thank you and thank you for this
introduction and for this free our
introduction that you give to this topic
this morning the purpose of my free our
talk here will be to talk more about the
signal processing sites more into
detailed not in the sense that I would
like you to able to understand or grasp
every single detail but try to look into
how we do a number of things like that
blink detection the only precoding why
we select them in particular ways and
and show you some of the expressions
that we get in more general cases in
particular
covering spatial channel correlation and
we have a blog called the massive my
MOOC on this address and in that one I
actually wrote about spatial correlation
a few days ago so if you're interested
here I've tried to describe these topics
in some more simple terms in general
let's see if I can get my slides up here
so I call this presentation advanced
inner processing for massive MIMO and
the topics that I will talk about today
I will give you a short recap of what
mastermind was and then I will talk
particular about correlated Rayleigh
fading and what that is and provide you
an example how you compute the
correlation spatial correlation in the
communication system and then it is
correlated fading will go through all
the things we can talk about today so
one of them is the channel estimation
including powell condemnation one of
them is the uplink spectral efficiency
if we have time I will go through in
detail the downlink I will talk
something about asymptotic analysis when
you have many antennas what happens for
different types of processing schemes
and then at the end I will talk about
what I call spatial resource allocation
trying to differentiate that from the
classical type of resource allocation
time a frequency that has been going on
for a long time
slightly short thing about power
location and finish up with some
a case study and some open problems and
as Eric was indicating safety pointer
works yeah yeah this is based on an
upcoming book that I written together
with my colleague Jack appointees in
Lucas and Jeanette D which is currently
under reviews for something called a
foundation and trends in signal
processing and it's a monograph 450
pages is twice the size of the red book
that Eric was giving commercial about
this morning and it's intended for PC
students first year or so that would
like to learn its topic hopefully we'll
be appearing in the fall and there will
be printed books available for purchase
but the intention here also that you
will be able to download this material
for free as well along with simulation
code and everything
this slides will by the way be made
available for you as well so we will
start creating some kind of repository
for that sending out some link hopefully
in the evening today and that will
include more of the talks here as well
yeah so let's start with introduction so
when I talk about Massimo here I will
only focus on a multicellular system
where we have a number of a station
represented by this race here we have a
number of user
I call them user equipment so you East
in the LT fashion and we talked about
the uplink and downlink communication
downlink from the base station to the
users uplink from the users to the base
stations and trying to define what is
the maximum system the canonical maximum
system here is a multi carrier cellular
network with L number of cells that
selected I will repair many times and it
operates in secret as TD protocol which
the Cronus means that uplink and
downlink transmissions in different
cells synchronized so it's the uplink
that are interfering with each other and
the downlink interfering with each other
and this is something that you at least
can achieve in practice within the
neighboring cells which is the important
thing the base station number J is
equipped with the number antennas
J so it could be different for different
base stations and typically in order to
call the mass in Miami system we have a
large number of antennas and exactly how
many you have that depends a lot on how
you would like to define things maybe 50
or 100 is what you should expect in
order to have a mastermind system maybe
64 will be what they actually call Massa
mango in another D system for example
the base station communicates with a
number of juicers in itself and this
user M cell J is to notice KJ you see
this is the same kind of lettuce as in
the previous presentation but now I put
subscript here to indicate that it could
be a different number antennas different
num of users in each of the cells and
they are transmitted these users they
are single antenna users and they are
all sharing the same time or frequency
resources and we typically have the more
antennas than users exactly how many
more antennas that depends a lot on how
you would like to design the system and
we will look briefly on that later on as
well
and each base station operates
individually so there is no joint data
transmission from different base station
it might coordinate things on a
long-term like power location opposite
that in the same way as Eric was
describing but not in the short-term and
the operating using transmit precoding
that is linear in the way that we have
talked about earlier today and also
linear receive combining as well that is
how I will define MSM IRA system and we
saw a similar kind of figure earlier
today where we are taking the time
frequency resources and shove it up into
some kind of coherent interval so I
prefer to call them coherence blocks for
spanning two different dimensions here
spans the time dimension and we have the
coherence time TC and expense the
frequency mention where we have the
coherence bandwidth bc and depending on
what kind of multi carrier modulation
scheme you are using you might have
multiple sub carriers within one of
these clearance blocks
and each of these ones we are dividing
map in this following form in the uplink
we send data and out of this number of
samples tao-tsay being equal to be camtc
so that's the product at a time and
coherence time acquiesce bandwidth these
samples are divided into top you being
used for uplink data tau D be used for
downlink data and then we can have some
pilots in general these are tau P in
number such as how you tau D plus tau P
equals the total number of samples that
we have and today I will focus just a
tarik on having pilots only in the
uplink but you can potentially have
pilots in the downlink as well in terms
of numbering we have L cells and we also
don't have L base station because
there's one base station per cell with
number M from 112 there are KL use in
cell L these are number from 1 to K L
and if you look at intersection here
between two different cells I denote
this channels in the following way H I
use H as channels and not G as Eric was
having earlier today then as a
superscript I say the base station cell
number so that is one of these 1 2 L in
the subscript I'm identifying where the
userís and first arrived to users cell
number in this case the user is in cell
L so then will be L here then the user
is called certain number between 1 and K
L and that is number K in this locate a
in this case and then we will have h l k
and then when i looked at a channel 2
for example base station j I put J here
in the superscript this is an example of
a channel between the user K in cell L
and base station J in another cell and
by switching the different indices here
in superscript we can go through the
channel from this user to any other base
station in the Atty
network and what's important to notice
is that the length of this vector here
depends on who is actually the base
station so this case we're on vacation J
it is an MJ x one vector or matrix this
is the basic notation that we're gonna
be using and how do we now model this
kind of vectors these channels and then
there's the important thing or what's
called correlated or spatial correlate
channel correlation and had a following
definition a fading Channel H is
spatially uncorrelated if the channel
gain which can be written as the squared
norm of the channel vector so we just
take each element absolute value square
of it and send them up that's just n the
channel gain if we compare that one and
the channel direction which is just take
the vector direction and see we
normalize it so to rank yeah it's the
length 1 vector if these two things the
channel gain and shell Direction are
independent random variables then and
also the channel direction can take any
direction so it's anything being on like
a unit sphere in in this case in some
kind of complex M dimensional space but
anyway is any directions if oldham are
equal distribution equally likely then
we say that the channel is spatially
uncorrelated otherwise the channel is
spatially correlated and the point here
is that it's very detailed requirements
that you need in order for this to
actually be a non correlated vector that
means that if you should have an example
of something uncorrelated there is one
good example the unco literally ID
Rayleigh fading that Eric talked about
this morning where the channel is
complex Gaussian has zero mean an
identity matrix as the covariance matrix
so we can also cause call it the
correlation matrix because we have a
zero mean and in this case all the
eigenvalues of this correlation matrix
here are the same it's an identity
matrix all learn at the
the same that's one of the things we can
look at nor to see that something is
spatial uncorrelated you can probably
come up with some other example where
you have the same main properties Serie
mean and a correlation matrix being
identity but and maybe in something
that's non Gaussian but apart from that
this is what the case is where you get
uncorrelated fading yes yeah absolutely
so in general we can say that any model
where we have some kind of eigenvalue
variations in the correlation matrix
will have some kind of spatial
directivity so in certain directions it
will be more likely that channels are
pointing out in some other directions it
is less likely and that could for
example be that you have two different
levels some of them idea ballasts have
the same values some others are a much
smaller there are any type of shape
there on the eigenvalues would give you
some kind of spatial correlation and
when can we say the spatial correlation
is strong well then that is the larger
eigenvalue variations you have and that
you can measure in a number of different
ways you can of course compare their
larger and smaller side you value we can
also look at how they look like and try
to give some kind of evaluation is this
a strongly or weakly correlated to stay
close to the same or is it large
variations so it's hard to define
something in exact detail
well they will the you can potentially
have something that uncorrelated there
as well I guess it's not necessarily to
become independent in this case they
become independent but the important
thing there is that there is a coupling
if you have a certain direction then the
strength of the channel will also not
depend on which direction that you
actually have a channel so one key
example here is what ain't called
correlated Rayleigh fading where the
channel H here is complex Gaussian
distributed have zero mean and some
matrix are as correlation matrix or
covariance matrix these are the same
thing when you have a zero mean chopped
and as I said the larger eigenvalues
voracious aren't are the more
correlation we have in some sense and
based on this I'm now defining the
correlator Rayleigh fading that I will
use during the entire presentation today
so we say that this channel that we
looked at from user K s ll to some other
base station J it's distributed like
this it has a zero mean in the nasa
covariance matrix and we see that the
subscript and superscript are exactly
the same here so I'm only changing H to
R + RS than always used as a spatial
channel covariance correlation matrix
and in every coherent block we get a new
realization this is independent from
previous one that is the block fading
assumptions that we had previously today
as well the variations in H describe you
some kind of microscopic effects when
you're moving around a little bit in the
environment and you're moving a few
wavelengths or something like that that
doesn't change the big macroscopic
changes that are described by
discovering this matrix this queres
matrix is by the way assumed to be known
after receiving base station to the base
station that are at the top here and
estimating some like that could be
challenging that's the research talk by
itself I'm not going to cover it today
you can essentially create a sample
covariance me
you start in order to estimate it and
see the computer reacts the eigenvalues
and eigenvectors of this covariance
matrix is describing the spatial channel
correlation of the channel which then
means that the very high in values that
are strong their corresponding
eigenvectors will say something about
this is directions where the channel is
more likely to point in and maybe you
have five very strong eigenvalues and
the corresponding eigenvectors will then
hopefully your true meeting large
probability span the vectors it will lie
in that kind of subspace and this
covariance matrix you typically generate
it in some simulations using a model and
we'll discuss one example of that there
is a lot of effort being done into
channel modeling in general and how you
can create models for this ones in a
realistic environment and declares
matrix describes and macroscopic
propagation effects and one of those
things is the average channel gain from
an attend at this space station that
we're looking at to the user that we're
looking at and if you take the trace of
the matrix what is that well that is to
take either just some of the eigenvalues
or the sum of the diagonal elements at
the same thing and if you take that and
divide with the number of antennas then
you get some kind of average of the
value that you have along the diagonal
and that describes the average path
flows from one antenna at the space
station to this particular user and
this value you can model it in some
different ways you can say that in
decibel scale here there is some
distance D that describes the distance
between the transmitter and receiver and
then the path loss factor here a large
scale fading coefficient will reduce
become smaller smaller the larger the
distances in a typical way of doing this
in decibel scale say sir take 10 times
alpha where alpha is called a path loss
exponent log 10 are the distance and you
can measure a distance in different
units in this case I measure it in
kilometer which means that if I put in 1
kilometer well then this term goes away
and it is only this factor here that
describes the channel gain I would have
at a reference distance of 1 kilometer
in that number conquer for example B
minus 100 decibel something like that I
do typically gonna lose over 1 kilometer
distance and in bubbles of channels or
or this large scale fading we you should
also add a term here at the end that
randomized things it can be a Gaussian
distributed for example and it's often
called shadow fading often have zero
mean and some kind of standard deviation
that is describing how much channel
fading you have it might be 8 or 10
standard deviation for example and it's
called shallow fading and usually you
view this as being that the probability
that your shadow by big objects or
something like that which has some kind
of practical meaning what it really is
is if you take channel measurements and
then you try to fit this into a model of
this kind and then you see that okay
didn't fit perfectly but what if we add
some random perturbation to this model
and we put some Gaussian thing there
well uh now it fits much better because
now we get something that is also
varying it's not following a straight
line the logs distance figure so by
adding this thing and calling each other
fading you get something that athletes
slightly realistic
yes so in the first of all I think in
the millimeter wave I think the line of
sight is the most important cases there
where you would have a line of sight
component in addition to this now I've
assumed serum even it would have a mean
value but suppose that you you have like
one large reflection it goes from
transmitter to receiver bouncing that
wool that's the only way of reaching me
because everything else is blocked well
then this covariance matrix here would
have rank one that describes that
particular path and then how much you
lose on it that path that will be the
corresponding eigenvalue that describes
that and typically it is beat that
parameter here this channel gain changes
quite slowly on time with respect to the
channel fading in that you get from
coheres block to pierce blocks therefore
we in order to get some kind of capacity
result we always need to assume these
things to be fixed
- people are fixed in a small world when
they're moving around getting new
calories a ssin all the time while this
covariance matrices what is beat a
parameter that you can compute from it
it's fixed and how many other models
than this that you can use in order to
generate things I have no intention
today to claim that or to give you any
model that is a path description of
reality I'm going to use this type of
models with correlation in order to show
you how the correlation enters the kind
of expressions that Eric was describing
previously today for the uncorrelated
case to give you some insights about
what spatial correlation can do for us
so if you have a complex Gaussian
distribution here with the covariance
matrix are how do you generate template
that well you can for example create an
eigenvalue decomposition where you would
be the eigenvectors stacked in different
columns and D will be a diagonal matrix
containing all of the positive nonzero
eigenvalues it could be some eigenvalues
to the zero if you have a lower rank r
then you have antennas n for example in
that case this D only contains those
eigen values and this jus here is not
really JUnit error matrix because it
only have our columns we can omit the
eigen vectors that belongs to none to
zero eigen values then a way to generate
realization of air channels will be to
take the square root of the variance
matrix and multiply it with something
that is complex calcium with zero mean
and have an identity matrix so some kind
