Wireless Communications: lecture 11 of 11 - 5G communications

Summary

Highlights

• Henk discusses moving beyond textbooks to focus on cutting-edge 5G technologies! 📚➡️📲
• Dive into MIMO capacity and understand why massive MIMO is a hot topic. 🏆
• Explore the world of multi-user MIMO and how it helps create a rich channel environment. 🌎
• Learn about pilot contamination - a crucial challenge in the world of massive MIMO. 🛑
• The lecture concludes by summarizing the vital role of 5G in modern wireless communications. 🎓

Key Takeaways

• Get ready to dive deep into the world of 5G with Henk Wymeersch! 🌐
• Learn about the differences between single-user and multi-user MIMO and why they're crucial for 5G. 🤓
• Discover how massive MIMO technology maximizes using multiple antennas for increased capacity. 🛰️
• Understand why channel knowledge and preprocessing are game-changers for downlink communication. 📡
• Pilot contamination is a thing! Learn why it's important in measuring channel capacity. 🚦

Overview

Wireless Communications: lecture 11 of 11 - 5G communications Transcription

• 00:00 - 00:30 welcome to lecture 11 of the wireless communications course at Chalmers University of Technology my name is Hank Ramesh in this lecture we move away from the textbook and look at more current topics will focus on 5g in particular looking at the key technologies at my mo capacity multi user manual and finally massive MIMO the lecture is based in part on the book but also on the
• 00:30 - 01:00 following papers first of all 5gu tutorial overview of standards trials challenges deployment and practice by Mansoor Shafi at all-but overview of massive MIMO benefits and challenges by Liu Liu a tall and massive MIMO 10 myths and one critical question by email bjornsen at all learning outcomes for today are the following you should be able to describe the key characteristics of 5g difference between single user at multi-user MIMO Express multi-user MIMO
• 01:00 - 01:30 and uplink and I'm downlink as a standard mammal problem explain what is massive MIMO a wide is beneficial in terms of orthogonal channels and also describe what is pilot contamination last time we focused mainly on the narrowband MIMO model is given by the expression on the top where H is a channel matrix this is a complex matrix of size M R by M T where M R is a number of receive antennas MT the number of
• 01:30 - 02:00 transmit antennas when the receiver has an estimate of the channel age you can perform different detectors such as maximum likelihood 0 forcing or MMSE maximum likelihood is the optimal detector it requires searching over all possible X values that could have been transmitted and is thus exponential in the number of transmit antennas zero forcing tries to invert the channel however this fails when the channel is poorly conditioned as the inversion leads to noise enhancement finally MMSE
• 02:00 - 02:30 tries to approximately invert the channel and thus reduce noise enhancement when challenging information is available at the transmitter we can do much more we can use a singular value decomposition to create parallel non-interfering streams the expression for the singular value decomposition is shown here where u and V are unitary matrices and Sigma is the matrix of singular values Sigma is a diagonal matrix where all the
• 02:30 - 03:00 entries on the diagonal are non-negative the figure below shows how we use the SVD what we do is we take our data symbols X tilde weave pre-code them with V this then gives us the signals that we will transmit over the channel we apply them to the channel H we receive an observation Y and then we apply a so called shaping transformation to have our final observation X tilde Y tilde we
• 03:00 - 03:30 see that Y tilde can be related to X tilde using a concatenation of these matrices we can substitute H for its singular value decomposition this is shown here now since u and u U and V are unitary u hermitian u is an identity as is v hermitian v this means that y tilde is diagonal matrix times X tilde plus noise in more detail Sigma is the matrix of singular values it's a diagonal matrix with these entries this entries are by
• 03:30 - 04:00 convention ordered from largest to smallest they are all non-negative the final entry here Sigma RH there is no negative that is nonzero as this subscript R H which is the rank of the channel the rank of the channel is always less or equal than the minimum of the number of transmit and receive antennas we look now in more detail what is y tilde we see that Y tilde K so the gate entry and Y tilde can be written as
• 04:00 - 04:30 Sigma K the gate entry and matrix of singular values times X tilde K plus noise this is true for any K ranging from 1 to the minimum of MT NMR however when K is greater equal than RH Sigma will be 0 so basically on those streams you only observe noise this means that the size of X tilde is limit to our age it does not make sense to make X tilde longer than our age as an
• 04:30 - 05:00 extreme case one could choose to make X tilde of length one and this is called beamforming in that case we only send our information over one stream using the largest singular value we also briefly talked about my mole and the wideband regime in that case we would combine my mole with OFDM using OFDM transmitters and receivers if we then look at any given subcat have again a
