Lecture 19

[Music] okay welcome back to the class and we have been discussing this hard decoding okay how this hard decoding actually reduces to the nearest distance likelihood rule okay plus I will use this framework to develop soft decoding okay before we wrap this up a quick recap of what we have done till now okay so we have constructed we have considered this abstracted transmitter and an abstracted receiver and our awgn Channel okay so if we recall this transmitter takes an message signal Maps it to some quam constellation goes this quam constellation is transmitted over AWG and channel and then receiver receives this receive signal R okay and then based on this receive signal it has to come up with this estimate of M okay right and how this estimate of M is calculated in our case is basically looking at the map rule okay and what is this map rule what do it tell it estimates the the estimate of M is chosen as m0 if this probability okay which is the a posterior probability is is greater than this a posterior probability okay and what is this a posterior probability basically the posterior probability of row given M okay into the probability of M okay right so now what this reduces to is as follows okay that's what we calculated in the last class that the this PDF of the receive signal is the PDF of the noise and which is shifted by S not given m not was transmitted okay and then on right side similarly we have shifted PDF of noise shifted by S1 and then scaled by P of M1 okay and what is this now this is a PDF of noise shifted by S PDF of noise shifted by S1 okay right now using this we'll basically analyze few things and what are we going to analyze okay so what we I'm going to consider multiple examples okay and then see how that map varies okay how does that map rule change so what are we saying okay when s not is one okay we are what are we saying that s not of one is transmitted with a probability of 7 okay recall that we are considering binary phase shift key okay so there will be only two values m0 and M1 okay right and so we are transmitting s0 is one and what is the other value minus one and that is being transmitted by with a probability of3 okay correct and now if you look at it okay so the we have to consider the noise variance Sigma Square so this is the noise power I'm considering for Simplicity as one okay what is happening now is okay so if you look at those like uh the probabilities which we uh looked at in the last slide okay what were they okay so if I have them here okay no okay so I'll have to go back a little bit okay but what are we what is this okay this is PDF of a gan correct okay and you look at this what is this scaled by a Priory probability of message and not being transmitted okay similarly this is a PDF of another Gan okay and then B Escape by a Priory probability of M1 so what am I going to do in the next slide which I showed you just now I'm going to plot both of them okay plot this PDF scal by this probability plot this PDF scal by this probability and I just showed you right what is we know everything from here okay we know noise variance we know P of m not we know what s not is right okay then it is very easy to plot this PDF scaled by this constant number it's very easy to plot this PDF and scale by this constant number correct okay right so if you look at the plot in the next slide that's what is happening how many such PDFs are there there are two PDFs okay one on the left hand side and one on the right hand side okay and so what is this this PDF in blue okay where is it located it is located at S1 which is minus1 right this is a blue PDF okay and why this blue PDF has a lower height and the PDF in red because this is scaled by3 right this blue PDF what is the a prary probability it's 3 right correct and otherwise the PDF will be at one if it is not scaled okay P of m0 is basically if both of them are like if it is not scaled PDF it has to its value it Peak will be at one now we are scaling it with probability 3 so you see this it is reduced to the height is reduced to3 in this case it is scaled by 7 so the height is 7 right the PDF the red PDF is that value the height it is is at 7 okay right okay now so this is what is sitting on the right side and this is one PDF and this is another PDF now what does what do we need to calculate okay what does basically the map rule says when will you pick M hat as m not okay m not is basically this s not okay when will you pick this when this when the probability in red is greater than probability in blue right okay so what's happening okay if you look at this this is basically the first PDF okay and this is the second PDF okay now what is happening here so these two PDFs intersect at certain Valu okay I'm calling this value as a okay and whenever they intersect wherever they intersect these the value is the probability is same of course right that is basically this they are intersecting and that value will be same okay whenever they are intersecting now on this side okay this intersection point I'm calling it as a threshold a okay right if so what are we saying here on this side of threshold what is happening this PDF has a lower value than this PDF right you look at this what is happening to this blue PDF on this side has a lower value than the red PDF correct on this side of the threshold what is happening right that this blue PDF has a higher value than the red PDF correct right so and what was the rule so what is this rule when whenever this probability is greater row given s into P of M is greater what will you declare the the estimate of message will become M and where is it happening it is happening on this side of the threshold Okay so this is basically