of iid Rayleigh fading thing what does
it mean to take the square root of a
matrix
well that essentially means to take this
eigenvalue decomposition and take the
square root of each of the eigen values
instead and in that case when you have
this hermitian transpose of a matrix
containing the eigen vectors multiplying
that with a complex Gaussian distributed
variable with covariance matrix I when
you multiply like that
this thing here will just take away M
minus re dimensions where there is not
nothing here in view that spans it and
will reduce the dimension of this thing
here so this is you permission
e hat will be accomplished Gaussian as
well but with zero mean and identity
matrix are of a smaller dimension so
essentially in order to generate
something like this you just take a
complex causal disputed element with ID
elements with the rank of York variance
matrix
multiply it scale it with the respective
square root of eigenvalues and then
multiply it with the eigenvectors in ten
Jeff generated so this is the way to do
this if you would like to do it in yes
[Music]
yes so in theory a lot of people have
been preferring to work with iit
Rayleigh fading models and the reason
for that is you can get all this nice
expression that Eric was describing then
there is probably no real scenario where
you have exact uncorrelated fading and
some cases you might be close to it in
some cases you will be far from it and
the purpose today will be to show you
what this correlation can have as an
impact on the performance and one of the
answers I would like to answer or
question elect answer is also will you
get better performance or worse
performance by taking this correlation
to account is all this analysis that has
been done with ID Rayleigh fading a
conservative or an optimistic analysis
are the systems
yes so four different juicers I assume
that they're realizations are the
channels are independent but they could
have various matrices that looks very
similar and that is something that will
play a part later on a correlated
Rayleigh fading vector with coverage
measures are has rank of our degrees of
freedom that is the size of the subspace
which actually is living so in this
presentation here I'm focusing on having
one antenna at the user site and most
punt antennas at the base station side
so then you only need to model the
covariance matrix for the base station
and uplink that affects the receiver in
the downlink it affects transmit that
way we'll come to in a moment so you you
need to have some kind of model for
covariance matrices and in a real
freedom system or simulator that there
is advanced models how to generate these
channel realizations they're not even
Gaussian distributed but for your front
rows you can guess what the covariance
matrices are going to be I will give you
one simple example that's used a lot in
academia and that have some of the main
properties and then at the end I will
run a case study with real models from
free TPP and the model but I'm going to
introduce you now it's called the locals
capturing channel queries model the idea
is that you have an array at the base
station with uniformly linear array for
example so the distance between the
antennas is equally large then you have
a user being located somewhere else and
there is no line of sight path between
the base station the user and that gives
the result that when I'm going to send
my signal it's going to bounce on a
number of different objects here
and it will eventually find its way to
the base station array and we will get
number of different multipath components
being received and there is a main angle
nominal angle to the user and then there
is some kind of spread of angle so all
of the different things that are coming
in here comes in from slightly different
angles around this nominal angle which
is what you're going to get on a map if
you draw a line between them see if my
remote works not
and I call a number of multipath
components that are received by the base
station by n paths and each of this
paths reaches the base station from a
particular angle this is for ealth paths
we call it like this with the subscript
n and when you put to a uniform linear
ray receive a signal coming from the far
field with a particular angle like this
one you have an array response that
describes how the phase shift at the
array is gonna look like and there are
some standard expression for that this
is what you get from uniform Linde array
but for other types of array structure
you can get other things here the
important thing is that you get e to the
power 2 J something D H here that says
what is the distance between the
different antennas and then something
sinusoid here for example of the angle
from which these signals coming in and
we have normalized so when you say that
their face is one at the first antenna
you don't necessarily need to have that
you can add an extra random face on top
of this one you will make no difference
in my derivation here and the signal
that is coming in from this direction
have a particular random gain here G n
that's how much you have lost on the way
and this is now for one of the paths so
what we're really receiving is the
superposition of these components for n
capital n paths different ones and we
will have to understand how to model
something that looks like this
particular when you get a lot of
different multipath components and let's
now assume that the angles that is
coming in is independent identically
distributed and they have some kind of
PDF here that says what is the
distribution of the angles in angular
domain typically it will be around some
kind of main angle that is not
necessarily to be like that it could
also be uniform distributed then we have
the gains they also seem to be iid
distributed have ceará mean and some
variance that is right like that and I
now define the sum of these variances as
beta
as you can expect this is gonna be the
beta that was showing up before the
average path gain a path loss and then
from the multi-dimensional central limit
theorem when you have a lot of different
paths the improbability this channel
will behave as it's as if it's complex
Gaussian distributed with zero mean and
the covariance matrix are how do we get
district variance matrix we get it by
taking the expected value of the
received signal so there's going to be a
sum of the outer product between the
array responses from each of different
paths here we have a lot of them so this
is holding in probability when you have
a lot number of paths
if we now look at one of the elements
say row l column m and you expand this
expression here you will get the number
of different things here since I assumed
that the angles in gains independent I
get one expected value over the gains
and one expected value over the app the
part that contains the angles and this
part here some of the games was what I
called beta and this part here now
because I have identically distributed
angles I don't need to have a particular
angle here with from 1/2 n in general I
just can take away that submit mix I
will get the same thing and this
expected value now can be written as a
integral over my PDF of the angle
distribution and then this inner product
here can be simplified in this form and
this is now a general way of computing a
covariance matrix from a user to a base
station and it could be from any angle
distribution that comes in the only
thing is suing that we have ID angles
and gates if I now put in a particular
distribution of the PDF here for the
angles I can compute this in the
computer maybe in some case I can
actually get an expression
enclosed four so this is the expression
that we have and what is called the
local scattering model here is not this
general model but actually when we
assume that there is a lack of
scattering around the base station so
all the multipath components actually
coming in from a small angular direction
to me as a base station from the user
however there is a lot of scattering
around the user so if this was the
nominal angle to the between the user
and the base station all the multipath
components here from the enth path will
be equal to the nominal angle plus some
random deviation around it and in
literature you can find a number of
different distributions that people have
been looking at in this cases and this
goes way back to 90s for example people
have been doing these kind of models
when they started to talk about having
multiple antennas at base stations and
one choice of the angles would be to
have a normal distribution another will
be to have a Laplace distribution that
something that looks like a Gaussian
distribution but have a slightly larger
tails you can also have a uniform
distribution then you have a very strict
all the I the multipath component
remains from this angle interval and
nothing in the words and in that
particular case this is often referred
to as two one ring model I will choose
the normal distribution which i think is
also the most common one in literature
if you look back that's what I'm going
to use in some simulations later today
in this case is I have chosen this Sigma
here to be the standard deviations so
it's the same simulation in all these
different cases what time was now if I
put this distribution into the integral
expression and evaluate this numerically
all of integrals then I can create the
covariance matrix for example in this
case having a user with one base station
with 110 s and the nominal angle to the
user is PI over 3 I just took a number
there and I selected my
angular standard deviation 210 degrees
so that is the standard Asian of these
variations where the multipath
components comes into to me and here I'm
showing what happened with my eigen
values are the covariance matrix of the
user and I'm ordered them in decaying
order here is a normalized eigen value
so in reference case here of
uncorrelated fading all the eigen values
are 1 and therefore in decibel scale
they will be 0 and with the different
models here the blue one being uniform
distribution there are some large shiny
values here that almost tend to be
larger than in uncorrelated fading case
and then it goes rapidly down and some
of the eigen values are not even shown
here then with the Laplace and Gaussian
distribution it goes down a bit more
slowly but eventually you also get rid
of all of the eigen values and you have
a number of eigen values that are almost
zero so we can say that in the
uncorrelated fading case here you have a
100 dimensional space where the channel
can appear with high probability with
this uniform distributed you have
perhaps a 20 dimensional subspace of
your 100 space where the channel is
likely to be and with this Laplace or
Gaussian distribution you have a
slightly larger one maybe 30 40 size of
this subspace where it actually lies and
there is a lack of channel measurement
for maximum assistance so we don't know
exactly how this is gonna look like in a
real system but we can expect that it
got to be some eigenvalue variations in
those cases as well and I will in
particular work with the Gaussian
distribution
I will change around this angular
standard deviation as well in
simulations and sometimes I will put
yeah I will compare this with
uncorrelated fading case
so I would say that if you make this
model more general so that you don't
only have like one cluster here around
the user but single cone bumps and other
cluster and then come here then that
will add more and more components to
this model here and I think what that
will for example lead to is that the
reason that it goes down abruptly here
is that this cluster is is relatively
fixed in the angles so the number of
eigenvalues that are non zero here
depends on a largest angle a raised if
you add additional clusters around they
were also contribute and widen this area
where you will have get in multipath
components and that will probably push
down the top I give values here push up
the lowest ones and you will have
something that goes down more slowly and
more slowly one simple example of what
could happen if we have a lot of spatial
correlation in the system now I'm just
for a moment looking at the single cell
system with K users and the receive
single why it's a sum of signals sent
from each of the users si denotes a
signal from user I here's the channel
and then we add some noise here to it as
well and we using this correlated
Rayleigh fading model for all those
channels in particular I'm assuming that
the covariance matrix to use your cake I
have this form here I take a covariance
I take it unitary matrix u here but I
cut away a lot of the eigen values eigen
vectors here so I assume that I have
only M over K eigen directions where the
channel exists and then the rest of em
are 0 and within that eigen space I have
like an identity matrix here in the
middle with my eigen values so I have
like two values constant eigen value
zero
I can value and in order to make sure
that the trace is always the same here
for example equal to M I need to scale
with case so this eigen values that are
nonzero RK the other ones are zero now
look at two different cases the first
case is when the users have autonomy
variance matrices meaning that if I take
this Jew matrix for use to multiply it
with its own new matrix I get an
identity matrix if I take different ones
for different users I get zero that is
because they are having different eigen
vectors they are spanning two completely
different subspaces so their child will
live in different subspaces and this is
how we can generate something like that
the other case is when these G matrices
are the same for every user their
channels are now like also in an
emmerich a dimensional subspace but they
have the same values so UK is the same
for all users so when I multiply UK with
UI from another user I will get I into
the matrix all the time because they are
same if I now when I receive my signal Y
and are interested in the singing of
music K multiplied with this new matrix
from this particular user then I
multiply this with each other users
channels and if I expand H I as I have
in a previous slide as UI some vector V
I that was incorrect Rayleigh fading in
the signal here then I get this in the
product here between the different view
matrices from the UK user I and in the
first case where these two were either
an identity matrix when it was the same
user or ceará when there are different
users well all the terms in this sum
goes away except one that comes from use
of interest so the receive signal here
is only containing the signal of the
user interest plus some noise that is
also Gaussian distributed in the case
where they use always the same well
nothing happens in step that we replace
it with an identity matrix so we still
have a sum over different user signal so
the end
with each other so what we see here is
that the spatial calculation can be very
beneficial if the users have different
eigenspaces
if they have very different covariance
matrices then we separate here so each
user is like operating orthogonal to the
other ones like in there were in a
single user system however when they
have the same space well then they are
interfering a lot with each other and
they are not even getting all of the M
antennas they living in M over K antenna
world so this it will be a first
observation of how spatial correlation
can affect our system let's now define
the uplink transmission system model
that I'm going to use today the received
signal at base station J is a sum over
signals transmitted by all the users in
the entire system some of these users
are within my cell and therefore their
signals are desired so here is the sum
over all the users in my own cell the
one 2kj users there's their channel and
their signal SDK their signal I have
some power p lk or p JK
you just take s here replace it with p
then you get the power of that signal
and if it's a random data signal well
then I will assume that this data signal
comes from a Gaussian curve book I mean
this PL KS as the power and if it's a
deterministic pilot signal that we're
gonna send later on well then it's just
having this is absolute value square
then I receive signals coming from other
cells all the other cells all of their
users summed up here the channels from
user I in cell L to my base station J
and their signal like that users send it
and then at the end I'm adding noise
here at the side so this is the model of
a multi cell system with this notation
at a
and the receiver noise have indented
matrix a square is matrix times some
Sigma square and in this cases I have
not put the variance of the channel to 0
but or to 1 but I keep it here
particular because the book that era
written is dealing with energy
efficiency in addition to spectral
efficiency and then it's very important
to separate how you get an snr it's
because you have lot large amount of
power or is it because you have a low
path gain or if you have a lot of noise
variance what we will do in the uplink
similar to what Eric was describing is
that we apply a linear receiver and here
I call this linear receive combining
where I at my base station in cell J
would like to do the test detect the
signal from its caithe user and we're
selecting a vector which I call the
linear received combined vector or it
can also be called the linear detection
vector it's an M J dimensional vector
that I used to multiply or take an inner
product with the received signal that
means that the I take now the inner
product with each of the different terms
that they had in my uplink received
signal model now since I selected this
received combining vector for my Keith
user I have one term here that contains
the signal from that user then I have
one term that contains a signal from all
the other users that are within my cell
they are desired to me but right now
when I use this receive combining vector
I would like this to be small value and
then I have into cell interference as
well and then have some noise and we
will look later today on how to select
these combining vectors in a good way
we have seen maximal ratio zero forcing
as two options earlier today and we will
look into how what will be the actual
optimal thing and how you from those
ones can interpret what zero forcing in
maximum ratio is as well and briefly
what is linear transmit
precoding in a downlink well in this
cases in the downlink when the base
