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Coordinated Multiantenna Interference Managementin Dense Wireless Networks

Antti [email protected]

Department of Communications Engineering (DCE)University of Oulu

5 June, 2014

CrownCom’14 1 Antti Tolli

Outline

Evolution of multiantenna systems

General objective

A very brief introduction to convex optimisation

Linear transceiver design and resource allocationI Resource allocation with MIMOI Multicell system modelI Linear transceiver designI Coordinated transceiver optimisationI Coherent vs. coordinated beamforming

Minimum power multicell beamforming with QoS constraintsI Centralised solution via uplink-downlink dualityI Decentralised solution via optimisation decompositionI Large system analysis

Throughput optimal linear TX-RX designI Weighted sum rate maximisation (WSRM) via MSE minimisationI WSRM with rate constraintsI Decentralised solution via precoded UL pilot


Motivation

Conventional cellular systems are interference limitedI In-cell users are processed independently by each base station (BS)

I Other users are treated as inter-cell interference

I Interference mitigated by sharing and reusing available resources

Coordinated multi-cell transmission with multi-user precodingI Increased spatial degrees of freedom in a multi-user MIMO channel

I A system with N distributed antennas can ideally accommodate upto N streams

I Inter-stream interference can be controlled or eliminated by a properbeamformer design.

I Coherent multi-cell MIMO: user data transmitted over a large virtualMIMO channel


Evolution of Multiantenna Systems

!"#$%&$%$"'$(

)!)*( +!)*,)!+*( +!+*(

+-.+!+*( +-.+!+*(/(0"#$%.(

'$11(0"#$%&$%$"'$(

233%40"5#$4(6718'$11(

+-.+!+*(


General Objective

Goal: Design dynamic multi-dimensional radio resource managementacross time, frequency, and space (location)

Assumption: Heterogeneous network composed ofI Large macro cells with massive MIMO antenna arrays,I Small cells and relays with small or distributed MIMO arrays, andI D2D communication with macro cell coordination

Backhaul / controlData


Coordinated Multi-cell Transmission/Reception

Coherent multi-cell transmissionI Each data stream may be transmitted from multiple nodesI Tight synchronisation across the transmitting nodes (common carrier

phase reference)I A high-speed backbone network, e.g. Radio over Fibre

Controller


Coordinated Multi-cell Transmission/ReceptionNon-coherent multi-cell processing

I Dynamic multi-cell scheduling and inter-cell interference avoidanceI Coordinated precoder design and beam allocationI Each data stream is transmitted from a single BS nodeI No carrier phase coherence requirementI Looser requirement on the coordination and the backhaul →

Decentralized processing

Controller


A Very Brief Introduction to Convex Optimisation

A. Hindi, ”A Tutorial on Convex Optimization”, Proc. of the 2004American Control Conference Boston, Massachusetts, June, 2004

Zhi-Quan Luo, Wei Yu, ”An introduction to convex optimization forcommunications and signal processing,” Selected Areas inCommunications, IEEE Journal on , vol.24, no.8, pp. 1426 - 1438, Aug.2006


Constrained Optimisation Problem

Engineering designs are often posed as constrained optimisationproblems:

minimise f0(x)subject to fi(x) ≤ 0, i = 1, . . . ,m

hi(x) = 0, i = 1, . . . , p(1)

whereI x is a vector of decision variablesI f0 is the objective functionI fi(x), i = 1, . . . ,m are the inequality constraint functionsI hi(x), i = 1, . . . , p are the equality constraint functions

Hard to solve in generalI especially when the number of variables in x is largeI the problem might have multiple local minimaI difficult to find a feasible solutionI possibly poor convergence rate


Convex Optimisation ProblemIf f0, f1, . . . , fm in

minimise f0(x)subject to fi(x) ≤ 0, i = 1, . . . ,m

hi(x) = 0, i = 1, . . . , p(2)

are convex and hi, i = 1, . . . , p are affine (hi(x) = aTi x− bi), then

I any locally optimal point is globally optimalI feasibility can be determined unambiguouslyI can be solved efficiently using, e.g. interior point methods

incorporated in generic convex optimisation tools

A function f is convex if its domain dom(f) is convex andf(θx+ (1− θ)y) ≤ θf(x) + (1− θ)f(y) ∀ x, y ∈ dom(f), θ ∈ [0, 1]

then so are the sets

f!1(S) = {x | Ax + b ! S}f(T ) = {Ax + b | x ! T }

An example is coordinate projection {x | (x, y) !S for some y}. As another example, a constraint of theform

"Ax + b"2 # cT x + d,

where A ! Rk"n, a second-order cone constraint, sinceit is the same as requiring the affine expression (Ax +b, cT x + d) to lie in the second-order cone in Rk+1.Similarly, if A0, A1, . . . , Am ! Sn, solution set of thelinear matrix inequality (LMI)