• 05:00 - 05:30 narrowband Momo model so ZK is now the observation on subcarrier k HK is the channel matrix on subcarrier k HK is again of size M R by auntie in contrast to 4G which is mainly focused on providing high data rates this is just one of the application scenarios for 5g so-called massive broadband massive broadband involves very high peak data rates as well as very high rates
• 05:30 - 06:00 whenever it is needed we will also support many more devices and much more traffic however massive broadband is one of the area's there is also two others one is massive machine type communication for instance for sensor networks and Internet of Things here the focus is also on having many more devices but more specifically on low power consumption a third application scenario is critical machine type
• 06:00 - 06:30 communication here our focuses on is on reliability and latency this can include applications such as virtual reality and autonomous driving the requirements and use cases are here and this table described in more detail so again there are three main use cases enhanced mobile broadband ultra reliable low latency communication and massive machine type communication for Internet of Things it is important to note that the requirements are very challenging
• 06:30 - 07:00 depending on the application so for enhanced mobile broadband we can have downlink data rates of up to 20 gigabits per second for ultra reliable low latency communication we can have latent seas of about one millisecond which is much less than current latencies for Internet of Things massive machine type communication we can have many devices per square kilometers we will also be able to deal with very high velocities of the users very large bandwidths up to one gigahertz and very low error
• 07:00 - 07:30 probabilities in order to meet those very demanding requirements 5g will rely on a number of technological and neighbors it will use much more bandwidth than before by going to higher carrier frequencies above 24 gigahertz where multiple gigahertz of bandwidth can be made available this is so-called millimeter wave secondly we will use more antennas so rather that a base station having just a few antennas it can have hundreds of antennas this is
• 07:30 - 08:00 called massive MIMO this will be the main topic of today's lecture we will also have many more base stations so there are shorter distances between the users and the base stations this leads to so-called femtocells we will also design new kind of signals and new kind of codes frizzes for short packet transmission and things such as polar codes finally there will be many other technological enablers such as device to device communication and network slicing in order to understand massive MIMO we
• 08:00 - 08:30 should first look into multi-user MIMO in order to understand multi-user MIMO it is worth to look back to the memo capacity this slide provides an overview of the capacity for the narrowband MIMO system for a given channel the observation model is shown by the first bullet where we have a certain covariance matrix of the transmitted signal and a certain transmit power the channel can be decomposed using a
• 08:30 - 09:00 singular value decomposition where Sigma is the matrix containing the singular values along the diagonal we accept without proof that the channel capacity of a MIMO system for a fixed channel is given by the following expression B is the bandwidth H is the channel R is the transmit covariance and again we enforce a certain transmit power if we have one transmitted and one receive antenna this expression refers to the standard single-input
• 09:00 - 09:30 single-output channel capacity we now look at two different cases first of all when the channel is not known to the transmitter in that case the only covariance matrix that makes sense is a scaled identity matrix we just split the total available power across all the transmit antennas we can substitute back this expression here and make use of the fact that h times h hermitian can be written like this in the end we find the following expression of the capacity we
• 09:30 - 10:00 can now pull out u hermitian and u to the left and the right hand side these are just unitary transformations so they don't affect the identity matrix which allows us then to write the capacity as the bandwidth times the log of the determinant of this matrix here so note that the u hermitian u are gone this overall matrix is diagonal because the identity matrix is diagonal and sigma and there's also sigma squared are diagonal the determinant of the rag of a diagonal matrix is just the product of
• 10:00 - 10:30 the entries along the diagonal so this is given by this expression finally the logarithm of a product becomes the sum of the logarithms so in the end we have this final expression of the capacity of the Maya channel without channel State Information at the transmitter on the other hand when channel state information is known at the transmitter we can create parallel channels and do water filling across those parallel channels this is given by this