station is sending signal to the users I
have a very similar sister model they
receive signal at user K in cell J is a
scalar now because I only have one
antenna I'm using the same notation now
for your channels except that I put
hermitian transpose actually if it
should restrict it should be a real
transpose errors now conjugate or
something like that but we take this
permission or conjugate transpose just
to get the expressions to look nice s
you can always undo that complex
conjugation in the real practical system
and what I get is this vector channel
from the base station multiplied with
something that is the precoding vector
so the signal that base station is
sending this is some base station L
sends the sum over signals that are
intend to first different users and we
have a pre coding vector which has a
normal one that's pointing out the
direction of that signal and then we
have this Sigma Li that is describing
the data signal also being compressed
Gaussian code book and have a power Rho
L K in this case that means that the
received signal then in the desired part
contains my channel from my way station
my pre column vector and my data signal
then I have some intra sellin difference
so these are signals that are sent to
the other users in the same cell they're
precoding vector their data signals but
my channel from my base station then I
have the signals from all of the other
cells here are some of the cells here to
some of their users here's the channel
from that base station to me here is the
pre-cooling vector of each of the users
and their data signals and then at the
end we have noise that is basically the
model that's begun to use so model
properly Mulla for downlink and do you
have any questions at this point
otherwise I will move on and start using
this model for channel estimation let's
move on
so in uplink for channel estimation
remember that we describe the coherence
block practice uplink date the downlink
data then I have uplink and downlink
pilots and a we will only focus on the
uplink pilots part and this total cake
of pilots tau P is divided by tau P is U
for uplink tau P D for downlink in case
you use that one should help ease you is
what I'm gonna use for it a plane case
here and what I'm sending as a base user
is so called pilot sequence this pile
sequence is a vector of length tau
presume that's length a number sample
I'm using for uplink signals and Juicer
K in cell J using a power sequence
called EPS ijk and the sequence is
normalized such that if I take the
square norm of it is equal to tau PU and
now when the user is sending the signal
is just taking the signal our spiral
sequence and multiply it with its uplink
power which was PJ K that means that as
a base station I'm receiving over my
channel is sum over all my users here
are the case user my cells channel here
is its pilot sequence and here is the
power that it is using similarly from
all the other cells somewhere to sell
some of their interfering uses here are
their child to my base station in in
cell J here is their power here is their
pilot sequence and here is a matrix now
nice because this is what they receive
during the entire pilot signaling so
it's a matrix number antennas times the
length of the pile of sequences but it
still contains compressed Gaussian
distributed zero mean and Sigma square
uplink variance noise elements what can
I do now as a base station well I would
like to use this pilot sequences this is
something that is known to me and I will
use them to learn my channels to each of
my users say that I'm focusing now on
the k2 user in my cell then the first
step that the base station will do is to
take this receive
matrix and multiply it with the sequence
of the use of interest psy JK and this
is the first step in order to learn this
channel that they have from me as a base
station to this user and multiply it
means that I'm correlated or multiplied
from the right here so that I get the
vector dimension number of antennas and
what will it contain well it will
contain all of the same terms as before
but in each of terms I have multiplied
from the right with a pilot sequence and
then you see that all of the times here
we get in a product between a pilot
sequence of another user and on my use
of interesting and this is yeah and
first of all the reason that I'm
correlating this thing here is that
something that come from estimation
theory I'm essentially removing
everything that is not coming in from
the same direction as the pilot sequence
of these different users this is a quote
sufficient statistic for this city you
don't lose anything from doing this yeah
like
that we will see later on so far this is
done only a coherence block-by-block
manner so right now h-here have taken
one random realization doesn't matter
what kind of distribution it comes from
a that depends on if you have hardening
or not and as Eric was describing you
don't necessarily need to have any pilot
sequence this in the downlink in order
to learn the channels even under
imperfect channel hardening the
important thing is that you have a lot
of antennas or that you're sending a
long block of data so you can use that
to learn the channel anyway hey you can
have the same sequence in all
subcarriers for the reason that we
assume that the statistics are the same
in each of these coherent blocks
yes we have assumed that the whole siq
system is synchronized in practice you
can make sure that the your cell in the
neighboring cells are sufficiently
synchronized for example in an OFDM
system all of the variations that are
within the cyclic prefix where the
distance of that which is like two three
four kilometers depending on which
length of cycle prefix are using those
will easily be synchronized but further
way things will definitely never be
synchronized perfectly that is something
that we assume here to keep the math
simple
let me now look at what will happen and
we see that the interfering terms here
contains is inner product between the
pilots a my desired user and other users
in the system ideally we would like to
make sure that these in the product
between two different users are always
equal to zero but in a large system of
course we cannot do that because if you
would like to make two sequences of
length tell PU orthogonal well you can
only find a collection of tau PU
different sequences of that kind that
are like a basis orthogonal basis for
the subspace and that are all mutually
orthogonal so there is only tau p su
sequences that it can generate and if
you have more users than that in the
entire network that it's a problem for
you what we can try to do is to say that
my network have tell PU mutually
orthogonal uplink sequences that we put
as the columns in a so called pilot book
I just use the name codebook and call it
pilot book so the columns here are the
pilot sequences and that means that
since they are talking onto each other
if I take this matrix and take multiply
it with itself in this way we get an
identity matrix and it's scaled in this
way because I assume that each pile
sequence have a square norm on top of
you
now if tau PU is greater equal to the
maximum of number of users that they
have in any of my cells
I can at least make sure that we then
they sell my users are used in different
pilot sequences and I can create some
kind of way some manner we will look
briefly on that later where some manner
where I take my pal sequences and assign
them to different users first I design
them within my cell so that they have
different pilots and then I can think
about how to do this between cells and
Eric was describing some different pilot
reuse patterns
earlier today we'll come back to that as
well in the general form I can just say
that the there is a set of users that
use the same pile that's me and now I'm
user K in cell J still and this septum
users are those users I in cell l that
such that they're in the product between
their pal sequence and my parallel
sequence is not zero and actually it
will be for the Tau P G which is the
type of design in this case what will
happen with the expression I had before
well a lot of terms in the system will
be serum because a lot of users will use
different pilot sequences and me who
will not do it well this will be a sum
of users that use the same pile secret
as me and let's remove myself from it
and put it over here so that's here I
have my channel my power and then the
inner product between my pal sequence in
my own ones that is helping you and here
I have the interfering terms all the
other users that they use in the same
path that's me their channel to my base
station their power tell PU showing up
again and here I have the noise time at
a noise term it's actually also changing
its variants when I multiply like this
so it sounds variance Sigma square times
tau P you so in this simplified
expression and now see here who is
actually interfering with me in my pal
transmission well that is all the users
that use the same pal sequences at me
but no one else in system I can now use
this put it into some classical
expressions for
estimation theory and actually nice
thing is that estimating Gaussian
unknowns in Gaussian noise and
interference is like the basic case in
channel estimation you can open up the
book by stealing k and see that this is
one of the first things that are scribed
there and they're also covering cases
where you have correlated unknown
elements so this is how the MMSE
estimate is formed
you take your received signal Y that you
receive from from data signals in this
case I'm actually not only looking at
the channel from use decay in cell J but
from any user in the entire network to
my base station I I take the received
signal from the pilot transmission
multiplied with the pilot sequence to
this user then I multiply with the
matrix which is in this matrix it's the
essentially the inverse are the sum of
the covariance matrices of the same
notes that I receive so these are all
the users that use the same pilots as
this use I'm interested in here is the
covariance matrix of their channel here
is the power and the top view and here
is the Sigma square times in noise so
this is like the covariance matrix out
why does I receive me and then I
multiply with the covariance matrix all
my user of interest and with the power
and then I mask that that is what basic
estimation Theory is telling us that we
should do and for missing here one can
also compute what will be the estimation
error that is the true channel minus the
estimate here and this has a particular
structure for each covariance matrix as
well it's the original careers matrix
other thing of interest minus something
and this is a looks a bit ugly due to
all others different indices here what's
important is that when we have
established stuff like this there is a
nice way of generating estimation or
estimates of the channel and and
estimation errors we can just take the
and say that the estimates of my channel
here since why is a Gaussian thing and
multiplied with number thing it is still
accomplished Gaussian random variable
with 0 mean and in tastic covariance
matrix that is the original one
- this estimation error variance matrix
this is what for H this is an estimation
error and the estimation error
accomplished Gaussian zero mean and have
this covariance matrix and this estimate
estimation error they are independent
random variables well that's shumphlett
condition estimation theory so this is
how you can generate estimates and
estimation errors some things to look at
say that we are just having one system
user in the system and we are defining
some kind of effective SNR being the
transmitted power times tau pu times my
variance of the channel development
Sigma square noise here then I'm
plotting here what they called a
normalized mean squared error that is to
take the trace of this estimation error
divided by the trace of the original
covariance matrix why do I divide with
this well now I get number between 0 and
1 and by looking at something like this
you can compare the estimation quality
different users and say that when I have
a perfect estimate so the estimation
error is zero well then this normalized
embassy is equal to 0 when I have a
totally bad estimation procedure here
then the estimation error will be the
same had the same covariance matrix or
signal have from beginning and then I
will have a 1 so when I'm not plotting
estimation quality in terms of
normalized mean square error when they
have a very low as now we will start up
here at 10 to the power of 0 which is 1
and then as I'm increasing my setup the
estimation quality will improve it will
go down here in normalized mean square
and the interesting thing here this is
when I estimate with one antenna this is
one estimate with ten antennas this is
one estimate with 110
how do I get better normalized means
where as I add more antennas well this
has to do with what model I'm using i'm
using this locals capturing channel
model with a Gaussian distribution of
the angular spread and they have the
seer a 10 degree angular standard
deviation and that means that the more
antennas I have here the more this fact
that I am concentrating on some
eigenvalues or make it easier for me to
estimate the channels in a case where I
have an independent Rayleigh fading as
eric was looking at today the blue and
red curve here we'll all coincide with a
black curve well when I have correlation
it becomes easier to estimate my
channels the curves going down here it's
not likely going to go down to Sirius
and add more antennas but it becomes
simpler to estimate correlated channels
because they have a structure because I
essentially only have to estimate some
of the eigen directions and they are
stronger I have better estimation
performance error this is one of the
basic properties that come from
correlation easier to estimate the
channels sir yes in that depends on many
antennas you have a but that is
definitely one of the issues with the
channel estimation itself that in this
expression need to compute a if your
110s 100-500 matrix there research that
have been done on how to approximate
those things and in I'm not covering you
today but in our book we also have
something we call the element wise MC
estimation where we are estimating the
channels one at the time we didn't have
MC fashion and then you are reduced in
the complexity because your estimate as
if you were in a one antenna system but
you will lose in terms of performance
the blue and red two will end up at the
black one so you can essentially say and
this is an important thing you bring up
that if we can afford a couple
we can exploit this if we can't afford a
complexity it will not be worse than in
AIA derail a failing case because we can
always estimate each element separately
and not accuse the correlation between
elements one question what now this this
is actually not related to the Shan
Harlan but rather the opposite effect
that the channel is living in a smaller
sub space so that you only have a few
eigenvalue it's like you have say 100
tell us but the thing you're trying to
estimate it's actually only 20
dimensional or 40 dimensional because
that is how many large hiking bouts you
have and that will mean that this is
only 20 40 and tennis are contributed to
channel hardening so you have a less
hardening effect but you have a better
estimation quality because you only have
to estimate something or smaller
dimension approximately yeah
so the interference have come from
different uses is not covered here I
will talk about that in a few minutes so
let's save this question yes let's save
that question as well so if you increase
the standard deviation you increase in
the spread from where the sings are
coming in and so higher standard Asian
means less spatial correlation so as you
increase this one these curves will move
up towards the the black one and
eventually it will be very close to that
one and if you insert a decrease in this
number you get more correlation you get
fewer eigenvalues that are strong and
then these curves will move further down
yeah
so I am NOT exploiting any information
between coherence in blocks here but if
you do that I guess the better your
estimates are here the more prior
information you have these here will be
to apply a common filter stuff like that
but that's I haven't looked into that
myself in the correlated case the
question no it will not go to zero as
you add more antennas it depends of
course which channel model you have but
the occupational mall where it actually
goes to zero in that model time using
here the you're always gonna have say
30% or 40% of your eigenvalues being
strictly non zero so you will always
have those ones Testaments let's move on
yeah and yeah so everything here is
assumed that the user is living in the
small world here where it gets new
channel realization but same variance
matrix if in practice you need to
somehow learn this covariance matrices
in updated are some algorithms that
people have an emotional these things I
I don't think that is a mature topic yet
so I will list it later today on the
open problems but learning something
about the spatial structure of use
certain in order to to apply this to
simplify things is one of the things
that needs to be shown in practice if
you actually can do it okay so Sal looks
now at two users let's use the same
pilot user K in cell J and user I and
cell L the indices here is not important
the important thing is that their pilot
sequences are the same and that means
that the the set of use that use the
same pilot sequences as them it's also
the same they are sharing the same pilot
if you look closely into this expression
that I had that will mean that this
matrix with the inverse of the like
variance
of the received pilot signal because it
would be the same and that they received
passing when we are multiplied with my
pilot sequence is the same because you
have the same power sequence then one
can say okay but the estimate are
myspace tip of the channel from base
station J to user is LL had this
expression now I know that this sign
matrix is the same for two users so I
can switch the indices they received
pilot matrix when I multiply with the
path sequence is also the same so I can
switch here and then let's assume that
the variance matrix of my own user is
invertible so i can put it in here the
éclairs matrix times the inverse of it
then I can now notice but but look this