F (x) = A0 + x1A1 + · · · + xmAm $ 0

is convex (preimage of the semidefinite cone under anaffine function).A linear-fractional (or projective) function f : Rm %

Rn has the form

f(x) =Ax + b

cT x + d

and domain dom f = H = {x | cT x + d > 0}. If Cis a convex set, then its linear-fractional transformationf(C) is also convex. This is because linear fractionaltransformations preserve line segments: for x, y ! H,

f([x, y]) = [f(x), f(y)]

PSfrag replacementsx1 x2

x3x4

PSfrag replacementsf(x1) f(x2)

f(x3)f(x4)

Two further properties are helpful in visualizing thegeometry of convex sets. The first is the separatinghyperplane theorem, which states that if S, T & Rn

are convex and disjoint (S ' T = (), then there exists ahyperplane {x | aT x ) b = 0} which separates them.

PSfrag replacements ST

a

The second property is the supporting hyperplane the-orem which states that there exists a supporting hy-perplane at every point on the boundary of a convex

set, where a supporting hyperplane {x | aT x = aT x0}supports S at x0 ! !S if

x ! S * aT x # aT x0

PSfrag replacementsS

x0

a

III. CONVEX FUNCTIONS

In this section, we introduce the reader to someimportant convex functions and techniques for verifyingconvexity. The objective is to sharpen the reader’s abilityto recognize convexity.

A. Convex functionsA function f : Rn % R is convex if its domain dom f

is convex and for all x, y ! dom f , " ! [0, 1]

f("x + (1 ) ")y) # "f(x) + (1 ) ")f(y);

f is concave if )f is convex.PSfrag replacements

xxx

convex concave neither

Here are some simple examples on R: x2 is convex(dom f = R); log x is concave (dom f = R++); andf(x) = 1/x is convex (dom f = R++).It is convenient to define the extension of a convex

function f

f(x) =

!f(x) x ! dom f++ x ,! dom f

Note that f still satisfies the basic definition for allx, y ! Rn, 0 # " # 1 (as an inequality in R - {++}).We will use the same symbol for f and its extension,i.e., we will implicitly assume convex functions areextended.The epigraph of a function f is

epi f = {(x, t) | x ! dom f, f(x) # t }

PSfrag replacements

x

f(x)

epi f

5


A Simple Exampleinfeasible). A point x ! C is an optimal point if f(x) =f! and the optimal set is Xopt = {x ! C | f(x) = f!}.As an example consider the problem

minimize x1 + x2

subject to "x1 # 0"x2 # 01 " $

x1x2 # 0

0 1 2 3 4 50

1

2

3

4

5

PSfrag replacements

x1x

2

CCC

The objective function is f0(x) = [1 1]T x; the feasibleset C is half-hyperboloid; the optimal value is f ! = 2;and the only optimal point is x! = (1, 1).In the standard problem above, the explicit constraints

are given by fi(x) # 0, hi(x) = 0. However, there arealso the implicit constraints: x ! dom fi, x ! domhi,i.e., x must lie in the set

D = dom f0 % · · ·%dom fm %domh1 % · · ·%domhp

which is called the domain of the problem. For example,

minimize " log x1 " log x2

subject to x1 + x2 " 1 # 0

has the implicit constraint x ! D = {x ! R2 | x1 >0, x2 > 0}.A feasibility problem is a special case of the standard

problem, where we are interested merely in finding anyfeasible point. Thus, problem is really to

• either find x ! C• or determine that C = & .

Equivalently, the feasibility problem requires that weeither solve the inequality / equality system

fi(x) # 0, i = 1, . . . , mhi(x) = 0, i = 1, . . . , p

or determine that it is inconsistent.An optimization problem in standard form is a convex

optimization problem if f0, f1, . . . , fm are all convex,and hi are all affine:

minimize f0(x)subject to fi(x) # 0, i = 1, . . . , m

aTi x " bi = 0, i = 1, . . . , p.

This is often written asminimize f0(x)subject to fi(x) # 0, i = 1, . . . , m

Ax = b

where A ! Rp!n and b ! Rp. As mentioned in theintroduction, convex optimization problems have threecrucial properties that makes them fundamentally moretractable than generic nonconvex optimization problems:

1) no local minima: any local optimum is necessarilya global optimum;

2) exact infeasibility detection: using duality theory(which is not cover here), hence algorithms areeasy to initialize;

3) efficient numerical solution methods that can han-dle very large problems.