• 10:30 - 11:00 expression here so this is just the standard water filling problem we do a maximization of the power across different spatial streams and this is our objective function we explicitly and called the constraints here as before when the channel is random we will have different notions of capacity such as average capacity and outage capacity so now we know that the capacity in the case of channels in information to receiver has a following expression it have also sum over the nonzero singular
• 11:00 - 11:30 values now here we see the nonzero singular values squared there's no interesting to ask ourselves are there certain values for this nonzero singular values that lead to better or worse channels in order to answer this question in a fair way we need to fix the total energy of this of the channel this is equivalent to setting the trace of Sigma squared to M R by M T so when the total energy in the channel is fixed we can now look at different distributions of these
• 11:30 - 12:00 singular values the first case is when there's only one nonzero singular value with all of the energy this would correspond in practice to a line-of-sight propagation case with only a single path the other extreme distribution is uniform distribution of the singular values in which case that we have M R times empty equals singular values each with a small energy this would happen for instance in the really fading case with iid complex Gaussian entries in the channel matrix this is
• 12:00 - 12:30 the same as seeing a rich non-line of sight environment the capacity for the line-of-sight case is the bandwidth times log 2 1 plus an SNR where this SNR then increases linearly with the number of receive antennas for the no line-of-sight case we have the bandwidth but now we also have this scaling with the minimum of the number of receive and transmit antennas log 2 plus 1 plus an SNR metric metric which depends on the number of transmit and receive antennas
• 12:30 - 13:00 in general for an arbitrary distribution of the singular values with a total fixed energy the capacity must lie between those bounds well we see in regard to the upper bound is that this has a very favorable scaling with respect to the number of antennas so when we put more transmit or receive antennas the capacity scales linearly this is extremely powerful especially when we go to very high order mimal with many antennas this is exactly what we
• 13:00 - 13:30 try to exploit a massive MIMO so now we know that in my mo communication we want to have a rich scattering environment because this provides us with a linear scaling of the capacity in terms of the number of antennas so a natural thing to do would be just use standard my mo but adding more and more transmit and receive antennas however in practice this does not work because the propagation environment does not provide enough richness to generate uniform singular values in practice only
• 13:30 - 14:00 a few singular values will be available and all the rest will be zero zero or close to zero a way to avoid this is by going to multi-user mimo where we have transmitters and receivers many of them each with only a few antennas then we can also reap these MIMO capacity gains so multimode user memo we are trying to find a new way to create rich channels so in single-user mimo the benefits from adding more antennas is low because the channel rank tends to be low even under
• 14:00 - 14:30 large antenna regime in multi-user MIMO we create rich effective channels through the spatial separation of users the picture on the right shows we have a base station and a multi antenna receiver which comprises different users and because those users are separated in space the channel tends to be rich the mathematic the mathematical model is now as follows to downlink signal for user k
• 14:30 - 15:00 YK is obtained by a signal HK XK so where XK is a signal intended for user k HK is a channel from the base station to user K plus the signals intended for the other users multiplied by the same channel HK because the channel from the base station to the user is the same channel HK plus noise so this is the downlink communication signal in the uplink we have the following each user
• 15:00 - 15:30 has its own data XK send this over the channel HK transposed and then there is at the receiver all those signals add up and we have our observation Y here we have assumed channel reciprocity so we are operating in a time division duplexing mode so down like an up lake occur over the same frequency band that allows us to write that downlink channel and the uplink channel are just related to a transpose we now study the
• 15:30 - 16:00 operation of uplink and downlink in more detail let's first start with uplink because it is easier we can write the uplink observation Y as a large channel matrix which comprises the concatenation of all the uplink channels times a long vector which contains the transmitted signals for all of the users the channel matrix is of dimension M R so it has M R rows times K times empty columns so the M R
• 16:00 - 16:30 rows correspond to the M R receive antennas at the base station M T is the number of transmit antennas per user and K is the number of users we see that basically we now have the standard MIMO communication system right why is H times X plus n so we can apply standard MIMO detection methods such as maximum likelihood zero forcing MMSE and so forth so for this case we don't need channel-set information at the