party this is actually estimate the
expression for estimate of my own
estimate amount user H hat J subscript Y
a channel from base station j2 k2 user
in cell J so that means that I now have
a relationship between the estimates to
my own user and the user not a cell the
only thing I need to do is to take the
inverse of a covariance matrix multiply
it with that one so like remove my own
covariance matrix then I multiplied with
other users queries matrix and then we
scale the power started using
potentially I using different powers and
then I get the other estimate so what
does it mean it means that the estimates
are correlated if you have generated one
of them you can multiply it with the
sequence of deterministic matrices and
scalars and you get the other estimate
so they are definitely correlated random
variable and that is one of the result
of what we call pal contamination that
they are interfering with each other
however they are like rotated version of
each other because if you multiply with
this matrices here somehow you will take
the estimate of one user and rotate it
somewhere else in the vector space in
the special case
the covariance matrices here are scale
identities in the way that Eric was
looking at earlier today well then these
two things here will us be in identity
matrix times the inverse nor identity
matrix that will not change anything in
that case the estimate of the user now
the cell will be equal to the estimate
our my own user times some scalar
meaning that in that special case this
two-channel estimate are exactly same up
to scaling factor in general when this
the Molalla matrix x inverse or the
other matrix is not a scale identity
they are correlated vectors pointing in
different directions yes so which one ah
this one's okay so this one's eye i
defined one of them earlier here you
take the received pilot signal matrix
here at the base station multiplied with
the pilot sequence of my own user if
this sequence is the same for the two
years that I'm looking at
well then I get the same thing here even
if I change the indices here so I've
used Li here and Li here they will still
be have the same value but I didn't
define that properly here on slides so
look now at the first time or what we
call pilot contamination effect where we
say the two different users are using
the same pilots then the received signal
at base station is LJ is the some other
channels from two users they have their
powers we have a Tau P you like Philip
I'll stick with this term absent noise
and if you take this two estimates of
two channels obvious look at these were
correlated to each other so if we form
call
matrix will be see that there are zero
if we take the inner product between
them and say divide with number antennas
take the expected value you would get
some expression here that is zero when
they have using different pilots and non
zero when they are saying the exact
expression here is not so important but
it's this curious thing that these two
different channels are assumed to be
independent random variables so if you
take the true channels to multiply arm
like this the expected value of M is
zero but when you take the estimates
it's not zero due to the fact that the
base station observing the some of them
when in estimates hey when we send the
pilot signals and this is what we call
pilot condemnation it has two different
effects one of them is that it reduced
estimation quality and the other thing
because we have extra interference
during the power transmission and the
other thing is this that this channel
estimates becomes this istic aliy
dependent despite the fact that these
channels are independent to start with
and this correlation that we can observe
here you know we have an inner product
between the covariance matrices that
means that if the two covariance matrix
from a base station two different users
are small in some sense for example in
the sense that the eigen values of the
matrix get we multiply them together is
zero or they are really living in in
almost different subspaces well then the
interference was more this can really
happen when have different supports or
potentially when the other user is very
far away so that the all of the elements
this other matrix are small but part
confirmation will definitely make it
harder for the base station to mitigate
interference when is not used here
because we don't have a clean estimate
or other users channel they are
correlated with each other and the first
way of looking at this that hopefully
answer some of the questions that have
asked is it show my following
simulations here I am looking at two
different users one of them come sends a
signal to me as a base station array
here coming in from angle 30 degrees
here is 30 degrees
then the other user have an angle that
are very between minus 180 and plus 180
and here I'm plotting this expression
from the previous slide here that says
something about how correlated are these
estimates it's a bit arbitrary that I
selected that notion but in a product we
know scale with never antennas the
interesting thing is that if I have one
antenna at the base station I get the
same value all time so it doesn't matter
from which direction interfering user is
how correlated is my channel on that
users channel they are the same but when
we start to get more antennas ten
antennas 100 tell us we see that the
interference between this user is large
when the interfering user have the same
angle as my user of interest or when it
has it's something that's like the
mirror angle from my array if I get the
signal coming in from 30 degrees or from
minus 30 degrees or in this case 180
minus 30 degrees as it's the same thing
it would get the same angle they say the
same response vector at my array array
response so this is one of the issues
would have a linear uniform linear rate
you should not use something that is
that structure because then you can get
this mirror effects but the important
thing is that when they have the same
angles meaning that they have very
similar covariance matrices then the
correlation between the channel
estimates are large and when they have
very different angles meaning that the
covariance matrix are very different
lying different subspaces well then we
have almost no correlation between them
even if the math says that we should
have something so that is one of the
important things here that the spatial
correlation creates this behavior in the
idea Rayleigh fading case you would
always be in the same number here with
one
in a case you will ever see this
behavior that users that have similar
queries made reason to fear lost wind
shadow in the estimation face and those
were different ones have much less
impact on each other
so some additional remarks piled
condemnation exist because of the
practical necessity of reusing pallet
sequences with between cells and it's
often described as one of the main
characteristics a massive MIMO but it's
not unique to mastermind this is
something that you serve in any
multi-user system we need to reuse the
same time frequency resources act in
different cells different at the same
time but however it might be that power
contamination has a greater impact on
mastermind systems because you have more
users in the system so it's harder to
create large distances between use that
use the same pilot sequence and the
eigen structure of the channel coherence
matrix determines the strength or the
power contamination and for example one
can try to assign pilots to the users
somehow to make sure that users are in
different angles or more generally have
very different variance matrices and how
to do that is something that people are
spending research time on doing I will
talk briefly about that later today as
well so the key point for Angele
estimation is that the uplink pilot
sequences are very important they have
one or all in the sequence per user in
the cell and I didn't talk that much
about it but you saw in all of these snr
expressions the pilot length type is you
showed up so because you have a long
pilot sequence you actually get higher
SNR as well so if you would have an SNR
of say 0 DB and then you send a pilot
sequence of length 10 then you actually
have a 10 DB SNR in channel
installations that's one of the positive
things are sending sequences I talked
about
MMSE estimation that use the channel
statistics top 10 good estimates there
are other alternatives like the
element-wise MMSE or the least square
estimation I'm not covering it here
but its way of simplifying things here
we essentially only need to know the
diagonal in the covariance matrices here
we don't need to know anything at all
about the covariance matrices we are
limited by the shallow coherence that
makes it important for us to reuse piles
across cells inter-cell interferes
reduces the estimation quality and
estimates becomes correlated which is
one way of looking at power
contamination the correlation is small
for sufficiently different queries
matrices but that when you have
different path loss of different spatial
characteristics to become small and we
will see soon that power coordination
leads to coherent interference that is
harder to mitigate before we take a
break let me just briefly start with
uplink and what we were doing when we
receive data and uplink this was the
kind of signal that we had before where
I selected a uplink combining vector v
JK that I multiplied with my received
signal Y in order to try to decode the
signal SJ case that comes from my use of
interest and when I receive this thing I
have also a lot of interference plus
noise interference from all other cells
or actually the same one as long as I
don't include my own user and here I
have inner product between channel and
combining vector chantin from another
user and my combining vector and here I
have a signal from a loser
here are noise and when I'm not using my
MMSE estimator the channel can be
decomposed in the following way the
channel is the known part plus the
unknown estimation error that means I
can take this expression here and I can
replace H here with the estimate and say
this is what I know and then everything
else here including this term with the H
tilde unknown this is everything else as
interference noise and this is the
information that I have in order to try
to get what is to received what is sent
data signal or my user
and when I apply this kind of estimation
no apply some known information
theoretic results on how to compute the
computer lower bound on the capacity for
this particular user we talk like that
what I really get this we look here it
was my received signal of interest if I
take the absolute value square on that
and take what will that be it will be
something like this absolute value
square of the inner product between my
combining vector and my estimate or
channel and the power of the signal s
was P JK and then the achievable
performance now is what they call a
spectral efficiency is the expected
value of a log to a 1 plus something
that can be interpreted as an SI in our
that contains this in the numerator and
then Li contains a lot of interference
terms here
it contains the sum of all the cells on
all of the other users except my own one
a similar term with my combining vector
but their channel estimates and here is
the power that they are transmitting out
and here I have a sum of all the
estimation error covariance matrices and
the noise covariance matrices and here I
have my combining vector as well this is
a general expression for the uplink
spectral efficiency for this kind of
system and the pre log factor here tau e
over tau C comes from the fact that I'm
only using some of the signals tau G are
on the samples that are tau C in general
in the coheres interval for uplink data
now we will finally take a break
so I get the number of questions in the
break so I was planning to just briefly
mention some of those things
why don't I think of a question I got
what what is Qiang hardening and what is
favorite propagation because these two
are actually two different concepts even
if they are kind of related the channel
hardening was the property that Eric was
talking about that a channel when you
take this norm of this channel vector
and divide with number antennas this
will look more and more like a constant
value even if it's a random channel from
stops so in Eric's case this converges
to the or look very similar to beta no
value in our case it will look very
similar to Trey's other covariance
matrix divided with number around tenez
but it's the same kind of concept then
the follow question is do we get channel
hardening with a correlated channel just
usually you seem to get that but it
might harden more slowly because it is
really the number of eigenvalues that
you have that are strong and are at
around the same value that provides you
with the channel hardening effect so in
the examples that I had when you have
like 20 strong eigenvalues the hardening
will be as if you had an ID Rayleigh
fading case with 20 antennas so you
might need many more antennas to really
get a good hardening effect the other
question was what is favored
appropriation that is something that
usually means that two users channel
vectors are gonna be more and more
orthogonal to each other and in idea
really failing to get that all of the
time when you have many antennas in
correlated fading if the users have
channels that are in exactly the same
subspace well then it will yet once
again become you need very many antennas
in order for this to happen you
essentially need to have this 20 strong
eigenvalues that nominee to be
number four the channels are actually
starting to look at thuggin all while if
their uses out in very different South
spaces will then we get this favor
propagation much faster and that is
something that we will see as important
also when we go far forward here so
before the break we get this expression
when you use Mme C channel estimation
the uplink challenge capacity of a user
carrying cell J can be lower bounded by
this expression which country contained
an expectation over different channel
realization different coherence blocks
of a log to a 1 plus and SNR expression
which is a random quantity that changes
with our channel estimates here in
different coherence blocked and this is
something that applies for any choice of
my combining vectors and the question is
can we based on this somehow determine
how we should select our combining
vectors well there is a result saying
that the optimal thing is having the
following structure you should notice
that what we have here is a matrix its
that is the sum of the covariance matrix
of all the estimation errors and the
noise here and then here we have like
Vijay K times the estimates of each of
them we have absolute values so we can
write at st the h h v in a if you expand
this expression and that's essentially
what they say here we should select our
combining vector in the following way we
take the estimate of the channel of
interest we multiply with the matrix
with an inverse on and this matrix is
the sum of all the users in all cells
the power data transmitting on and this
is some kind of guess of what the true
channel times itself is going to look
like the estimate times estimate
hermitian plus the covariance matrix of
the estimation error we sum this up and
then we add the noise in this is what is
maximizing that assign our expression
and if you put it back into the
expression you
simplify your sign our expressions gonna
look in this form here when you get take
an inverse of something that's
essentially the same messy air except
that from the first term here I'm
removing my own juices channel and then
when I take an inverse of this matrix I
multiply with H here from different
sides you don't need to read this exact
detail and try to understand exactly
what it contains I will give you a nice
example of what this actually means we
call this the multi cell MMSE combining
vector what is the reason for that the
reason is that this choice is also
minimizing what you can call the mean
squared error in the combining part here
to say that you're very interested in
this signal here sjk if we subtract the
received signal Y times my combined
vector and we like the difference here
in mean squared sense to be small
conditional estimates that we know then
this is also what minimizes the string
here so that's what's called the minimum
mean squared error combining why do we
call them multi-cell minimum mean square
well that's because the word minimum
mean square or MSE combining was already
used in the literature for single cell
systems and in that literature the MMS
sees changes when you change the set up
but it's easier to get the feeling that
you have combined computed at once and
then you change the set up but you still
call it and in the C and that is what
happened in literature therefore we had
to invent this named motifs lmmse had a
question sorry what
no there is no constraint at all on the
combining vectors the interesting thing
here if you go back to this expression
is that V is showing up in all of the
terms here so you can scale V by a
factor X and then the same scaling will
appear all the ways and they will cancel
out each other yeah exactly so it's
being done at the base station and you
are scaling the signal interference and
noise with the same factors so the
actual scaling and the vector is
unimportant here but it's what's
important this which direction it has
and this is giving us the right
direction and when in terms of the SNR
you can take any scaling factor in front
of this and get the same result but in
terms of minimizing this mean square
there it's only this thing here that
does that this is what you really should
use otherwise you in a real receiver you
need to do another step for your scale
things yes and we will come back to that
in a few moments and let's first try to
understand what does this really mean so
the MMSE receiver combines nning
consists of actually two different steps
the first thing is what you can view as
a whitening are the signal so say that
we take the received signal YJ that is
what we receive during the data
transmission and we compute a
correlation matrix out of f1 or
covariance matrix Y J YJ hermitian they
expected a value of that given our
estimates we call this c YJ then if you
take this thing and you multiply your