becoming the threshold okay so if this received value is greater than this threshold what does that mean okay that this PDF is greater then what will happen if this PDF is greater which message was transmitted m0 right m is equal to declare M hat as m0 if the received value row is greater than a and it is essentially the same what we saw on the last slide okay that uh probability of this is that scale PDF look at this right P of n row minus what is this s given M and if this value is greater than P of row minus S1 into P of M1 you declare the transmit signal as m0 and where is that happening as I said it is happening on this side of the threshold okay right so if row is greater than a m had becomes m not okay so now to calculate this threshold you should know this threshold where to like basically declare this right if row is greater than a you to calculate this threshold okay and how will you calculate this threshold both these PDFs are equal correct okay and if you calculate this if so what are we saying on this threshold both these PDFs are equal so I've said that instead of inequality I made that equality okay and if you solve this okay right so what have I said okay at this value what is row row is basically a right so what have I done here I've taken those was last equation okay P of it was row minus s squ but whenever we are calculating a threshold row is basically we are treating that as a so row has become a minus s not this is the PDF 1 by < tk2 pi Sigma square that gets cancelled on both side and then we have P of m0 okay so if you solve this there is a threshold which we are going to get okay right so I'm not solving this this is one step exercise which can be done very easily okay now you see this okay this will immediately tell you how this map is basically equivalent to nearest distance rule okay just to make our life simple what are we doing here I'm setting P of M1 is equal to P of m0 okay so what happens here lawn of 1 is zero so this term goes to zero correct what will this threshold be S not as 1 and S1 as minus one this threshold will be zero right okay so what are we saying here okay choose M hat as m not if row is greater than a if the receive signal is greater than zero right so the threshold becomes zero okay so what is this if received value is positive right you are declaring that as s not if row value is negative you're declaring that as S1 isn't this the minimum distance rule okay what is this if row is greater than Z which basically the received signal will be close to s0 if row is less than zero the receive signal is close to minus one that's essentially the minimum distance trueu okay and it is basically coming from the map rule okay so that's that map rule is basically reducing to the minimum distance rule which we looked at in our original demodulated discussion okay and what we did not look at at that discussion is basically the effect of this a Priory probabilities okay right I could show you this minimum distance very easily with this by taking this a Priory probabilties same okay but actually in this case also a Priory probabilities are different you'll get a different threshold but at the end of the day that is also a minimum distance rule with a different threshold okay that's what we are going to see now okay so let's see this okay so in this case if P of M1 and m0 as what we have taken in this Slide the minimum this threshold comes out to be 4236 okay right and you see this the threshold is moving towards moving away from s right instead of now it has become minus 44236 okay so what are what is happening here that the the probability of transmitting s not is greater than probability of transmitting uh S1 okay so the you're giving more region to declare S1 okay basically as as the correct decision okay so you look at this minus. 4236 so it is the decision region has become here okay the threshold has moved towards here okay so if the decision region is basically moving towards the negative side okay there are a more chances of declaring the receive signal as s not right because the decision region is moving towards the negative side okay correct and why is that happening because probability of transmitting s not is greater than probability of transmitting S1 okay so now what happens if probability of m0 I mean s not and probability of s S1 are same okay in that case so this noise variance Still Remains same okay see this both of them have like equal Heights and where are they intersecting we just did this exercise in the last slide also they'll intersect now at zero okay intersection threshold is zero okay so this is basically what students normally do is they like uh when we give them an question like this they intuitively uh calculate this threshold okay that if you given a system like this binary system okay it is declare a value receive signal as like basically s not if it is greater than zero okay otherwise declare it as S1 but that intuitive probability that intuitive calculation of decision threshold is wrong because you have not Incorporated the a Priory probabilities okay right so that is another takeaway but what are we saying here when this the a Priory probabilities are same the decision threshold is balanced okay is balanced between s not and S1 okay in the last case P of S1 was greater P of s not was greater right okay so the decision region basically was moving towards the left okay now both of them are same it has come somewhere in between okay right so what is the next case Okay so both the probabilities are basically the same okay now and what have I done here is I have reduced the noise variance okay so what happens in this case you see this what will this noise variance do okay noise