Note that often seemingly ‘slight’ modifications ofconvex problem can be very hard. Examples include:

• convex maximization, concave minimization, e.g.

maximize 'x'subject to Ax ( b

• nonlinear equality constraints, e.g.

minimize cT xsubject to xT Pix + qT

i x + ri = 0, i = 1, . . . , K

• minimizing over non-convex sets, e.g., Booleanvariables

find xsuch that Ax ( b,

xi ! {0, 1}

To understand global optimality in convex problems,recall that x ! C is locally optimal if it satisfies

y ! C, 'y " x' # R =) f0(y) * f0(x)

for some R > 0. A point x ! C is globally optimalmeans that

y ! C =) f0(y) * f0(x).

For convex optimization problems, any local solution isalso global. [Proof sketch: Suppose x is locally optimal,but that there is a y ! C, with f0(y) < f0(x). Thenwe may take small step from x towards y, i.e., z =!y +(1"!)x with ! > 0 small. Then z is near x, withf0(z) < f0(x) which contradicts local optimality.]There is also a first order condition that characterizes

optimality in convex optimization problems. Suppose f0

is differentiable, then x ! C is optimal iff

y ! C =) +f0(x)T (y " x) * 0

So "+f0(x) defines supporting hyperplane for C at x.This means that if we move from x towards any otherfeasible y, f0 does not decrease.

9

x1


Linear Transceiver Design and Resource Allocation forCoordinated Multi-cell Transmission


Resource Allocation with MIMO

Multiuser MIMO: base station and users are equipped with multipleantennas

I The fundamental idea: inter-user interference is minimised

I Requires channel knowledge of all same cell users

Multiple users – only a subset of users selected at a timeI Scheduling/resource allocation

In general, a difficult non-convex combinatorial problem

1 Select a set of users for each orthogonal dimension(frequency/sub-carrier, time)

2 Optimise transceivers for the selected set of users per dimension.

Greedy allocation: Select a set of users with best channel conditionssuch that their spatial signatures overlap as little as possible

I Often unfair, users with weak channel conditions suffer



User 1

User 2

User 3

User 4

MIMOBS

Time instant t



User 1

User 2

User 3

User 4

MIMOBS

Time instant t+1


System Model

Coordinated multi-cell MIMO system1:I B BSs, NT TX antennas per BS and NRk

RX antennas per user kI A user k is served by Mk = |Bk| BSs from the joint processing setBk, Bk ⊆ B = {1, . . . , B}

yk =∑

b∈Bab,kHb,kx

′b + nk (3)

=∑

b∈Bk

ab,kHb,kxb,k +∑

b∈Bk

ab,kHb,k

∑

i 6=kxb,i

+∑

b∈B\Bk

ab,kHb,kx′b + nk

I ab,kHb,k ∈CNRk×NT channel from BS b to user k

I x′b ∈CNT total TX signal from BS b, and

I xb,k is the transmitted data vector from BS b to user k

1Extension to multicarrier systems is straightforward – add sub-carrier index c to every variable


System Model

xb,k = Mb,kdk ∈CNT is the transmitted data vector from BS b touser k, where

I Mb,k ∈CNT×mk pre-coding matrix,

I dk = [d1,k, . . . , dmk,k]T vector of normalised data symbols,

I mk ≤ min(NTMk, NRk) number of active data streams.

The receiver is equipped with a LMMSE filter, dk = UHk yk:

Uk =

K∑

i=1

∑

b∈Bi

a2b,kHb,kMb,iMHb,iH

Hb,k +N0INRk

−1

∑

b∈Bk

ab,kHb,kMb,k (4)


Linear Transceiver Design

Entire capacity region of multiuser MIMO DL has been recentlydiscovered

I Also with individual peak power constraint per BS antenna23

I Require complex nonlinear precoding based on dirty paper coding

I Sub-optimal but less complex transmission methods are needed

Linear beamforming is usually remarkably simpler in practiceI Dimensionality contraint per BS:

0 ≤∑

k∈Ubmk ≤ NT, 0 ≤ mk ≤ NRk

. (5)

I Dimensionality constraint in the multi-cell network: Upper bound∑k∈U mk ≤ BNT

I Very difficult in general (feasibility conditions for interferencealignment in high SNR)

2W. Yu and T. Lan, ”Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,” IEEETransactions on Signal Processing, vol. 55, no. 6, part 1, pp. 2646–2660, Jun. 2007.

3H. Weingarten, Y. Steinberg, and S. Shamai, ”The capacity region of the Gaussian multiple-input multiple-outputbroadcast channel,” IEEE Transactions on Information Theory, vol. 52, no. 9, pp. 3936–3964, Sep. 2006.