received signal with the inverse of the
square to this thing you have whitened
your received signal the receive signal
will now have an identity matrix as
correlation matrix so if you take the
hwhiteland here in the product then you
do the math you see that this becomes an
identity
and the reason that I write this up here
is this that is actually what we are
doing in our max to me in the MMSE
receiver so first we are taking a
receive signal and whiten it like this
and then when we have whitened it we
select a combining vector in the fall
for the whiten ones that is like a
maximum ratio combining of the same way
as Eric was describing before what is
the channel where we receive our desired
signal is the original channel times the
whitening factor that we get from here
so the resulting combining vector is
this thing here and then we need to undo
the whitening procedure once again so it
is estimate our channel of interests
times the inverse of this received
covariance matrix receive signal square
matrix given our estimates and this it
becomes even more clear if we look at
nice simulation example in this example
I assume that there are two antennas and
we have real valid signals I can plot it
like this then I receive a signal and I
can select my combining vector as two
dimensions and only real value there is
only an angle that is important to us
and this angle I am showing here 0 45 90
%
degrees and we moved around like this
and since my receive signal typically
contains different things I can say that
as I changed my combining vector the
desired signal which is like the
estimate times my combining vector
absolute value squared it will have this
value here represent the black one here
this is to call polar plot so the
further from the origin we are the
strong
cygnus are so if we we select this
combining vector which is maximum ratio
combining we will get the largest value
of the desired signal and then as I'm
changing my angle I will get smaller
smaller values until here where I
actually get 0 left on my desired signal
and then if I change the angle again I
can get the stronger signal again this
looks like this if I instead look at the
interference plus noise part if I select
this angle minus 30 degree or something
like that you will get the largest part
of interference and noise and then over
here you get the smallest part and then
here and this for this angle you get the
large value again and then it works
around like that and if you sum up the
desired and interfering signals you get
the red part here so you see in some
directions you get more total received
signal written like that is some you get
less and here it is not really clear
which direction should we put the signal
in should we take something that points
in the direction here where we get the
larger signal should we take something
that points in the direction where we
get to less least interference or should
we take something in between as the
multi-cell and missy is doing and how is
it doing that and what the complicating
thing is is that this red shape here the
total received signal is angle dependent
and if I wipe them the signals as I
describe now it doesn't matter in which
direction I put my combining vector in
the whitened case because I should
always get the same receive signal i
whiten and so the coup relation matrix
personal identity matrix and since the
total received signal power is signal
plus interference and now that sum is
the same umber irrespective which
direction I look well then it must be
that in the direction where the signal
is our strongest the interference now
must be as in weakest otherwise they
cannot add up to the same value all the
time
and that is what we see here as well the
black thing is once again the power
desired signal in the whitened case and
now here I have the strongest signal the
blue one here is their power of the
interference plus noise and it has this
smallest value here where the search
English at least strongest and tested
largest value in this angle directions
where my desired chain list at the
strongest it says the weakest values
when small what zero and that means that
in this case it is very easy to put
select the best combined vector this is
what the cell let me see so that is what
the multi cell MMSE is doing it is
whitening the received signal it is then
applying a maximum ratio combining
filter and then we have the MSE the best
solution so that would be the
essentially the same thing so it is we
can one play around with the matrix
inversion lemma you can see that you get
the same kind of behaviors it is the
same combining vector it is only a
matter of how we illustrated in the that
case it would look more complicated
because you don't get the that round
circle but it's the same thing
a yes so so this part this is the this c
YJ is actually this matrix here and in
this one going back to the questioning
your here i include all the users
including myself so it's both the single
and interference you can actually take
out the term here that contains my own
desired signal that will change the
vector scaling but it will not change
the direction of it and one can show
this with the matrix inversion lemma
number can take out one of the terms
that will only create the scaling
obvious vector M and there is this lemma
for mathematics if you have a matrix
with columns x1 to xn like this and you
have a diagonal matrix D 1 to D n and
you write X D X our mission this is the
same thing as taking the sum over the
columns of x1 column times itself
permission times the diagonal amount
here and the reason that are writing
this up is that we get a lot of
expression of this kind previously where
in the cases we had the power here are a
user and here we have an estimate or
that user here another estimate of that
it has its own estimate on channel and
an estimate of channel again here with
the hermitian on it and in that way we
can write something in this form and
this is something that appears in this
term here this is a matrix instead of X
I call it H this matrix contains the
estimate from bases in J to all other
users in cell L so used to one up to
user K L I put them in columns and that
and I put the matrix here and then I put
the same matrix over here and then I
have a diagonal matrix PL with all of
the powers of these users and this way
we can write it in a compact form we see
you also hear that now it actually
doesn't depend on which other users that
I am certain transmitting to it's the
same expression in this inverse of the
matrix
that means I can take them and Missi
combining for use one up to you see KJ
in my cell the only thing that is
different is that I was multiplying from
the right here with my own channel
estimate and then I had put on the left
hand side the power of that user and I
can combine put them together here as
the columns in this form and I can put
the power here in diagonal elements so
this is one way are writing this multi
cell embassy in the matrix form so you
just have to compute the same - once and
see that this is the expression of to
get okay so that is the optimal phase
but it's rarely juice in literature and
one of the reason for that is that you
might have a high complexity because you
have a matrix inverse that is
proportional to number of antennas that
dimension and need to estimate the
channel to every user in the entire
network from your base station you can
do that with the pilots that you have
but you still require quite some effort
in terms of scene processing or
computations and it's also difficult to
analyze the performance mathematically
so I will show you some simulations on
that and it's also I think that
combining schemes that people have been
developing for seeing a certain areas
have often gistic only being applied to
motor cell scenarios that is my guess on
the motivation why under schemes have
been used of course you cannot prove why
people using the combining schemes today
have been doing but I will talk now
briefly about other schemes that are
kind of simplifications on mobile cell
MMSE that will achieve lower spectral
efficiency but reduce complexity lets
you estimate fewer channels reduce the
amount to require channel sistex and
they might also be more mathematically
tractable so this is the expression that
we have for multi cell MMSE and say now
that I can only afford to estimate two
channels to the uses within my own cell
so it's only the patent is first sum
here where L and J are equal
that I can can compute what should i do
then well then there is what is called a
single cell and Missy wherein I take
away this sum here and I put the term
where L and J are the same so this is my
estimate for my own users in the same
way what should I do with the remaining
terms here for the other cells and all
the users well if I can't estimate their
channel I could replace their estimate
plus estimation error with the actual
covariance matrix of their challenge
instead and this can be viewed as the
MMSE combining in the case where happen
to only know the estimates on a channel
but in my cells then you will get
something like this this has been
applied a number of cases times in
literature but one of our problems is
that I still have an MJ dimensional
matrix that I need to invert here and if
I would like to get rid of that one
thing I can do is to say let's take away
all the covariance matrices here this
estimation I have a good estimate so
let's take away this estimation error
let's take away this covariance matrix
from the users in other sounds they are
probably far away let's skip them then
we get only this term for my own signals
estimates and the estimation noise here
or in the noise part here that is left
have this expression and then if you
play around with some matrix algebra one
can actually move around things in this
expression I can take what is put on the
right hand side here move it to the left
hand side switch the order of things
here in the middle this is a classical
result and now all of a sudden I have a
channel on the other side I have the
inner products in the other direction
and that means that now this matrix here
which was before and identity matrix
where size MJ now it is the inverse of
this matrix with the powers from the
producer so this is say K J dimensional
matrix so all of a sudden the thing
I need to invert here is only KJ
dimensional and this is something I call
regular rise zero forcing then the
reason that I call it regular I see
reporting is that if you take away this
term here this is zero forcing and here
I have regularized that it means that
you haven't you invert the matrix and
you want to regularize something you add
a scale diagonal matrix to try to
improve the smallest eigen values
question hey this one I don't know if
there is any particular performance
metric that it I I chose this actually
mean this has been around the literature
for many many years the regular eyes
Sarah forcing and I try to just show
here how you from the optimal thing can
come to it and I think where it comes
from the beginning is that people have
been saying that Sarah forcing is a good
strategy as eric has talked about before
but sometimes could be a bit unstable
and in those cases you could regularize
it and then it works better yeah it's
actually you're also right that if you
have this senior cell and Missy and all
of this make covariance matrix here are
identities so in the uncorrelated really
fade in case that will happen then you
can some more everything up and say this
is a scale identity matrix and then it
will look essentially like this so the
data is also one way of defining a
regularizer forcing so in a way seeing
as I see in regular zero forcing is more
or less the same thing in the
uncorrelated Rayleigh fading case
I think you can find such special cases
and I think in general for if you have
uncorrelated fading they will be very
similar to each other the next
simplification now since I quote regular
i0 forcing is to just take away this
regularization term and then you have
the zero forcing as Eric have described
it previously this is called zero
forcing because if I multiply with my
channel matrix the matrix all of the
users channel from the left here as hjj
hermitian well then you will get a nice
term here times its inverse so it
becomes an empty matrix so in the case
when you have if you apply this thing
you will remove all the interference
that comes between you users within the
cell and with the perfect CSI case that
is really removing all interference
willing to sell in the case when you
have estimated channels you only remove
the interference that comes in along
your estimated possible channel and then
the last simplification would be the
same let's take away this matrix here
then we could make some ratio combining
what what would be the reason to take
away this matrix well one motivation
might be that the matrix here on inverse
is already any scaled identity matrix
that could happen when you have very
many antennas that the name the diagonal
here becomes large
because of channel hardening effects and
the off diagonal elements are the kind
of terms that appears in when you talk
about failure propagation they will
become small so then it will almost
identical in other case will be if you
start from regular i0 forcing and the
states say I can take away this term
over here because my Norris is much
stronger than the signals well then I
can also end up with something look
in this form but this is the optimal
thing and this is in some way jurist
exemplification of it that will have
lower complexity we simply compute but
we'll also provide you with lower
performance and how low we will look
into now in order to do that I will
define what I call the running example
which is set up with 16 cells each of
them have a square size they contain a
certain number of users and the base
station in the middle and then I have a
wraparound topology so for a user in
this cell here one it might also appear
here and create interference to cell
number 16 here I am M antennas per base
station I have K uses that a deploy
randomly within the cell
I generate this beat large scale
coefficients in the way that I talked
about before I select this parameter
here that said what is the path loss at
1 kilometer distance as minus 148 to me
there's something I've taken from a free
GP P model I select the the path loss
exponent as 3.76 and the child of fading
have a standard deviation of 10 and
actually when you throw out users in
this particular way it could happen that
the user in one cell due to this random
channel fading even if it's closed at
one base station physically it will get
the better beta parameter from another
base station and I wanted to keep the
cell notion as being a square here so
every time that happened I instead of
moving that user to the other cell I
just we generated the shadow fading
realizations that's just another way of
getting this type of result and this is
some kind of macro diversity you can say
that if you are at a bad location maybe
another base station might serve you
instead but it's just simplification
that I did
and I selected some values for the
transmit powers 20 DB per user noise
power minus 9 to 4 DBM this was also DBM
DBM is by the way DBM millon of merely
what said this is 100 milliwatts and
have bandwidth for 20 megahertz the
exact numbers here is not so important
the important thing is what kind of
sings noise ratio gonna get and for a
user that are in the cell center very
close to the base station I allow them
to be 35 meters from the base station
you will get around 20 DB SNR and when
you are at the corner of the cell you
will get around minus 6 DB and both of
these numbers are before I add this
random shadow fading because with that
one you get any number and I will
compare this Gaussian local scattering
model with different values on the
angular sand deviation with the
uncorrelated Rayleigh fading model so
you can see some of the differences
between these things and how do we reuse
the piles between cells I define a pylus
reuse factor in a similar way as Eric
did where we can have these cases piles
reuse factor 1 that means that all the
same pilots are used in all cells that
means that I need 2 1 times K uplink
pilots and totals the topping U is equal
to K I can also say that every other
cell here is using the same one blue one
uses one and this blue yellow one are
using others color then I need two times
K pilot sequences so more pile sequences
but fewer cells are using the same ones
and here I have four different patterns
so then I need four times K uplink pilot
sequences how do we simulate the
spectral efficiency in the system like
this well first if you would like to
create a simulation environment you look
at the macroscopic propagation effects
you drop the users randomly in yourself
then you compute the distances and
angles to the base stations then you
generate chatter fading realizations and
compute the average path loss then you
create your channel covariance matrix
based on the model
I'm using and then when you have that
one you can also get the estimation
error covariance from the expression
that we have next case you go to the
microscopic propagation effects and
generates randomly estimates of the
different channels and when you have
those ones you can compute some kind of
instantaneous spectral efficiency that's
the same thing as we have before but
without the expectation to compute what
you get for one particular channel
realization and then you take this
expression and sum it up over all the
users and then you take the average of a
different channel realizations and you
can also take your different shadow
fading realizations and use some
locations to get some kind of average
sum cell over the system this is how a
simulation would work here is the
expression or an example of what you get
so this is when you have ten users in
your cell you have coherence blocks of
200 symbols similar to what Eric hat and
this is with universal paltry use so we
have a podrace factor of one so you have
K pilots we will look only on the uplink
and we get show the sum rate there's
some spectral efficiency here and here
is the number of antennas the top curve
is with multi cell MMSE gives you the
highest performance and it's increases
with number of antennas then we have the
single cell MMSE it gets also higher
performance with never attend us but