variance is basically reducing the width of the noise PDF okay so we have lesser noise in the system okay right what does this tell okay so if you transmit this symbol okay right it will basically that what is the probability that it will B basically move towards move away from zero okay what is that probability this is that probability right okay why because what are you doing you're adding this you transmitting this s0 okay and you're adding noise to it the noise has a lesser variance okay there's a lesser probability that it will cross the threshold right okay Ral to S + n okay if the noise has lesser variance okay that then with a lesser probability this signal this transmit signal will move towards will cross the threshold and that's what you are seeing over here in this in this region right or right if you sum this probability okay this is the probability basically wherein this s not will will move will cross the region will cross this decision boundary and move towards S1 okay right and you see this this probability is basically come down when compared with this case look at this right and why is that happening because the noise variance is higher in this case right okay correct what is this probability this is basically probability of error okay you transmitted this s not okay right the noise has taken it toward cross the threshold and move it towards the left side okay and what is the probability of error you basically sum this up correct and with Sigma Square equal to 1 this is basically the probability of error okay when Sigma square is equal to 31 this is the probability of error you sum this up okay so the probability of error has reduced because the noise has reduced okay right so intersection threshold Still Remains zero okay because we are not changing the a Priory probabilities okay but what are we showing over here the effect of noise on how this like the trans how it is going to affect the transmit signal because noise has a lesser variance which essentially means what it receive signal has a higher SNR right okay so and this gets reflected like this wherein the probability of this s not becoming S1 is only this right and similar is the case with S1 becoming s not this is the probability of error for the right okay now you further reduce the variance point1 you see what happens look at this because noise is further reduced okay you basically like this is a transmit Point okay you're adding this Goan noise and but the Varian is of the noise is low okay so what is what is it telling okay that when you have a noise like this what happens how will your receive symbols look like okay this is bpsk right the noise is is a lower variance what will happen you'll receive points only in this range they'll move like this right okay so they'll move little because the noise variance is low correct okay and that's what is showing you over here okay like look at the this is the error probability then close to zero right so whatever you are transmitting you'll be able to receive that signal with the very high probability you'll be able to correctly decode that signal okay that's what this this plot is telling you we only look at the constellation plot but this is another way of looking at it from noise perspective when you add basically to when you plot it with the with the goshan PDF okay right what is Ral to S + n n is goian and that's what we are plotting over here right and we are plotting them for two different values of s okay one is+ one other one is minus okay right intersection threshold remains same but this is the key point okay that the with very low probability you are going to make an error at the receiver okay then you further reduce it okay right you see this intersection threshold remains zero look at this whatever you transmit that's what you're going to receive even here so you'll they'll move very little you stay in this because look at this okay right because the noise is reduced you're not never going to make an error right okay so there are a noise variance at which communication system would like to operate because you'll not make errors but that's not guaranteed and that's why we have all this ldpc codes to help you out wherein the noise is high but you can by sending parity bits you can recover the correct signal okay right so this above thresholding based receiver as I said is called a hard decoding because you are already declaring that the receive signal is either + one or minus one that's what we do right so if it is we did it for bpsk right same idea can be extended for qpsk also why because qpsk is bpsk on xaxis and bpsk on Y axis okay so if it is a qpsk symbol you look at you do this hard thresholding for the real part of it then you do the hard thresholding for the imaginary part of it right okay so same idea can be extended okay right so for other higher order modulation scam we'll have to do a little bit of more homework but the idea Remains the Same okay right so this was basically the hard decoding and we basically threshold and the threshold is we can calculate it very easily okay right and typically this the a prior probabilties are same okay for any message signal they are same but you should if they are not same you should be able to incorporate those probabilities in your decision okay because they change the decision threshold right okay so after studying this what is left for us to understand is the soft decision values okay so what are what happens here is what is our map rule again we come back to this map rule okay and you'll quickly see that it is the soft decision is only basically rewriting it in a slightly different form okay and that's what I'm going to show you what is our map rule fine what did