• 16:30 - 17:00 transmitter at the users in order for the base station to recover the data so this one is relatively easy and we can use all the techniques from standard manual multi-user MIMO and the downlink is far more interesting in case the base station does not have challenged information all it can do is sent to different users at different times or over different frequencies so we revert back to TDMA and FDMA if the base station does have challenged information
• 17:00 - 17:30 it can do something much smarter called multi-user precoding it will basically send signals to all the users at the same time over the same frequencies and still allow the users to recover their signals with all complexity the observation at user K is given as before YK it involves signal antennas for user K times the channel plus signals intended for other users plus noise now rather than sending data in the XK
• 17:30 - 18:00 itself we will pre code data so here the SK is our qualm data and WK is a pre coding matrix so WK x sk is equal to xk the same goes for the other user so the data of the other users is also pre coded with individual precoding matrices and of course there is noise this may seem a bit complicated but let's look at
• 18:00 - 18:30 a most simple scenario where we have users with one antenna so we will send one stream of data to each of the users then the model is simplified to this YK is now the scalar observation of user K on it's single receive antenna SK is a transmitted qualm symbol for that user and this is the effective channel seen by that user it is a pre coding vector multiplied by the downlink channel
• 18:30 - 19:00 similarly for the other users there is interference of this form which includes the symbol antennas for the other users their pre coding vectors and then the channel of user k the noise is now scalar we can now stack the observations for the different users and we obtain this matrix model so the observation for all users is a matrix h times W times s plus noise this is written here in more
• 19:00 - 19:30 detail observationally users the channels for all of the users the precoding factors for all of the users and the data for all of the users so what the base station can do now is to design W so to choose W in a smart way to optimize a certain performance criterion such as zero forcing or MMSE the number of users that we will transmit to can change over time and the best K users can be selected out
• 19:30 - 20:00 possibly out of a large set of users in order to have a good effective channel so suppose we have a hundred users available we will choose maybe the best ten to send to in order that we can design a good precoding matrix the selection of the best users is called user scheduling there is also a version of pre coding that does not rely on instantaneous channel seed information but rather statistical channel State Information for instance the covariance matrix of the channel
• 20:00 - 20:30 so now we should look in how we should design W in order to make life easy for all of the users in multi-user mimo precoding with again focus on a simple example with K users each with one antenna and one base station with M antennas the downlink channel is an ax K by n matrix where each row corresponds to the downlink channel of a single user we consider zero forcing pre coding in which the pre-cooling matrix is the
• 20:30 - 21:00 pseudo inverse of the channel we should be careful because when the matrix the channel matrix is poorly conditioned w may be very large and we will transfer it with too much power so we need to enforce the following constraint that the norm squared of w satisfies the total power constraint one way to do this is by scaling all the entries in W with a low enough number such that the constraint is met by doing this we guarantee an equal SNR per user which is
• 21:00 - 21:30 desirable in some applications on the other hand we will then be limited by the worst user we can be a little bit more aggressive by multiplying the columns of W with different scalars and this will lead to different SNR per user so we do this the following we take the pseudo inverse of the channel as our pre coding matrix s is the transmitted data and that in between we have a matrix of square roots of powers this is a
• 21:30 - 22:00 diagonal matrix where the first entry is the square root of the power assigned to user 1 the last entry is the square root of the power divide assigned to user K we can then play around with these powers so if all the powers are equal we get back to the first case of equal SNR per user we can also adjust the powers to optimize some objective function for instance the sum of the rates to all of the users so here we have a generic objective function f of P the total
• 22:00 - 22:30 power constraint of course needs to be considered and this can be expressed as a function of the powers allocated to each of the users this is this constraint now if F of P is some rate over all of the users then we find back our traditional water filling solution a question we ask yourself is why didn't we do SVD I leave this for you to think about finally now