there's a small gap here we have the
regular eyes zero forcing we have the
zero forcing case and then we have the
maximum ratio case so what we can see
here is that for every approximation
that we do we lose something in
performance but we gain something in
terms of complexity and there are
particular these different cases to move
to cell and we see here at the top we
have like the regular eyes zero forcing
as this slightly better than the zero
forcing because it doesn't run into
problems when you have few antennas zero
forcing have problems when you have few
antennas as compared to a number of
users because then it's difficult invert
that matrix
so we had a top curve we had the middle
curves and you have the lowest curve
with maximum ratio this is for the
Gaussian lock of scattering model by the
way with 10 degrees angular spectrum
this angular standard deviation what
happens now if we are adding some more
pilot reuse factor so let's remember
this cap point here and we increase it
to two we see that all the curves except
the maximum ratio curve are moving up in
particular the curve with the the
multi-cell and missy is going up so when
you have more pilots in particular you
can estimate you reduced interference
towards two users in your own cell when
you estimate the channels but you can
also get better estimate to the user in
a neighboring cells and that is what the
multi cell MMSE scheme is using in order
to remove the interference from reduced
interference from those uses for the
other ones the difference is much
smaller and if we go up to pilot reuse
factor of four we can see that the term
here for multi cell MC goes up slightly
but actually the curves here they were
going down again but otherwise the same
behavior so if we just summarizing this
in the table this is for the case when
you have 110 s 10 users then here is the
pilot risk factors here is the different
schemes and how bold faces where we get
the highest performance so multi cell
embassy prefers a very large reuse
factor and the other free schemes here
in the middle that inverted matrices but
not as advanced at Mozilla we see they
get around the same values and they
prefer a pilot reuse factor too and then
max ratio it actually is not trying to
reduce any interference in time network
it prefers a paltry factor one because
yeah if you increase the pilot we use
factoring it better estimate to use away
in the cells but you will also consume
more resources in order to do that and
that doesn't pay
because you have quite good estimates
you use in the cell it is the
interference that limits you nothing
else so from that I will say that there
are free schemes that are really useful
in practice it is multi celibacy because
it gives you the higher spectral
efficiency but the downside is that it
has the highest complexity this is the
maximum ratio that have the lower spec
literacy but at least has the lowest
complexity as well so if you cannot
afford anything else then use that one
and then in between I would pick the
regularizer foursome because it inverts
much smaller matrices than singles lmmse
and it's more stable than zero forcing
that could run into problems when you
have few antennas so raver I still
forcing some kind of good balance
between these two different cases so
these are the free schemes that you
should really care about from a
practical perspective yes
so if you have a lot you reuse factor
you move the users that use the same
pilots as the users here further way
over here there are cells all around
that the fear with you here is only the
ones that are longer diagonal and here
they are even further away so we're
using a higher pancreas factor you will
get less pilot contamination but you
will need to spend twice or four times
the number of samples per coherence
block to estimate the channels and that
reduces the number of samples you have
available for data transmission so
reduce the factor in front of the log
but you increased as an art inside of a
log and there are some kind of balance
when one of these effect is dominating
and for multi cell and Missy it seems to
always dominate to actually increase the
SNR at inside even if you have to pay a
lot for the part reuse factor for the
spending more resources on the pilots
but for maximum ratio you are fine with
having short pilots and get more
occasion to send data simple support
challenge epic winners block them yes
so we should not say so far if it's
saturates with number of 10s we will
look at that before we end today as well
but for any scheme here even if you
would to be no interference or anything
like that as you add more antennas in
the best case the SNR would grow
linearly with number antennas and then
it put a log two on top of that so in
the best case it will grow us
logarithmic number antennas and this
have this kind of shape so far it could
do that and the hint is that some curves
here will saturate some will not but
we'll come back to that one can also
compute other expressions using the kind
of techniques that Eric was describing
in order to get clothes from expressions
I didn't intend to go into details on
that there is a matter bound called the
use and don't forget bound that here
he's talking about in this book and if
you go with that one you could compute a
closed form expression for the SNR that
you will achieve with maximum ratio
combining Eric presented some formulas
for that from extreme ratio and zero
forcing in the uncorrelated case in the
correlated case you can compute such
formula using the same bound technique
but only for maximum ratio this is the
expression you're going to get which
looks much more daunting than what Eric
was presenting but it's actually have a
very simple implementation so so now I
have this formula part it has a signal
part a non coherent interference a
coherent interference in a noise part
here let's go through each of them the
signal part up here it is proportional
to the power that I'm spending actually
squared because it also enters things in
the estimation
I should not overstate that because the
power is also in this sign matrix here
but the important thing here is that I
get a trace of a matrix and this matrix
contains the covariance matrix of the
user and this side that has was an
inverse of the covariance mix that we
receive of the receive seeing your pal
transmission then the exact formula that
there is not important thing the
important thing is that take a trace of
an MJ dimensional matrix now means that
we sum up the diagonal elements of an MJ
matrix and if nothing strange happens
that should be proportional to the
number of elements that we summing up so
this term will be proportional to the
number of antennas so that means that
the signal part grows with number
antennas more antennas higher signal in
the same respect if you look at this
interference term here that I called mom
coherent interference here is the sum of
all users in all cells and you have some
term here contains one trace of
covariance matrices divided with another
trace of covariance matrices and every
time I take a trace of a matrix of
dimension MJ is to get something
proportional to MJ proportional to MJ /
proportional to MJ well this should be
some constant we don't know exactly
where it is but it should not grow with
number antennas so that is what we call
this noncoherent interference has some
number but does not increase with number
of antennas then we have one additional
term where we sum up the users that use
the same pilot as my user and I remove
myself from that set remember this was a
set of users using the same pilots and
here in the numerator I have the trace
of a matrix squared divided by a trace
so I get MJ proportional MJ square / MJ
again so this term should grow with the
number of antennas and this is then
something that I called coherent
interference it grows with their
antennas and yeah it's visible from this
sum here that it is something has to do
with pilot contamination it has to do
with did you seduce the same pile does
me
that their estimates or the base
stations estimate above channels are
correlated with this to it to my own
channel and that we have a noise term
their attempt and what's important here
is that somehow in this term that says
something about the coherent
interference the covariance matrix on me
from my base station to my own user and
the covariance matrix from my base
station to the other user is showing up
here again so if those two various
matrices are very different we could
guess that this term should be small and
when they are very similar they should
somehow be larger and let's look at that
term what happened with the coherent
interference and the same kind of setup
as before when we look at two users
we had one user at angle 30 this was my
desired user and then I varied from
which angle my interfering user is and
here I showed the size of this term from
that you set divide it with the single
term here so I see how large is the
coherent interference with respect to my
desired chingum if I have one antenna
doesn't matter in which direction the
interfering user is because I would get
the same amount of interference all the
time when I add more antennas the
interference depends a lot on which
angle we are in so when two you step the
same angle or we had this strange mirror
angle as well for this type of race the
interference is not sure but it's not
larger than in the in relative terms
it is never larger than it would have
been in a singlet in the case and then
we see that as the angle difference
between them increases the incoherent
interference also decreases and when
they are very different like here there
is barely no interference between these
two uses whatsoever
so here we see clearly what coherent
interference has what impact that has on
our performance in system
so it's mainly when users have similar
covariance matrix when that they will
interfere a lot with each other and what
is also important here is to notice that
okay this is what we get in singing
until a case but also in a uncorrelated
to Rayleigh fading case you will be here
all the time because it's only the path
loss is important not to angular
differences so the coherent interference
that you get here is never worse than in
the ID relief any case it can only
become better here because of you having
different various matrices and this
starts to reveal something about what
kind of correlation impact you will have
will it be better or worse to have
correlated fading here is a graph that
try to illustrate that here I look at
the average some spectral efficiency and
I'm changing now this angular standard
deviation from zero or actually very
close to zero up to say 50 degrees which
is a large area and we looked at 10
degrees previously and I look at the
motifs of MSE the regular SEO forcing
and the max ratio case and this dotted
lines here is actually what you're going
to get with uncorrelated Rayleigh fading
so no correlation at all 110 has ten
users and I think I chose the paltry is
factor of two here and what you see is
that when you have a lot of correlation
it is clearly better to have a
correlated fading than the uncorrelated
fading case then after a while it starts
go down and comes to something that is
quite similar to down correlate a
Rayleigh fading case and in some of
these cases becomes slightly worse some
on occasion it still keeps a little bit
better and actually what is interesting
with the student for only linear rate
that I was looking at is that asks you
since this is uniform linear array there
are some directions where the pointing
away from their a that where you can
resolve use is much better than a long D
array
so for that reason even if the you have
put this one here in the case where you
get multiple components from uniform
district for directions as you start to
get in these cases here it's actually
not seeing an IND Rayleigh fading case
because they are a have different to
resolve ability in different directions
that is why they are not really
conversion to each other
one can also ask the question okay this
is for some rate what happens for for
particular uses and I can talk about a
long time about that on the average as
you move around in the world you will
actually in this case is here B with say
10 degrees angle stimulation you will
always be better in the probabilistic
sense but it might be that you can find
a location I'm here my users are at this
particular locations in this case I will
be better off with incorrect availing
than with quality fading but if you are
looking at many different snapshots you
will on the average or in terms of a CDF
curve as you move around in the world
you will be better off with correlation
one final graph here from the uplink
where what I've tried to do is to take
the received signal with max ratio with
very large they are forcing in
monticello ms see and try to separate it
in this different part a single part the
query noncoherent interference and the
coherent interference and for the case
with maximum ratio we have this nice
separation in the formula and Here I am
numerical it done the same thing for the
other schemes and I have plotted the
powers I divided with the noise power so
that we get something in DB relative to
the noise so if we are down at zero dB
well something is actually equally
strong as noise and I show you result
for different pilot we use factors and I
have Gaussian local scattering 10
degrees Angeles
Aviation hundred antennas ten uses and
if you look at something like maximum
ratio what we can see is the desired
signal is at the top here and it may be
gross a little bit as you add higher
paltry use factors because you get
better estimates but you barely see it
in this decibel scale however the
noncoherent interference it is here and
it's very large so it's only is less
than tend to be smaller than the desired
signal and that has an impact on our
performance and down here we have the
coherent interference and we see that as
we are increasing the pine-trees factor
the coherent interference goes down
because now there are few uses that are
actually creating piled confirmation
while the coherent noncoherent
interference is more or less the same
all the time here it's ready right here
forcing we sacrifice a little bit in
desired signal we cut down on the
noncoherent interference a lot because
now we are actually trying to reduce
interference between use in their own
cell the zero forcing will do the same
thing
the difference in coherent interference
is almost invisible with multi-cell MMSE
we are further reducing the interference
non-cleared interference and now we see
that it actually goes down a lot as we
increase the paltry Deus factor as well
because now we can use more and more
good estimates to use as another cells
to reduce interference as well and you
see that the coherent interference is
also going down war here as well so we
can see that the useful signals are
going down as we use more dissonant and
sophisticated
combining schemes but also noncoherent
interfering goes down and coherent as
well so this is really the behaviors are
these different combining schemes
so when it comes to the uplink here we
have a lower bound on uplink capacity
that was based on minimum mean square
estimation and this is an achievable
spectral efficiency meaning that is
something that we can achieve in
practice with Gaussian codebooks if it's
not that and we maximized it using multi
seller music combining there were also
other choice to receive combining and a
low dose once it was regular a seer
force a maximum ratio that are the
practically useful once because they are
looking two different balances between
complexity and performance factors that
affect the spectral efficiency is
different in transmit power also the
pilot reuse factor the spatial
correlation and the pilot contamination
in size that we get from the spectral
efficiency analysis and running example
is in following that the received signal
power and the coherent interference
grossly in the limited number antennas
noncoherent interference and noise is
independent of number antennas and
coherent interference it's actually
negligible when you use large pile to
reduce factors I've always maybe I
didn't say that explicitly but that's
what you can see here for all of the
schemes this coherent interference is
much more than the noncoherent
interference and this is in DB scale so
it's much smaller yes and then I didn't
talk too much about this additional
bound use and then forget about now I
have a part on down long spectral
efficiency but I was planning to
actually skip over that one and you will
get the slides with all this material
but when you have done the uplink it
becomes a bit boring to do the downlink
because there is something called the
uplink downlink duality that says that
these expressions that you get if you
compare an uplink expression for a
speculation with the downlink expression
they will look very similar
this will just be that interference
terms will have switched some indices so
in the uplink the interference that
comes in transmitted from users
arriving at the base station and they're
dialing the interference come from base
stations and to users so you have to
switch some different indices and things
like that but essentially the analysis
of thing
very similar and which noncoherent
incoherent terms shows up at which users
is also switching to some extent but we
still quantified the size of them and
that was still hold so I think I will
jump over that part and jump is Delta
asymptotic analysis so this is something
that we very popular to do in Massa my
mu already from the start with
marchetta's 2010 papers with an infinite
number of antennas so let's do that as
well in the correlated case and that
really means that we look at what
happens performance wise when you add
very many antennas one should not
interpret this as something that I
propose that we should put up an
infinite number of antennas this is more
of an exercise of seeing what really is
limiting us when you have a lot of
antennas and whether or not we can
achieve any performance that you are
asking for if we just add sufficiently
many antennas and of course not
physically possible to add more and more
antennas I mean if you put them on the
line sooner or later you have reached
around the whole globe and then there's
no more space for it so so it's not
really a practical way of doing and when
