Lecture 55 : 5G MIMO codebook design – part II

Okay, welcome to the class and what we have been discussing is this codebook based MIMO okay and we will continue discussing this today, in this class we will wrap it up. So, what we learnt yesterday was that how codebooks are used to feedback the channel information for nl is equal to 1 layer okay wherein base station had 2 antennas and then UE had 1 antenna okay right and then what we are going to see today is how does this happen for nl greater than 2 layers okay greater than equal to 2 layers, we will take an example of nl is equal to 2 layers and then we will see how what additional information UE needs to feedback when the number of layer is greater than 1, how does this happen and then the additional information okay. So we will quickly recapture what we learnt here and then we will move on to this aspect okay. So, what we saw here was that we considered this example case wherein base station has 2 antennas, we are looking at the downlink okay, user has 1 antenna, so the rank of the system is 1 right, it does not matter how many ever antennas base station has okay, we are taking example of 2 antennas but the same idea is applicable for any number of antennas okay. So what I but whatever what have I given you is the codebook for 2 antennas okay, if there are more number of antenna the codebook will accordingly will have to be different okay and these codebooks are defined in the standard okay for different number of antennas, so the idea is readily extendable to any number of antennas at the base station but the UE only has 1 antenna okay right. So this is the example codebook which we saw and what we saw here was in this codebook, this codebook is giving information only about the phase of the channel right okay, so because it does not the way this codebook is designed all the channels if you look at it okay, this is both of them have the unit magnitude in all these codebook entries okay and what is different across these codebook entries is only the phase okay and what this codebook helps us in feeding back is the indirectly the channel information okay right and why it is indirect because as we saw yesterday that UE picks up the codebook which maximizes the SINR okay and which pre-coder entry this is this codebook will be used as the pre-coder at the base station okay right and UE picks up this entry such that it maximizes the SINR okay and we know which pre-coder maximizes the SINR for this kind of system okay, it is maximum ratio transmission okay. So in practice what is UE doing, it is UE has estimated the channels okay instead of directly indicating what is the channel it says it is the base station please use this pre-coder so that my SINR gets maximized correct and which is basically indirectly telling the channel and but there is a clear limitation on the way this channel is being indicated and there is only the channel information has to be indicated only using two bits right because UE can indicate only one of these entries and it needs only two bits to do that okay right. So if you need to send more information basically if the codebook size is bigger then you need more bits and those codebooks are also defined in the standard okay and those codebooks are typically used for multi-user MIMO okay and I have not taught you that but the feedback required for sending those bigger size codebook is much more than the simple example which we are taking now okay right. So and feedback runs into hundreds of bits okay you see only two bits this feedback runs into hundreds of bits for different kind of codebooks okay and that is as I said for multi-user MIMO. So now what does this codebook contain? It contains only four pre-coders okay as I said these four pre-coders and UE will choose the pre-coder entry which maximizes the SINR, SINR of I and how did we define this SINR of I that is defined in the next slide okay and how is this calculated? UE assume that base station is using the pre-coder V of I because that is what it is indicating okay. It assumes this is a system model okay right if whatever in pre-coder entry it will feed back base station is going to use that entry as the pre-coder okay so and it knows that it knows the system okay that the base station has because the base station has two antennas because it is picking out the entry from two antenna codebook right and it has estimated user has estimated these two channels only and they are un-pre-coded channels right. So all this information is not hidden from the user that how many antennas base station is using to transmit the signal in this case right okay correct. So what UE assumes is that GnodeB is using this pre-coder and GnodeB will transmit if it is using this pre-coder what will GnodeB do? Pre-coder what is the first entry of the pre-coder VI1 that is okay so what will be sent from first antenna VI1 into X if X is the transmit signal because you will transmit only one symbol it is a layer one transmission okay right and so GnodeB will transmit this from first antenna and this is from it will transmit this from the second antenna because this is a second entry multiplied by the symbol okay right. You remember this where this is coming from okay you what happens VI1 right this is what we what is being transmitted and so this is from first antenna and this is from second antenna and the received signal is this thing correct this is the signal from first antenna gets multiplied with the channel from first antenna similarly from second antenna and the noise right and so this is what user is assuming that is what it is that is how the signal will be transmitted and it knows this channel okay and what it needs for calculating the SNR is basically the power of X right and then what it does is it knows all the pre-coder entries okay HVI VI it knows and what it does is it calculates the SNR like this. Whichever pre-coder entry maximizes the SNR it feed back that pre-coder entry to the base station okay and this entry is information is called PMI okay pre-coder matrix indication okay and you see this, this is because UE has one antenna and then it is it implicitly knows that the rank of the channel is 1 right and if it has two antennas you will see that I am going to discuss that it will have to feedback something on rank information also okay so along with pre-coder matrix indication it feedback RI this is called rank information I will tell you how all that is done so we will have to extend that system model because in this case it is a simple model