• 22:30 - 23:00 we understand the my ammo capacity and the scaling with the number of antennas in a rich scattering environment as well as that in multi-user mimo it is relatively easy to generate such a rich scattering environment we are ready for our final topic massive MIMO while multi-user MIMO is beautiful in theory it never worked out in practice the reason is that to have many antennas on a mobile device is very costly also when we only have very few antennas say less
• 23:00 - 23:30 than 10 the gains are relatively modest also the signal processing is complex mainly at the base station on the other hand when we look at an extreme case of massive MIMO we will have one antenna per user and a very large number of antennas at the base station the figure here shows the difference between 4G MIMO communication and 5g MIMO communication so in 4G we have a relatively limited number of antennas in
• 23:30 - 24:00 5g a massive number of antennas far exceeding the number of users and as the picture handset we can basically send distinct beams to each of those users so we will now try to understand what it means to have such beams and how this leads to capacity improvements the capacity of the mine would channel is given by this expression we will consider here only the case where there's no channel State Information the transmitter which gives the transmit covariance is a skill identity matrix
• 24:00 - 24:30 our capacity bounds are shown here on the left hand side for the line-of-sight channel on the right hand side for the rich scattering environment we now have two cases is one where ever have a very large number of transmit antennas and AM channel matrix with iid complex Gaussian entries the channel matrix will look down something like this the number of rows is the number of receive antennas which is small and the number of columns is done with
• 24:30 - 25:00 transmit antennas which is large so the matrix will be in so-called fat matrix because we have complex Gaussian entries in a very high dimensional space along the different rows each of the rows will be almost orthogonal this implies an h h hermitian which is of course a small matrix will be a scaled identity matrix where the scaling depends on the number of transmit antennas this means with
• 25:00 - 25:30 very large number of transmit antennas this will be a large number the capacity then we find easily that the capacity is the bandwidth now times the number of receive antennas log 2 1 plus snr so here mr is a number of receive antennas which is a relatively small number and we can also consider this to be the number of users in case each receiver has a single antenna but there are many receivers and more of them in case 2 we
• 25:30 - 26:00 become we consider the case we have a very large number of receive antennas and the channel is again of the same form in that case H will be a so-called tall matrix with many rows each corresponding to a single receive antenna and few columns because we will have few transmit antennas you can apply the same math we find that h hermitian h this is different from before where we
• 26:00 - 26:30 had HH hermitian is a skilled identity matrix a small matrix with a scaling mr that is according to number of receive antenna so for lots of receive antennas this number can be very large we apply the following matrix equality to the standard capacity expression and we find that the capacity is the bandwidth times the number of transmit antennas and the log 1 plus an snr again this expression
• 26:30 - 27:00 is in general written for a number of transmit antennas but we could also consider this to be a number of users when each user has a single transmit antenna so this interpretation as number of users is interesting for our massive my more case but you don't necessarily in to consider this for general my mo all right so what's important here is that in both cases where we have a very large almost transmit antenna or a very large number of receive antennas we see this
• 27:00 - 27:30 linear scaling of the capacity with a minimum of the number of transmit and receive antennas we can only reach this under so-called favorable propagation conditions for which the channel matrix has almost orthogonal roles in case of the number of large transmit antennas or almost orthogonal columns in the case of the large number of receive antennas now we go back to our massive MIMO scenario we will change notation slightly so please beware we have a single base
• 27:30 - 28:00 station with M antennas where m is a large number and K users where K is much smaller than M each user has only a single antenna so this is shown in the figure in the bottom right in order to have favorable propagation conditions we want users to be well separated so they see independent channels we will use multi-user MIMO at the base station with channel reciprocity so this means that we use time division duplexing when
• 28:00 - 28:30 users are sufficiently far away from each other we have nearly orthogonal channels the H matrix which is the channel in the uplink is an M by K matrix so it is a tall matrix or many rolls and a few columns we will decompose the H matrix into two parts the Part D corresponds to the path loss so this part of the of the channel is a diagonal matrix of dimension K by K and this is the per user pattle so this