we are doing this in a case with
arbitrary spatial correlation we need to
impose some kind of conditions on what
happens with our covariance matrices as
we adding them more montana's how do
they scale and if there are scale of
empty matrixes well that is not a big
issue we just add more atom elements
along the diagonal having the same value
but that's it's not the case that we are
looking at here and then we have two
different conditions that looks I mean
daunting because they're written in the
mathematics and mathematics often it's
much more complicated to read and
actually understand the impact
the first thing here says that if you
take the trace of the covariance matrix
divided with the number of antennas then
the limb in for that is extricating zero
what does that mean
I mean if we sum up the eigenvalues
divided with the number of them so the
average value of the eigenvalues should
remain strictly greater than zero
practically that means that as you add
more antennas you will gather more and
more energy and that an energy will
always be proportioned to a number of
antennas you have if I put antennas here
and then start to put them every 10
meters in that direction sooner or later
it will leave Gothenburg and then of
course those antennas will not
contribute very much to assist them so
dad is not probably doing weave in those
case it will not be satisfied but if you
put them on one array close together
adding more antennas should gather more
energy the second condition is that the
spectral norm which is the largest
eigenvalue are the matrix the the
largest value it were gonna take when we
add more more antennas would always be
smaller than infinity so that means that
that this more more energy that we are
gathering from our array will not be
just focused in one or a few directions
the number of eigenvalues at a large
that we show before those were 30 out of
100 if I had more antennas they will
still be say 30 percent of all the
elements it will be a fractional and
perhaps at a large and not a fixed
number in those cases this satisfies the
first condition expected channel gain
grows with number antennas second
condition no I invalid grows without
bound and actually one can show that
these two condition are sufficient to
get this channel hardening effect that
the norm of the vector squared divided
with number antennas goes towards a
constant value under this condition you
can show that the purpose by looking at
the variance
a so I write this as a sufficient
condition because and if you want to
show that the variance of your square
norm divided with number antennas goes
to zero then these two assumptions will
be helpful for you but maybe you can
find some other clever case where that
is not the case where you don't need
these conditions but you still get
channel hardening yeah so I don't know
what this is a necessary condition but
these are sufficient conditions I also
define the notion of linearly
independent covariance matrices and you
probably have heard about linear
independence before something is linear
independent when you can't write it as a
linear combination of the other ones so
in here we say that the covariance
matrix R is linearly independent of some
other covariance matrices if we can't
write R as a linear combination or R 1
to R M here and mathematical that means
that R - a some of the other covariance
matrices multiplied with some scalars
will be strictly greater than 0 for
whatever values are these scalars that
we are selecting so for all such values
and I also define asymptotically
linearly independent that means that
more less the same thing but now I
divide with number antennas and I look
at what happens with this expression
when you look have very many antennas in
it and these two things are actually not
really the same you can have something
where this R here is different from the
sum of the other ones in only one
element then they will be linearly
independent but it will not be
asymptotically independent because if I
divide with number antennas that one
element where a difference will be
divided by n and then that will go
towards 0 so this is a stricter
condition where
I'd need them to be different in in a
larger way and here is an example that
look at the two covariance matrices R
and R 1 the first one
both of them are diagonal for simplicity
here one of them contains two times and
then T matrix and then an identity
matrix and the other one is just pure
identity matrix and the first part here
is M Prime and the other one is M minus
M prime so both matrices are M by M and
M prime is so far just an integer number
the constant then if we compute this
difference between R and R 1 multiplied
with some scalar C 1 you can expand and
compute exactly what it becomes for this
particular case and you can show that
this expression becomes smallest by
taking the derivative if you put C 1
equal to this expression here and the
expect values here are not so important
but this is smaller we quote to this
expression M minus M prime times M prime
divided by m square if M prime is a
constant then it says this is
proportional to 1/r M so you have M the
value of M square so in that case this
will go down towards zero as you add
more and more antennas into it so in
that case we are not asymptotically
linearly independent they are different
between each other but the part the way
they are different is all only a small M
prime sized part of the matrix however
if I say that M prime is now
proportional to n up to some factor
maybe 20% of eigenvalues are different
in that case here I get something that's
proportional to M proportional to M
divided by proportional to M square so
then we have something that is constant
and you can work out here get 1 minus a
times a that's what it's gonna become
and in that case this is not depending
on M in the middle and now we get
something that is depend
so we can see here subspace where the
matrix are linearly independent must
have a dimension that is proportion
number of antennas that is what it
really means and they can still be there
another matrix like this and here is one
important example I think where I
compare two different diagonally
elements diagonal covariance matrix this
one is a diagnose an identity matrix and
one is scaled identity matrix or now
it's a diagonal matrix with different
elements here what I put on the diagonal
here are iid positive random variables I
put just something random into that
matrix and then I put something else
here that's a deterministic value so
this is one way of generating the
coherence matrix and here is another way
then if I look at this condition are
linear ly asymptotic linear independence
and I sought to rewrite things well I
get one of the elements here minus c1
times and one from this matrix and then
take the absolute value square of that
sum it up we'll all the antennas and
divide with the number of them and then
by what is called the law of large
numbers one can show that this converges
when we have many antennas to the
expected value of the difference between
100 elements minus c1 and it could be
any of the elements because they have
the same distribution and if you should
pick c1 tomatoes as small as possible
well then what you should pick is c1
being equal to the mean value of epsilon
then you get this expression over here
and this is actually variance of each of
these elements meaning that if you just
create one of the matrix randomly
somehow any distribution as long as the
variance is strictly greater than zero
which any random truly relevant random
variable will have you will get
something here that is equal to the
variance and therefore be strictly
positive meaning that I would say that
if you take any
linearly dependent matrices and then you
add some small variation like this
randomly then you can use this type of
proof to show that it will be
asymptotically independent so someone
said I know to covariance matrices that
are linearly dependent here are them and
then I will say yeah sure but add a
small random variations to the other one
and now you have something that is
linearly independent so any case where
something is linearly dependent is
unstable and therefore we can expect it
will never happen in practice that you
have something that's exactly linearly
asymptotic a linearly independent
linearly dependent everything will be
practically where's Michelle
asymptotically linearly independent and
thatis the takeaway message and which
will be very important when we now look
at asymptotics the first asymptotic is a
bit classical what happens under this
assumption one which was what happens
when we scale the covariance matrices
the two conditions if we use maximum
ratio combining using this closed form
expression that we have then you can
show that the difference between that
expression and the following expression
goes to zero as we add more and more
antennas and this expression is more or
less expression without taking away the
noise term taken away the coherent
non-parent interference I only kept the
single part and the non-quick and the
coherent interference here the pal
quantization terms so this shows that
the impact of noise are not weird
interference vanishes while the coherent
signal and the coherent interference
terms remain and therefore in most cases
this will be come a upper spectra
sufficiently limit as you add more more
antennas you converge worse a limit that
is determined by the various matrices
abusers how they are generated you need
to put in larger larger matrices here to
determine what it becomes in the Ivy
Rayleigh fading case you can compute put
in these scale identities here and get
this values in closed form
no you yeah so you should so you're
never gonna do that yourself I'm just
saying that that you will never be able
to measure users with linearly dependent
covariance matrices it doesn't exist in
practice it's only exist in very extreme
special cases where someone have created
that scenario because if you add this
small location would get this linearly
dependent variance matrix with someone
sitting here in the audience then if I
just move a small small bit that would
be like getting some kind of
perturbation some small variations in
the covariance matrix and that will be
enough to break the linear dependence so
that's a very extreme thing it will
never happen so I'm not change the
should create the the perturbation
you're a reality of course you can drive
a similar type of things for the
downlink expression that I didn't show
you with maximum ratio at least in
takeaway messages and there is an upper
limit under spectral efficiency yes you
add more antennas into system if the
base station is instead using this multi
cell and Missi combining using this MMSE
channel estimates that we have then the
uplink spectral efficiency of user K in
cell J grows without bound as we had
more antennas to a system the totally
opposite result no limit and this holds
under this assumption one and under is
and the requirement that the covariance
matrix of my own user from my base
station is asymptotically linearly
independent of the set of covariance
matrices of the other users has used the
same pilot as me so you get rid our pal
contamination or or not pal going
between its radar the coherent
interference it doesn't limit you if the
covariance matrices are asymptotically
Linnaean
and that is something that always
happened in practice and the reason for
this is the following if your covariance
matrices are asymptotically linearly
independent well then if you take one
estimate minus the constants time the
other estimate you can write it in this
form this part was the same we looked at
before the received pile signal
multiplied with you you are a pilot
signal here was the same sigh and then
you have different covariance matrix
here in different scaling factors and if
this are linearly independent well then
these estimates will be that as well so
even if they are a correlate to each
other they are pointing in different
directions and therefore you can't write
one of them as a linear combination of
the other one or scale version of other
one similar things can be proved in
downlink and here is in description of
that say that this plane here is the
subspace spanned of all of the other
users that use the same path does meet
their channel estimates so for my base
station I just gather those channel
estimates I see here is the space where
that they are spanning and then base no
matter my channel is pointing slightly
in a different direction it's not
exactly in that plane due to the
asymptotic than the independence of the
covariance matrices that means that I
can pick a combining vector that is
orthogonal to this subspace that scale
span but the uses that's interfering
with me in the pile transmission and by
doing that I will get no interference no
coherent interference from them and this
vector will still be not orthogonal to
my own channel that means that I can
make sure that all of the interference
terms are zero well my desired signal
term is non zero and this I can only do
if they are linearly independent
covariance matrix otherwise this channel
will also lie in this subspace
a simple example where take try to take
this to the extreme I have two base
stations at different locations and yeah
I put out two users in each cell they
have only a free 0.6 degree difference
they're almost the same covariance
matrices but not exactly the same here I
have used one in one cell here I've used
it too and the other in the other cell
your Suwannee 1 cell 1 user 1 itself -
they are at slightly different angles
used to - in itself - you should - in
cell 1 and I have two users I have pilot
sequence negative - I have - - to be SNR
for my server base station and almost
the same as an affirmative hearing base
station and I once again use this
Gaussian local scattering mountain but I
neglected shadow fading just so I don't
need to impose any additional variations
and they setup because it's difficult to
simulate here is what happens I have the
average some spectral efficiency in my
cell with respect to a number on tennis
and no T right that I put logarithmic
scales over ten hundred thousand
antennas and what you can see here is
with max ratio Sarah for stingray right
zero forcing single cell and Missy all
of them seems to converge to a limit
while the multi cell and Missy case goes
up to infinity and it starts to grow
linearly here and linear in log scale
means that it's gross with log of number
antennas as we mentioned earlier then at
practical number of tests you might
still be always here in practice
so maybe this is unimportant for
practical purposes but from an
asymptotic respect this is really what
happens so asymptotic behaviour key
points here is that the impact of noise
and noncoherent interference always
vanish irrespective of which method you
are using also vanish with maximum ratio
the coherent interference caused a pile
contamination can be a challenge and it
only vanishes if you use this multi cell
and Missy
scheme and then that case a spectral
efficiency crouched a portion of two
number antennas and spatial channel
correlation is very important here it
enables this unbounded spectral
efficiency when using multi LMC and it
determines the upper limit that you will
get with the other schemes and knowing
of course the channel covariance
matrices system very important yeah
however in our recent work that I'm
referring to here we show that you'll
actually only need to know the diagonal
of the covariance matrix as we talked
about earlier as well in order to get
some like this you can do some
variations on this schemes and how do we
in general estimate covariance matrices
well we said we use the pilots the piles
that we have we can use them to form a
sample covariance matrix and get an
approximation all covariance matrices
from that one but that's not the topic
of today any questions at that point
then I will briefly talk about spatial
resource allocation
before we end for today so what does
that mean well we actually in practice
out say thousands of users residing in
one particular cell but it's only a few
of them that is actually active at the
given point in time and which also be
varying number of users per cell so how
does this AK and these two facts affect
what we do in our systems well we know
that we have multiple use that are
sharing the same time time frequency
resources and that the policy quizzes
are keep two separate uses and we cannot
assign these pal synchronous once and
for all to you sir because most of the
time it user is not active so we should
only give it to the users when they're
active and then we have this assignment
problem who should get it and and when
so there are three important subtopics
here that I will briefly mention one of
them is the pilot assignments one of
them is should we always schedule all
users and separate and spatially by pre
kernel combining or credits sometimes be
good to separate users in time a freak
and how does the traffic load variations
in terms of having varying number of
users affecting the system so the first
thing was the pilot assignment that we
have looked at this in a simple way
where we had pilot reuse factors so
there are mainly two different
approaches that you can find in
literature one of them is to try to
allocate sequences directly towards uses
and that is a combinatorial problem you
should take all of your policy because
you give them out to use or try to find
the best way of doing that and that is
and it totally crazy try to do that over
a large system in an optimal fashion so
therefore you can find people in
literature that'll try to do that in a
greedy manner for example and the other
option is to do this pilot reuse factor
where we take your taupes you pilots
divide them into F groups and then you
have tau P you divide them by efforts in
each group and here is then in an non
regular setup how you could potentially
create two different groups and the
second approach is definitely the
simplest one because then we can often
look line when you create you or deploy
your base station you select which base
stations and that you use which are
these different patterns so in each of
these areas this one set but you can on
top of that always apply some kind of
green the allocation as well these are
not two different competing methods what
I think is the important is the
following