A generalised method for joint design of linear transceivers withI Coordinated multi-cell processing

I Per-BS or per-antenna power constraints

I Subject to various optimisation criteria and Quality of Service (QoS)constraints

The proposed method can accommodate any scenario betweenI Coherent multi-cell beamforming across virtual MIMO channel

I Single-cell beamforming with inter-cell interference coordination andbeam allocation

The presented methods require a complete CSI between all pairs ofusers and BSs

I The solution represent an upper bound for the less ideal solutionswith an incomplete CSI.

Centralised and decentralised mechanisms to perform scheduling andprecoding



Per data stream processing: B BSs send S independent streams,S ≤ min(BNT,

∑k∈U NRk)

For each data stream s, scheduler associates a user ks, with thechannel matrices Hb,ks , b ∈ Bs.

I In some special cases Bs ⊆ Bks . For example, a user may receive datafrom several BSs, while |Bs| = 1 ∀ s.

Let mb,s ∈CNT and us ∈CNRks be arbitrary TX and RX

beamformers for the stream s

SINR per stream:

γs =

∣∣ ∑b∈Bs

ab,ksuHs Hb,ksmb,se

jφb∣∣2

N0

∥∥us∥∥22

+S∑

i=1,i 6=s

∣∣ ∑b∈Bi

ab,ksuHs Hb,ksmb,iejφb

∣∣2(6)

φb represents the possible carrier phase uncertainty of BS b


Coordinated Transceiver Optimisation

The SINR expression can accommodate several special cases formulticell coordination:

1 Coherent multi-cell beamforming (Bs = Bk = B ∀ s, k) with per BSand/or per-antenna power constraints

2 Coordinated single-cell beamforming (|Bs| = 1 ∀ s): the other-celltransmissions considered as inter-cell interference

3 Any combination of above two, where |Bk| and |Bs| may be differentfor each user k and/or stream s.

γs =

∣∣ ∑b∈Bs

ab,ksuHs Hb,ksmb,se

jφb∣∣2

N0

∥∥us∥∥22

+S∑

i=1,i 6=s

∣∣ ∑b∈Bi

ab,ksuHs Hb,ksmb,iejφb

∣∣2



The general system optimisation objective is to maximise a functionf(γ1, . . . , γS) that depends on the individual SINR values

maximise f(γ1, . . . , γS)

subject to

∣∣ ∑b∈Bs

ab,ksuHs Hb,ksmb,s

∣∣2

N0

∥∥us∥∥22

+S∑

i=1,i 6=s

∣∣ ∑b∈Bi

ab,ksuHs Hb,ksmb,i

∣∣2≥ γs,

s = 1, . . . , S∑s∈Sb

∥∥mb,s

∥∥22≤ Pb, b = 1, . . . , B

(7)

Additional Quality of Service constraints (QoS) can be alsoincorporated in (7), e.g., minimum/maximum SINR or rateconstraints per stream or per user


Coordinated Transceiver OptimisationOptimisation criteria, e.g.,

1 Sum power minimisation with fixed per stream SINR constraints:

f(γ1, . . . , γS) = −B∑

b=1

Pb

2 Weighted sum MMSE minimisation:

f(γ1, . . . , γS) = −S∑

s=1

βs(1 + γs)

3 Weighted sum rate maximisation:

f(γ1, . . . , γS) =

S∑

s=1

βs log2(1 + γs) = log2

S∏

s=1

(1 + γs)βs

4 Max min weighted SINR per data stream, i.e., SINR balancing [?]:f(γ1, . . . , γS) = max min

s=1,...,Sβ−1s γs

5 Maximisation of weighted common user rate:

f(γ1, . . . , γS) = ro = mink∈A

β−1k∑

s∈Pk

log2 (1 + γs),

Pk is a subset of data streams that correspond to user kCrownCom’14 23 Antti Tolli


Linear MIMO transceiver optimisation problems cannot be solveddirectly, in general – iterative procedures are required

I No cooperation between usersI Transmitter and receivers optimised separately in an iterative mannerI Some controlled inter-user interference allowed

Guaranteed

bit rate users

Controller

Best effort

users



Transmit beamformers

optimised

Iteration t+1

Controller

Receive

beamformers

fixed


Coherent Multi-cell versus Coordinated Single-cellBeamforming

A. Tolli, H. Pennanen and P. Komulainen, ”On the Value of Coherent and CoordinatedMulti-cell Transmission”, The International Workshop on LTE Evolution in conjunctionwith the International Conference on Communications (ICC’09), Dresden, Germany,June 2009

A. Tolli, M. Codreanu, and M. Juntti, ”Linear multiuser MIMO transceiver design withquality of service and per antenna power constraints,” IEEE Transactions on SignalProcessing, vol. 56, no. 7, pp. 3049 – 3055, Jul. 2008.