• 28:30 - 29:00 relates to the distance the matrix G is the multipath fading so this we will make as Gaussian matrix with iid entries it then follows that G hermès G hermitian G is an identity matrix of small size a K by K matrix with a scaling of factor M the number of antennas in the downlink we have the transpose of this channel so the dialing channel from the base station to all of the users is a K by M matrix so this is a fat matrix with a few roles
• 29:00 - 29:30 and many many columns and we find that this must be equal to the pathless matrix times the Gaussian matrix and this Gaussian matrix is of course of size K by ax if we compute G transposed chief star then we find that this is a skilled identity matrix of size K by K with a scaling factor M again the number of antennas so you can verify this by plugging in these matrices here it's
• 29:30 - 30:00 important to know that G in both cases will have iid Gaussian entries 0 mean and variance 1 in the figure I also show the channel between a single user and the base station and this is H I so H I is one of the columns in the big h uplink matrix so H is a column vector of length and by one H I can also be written as square root of D so this is the partly due to the path loss times G a vector of length M by 1 with iid
• 30:00 - 30:30 Gaussian entries now that we understand the uplink and downlink model let's look at each of them in more detail first the easy case uplink the observation observation model is of this form where H is a tall matrix with many roles and few columns X is a K by 1 vector containing the signals for each of the K users we can now compute h hermitian h and we find that this is equal to M times D M is the number of antennas at the base station
• 30:30 - 31:00 D is a diagonal matrix containing the path losses for each of the users this inspires the base station to use the following combining matrix h hermitian x y is a small vector Z of length K by 1 and turns out to be equal to the following expression M times D times X plus W the W has known statistics independent across users D is a diagonal matrix so this allows us to write a user per user observation model ZK given by M
• 31:00 - 31:30 times the petals of the cage user the signal sent by the gate user and noise the noises independent user by user this is important as an R per user is an SNR K which is found by computing the signal power and dividing by the noise power where we need to of course account for this noise variance we find that SNR has this expression we can compute the rate for the caged user by just substituting
• 31:30 - 32:00 the SNR in the rate expression we find the following rate for the caged user and that we can add up all the rates for all of the users because this is the sum rate that the base station will see from all the users combined and this is of course only the sum of the individual rates and we see that this expression on the right is exactly the capacity expression that we could reach for this system so this means that at least asymptotically for a large number of antennas at the base station a matched filter is the best thing to do so this
• 32:00 - 32:30 means that even though we have multiple multi user interference simple linear combining at the base station can lead to optimal results and the downlink things are a bit more complicated because the base station will need to do additional processing so that the users don't see interference from other users the channel in the downlink is given by H I transpose for user I note now that this vector is a row vector of length M
• 32:30 - 33:00 the base station will do some form of precoding so the base station will send W times s as X where s contains the data for each of the user so s is a vector of length K by one W is a suitable precoding matrix and this is the downlink channel H transposed and is the vector of noises seen at each of the users we know from the multi-user MIMO
• 33:00 - 33:30 slides that in general a good recorder is to do the pseudo inverse of the channel so here we'll do something similar we will choose a following 3/4 of the following form so it will be the complex conjugate of the channel times a diagonal matrix which contains the square roots of the powers that we will use for each of the user and a scaling with a square root of n this diagonal matrix here is to ensure that we meet
• 33:30 - 34:00 the total power constraint that we don't send with too much power so we need to ensure that the norm squared of W satisfies the total power constraint we find then that W hermitian w can be written after some steps as DP times D so here DP is the matrix of the powers along the diagonal D is the matrix with the path loss X the petals values along
• 34:00 - 34:30 the diagonal so why did we go through this math let's look again at the observation seen by the users so this is the aggregate observation across the different users given by H return h transpose WS plus noise we substitute what W is so we define W before and we find this for W we know that H transpose
• 34:30 - 35:00 H star has a well-known expression this is from a few slides ago and that in the end we find that Y is equal to this so here D is the diagonal matrix D P is diagonal matrix and n is independent noise per user so this means that your observation just breaks down into different observations per user as false so the gate entry in this vector Y we call this YK this is the signal seen by