listen in this case I have been looking
at the case with 110 s 10 users the same
Gaussian local scattering model as
before and I have taken the same setup
and I have given different assignments
of the pilots the users so I let the
users stay still and I randomized the
pilots allocations and I do this
thousand sometimes so I can plot some
kind of cumulative distribution function
as some spectral efficiency that you're
going to see per cell and the black
curves is with pilot we use factor 1 and
the
the blue curves is with pathways factor
of two and I'm using once again this
square cells in the running example and
what you can see is that there's barely
no variations with maximum racing
regular zero forcing may be some
variation with multi cell and messy but
there aren't almost flat so the the
conclusion of that is that in terms of a
some spectral efficiency perspective it
doesn't really matter who you gives two
pilots to the some specialist which is
almost the same you can gain few percent
in performance but that's it however if
we not zoom in at one cell and look at
the users that are the farthest away
from the base station and then we look
at what happens to its performance as we
are changing the pilot assignments here
is to see the F of the SPECT efficient
of the weakest you sir and we see that
that one is varying much more we have
actual ratio regular C for seeing
multi-cell MMSE are the two different
paths we use factors and it's not even
the smooth curve here there's happening
something strange here around zero point
one why is that well probably it's like
there is one other user in the system
that is interfering with you a lot if
you happen to get the same pilot that
that guy and with 10 percent
probabilities you will get that if you
give something at random so as long as
you stay away from that you are are
happy so from the weakest user
perspective is important how you
allocate your pilots from a some
perspective it's not so important and
this is something that you barely seldom
seen literature because people that have
been looking into power assignments has
focused on the worst uses the other
thing I wanted to talk briefly about is
about scheduling and I would claim that
this like this here I showed the Sun
spectral efficiency and here has showed
the number of users that I'm serving or
actually I
Eric promised that I were going to say
something about multi-antenna terminals
so let's do that say that you have a
user with multiple antennas one of the
thing you can use so some tennis do is
to send
multiple streams to them so instead of
having ten users getting one data stream
each you can have five users hitting two
data streams in two different antennas
or there's to this user so here I'm
actually having the number of data
streams and I'm sending and then I'm
claiming that they're gonna be like this
as I add more and more users data
streams in the system the some spectral
efficiency will grow up to a certain
point where it has its maximum value and
then it will go down again and based on
that claim I will defy three different
operating machines one of them is the
multiplexing regime where the some
spectral efficiency is almost growing
linearly with the number of users so
data streams so that means that I add
more used into system they get almost
the same spectral efficiency as to
existing users are so therefore it grows
but after a while I'm coming into a
saturation regime where okay I'm adding
more users into a system but I have now
so much interference in the system
between uses that as I'm adding more
users I will reduced interference on
everyone and I will not increase the
rate this spectral efficiency cake I
will just divide it between the users so
I can still add more users into the
system and we get the same specs some
spectral efficiency but less per user
and then eventually I have so many used
into system that the suspect efficiency
starts to go down I call this the
scheduling regime because you never want
to be there if you have so many users
that you ask for being over here well
then you should operate here around here
instead and let users take turn on
accessing the system these are the three
different schemes let me show what
happens if we look at for example a
downlink with this model that we had
before 100 antennas varying number of
users 200 as the coherent block length
same model as before and we see this
type of behaviour as you add more
antennas
these sound speckle efficiency first
goes up then it reaches a maximum with
regularity of forcing a maximum ratio is
here around 40 with multi let me see it
is 50 and then it starts to go down
again so you seem to be 40 50 somewhere
there more users than that you should
not schedule in this case we see that
you have 110 s so it seemed like you
should at least have twice the number of
antennas and users and some people maybe
myself have written in papers a mass M
I'm a system is in system we have an
order of magnitude more antennas and
uses this is showing that you don't
really want to operate in such a case
you should make perhaps say it's a
system with twice the number of antennas
or more than number of uses because from
a suspect officially perspective it's
better just load system a lot until you
reach this point third thing I wanted to
mention was what happens if you are
varying the number of users in your
cells and one way of doing that is to
say that instead of having K users all
the time I put K into random number
generator and that will give me how many
uses at half of the moment and a
classical way of doing that is to put it
into a Poisson random number generator
and if I put the Poisson number which KS
parameter I get the mean value of K but
I get a random number of integer uses
and here I have 10 users then it looks
like this 20 users and I have this
distribution this is a probability a
number of active users and then I have
actually cut off here I say I'm not
allowed to have more than 40 users so in
the case I have 40 users on average I
will most of the time have more and then
I limit things here but you can see that
as you have more users not only that a
mean is increasing also the standard
deviation is increasing with number
antennas so what happens now if I would
look at what happened with on the some
spectral efficiency for a case like this
and this is what happens I'm comparing
the some spectral efficiency
we're versus number of uses here with
having exactly K users or on average K
users and then I generate a number user
with a Poisson random number and I say
that if I get more than 40 users
I have only 40 users no more and the
blue curve is exactly the dashed one is
on average and they are more that's the
same from multiple cell embassy regular
eyes - you're forcing maximum ratio more
or less the same so we can and these are
random number different different cells
that means we can interpret all of these
result within given number user per cell
as the some special efficient you would
get within this number on average and
the reason for that is just because
things are growing kind of linearly so
how for a time you have less half-hour
time you have more on average since it
grows almost linearly you get average
number that is more or less the same
over here you lose a little bit because
half of the time you are more than
fourteen I didn't allow for that'sthat's
where you when you saturate you can lose
something so pilot sequences are key to
separate uses spatially some spectral
efficiency is little affected by the
pilot assignment but individual juices
can be greatly satisfied greatly
affected too that is very important to
keep in mind when you assign pilots
terms of scheduling all users use the
same time frequency resources whenever
needed is one claim however when you
have a very large number of users so
that you have reached the top some
spectral efficiency value then you
should start to do time frequency
scheduling and let some of the users
take terms in terms of traffic load
variations it's easy to simulate average
performance on the traffic variations
because on average K or always exactly K
brings you almost the same results I had
one part about power allocation as well
and I think I should jump over most of
these parts but it's really about is
that we somehow should select the powers
so far actually have equal power
for everyone and the CEREC was pointing
out that might not be a good choice if
you want to maximize something like the
worst useless performance in the system
and this is not an easy problem because
you have KL and the sum of the sales
number of use in the entire system and
you have a spectral efficiency in the
uplink and downlink for all of them and
you should select your combining and
precoding vectors and the transmit power
is above uplink and downlink to somehow
try to optimize something like this and
that is not very easy to do so what one
can do is to try to yeah if you look at
for example the expression that I didn't
show you what the expression looks like
in a downlink but it has this form in
general a factor of how many samples per
coherence block to the use for downlink
data log to a 1 plus your downlink power
some factor that depends on your channel
and your pre coders is sum over all the
interfering users some power here some
value that depends on who you are if
you're interfering with different users
they're precoding vectors and we said
that some noise power here and giving
something like this one can define a
utility function where we somehow try to
maximize the performance one choice will
be the max some spectral efficiency so
the sum of the spectral efficiency for
all the users one will be the minimum of
all the spectral efficiency that's some
kind of network wide speculation maximum
fairness and the thing that Erik was
describing calling max min fairness is
not network wide is something more
turistic where you try to make sure that
to maximize this the lowest one within
each cell but then balance somehow
between the cells with a particular
heuristic algorithms that you find in
this book and another option is to
maximize the product of the SNR which is
essentially in higher sign our
approximation of maximum spectral
efficiency and if you formulate a power
optimization problem in terms of
maximize this utility function one of
these three different choices that I was
describing by selecting your downlink
power like
with respect to some some power
constraints saying that for its base
station I have a maximum power that I
can use and I sum up that powers between
for my different uses then I have an
optimization problem and if you're
familiar with optimization problems you
know how to look into these if you have
a particular structure that is well
known or not and for these different
cases that I mentioned the maximum sound
spectral efficiency it is known that
this is a non comma which program that's
very hard to solve in general for the
maximum fairness case is the question
linear program is relatively easy to
solve with the max product SNR is just
called geometric program which is also
relatively easy to solve and that is
what we have done and then I will just
show you for to wrap up here a case
study where we looked at the similar
setup as before but now we use a 3d PP
3d model for non-line of sight cases in
an urban micro cell system so this is
based on the quadriga implementation
that anyone can use we optimize the
power of occasion as I mentioned we used
least square channel estimation I
haven't talked about that but I wanted
to do this without assuming any chance
that this is just to show what you can
achieve in performance you smack some
ratio or regular i0 forcing processing I
put a cylindrical array with 100 or 200
10s and if I have 110 us I can either
have 10 antennas around and 5 different
levels and two different polarizations
so on each antenna i-i've
cross-polarized them or I can have 20
antennas around five vertically and only
one polarization and in the case where I
have 210 eyes I look at the same twenty
around five vertically and two
polarizations I assume this base station
is 25 meters in public route do you sirs
have a height of 1.5 meters and the
reason to look into these three
different cases is that in the first
second case do you have the same number
of RF chain
so then you can look at you would you
like to have a larger first molar II
really like the polarization or not and
in the two cases here you have same
physical size this is the physical size
not the hardware it's limiting you have
some assumptions you can look into
exactly how they're so work working but
it's a quite enough similar to we're
happy for now we're dropping users
randomly and associated with the base
station that gives you the best path
loss and if you optimize the max product
si not this is what you can get as a
cumulative distribution function for a
downlink throughput it varies between
different users in the cells when you
maximize the profit max product SNR and
we can see if we compare the different
types of a race here the more antennas
we have the more happy we are as we will
expect and if you are limited to having
only 100 antennas then it is better to
have a larger rate with only one
polarization or you can switch
polarization between antennas doesn't
matter then having a smaller array with
two polarizations at the same locations
because it is the bigger aperture that
helps you to see more things and you can
see regular eye serum for some performs
better than actuation if you apply
maximum fairness network-wide
you get more of the same values for
everyone if you look at some kind of
graph here are the performers for
different users the 95 percent likely
performance that 90 percent of user will
get it is higher when you use max min
fairness then if you use probably a sec
now well if you maximize the max product
as saying now you're gonna get much
higher performance for the stronger
users here you can get twice or three
times to performance for for the best
you system system so it depends on if
you would like to help the weakest or
the strongest users or get some kind of
balance between these things and I skip
over the uplink part here as well so to
summarize in this case study we looked
at 20 megahertz channel I didn't say
that the downlink performance that we
get her cell
is 485 megabit per second over all my
users it's like 10 years that are
separating that Patrick gives you some
kind of area throughput by certain
numbers the uplink have a 250 to make a
bit per second and I have another Valley
for area who put the reason for these
two times different series just because
I allocated 2/3 of the data symbols for
a downlink and 1/3 for the uplink and
someone can see clearly that there's a
trade-off between high average
throughput which gives you this number
and use a feminist in system if you
compare that with an LTE system it will
in a similar system get dis numbers in
area throughput in megabits and here we
have gigabits just like a 1332 5% higher
throughput that we get in these cases
and the game comes from the coherent
precoding and combining that gives you
better performance per user and from
multiplexing many users in the system so
I'm reaching the end and before we wrap
up I just want to briefly go through
some open problems in this topic area
and you can definitely have a look in
the slides when I send them out later
today and come and talk to me later
during the conference and the school as
well about these things one of the
things that's very important is to make
mastema working after D mode we're going
to hear more about that tomorrow this
has been a long-standing problem people
have been hoping to do this efficiently
already in the 90s Chawla measurements
modeling of channels data traffic
modeling a lot of different things that
need to be model and in an appropriate
way
for a future master - system in order to
really be sure that we are measuring
performance in a way that is realistic
not of implementation aware algorithmic
design that is important for example to
implement zero forcing regularize zero
forcing or these multi seller may see
things with the complexity that is more
like maximum ratio and to try to get to
maybe lower solution energy efficient
hardware into it maybe we can cut down
on the hardware components and do some
clever approximations in order to get
good
in an efficient way cross layer system
design is also important we have focused
a lot on the physical layer under signal
processing here but they're also
protocols for random access for system
information broadcasts that are very
important to make a system work
I touched briefly on the spatial
resource allocation and on the power
control where it's important to balance
between some performance and fairness
between uses estimation of the spatial
properties particular on the mobility
was something that people asked about
that that's definitely a open problem
and we terms of information theory if do
you into that finding tight your bounds
on the capacities for example without
relying on channel hardening could be
very important and interesting and to
find some kind of non-trivial upper
bounds you can always neglect lot of
things and get a huge value for an upper
bound and something that says it's a
reasonable upper bound on the
performance as well will be of interest
new deployment characteristics like
having multiple antenna users which we
definitely will have in practice you
already have to receive antennas in your
phone so there's no reason not to use
those ones in the real system
distributed a race in the extreme case
if we call cell free massive MIMO or new
distributed a lot of antennas over each
area and essentially don't have any
cells anymore and you can try to think
about if you can explore it having a
large antennas in a new and clever way
we have seen it we can do it to cut away
down like parallax for example maybe we
can enable simple hardware and
algorithms and something else that we
haven't thought about yet and finally
the Graham question for later on is what
it will be the successor or massive MIMO
supposed to be in five years time or 10
years time have massive MIMO system that
performs as well as theory says how can
we move forward from that how can we get
another 10 times performance boost
that's an interesting thing to look into
if someone have an idea about that so
that's what's everything for today thank
you
[Applause]