A. Tolli, M. Codreanu, and M. Juntti, ”Cooperative MIMO-OFDM cellular system withsoft handover between distributed base station antennas,” IEEE Transactions onWireless Communications, vol. 7, no. 4, pp. 1428–1440, Apr. 2008.


Coordinated single-cell beamforming

Each stream is transmitted from a single BS, |Bs| = 1 ∀ sA user ks is typically allocated to arg max

b∈Bab,ks

Near the cell edge, the optimal beam allocation strategy depends onthe the channel Hb,k.

Large gains from fast beam allocation (cell selection) availableI A difficult combinatorial problem → exhaustive searchI Sub-optimal allocation algorithms

Allocation objectives

I Generate the least inter-streaminterference

I Provide large beamforming gainsController


Heuristic Beam Allocation Algorithms

1 Greedy selection: Beams with the largest component orthogonal tothe previously selected set of beams are chosen.

2 Maximum eigenvalue selection: The eigenvalues of channel vectorsare simply sorted and at most NT streams are allocated per cell.

3 Eigenbeam selection using maxmin SINR criterion:

I A simplified exhaustive search over all possible combinations ofuser-to-cell and stream/beam-to-user allocations

I Beamformers matched to the channel, i.e., mb,s = vb,ks,ls√PT/|Sb|

I For each allocation, the receivers us and the corresponding SINRvalues γs are recalculated

I The selection of the allocation is based on the maxmin SINRcriterion, i.e., arg max

b,k,lmin

s=1,...,Sγs.


Simulation Cases

1 Coherent multi-cell MIMO transmission (Bs = B ∀ s) with per BSpower constraints

2 Coordinated single-cell transmission (|Bs| = 1 ∀ s)I Exhaustive search over all possible combinations of beam allocations.

The SINR balancing algorithm is recomputed for each allocation.

I Fixed allocation, i.e., user ks is always allocated to a cell b with thesmallest path loss, arg max

b∈Bab,ks .

I Heuristic allocation methods

3 Non-coordinated single-cell transmission (|Bs| = 1 ∀ s), where theinter-cell interference is neglected at the transmitters

4 Single-cell transmission with time-division multiple access (TDMA),i.e., without inter-cell interference


Simulation ScenarioA flat fading multiuser MIMO system

K = 2− 4 users served simultaneously by 2 BSs

{NT, NRk} = {2-4, 1}Equal maximum power limit PT for each BS, i.e. Pb = PT ∀ bSNRk = PT max

b∈Ba2b,k/N0

1=k

1=b 2=b

22,1

21,1 aa = 2

4,223,2 aa =

3=k

23,1a

23,1

21,1

a

a=α2=k 4=k


SINR Balancing - Full Spatial Load

0 3 6 10 20 Inf0.5

1

1.5

2

2.5

3

3.5

4

Distance α between different user sets [dB]

Erg

odic

sum

rat

e [b

its/s

/Hz]

Coherent multi−cell TXCoord. single−cell TX (ex. search)Coord. single−cell TX (fixed)Coord. single−cell TX (MaxMinSINR)Coord. single−cell TX (MaxNorm)Non−Coord. single−cell TX (ex. search)Non−Coord. single−cell TX (fixed)TDMA (ex. search)TDMA (fixed)

Figure : Ergodic sum of user rates of {K,B,NT, NRk} = {4, 2, 2, 1} system, 0

dB single link SNR.CrownCom’14 32 Antti Tolli

SINR Balancing - Full Spatial Load

0 3 6 10 20 Inf0

2

4

6

8

10

12

14

16

18

20


Erg

odic

sum

rat

e [b

its/s

/Hz]

Coherent multi−cell TXCoord. single−cell TX (ex. search)Coord. single−cell TX (fixed)Coord. single−cell TX (MaxMinSINR)Coord. single−cell TX (MaxNorm)Non−Coord. single−cell TX (ex. search)Non−Coord. single−cell TX (fixed)TDMA (ex. search)TDMA (fixed)

Figure : Ergodic sum of user rates of {K,B,NT, NRk} = {4, 2, 2, 1} system, 20

dB single link SNR.CrownCom’14 33 Antti Tolli

SINR Balancing - Partial Spatial Load

0 3 6 10 20 Inf2

4

6

8

10

12

14

16


Erg

odic

sum

rat

e [b

its/s

/Hz]

Coherent multi−cell TXCoord. single−cell TX (ex. search)Coord. single−cell TX (fixed)Coord. single−cell TX (MaxMinSINR)Coord. single−cell TX (MaxNorm)Coordinated single−cell TX (Greedy)Non−Coord. single−cell TX (ex. search)Non−Coord. single−cell TX (fixed)TDMA (ex. search)TDMA (fixed)