wherein the rank is 1 pre-coder matrix indication is whatever is out of those four entries okay right. So now from the last slide to this slide where are we moving we are what we are saying here is this is for a single antenna system now let us say I want to extend it to a generic case wherein transmitter has multiple antennas and user has also multiple antennas receiver also has multiple antennas and we know this system we have looked at it multiple times okay right so and we looked at it in context of TDD system wherein base station knows the channel and it can design the pre-coder correct okay but and this is base station uses pre-coder V NL layers pre-coder maps this to the NT antennas goes to a channel multiplied the receive signal gets multiplied by W of H and NL layers are basically recovered at the receiver okay. So this is a generic framework which can help us in transmitting NL data layers okay right and if what happens if NR is set to 1 it reduces to the framework on the last slide because number of receive antenna is 1 okay fine okay now how does this work then we write it again mathematically what happens here. So mathematical model for this NR cross NT system with how many layers this is the number of layer okay what is the system model what is this H X tilde X tilde is a pre-coded signal which is V X tilde is a pre-coded signal and what have I done I have just written H as W lambda V Hermitian okay SVD and then what happens here I pick the pre-coder as some M okay I am not writing it as V because base station does not know that right okay. So this pre-coder base station should find out what is the pre-coder and who is going to feedback this information user correct okay right and W lambda V Hermitian and if it finds out that this is the base user feeds back this information okay then the pre-coder M becomes V okay I am not directly writing it I am saying that who is the user has to feedback this information and user will have to calculate this V and then feed it back to the base station then base station is going to use this pre-coder okay right that is why this two step process and then correct if then base station later if base station uses this pre-coder okay then what will happen user will use a particular receiver and if in this case everything is fine it knows that channel is stationary nothing has changed it will design this receiver will be W Hermitian and will be able to recover lambda of X it is a diagonal system okay if you look at Y tilde all the streams are decoupled together okay and will be able to recover the data okay but the clear problem what we are looking at is how base station find out what pre-coder M to use correct because it does not have a channel information only user has the channel information correct okay. So, if it is a NT cross NR system or NR cross NT system what happens base station will first transmit NT port CSIRS right okay because user has to estimate the channel and what is the channel dimension NR cross NT right it is a un-pre-coded channel okay correct and then in order to estimate this complete channel these many ports of CSIRS should be transmitted let us say that has happened okay it is assumed that base user has estimated the channel okay and after that what happens okay user has a channel information right how does it design this pre-coder what is the code book and then the next step of estimating the rack how does that happen that is what we are going to look at okay fine. So, for GnodeB to use this pre-coder UE has to either feedback H or V right either feedback this H or feedback this V okay and we know now that it will never feed H okay it will feedback V okay and that feedback and then this V will be one of some code book entries right okay we saw that for a simple case system now we are doing it for some like we for some other system where in the rank is not 1 okay and we are going to do it for rank 2 system okay right. So, but the same idea will be applicable for higher rank system also okay what are we saying here it will neither feedback H okay nor feedback V also directly it will feedback some of the code book entries which are defined which are known both to the base station and the user okay right. So, now what are we saying here what I am so I am looking at a very specific case again simplifying the system and if you understand what is happening here they are applicable for any generic NR cross NT system what am I what is this what are we considering here I am considering a 2 cross 2 system okay 2 transmit 2 receive okay you see this is a code book I am taking a simple example this is how like I have simplified it because if you look at 5G the number of base station antenna is 64. So, I have reduced the system for us to understand what is happening okay correct so this is 2 transmit antenna code book only 2 entries so 1 bit feedback fine okay one of these entries need to be transmitted back to the base station by the user okay and what user has is has done is it has estimated the channel H okay fine right so what will it do so first thing you notice is rank 2 code book has only 2 decoders and compared with 4 of rank 1 okay so this is I have simplified it and now so the system model for this code book based MIMO what we are doing okay this is called spatial multiplexing also because you are multiplexing multiple streams and parallel just for the record okay so what is this is our system right okay y is equal to Hx tilde plus n okay then user assumes that one of these decoders entry is transmitted okay something similar to what we did in the last case okay Vi okay user has the channel information right so what will happen I call this as H tilde of I Hv of I what is I either of these code book entries correct okay right and then I call this H tilde of I and then we will have to design some receiver correct because what is happening here it is this is one thing which is different from rank 1 case okay you look at this 2 layers are being transmitted okay these 2 layers need to be separated right then the receiver needs to be designed by the user okay so there is some receiver W which user is designing it can be SVD based it can be zero forcing it can be any MMSC based receiver okay that is being done and once that is done right what happens okay so this is this is the equivalent system WH of Y it becomes what is this we can call this as Y tilde okay I am not right so this is this is a vector okay and user looks at this vector so we are assuming that this once this receiver design is done user