• 35:00 - 35:30 a certain user K is some constant which relates to the number of transmit antennas the path loss seen by user K the power allocated by u to user K and the signal intended for user k plus noise there is no longer any multi user interference we can now compute the SNR seen by user K this as this expression and then we can go again through the exercise of computing the rate and the sum rate and we can conclude that also we have an optimal scheme so in downlink
• 35:30 - 36:00 the matched filter is asymptotes optimal I mean here the match filter pre colder and this means that we can do simple linear processing at the base station to remove all the interference in the downlink among different users so this is very powerful of course all of this hinges on this favorable propagation conditions let's recap what we know so far first of all in the uplink so from the user's to the base station when we have favorable propagation conditions there
• 36:00 - 36:30 is no need for any channels that information at the user's and the base station can do simple linear combining for instance using maximum ratio or zero forcing or MMSE in order to have an optimal sum rate so this means that the base station can get the optimal rates even without any collaboration among the users there is no inter sim enter user interference in the downlink again under those favorable propagation
• 36:30 - 37:00 condition the base station now needs channel seed information in order to find a good pre coder and practice the pre coder can be maximum ratio or zero forcing or MMSE and this again leads to optimal some rates there is no inter user interference this means that life is very easy for the users they just see their own data stream without any interference from other users so some of the benefits of massive MIMO are that we have simple processing at both the base station and at the user side we are
• 37:00 - 37:30 asymptotically optimal in terms of some rate and the fear ins goes away both an uplink and downlink and we can serve all the users at the same time over the same frequency on the downside we can have increased hardware cost and increased power consumption at the base station this is because we need many antennas and each antenna needs their own RF chain and analog to digital converter we have the need for challenging information at the base station we will
• 37:30 - 38:00 talk about this later how this can be achieved and we also need favorable propagation conditions in a relatively scattering environment each user will have an independent channel a question that becomes how many antennas do we need to put at the base station to see such favorable propagation conditions so we want at the base station site that we have the following property consider in our case with ten users and M antennas
• 38:00 - 38:30 at the base station the figure on the Left shows the matrix h hermitian h visualized in 3d we see that this is far from an identity matrix however when we go from 20 to 200 antennas we see that the diagonal is approximately the number of antennas and the off-diagonal elements are approximately zero it is possible to include compute the scaling of the off diagonal elements as a function of the number of antennas so we
• 38:30 - 39:00 see that in practice we would need in fact a quite large number of antennas at the base station to have these favorable propagation conditions we can also create favorable propagation conditions and line of sight provided we have enough antennas to do this we need to look at the geometry of the antenna array at the base station let's consider a uniform linear array where the antennas are on a line equidistance with in turn 10s spacing D the signals from the user is arriving at a certain angle
• 39:00 - 39:30 theta this means that the signal arrived at one antenna element at some time and at the next antenna element sometime later or this time later depends on D times sine of theta which we will abbreviate by Delta so let's look at a singular that is received by the first antenna element this will be equal to the transmitted complex baseband signal up to some propagation delay an amplitude due to the fat loss and the delayed carrier we can encounter the delayed
• 39:30 - 40:00 effect of the carrier in a rotation of the amplitude so we call it as G or we can change the time basis so that we have the following observation at the next on tala at the next antenna element and a new time basis the signal will arrive a little bit later Delta over C and we see this also in the carrier we assume that we cannot see this effect in the signal itself only in the carrier so this effect goes away and then we can
• 40:00 - 40:30 write the exponential of the difference as the product of the two Exponential's we can now go back to be too complex baseband so the receiver will down convert that means it will wipe out the carrier here so then what a receiver sees on its antenna element M is an observation ym which is G some complex number that's the same for all the antenna elements the transmitted signal and then a rotation this rotation depends on D on Delta also on the