Figure : Ergodic sum rate of {K,B,NT, NRk} = {2, 2, 2, 1} system at 20 dB

single link SNR.CrownCom’14 34 Antti Tolli

SINR Balancing vs. Rate Maximisation

0 3 6 10 20 Inf0

5

10

15

20

25


Erg

odic

sum

rat

e [b

its/s

/Hz]

Max rate

Max min SINR

Coherent multi−cell TXCoord. single−cell TX (ex. search)Coord. single−cell TX (fixed)Non−Coord. single−cell TX (ex. search)Non−Coord. single−cell TX (fixed)TDMA (ex. search)TDMA (fixed)

Figure : Ergodic sum rate of {K,B,NT, NRk} = {4, 2, 2, 1} system at 20 dB

single link SNR.CrownCom’14 35 Antti Tolli

Minimum Power Multi-cell Beamforming with UserSpecific QoS Constraints

D. N. C. Tse and P. Viswanath, Fundamentals of Wireless Communication. CambridgeUniversity Press, 2005, Chapter 10

H. Dahrouj and W. Yu, ”Coordinated beamforming for the multicell multi-antennawireless system”, IEEE Transactions on Wireless Communications, vol. 9, no. 5, pp.1748–1759, 2010.

A. Tolli, H. Pennanen, and P. Komulainen, ”Decentralized Minimum Power Multi-cellBeamforming with Limited Backhaul Signalling”, IEEE Trans. on Wireless Comm., vol.10, no. 2, pp. 570 - 580, February 2011

H. Pennanen, A. Tolli and M. Latva-aho, ”Decentralized Coordinated DownlinkBeamforming via Primal Decomposition”, IEEE Signal Processing Letters, vol. 8, no.11,pp. 647 - 650, November 2011

H. Pennanen, A. Tolli and M. Latva-aho, ”Multi-Cell Beamforming with DecentralizedCoordination in Cognitive and Cellular Networks”, IEEE Transactions on SignalProcessing, vol. 62, no. 2, pp. 295 - 308, January 2014


Simplified Downlink System Model

Assume now downlink channel with B BSs each with NT transmitantennas serving in total K single-antenna users

Focus on the coordinated beamforming case |Bk| = 1, Bk = bk ∀ k.

The system model from (3) is simplified to

yk = hbk,kmkdk +

K∑

i=1,i 6=khbi,kmidi + nk (8)

where

I bk is the index of the BS serving user kI mk =

√pkuk where uk ∈Cnt , ‖uk‖ = 1 is the normalised

beamformer, and pk the corresponding power allocationI dk ∈C is the normalised data symbol, E

[|dk|2

]= 1

I hb,k ∈C1×NT is the channel row vector from BS b to user k includingthe pathloss ab,k, assumed to be ideally known at the transmitter


Coordinated Minimum Power Beamforming

Minimise the transmitted power

Subject to user specific SINRtargets

Controller Centralised optimisation problem:

min.NB∑b=1

∑k∈Ub

∥∥mk

∥∥22

s. t.

∣∣hbk,kmk

∣∣2

N0 +K∑

i=1,i 6=k

∣∣hbi,kmi

∣∣2≥ γtargetk , k = 1, . . . K

(9)

where the variables are mk ∈CNT , k = 1, . . . ,K.


Solution via Uplink-Downlink Duality451 10.3 Downlink with multiple transmit antennas

User Kydl, K

x dl

uK

H*

User 1ydl,1

wdl

u1~x1

~xK

User K

User 1

xK

x1

yul

wul

uK

u1

H

xul,1

xul, K

subscript “dl” to emphasize that this is the downlink. The dual uplink channelhas K users (each with a single transmit antenna) and nt receive antennas:

yul!m"=Hxul!m"+wul!m"# (10.40)

where xul!m" is the vector of transmitted signals from the K users, yul!m" is thevector of received signals at the nt receive antennas, and wul!m"! !N $0#N0%.To demodulate the kth user in this uplink channel, we use the receive filter uk,which is the transmit filter for user k in the downlink. The two dual systemsare shown in Figure 10.16.

In this uplink, the SINR for user k is given by

Figure 10.16 The originaldownlink with linear transmitstrategy and its uplink dual withlinear reception strategy.

SINRulk &= Qk " u#khk "2

N0+!

j $=k Qj " u#khj "2

# (10.41)

where Qk is the transmit power of user k. Denoting b &= $b1# ' ' ' #bK%t where

bk &=SINRulk

$1+ SINRulk % " u#khk "2

#

we can rewrite (10.41) in matrix notation as

$IK %diag(b1# ' ' ' #bK)At%q= N0b* (10.42)

Here, q is the vector of transmit powers of the users and A is the same as in(10.38).

Figure : Linear downlink TX and its uplink dual with linear RX.