does this operation all these 2 streams are decoupled together or these 2 layers are decoupled together right okay fine then what does it do okay. So now there are 2 entries okay what will it do so it has to design what are you saying here it has to decide which pre-coder information should be fed back right okay how is it going to do to select this particular pre-coder V of I UV does the following okay for both of these entries right I will run from 1 and 2 the 2 pre-coder entries okay for each one of them right because it does not know what is the best entry so it has to run this operation for both of them correct okay so it does what it does is it does HB1 okay designs a receiver W for that correct okay right I call it WI okay what is this HI tilde this is this is right okay so step number 1 pre-coder 1 you calculate H1 tilde design W1 for that okay so it can be any receiver based on what UV capabilities are okay can be like MMSE receiver it can be SVD receiver it can be zero forcing receiver any receiver okay but the receiver design will be different for H1 tilde and H2 tilde why because in one case you are multiplying H with V1 in another case you are multiplying H with V2 right. So the matrix become different the inverses will be different that is what I am saying if you are using zero forcing receiver inverse of HI H1 tilde is different from inverse of H2 tilde correct okay so this step has to be done for both these pre-coders okay right so you idea is clear okay H is estimated then H V1 tilde is H1 tilde is calculated okay and W1 is calculated then this operation is done right these two streams are decoupled now for pre-coder V1 similarly you have to do it for pre-coder V2 as well right okay so once this is done what happens okay that what will user do after that okay this is the first step okay. Now second step will be see these two streams are decoupled together okay right what does what will it find out in this case in the last case it found out SNR there was only one layer okay and in this case it will find out SINR signal to interference plus noise ratio okay right what will it find out in each case let us say it does this operation for H1 tilde okay how many layers are there two correct each layer first layer okay what look at this what am I saying here SINR I1 okay SINR when the pre-coder entry is one correct okay what will be the SINR P is there because X is by and not and what am I writing over here some function of the channel okay that because it will if you do zero forcing correct the signal will not have any channel term but the noise will have right H inverse will multiply the noise correct. So it will be some SINR will be some function of the channel of course okay we do not know what it will be based on the receiver design so this is something you can say that because if we do the SVD what will come here singular values you remember lambda 1 square P by and not lambda 2 square P by and not if it is zero forcing those lambdas will be different okay because those coefficients will be different because you have done a different receiver design right okay so SINR is a function of receiver design plus the pre-coder also because what are you doing you are looking at a H tilde H1 tilde H2 tilde right okay. So what has happened it is looking at pre-coder I calculating the SINR of first layer calculating the SINR of second layer right okay and what it does is computes the sum rate okay right okay it is interested in maximizing the rate log 1 plus SINR right you remember in the last case what did we maximize SINR that was indirectly saying as it is maximizing the rate because there was only one layer log 1 plus SINR right okay here what am I saying here is that this is SINR okay there are two streams each of this stream will have a different rate okay correct so what it does is it calculates the rate of first stream it calculates the rate of second stream for each of this pre-coder I right and calculate a sum rate look at the index it is SRI because you are calculating for a particular pre-coder I so you will calculate this for you will have two such numbers of some rate okay for each of this pre-coder right it will become sum rate 1 and sum rate 2 what happens after that feedback the index of pre-coder matrix VI with which maximizes the sum rate SRI two numbers whichever is greater is basically if right you will feedback that pre-coder index right V1 maximizes the sum rate you tell the base station that please use this pre-coder for me it is maximizing my sum rate right okay similarly here okay whatever pre-coder entry maximizes sum rate that is fed back by the user to the base station right okay so this step you should remember these two steps these steps are done before data is transmitted because what is what are we doing okay that look at the look at the flow of the system okay it is a FDD system or even TDD system when base station does not have the downlink channel information right it wants to pre-code the signal it is asking the user to feedback that information back to the base station okay so data has not been transmitted correct okay so this is only the pre-coder calculation stage and it is done when CSIRS is transmitted okay and when why will CSIRS be transmitted because base station wants to find out the pre-coder information right and in this case in whatever we have learned the CSIRS which is being transmitted is un-pre-coded CSIRS correct right okay and in fact if you look at it then so this is sum rate okay now during this step if this is a sum rate user can feedback the MCS information also right what is the MCS information rate only right okay so it will feedback PMI and feedback MCS correct because what is MCS the sum rate which is being observed now look at it if there is the way I have taught you the way it is done in the standard right how many MCS indices user will feedback in this case only one because only one transport block is sent from the MAC to the physical layer when the rank is less than or equal to 4 right okay correct so and that MCS information will correspond to this particular rate right okay so fine okay so PMI and MCS okay is this clear okay how this is done and please remember that this channel information is fed back indirectly right and base station is also not interested in finding out the channel it is interested in finding out what pre-coder it should use and if you tell that you use this pre-coder it does not care about the channel after that why should it you need that channel information to as I said