterior probability right okay see there are two different probabilities one is a Priory probability which is basically the probability of a transmitting a message okay even before any event has happened okay this is which are given as P of m0 and P of M1 and what is this probability this is a posterior probability which is the probability which is being calculated after an event has happened and what event has happened basically the receiver has received the signal okay as I said these a Priory probabilities are known to both transmitter and receiver okay that these messages are being generated with this probability but what is not known is basically whether not was transmitted or M1 was transmitted okay and how is it calculating and this is calculating receiver is calculating the a posterior probability okay and maximizing it it's only picking up m0 if this probability is greater than this which is basically maximizing it it is picking up the maximum value okay so now what is this from here to here okay how have we written this if you remember the basic base rule okay right that's what we are using over here here on both sides let's explain this first this part is parall to what is written over here okay correct so what is if we write what is If X and Y are two event or two uh what okay so how can we basically write this okay so what is can we write it like this right what is this P of X comma y right this is also P of X Comm y That's what I'm writing okay so if you look at this so how can you write this P of X comma y is equal to P of Y given x p of x divided by right okay so if you carefully look at this what are we doing here we are doing the same thing okay so uh if you treat this as X you treat r equal to row as y okay what am I doing here x given y okay I'm writing as y I'm given X probability of x divided by probability of Y correct okay same thing I'm applying over here okay so and I'm R is equal to instead of writing r equal to row I have written row over here okay same we have done over here okay similarly this will also follow you'll basically write the you'll use the base rule to expand it like this okay fine then if you look at this now carefully what is sitting in the denominator okay is same on both sides okay and it is both of them are positive okay so we can conveniently cancel them without changing the inequality okay so that's what I've done now okay so what have I done I have basically simplified this and I have written it like this okay why have I simplified it like this because I know this this is given to me P of m0 I know PM of M1 and next I'm going to show you that this is very easy to calculate okay P of R given row given m not okay right so what is we use Bas rule okay to move from here to here okay what is this you should also notice this this is capital P this is discrete probability discrete probability because messages are discrete okay now what is this from here to here when I write p this is row if you look at this row okay what is this this is a receive signal and what is happening over here you're adding noise goshan noise wasian noise is a continuous random variable okay from here to here it has become discret from it was discret and it has become continuous okay so this notation is for continuous random variable this notation is for discrete random variable okay and this is binary P of m0 and P of M1 okay now I'll have to calculate this okay and once we calculate this the calculation of all of this becomes very easy okay and that's what I'm going to show you okay let's so now one more fact okay that this basically the map rule this is the map rule right okay it becomes maximum likelihood rule when messages are equ probable what happens when message messages are equ probable what does that mean p of m0 is equal to P of M1 okay and that also gets cancelled so then what are we maximizing this and what is this this is called the likelihood okay right from a posterior probability we move to the likelihood okay so this is if you maximize only this this is called the maximum likelihood rule okay so another key thing is map reduces to ml when messages are equally equally probable and the reason is this this is map and this is ml likelihood okay so now what are we left with okay if you look at here going back we need to calculate this okay this PDF this PDF of a continuous random variable R what is r r is basically s + n right and I have to calculate this continuous distribution for row given M and row given M1 okay so you'll see this I'll like combine both of them basically use the same notation when I say combine I'll use the same notation and what is this notation row given m I okay and m i is basically okay it should be for Zer and one okay that's the type it should be row given m0 and row given M1 okay right so we need to calculate this okay so how do we calculate so what are we saying you if you look at this signal okay so how do you read this what is this probability this is a conditional probability right condition on given whether the transmit message was m0 or M1 right okay so what is it saying Mi is given to us okay now if Mi is given we of course know s also whatever s was transmitted okay so what are we saying okay so for with a given r with a given Mi if you look at this how will you calculate this okay r equal to row is obtained from n by adding a known constant s of I okay so let's look at this I'll just explain it to you so what what is being said in this bullet is as follows okay what are they saying that given Mi okay what is what is our signal model Ral to S + n right okay correct now what do we need to calculate we need to calculate this PDF right and what is this this is a conditional PDF and condition on M ofi