Solution via Uplink-Downlink Duality

Downlink SINR γk for user k:

γk =pk |hbk,kuk|2

N0 +∑

i 6=k pi |hbi,kui|2, k = 1, . . .K (10)

Denote a = [a1, . . . , aK ]T where

ak =γk

(1 + γk) |hbk,kuk|2(11)

Rewrite (10) as

(IK −DaG) p = N0a (12)

where p = [p1, . . . , pK ]T, (k, i)’th entry of G ∈ IRK×K is equal toGk,i = |hbi,kui|2 and Da = diag{a1, . . . , aK}


Uplink-Downlink Duality

The RX signal for user k in the dual UL is

dk[m] = uHkhH

bk,k

√qkdk +

∑

i 6=kuHkhH

bk,i

√qidi + uH

knbk

where qk is the TX power of user k

The dual uplink SINR γulk for user k (∣∣uHkhH

bk,i

∣∣ = |hbk,iuk|):

γulk =qk |hbk,kuk|2

N0 +∑

i 6=k qi |hbk,iuk|2, k = 1, . . .K (13)



Denote b = [b1, . . . , bK ]T where

bk =γulk

(1 + γulk ) |hbk,kuk|2(14)

Rewrite (13) as (IK −DbG

T)

q = N0b (15)

where q = [q1, . . . , qK ]T and Db = diag{b1, . . . , bK}

Solve p from (12) and q from (15)

p = N0 (IK −DaG)−1 a = N0

(D−1a −G

)−11

q = N0

(IK −DbG

T)−1

b = N0

(D−1b −GT

)−11

(16)

(17)



To achieve the same SINR’s in both DL and dual UL γk = γulk ∀ k,set a = b (Da = Db)

K∑

k=1

pk = 1Tp = N01T(D−1a −G

)−11

= N01T(D−1a −GT

)−11 =

K∑

k=1

qk (18)

The total transmit power is the same in both DL and dual UL


Solution for Fixed Beamformers

Find power allocation p (similarly for q) that satisfiesγtarget = [γtarget1 , . . . , γtargetK ]

minimise∑

k pksubject to γk(p) ≤ γtargetk , ∀ k (19)

where the variables are p and (Gk,i = |hbi,kui|)

γk(p) =Gk,kpk∑

i 6=kGk,ipi + wk(20)

Equivalent to a linear program (LP)

minimise 1Tp

subject to(IK −Dtarget

a G)

p � N0a(21)

For a feasible γtarget, the closed form solution is given by (16)-(17).


Iterative Solution

Transmit beamforming and power loading can be solved optimallyvia dual uplink formulation4

min.qk,uk ∀ k

∑k qk

subject to γk(q) ≥ γtargetk , ∀ k(22)

Iterative solution1 MMSE filter uk = uk/‖uk‖, uk = (

∑i qih

Hbk,i

hbk,i +N0I)−1hbk,k isthe optimal power minimizing receiver for fixed powers q

2 Eq. (17) is optimal for fixed receivers uk ∀ kAlternating optimization between (1) and (2) until convergence

Joint update method

qk[t+ 1] =γtargetk

hHk

(∑i 6=k

qi[t]hHbk,i

hHbk,i

+N0I

)−1hbk,k

=γtargetk

γk[t]qk[t] (23)

4More details in: M. Chiang, P. Hande, T. Lan, C.W. Tan, ”Power control in Wireless Cellular Networks”, 2006


SOCP Reformulation

Second order cone is associated withthe Euclidian norm ‖x‖2Important constraint in manyprecoding design applications

2.2 Some important examples 31

x1x2

t

!1

0

1

!1

0

10

0.5

1

Figure 2.10 Boundary of second-order cone in R3, {(x1, x2, t) | (x21+x2

2)1/2 !

t}.

It is (as the name suggests) a convex cone.

Example 2.3 The second-order cone is the norm cone for the Euclidean norm, i.e.,

C = {(x, t) " Rn+1 | #x#2 ! t}

=

!"xt

# $$$$$

"xt

#T "I 00 $1

#"xt

#! 0, t % 0

%.

The second-order cone is also known by several other names. It is called the quadraticcone, since it is defined by a quadratic inequality. It is also called the Lorentz coneor ice-cream cone. Figure 2.10 shows the second-order cone in R3.