• 40:30 - 41:00 antenna index here lambda is the wavelength which relates to the carrier frequency and the speed of light we can now stack all of these observations into a vector Y of length M and we can write this vector Y as the transmitted complex signal times the channel and now we also add a noise I've also removed here the notion of time because we don't need it so this means that for a given user k the channel would look something like this right it is a gain for that user and
• 41:00 - 41:30 then the delays across the elements for that user each user will have a different theta k because different users will come in when the signals from different users will come in from different angles we see when two users have two different arrival angles theta allan theta k when those are sufficiently different this product between the two channel vectors will be close to zero typically the spacing
• 41:30 - 42:00 between between the antenna elements is about half of a wavelength so now let's look at this product to see that this is actually the case we evaluate here this function for two different users k and now we will take the norm squared because that's one real number we can plot we can pull other complex gains over then in the end we end up with the summation over all the antenna elements
• 42:00 - 42:30 and the difference of the signs of the angles we call this difference Delta L case so this is some type of metric as to what extent users are separated in the angle domain after some straightforward manipulations we obtain the following expression which we visualize here as a function of Delta alakay the separation angle between the users so here we have Delta alkane we see that when Delta L K is zero no matter how many antenna elements you use at a base station the function will be
• 42:30 - 43:00 large so this means we cannot distinguish between the users so we cannot separate the signals from those users basically we have no whole when the number of antenna elements is small let's say 10 then the function only decays slowly so this means that for users with small Delta L K we still will not be able to distinguish them when M is equal to 20 the func is much more steep this means that when the Delta L
• 43:00 - 43:30 key is relatively small the product will be close to zero and we can distinguish the users in other words with more antenna elements it is possible to distinguish users that are nearby in the angular domain when the number of antennas goes to infinity then any pair of users can be distinguished and the channel exhibits favorable propagation or conditions a last topic we will cover is how the base station got channel State Information in this figure here we
• 43:30 - 44:00 consider one coherence interval of the channel over a certain coherence bandwidth the coherence time we call TC the bandwidth bc during this time frequency block the base station needs to do a number of actions so first of all the users will send uplink pilots to the base station from which the base station will estimate the channel and compute downlink 3/4 then we have some time for downlink transmission and also we have later than the uplink transmission so of this whole block a
• 44:00 - 44:30 certain time is dedicated to the pilots the more pilots we send the better we can estimate a channel so this is beneficial on the other hand the more pilots we send the last time we have available for data transmission so for this uplink training for certain coherence bandwidth BC and a certain coherence time TC we can send a maximum of BC times TC complex numbers over this fixed channel if we use a fraction a for pilots let's say 50% for pilots then the
• 44:30 - 45:00 rate that we can achieve will be of course 50% of the maximum rate we could have achieved if a genie told us the channel so we want to limit as much as possible the time we spend for uplink pilots one way to do this is by allowing all the users to send simultaneously to the base station over this coherence band we can do this when all the users have orthogonal sequences for pilots and then the basis you can resolve all of
• 45:00 - 45:30 those signals if we have more users what we then want to do is give some of the users the same pilots so now let's suppose the two users have the same pilot and send the same uplink signal to the base station then what the base station will effectively estimate is a superposition of the challenge of those two users so this then in turn leads to a wrong pre coder which leads to a loss in rate this in turn means that the SNR in both uplink and downlink our interference limited so fundamentally
• 45:30 - 46:00 the fact that we have limited time for the pilots creates an interference limit for the massive MIMO communication this effect is known as pilot contamination because the pilots from different users contaminate each other in terms of the channel estimate let's revisit the learning outcomes for today you should be able to describe the key characteristics of 5g the differences between single user and multi-user mimo Express multi-user MIMO and uplink and downlink as a standard MIMO explain why
• 46:00 - 46:30 massive MIMO users can have nearly orthogonal channels are why this is useful and describe the effect of pilot contamination this ends both this lecture and the entire course of wireless communications at Chalmers University of Technology thank you for your time in your attention