what is the meaning of conditionality it is basically given to you okay now so what are we saying we have been told to calculate this PDF that the message was either m 0 or M1 so what is the PDF of R PDF of R is basically how do you what is r r is being generated by adding known constant s to some random variable n because it is given right okay it is given so then what is r to summarize R is basically derived by adding a known constant s of I to n what is what are we referring to S of s of I is basically this right for a given M of i s of I is generated okay so we know what s of I is being transmitted if we are calculating this probability okay right so what are we saying here now if this is the case okay what will be the PDF of R how are we generating R it is being generated by adding a constant s of Y okay correct so what will the PDF of R if you basically this simple random variable okay transformation of random variable that if you add a constant okay to another random variable what happens the PDF actually shifts okay right and how much does it shift it shift by the known constant okay and so what are we saying here okay that the this PDF read this carefully this is the PDF of R right and this is what we intend to calculate okay and what is this PDF this PDF is equal to the PDF of Noise Okay which is shifted by a constant s ofi this is the PDF of noise shifted by this constant s ofi okay right and why are we able to do it because we are given Mi correct so this given Mi is coming here this given Mi is coming here okay right so this PDF of this received signal is equal to the shift shifted PDF of the noise okay and how much it shifted it shifted by S ofi okay right so now what happens is as we said in the beginning of the class that this message f m ofi is independent of the noise or the noise is independent of M ofi okay right so from here to here what have I done basically because m ofi is independent of N I have dropped it okay right so if two random variables are independent P If X and Y are independent P of x given Y is equal to P of X that's what I WR n and M are independent so I've dropped m m of I so this PDF is basically equal to the PDF of this noise right okay and we know the PDF of the noise is a goian PDF okay right basically calculated this PDF P of R and how it calculated it is basically the PDF of the noise so let's use that okay what is our map rule this is choose M hat is equal to M okay how will you choose this P of row given M into P of M okay this is map okay I'm still not assuming that P of M is equal to P of M1 right okay so what are what does map rule say choose M if this probability is greater than this probability correct and what was our objective our objective was to calculate this PDF okay and I calculated exactly that on the last slide okay this is that PDF which had to be calculated and this is the PDF of the Noise Okay so if you look at it what is it coming out to be P of n PDF of noise shifted by what will happen here it'll be shifted by S not and this is that probability what will happen here PDF of noise now shifted by S of one into multiplicate and this is the a prior probability right okay so what do we have now okay just substitute it over here correct okay this is what was our PDF of noise it was a PDF of it was a gan PDF now it is shifted by S not okay right if you recall what was the original PDF it was zero mean right okay and how did we write it exponential minus row s by 2 Sigma Square now what are what has happened when while calculating this PDF it is now shifted by S not so this shift this is the mean correct so this is row minus s Square by two so this is come out to the mean this is s not is mean over here and S1 is mean over here right okay so what is what have we achieved till now is basically what did we do over here we basically looked at this goian channel okay and we have abstracted this transmitter and receiver okay and what is this transmitter doing basically it is taking in some symbols and then it is giving it is taking in some some message bits and then it is basically giving us some symbols and then what is this receiver doing what is the job of the receiver it looks at this this receive signal R and then tries to estimate M hat okay and what is the rule we are looking at which we claim that it will maximize the probability of correct decision which that is basically the maximizing the a posterior probability okay right and what we have done till now is basically simplifying this the PDF the PDF of the receive signal okay in terms of the PDF of the Noise Okay correct and it turns out that if you use simple base rule it this PDF of the receive signal becomes the PDF of the noise and shifted by whether m0 was transmitted or M1 was transmitted okay now in the next class I'm going to simplify this for you and show you how this rule actually reduces to the nearest distance rule okay right and then we will build on top of that can spend some time for us to understand how that this framework leads us in giving in calculating the soft values of the transmit bits okay so with that it'll like close the discussion on the demodulated design okay right and why are we doing this exercise because two things last time when I we discussed about the modulator I just like told you without going in detail that like this basically the demodulator works on the principle of uh minimum distance rule but we never came to we never discussed how that minimum distance rule is actually derived and then how this soft decision rule is implemented okay so we are doing all this exercise to understand those two aspects okay so I'll stop here then we'll continue discussing this in the next class thank you [Music] [Music] [Music]