we do for this hard decision decoding we basically simplified this okay correct and now but at the end of the day right what is this this is basically a probability okay right there's also another probability this correct okay and we compare we calculate this probability we set it equal both of these probabilities to be equal and then we come up with the threshold correct okay and we use that threshold to declare whether the transmit bit was z+ one or minus one if we don't do that okay right if we basically stick to these probabilities okay right and then feed these probabilities to the decoder that's what is called the soft decision rule let's see that okay so what are we saying here so earlier we set them equal okay right and then we basically came up with a threshold I'm not doing that now what are we saying I moved it over here okay right now what is this P of R this is this is a PDF okay right so and then I have substituted this okay what am I saying here okay what are this I'm not simplifying it okay right so what is what is coming on top what is this this is a probability that M was transmitted and this is the probability that M1 was transmitted okay right I'm not thresholding that whatever probability I have basically calculated I keep as I keep that probability as such okay and calculate its ratio okay now if I want to threshold this ratio okay so what will happen is that will become my hard decision rule how will that happen if this ratio is greater than one what will the message signal be message signal is s if the this is less than one the message signal will become S1 right okay because it's rewriting the same thing in a different form but instead of thresholding this what am I doing here I'll keep these probabilities as such okay and what are this what are this we do not threshold but calculate the above probability okay and what is this first of all this is the likelihood right and this is the ratio of of to likelihood okay correct so what is what can you call it as this is likelihood ratio okay now there's an exponential coming over here so typically you can further simplify this by taking log of it okay right so that ratio is called log likelihood ratio okay and what is that ratio log likelihood ratio it is a ratio of two likelihoods scaled by the a Priory probabilities correct if let to you want to simplify this you can set these like a prior probabilities to same they get they'll get cancelled okay so what is happening here in order to understand it better the ratio of log likelihood ratios okay you have the likelihood you take the log of that and then you calculate a ratio and what is that ratio that ratio tells us what is the probability of s not being transmitted divided by the probability of S1 being transmitted method okay so if you look at it what are we doing now we are calculating the ratios and feeding that ratio to the ldpc decode right so at this stage we have not decided ourself whether the transmit bit was 0 or 1 or plus one or minus one okay we only calculated the ratios okay and then you can how is it going to help the decoder I'll give you an example and what is that example as I just showed you that what happen happens at the through this awgn channel okay right that this is what you transmitted okay noise basically moved it to other side this is what is received okay correct now if you threshold it youve already moved it to minus one correct okay you already pushed it towards the minus one you have declared it with so much of confidence that the transmit signal was minus1 okay so the decoder will have to do lot of work to bring it back to plus one to correct this error right but instead of that if I pass on this probability right which I'm calculating here probability of this bit being zero or this bit being one okay I only pass this ratio I don't declare it correct and with those probabilities it'll be easier for the decoder to decide to make it to push push it towards the to towards the actual transmit symbol and correct the error okay because you not pushed it you not thresholded it you just retained these probabilities and these calculating the probability like this is shown to give you 2.5 DB of gain okay you can improve the SNR okay by a factor of 2.5 DB okay by only making this change okay and actually if it is you get with this kind of improvement okay you can it is a 3db Improvement in SNR the uee can go twice the distance okay with this 3db SNR improvement from the base station okay and that Improvement is huge okay typically what happens is that in any receiver you never do the hard thresholding you always calculate these log likelihood ratio and then feed it to the decoder which makes the decoder life easier and then it is able to give you that gain okay so this brings to the end of discussion on the demodulator aspect and what did we learn in this lecture in the last lecture was two things one is that how does this actual hard decision decoding is a result of maximizing the a posterior probability okay and how do you define this a posterior probability this is the probability of a given bit being transmitted of probability of a bit being transmitted given that you have received some signal okay right so you maximize that probability and if that lead first leads to hard decision rule but if you don't use that if you simplify it like this if you rewrite it like this okay you can calculate the ratio of the probabilities or the likelihoods okay and those ratios if you feed to the decoder directly then that will give you a much better performance then when compared with the case when you are thresholding and then giving feeding it to the decoder okay so I'll stop here thank you [Music] [Music] [Music]