to calculate the pre-coder right okay so raw channel as I said is never fed back by the user to the base station it is only one of the pre-coder information is sent and that to also through code book entries which is which are known both to the base station and to the user right okay fine there is one more thing left and that is a rank information okay because if you remember okay so that is that is in this slide we will discuss about this idea of rank calculation okay and then in the next class we will look at in all in details what how this happens okay. So even so what is this rank we are looking at a channel H right for the system which we considered earlier transmitter has two antenna receiver has two antenna okay so it is a what is that how does the channel metric what is the dimension of the channel metric 2 cross 2 right now what is the maximum rank of a 2 cross 2 matrix 2 what is the minimum rank of a 2 cross 2 matrix 1 right okay because if two columns of this channel are dependent on each other what happens the rank becomes 1 the basic fundamental linear algebra okay right that rank can be 1 or 2 okay and the rank of such a matrix is 2 if two columns are linearly independent of each other rank of this matrix is 1 if the columns are dependent on each other right now I am bringing this notion for the first time that this rank can be 1 or 2 okay and it could happen in practice also right that user is observing a channel whose rank even though it transmitter has two antennas and receiver has two antennas it is not guaranteed that the rank will always be 2 okay now the channel could be such that these two columns are dependent on each other could happen okay right how can it happen I am not going to discuss that in detail here okay we do not have time but the rank can be 1 or 2 that is the problem okay right if the rank can be 1 or 2 user need to find out that also because user only has the channel information okay if the rank is 1 then it will be how many layers user base station can transmit only 1 the rank is 2 can transmit 2 layers right okay fine and you will feedback that information also back to the base station there is so in the control information which is fed back as I said user has to feedback ri also okay so that is the problem okay that channel rank is 2 but the channel rank can also be 1 there is a maximum minimum is 1 fine okay we are bringing in this notion for the first time because we have to discuss about rank information till now we are we have always been saying that the rank is 2 if it is a 2 x 2 system okay we did not have to like focus on this rank 1 because there is no point in doing that but rank can also be 1 and that is what we are going to see so this rank has to be indicated to the g node be by the user because user only has the channel information okay this parameter is called rank indicator okay right and how will you will decide which rank to indicate correct okay there should be some recipe again that comes from I will teach you very basic principles okay if you have understood what has happened on the last slide how to calculate the sum rate okay this rank calculation recipe will also be easy to understand okay so we will next discuss an algorithm that you can employ to calculate the rank the recipe I will give you for this system how to calculate the rank okay then what will happen with this information is that you feedback you has to feedback what will it feedback feedback ri will feedback PMI will feedback MCS to the g node be okay right if it is so this always needs to be feedback because what we are the way 5G or 4G systems were designed a station always has multiple antennas okay correct so whenever the control information is sent right by the user to the base station these three parameters will always be sent by the user because there is always a MIMO transmission the number of layer could be 1 right okay because base station is deploying 64 tens of antennas correct okay right so whose if this is the case which we are discussing wherein like base station does not have the channel information okay user will always have to feedback three parameters not only MCS and because without these two parameters base station cannot precode the signal and we know without precoding a signal those multiple antennas at the base station okay serve very limited purpose okay right so this is always done correct okay so we will in the next class we will immediately begin with discussing this rank indicator recipe okay very simple okay but before we discuss that and to conclude this class what have we learned in this class we have what we have seen here is that we have extended the idea of feeding of UV feeding back this precoder information from nl is equal to 1 layer to nl is equal to 2 layer okay right and if it is for nl is equal to 3 layer what will happen same idea right okay so it will be let us say it is a 3 cross 3 system or 4 cross 3 system or 64 cross 3 system correct user will have the complete channel information through the CSIRS right okay and it knows that we see UV knows that it has three antennas so what will it do right there will be a different code book not this code book there will be a different code book which is defined for this system correct okay we will do all these operation whatever we did in the last slide okay so it will calculate basically the depending on how many entries that code book has okay let us go back just to how many entries code book has right okay it will feed back instead of one or two it will feed back let us say there are like 16 entries in the code book in a for a higher dimension system you will have to do this operation right for all 16 entries of the code book correct and then if it is a rank 3 system right it will have to do so it will figure out how many such summaries will be there there will be 16 different sum rate corresponding to 16 different entries of the code book right and how will that sum rate these sum rate be calculated because maximum rank is 3 we are assuming that okay i1 i2 and i3 sum all of them for 16 different code book entries okay whichever code book entry gives you the maximum you feed it back to the base station right okay so the idea remains same estimate the channel right design the pre-coder for design the receiver for each of this pre-coder entry calculate the SINR of 3 layers sum them there will be 16 different entries 16 different sum rate corresponding to 15 16 different entries of the code book feedback the maximum that is all okay we stop thank you.