2.2.4 Polyhedra

A polyhedron is defined as the solution set of a finite number of linear equalitiesand inequalities:

P = {x | aTj x " bj , j = 1, . . . ,m, cT

j x = dj , j = 1, . . . , p}. (2.5)

A polyhedron is thus the intersection of a finite number of halfspaces and hyper-planes. A!ne sets (e.g., subspaces, hyperplanes, lines), rays, line segments, andhalfspaces are all polyhedra. It is easily shown that polyhedra are convex sets.A bounded polyhedron is sometimes called a polytope, but some authors use theopposite convention (i.e., polytope for any set of the form (2.5), and polyhedron

Boundary of second-order cone in IR3,

{(x1, x2, t) |√

(x21 + x22) ≤ t}Canonical form of SOCP

minimise cTxsubject to ‖Aix + bi‖2 ≤ cT

i x + di, i = 1, . . . ,mFx = g,

(24)

where x ∈ IRn is the opt. variable, Ai ∈ IRni×n and F ∈ IRp×n


SOCP Reformulation

By rearranging the constraint in (9) as

N0 +K∑i=1

∣∣hbi,kmi

∣∣2 ≤(

1 + 1γtargetk

)|hbk,kmk|2, k = 1, . . . K

Eq. (9) can be reformulated into epigraph form

min. p

s. t.

∥∥∥∥∥∥∥∥∥

hb1,km1...

hbK ,kmK√N0

∥∥∥∥∥∥∥∥∥2

≤√

1 +1

γtargetk

hHbk,k

mk, k = 1, . . . K

‖vec(M)‖2 ≤ p(25)

where the variables are mk ∈CNT , k = 1, . . . ,K, and whereM = [m1, . . . , mK ].

Standard form SOCP


Decentralised Solution via Optimisation Decomposition

TDD assumption:

each BS is able to measureat least the channels of allcell edge users

1

2

1 BK-1

K

Implementation alternatives

1 Simple ZF solution: inter-cell interference nulled whileoptimising the served users

2 Interference balancing: allow some controlled inter-cellinterference, and design the precoders in the adjacent BSsaccordingly



Proposed distributed solution

Beamformers are designed locally relying on limited informationexchanged between adjacent BSs

The coupled terms are decoupled by a dual decomposition5,Alternating direction method of multipliers (ADMM)6, or primaldecomposition7 approach

I Decentralized algorithm

The approach is able to guarantee always feasible solutions evenwith low feedback rate

Allows for a number of special cases with reduced backhaulinformation exchange

5A. Tolli, H. Pennanen, and P. Komulainen, ”Decentralized Minimum Power Multi-cell Beamforming with LimitedBackhaul Signalling”, IEEE Trans. on Wireless Comm., vol. 10, no. 2, pp. 570 - 580, February 2011

6C. Shen, T. H. Chang, K. Y. Wang, Z. Qiu, and C. Y. Chi, ”Distributed robust multi-cell coordinated beamforming withimperfect CSI: An ADMM approach,” IEEE Trans. Signal Processing, vol. 60, no. 6, pp. 2988 - 3003, Jun. 2012.

7H. Pennanen, A. Tolli and M. Latva-aho, ”Decentralized Coordinated Downlink Beamforming via Primal Decomposition”,IEEE Signal Processing Letters, vol. 8, no.11, pp. 647 - 650, November 2011



For the coordinated single-cell beamforming case (|Bk| = 1 ∀ k),SINR formula can be written

Γk =

∣∣hbk,kmk

∣∣2

N0 +K∑

i=1,i 6=k

∣∣hbi,kmi

∣∣2

=

∣∣hbk,kmk

∣∣2

N0 +∑b6=bk

ζ2b,k +∑

i∈Ubk\k

∣∣hbk,kmi

∣∣2 (26)

where the inter-cell interference term is

ζ2b,k =∑

i∈Ub

∣∣hb,kmi

∣∣2 (27)



Now, (9) can be reformulated for the special case |Bk| = 1 ∀ k as:

min.B∑b=1

∑k∈Ub

∥∥mk

∥∥22

s. t. Γk ≥ γk,∀ k∑i∈Ub

∣∣hb,kmi

∣∣2 ≤ ζ2b,k, ∀ k 6∈ Ub,∀ b(28)

where the variables are mk and ζb,k.

Inter-cell interference generated from a given base station b cannotexceed the user specific thresholds ζb,k ∀ k 6∈ UbBSs are coupled by the interference terms ζb,k. For fixed ζb,k, theproblem would be decoupled between BSs


Decentralised Solution via Dual Decomposition89

Introduce local copies ζ(b)b,k of the interference terms ζb,k

Introduce additional equality constraintsI Each ζb,k couples exactly two (adjacent) base stations, i.e., the

serving BS bk and the interfering BS b.I Enforce the two local copies to be equal ζ

(b)b,k = ζ

(bk)b,k ∀ k, b ∈ Bk,

where Bk = B \ bk.

min.B∑b=1

∑k∈Ub

∥∥mk

∥∥22

s. t. Γ(b)k ≥ γk,∀ k ∈ Ub ∀ b∑

i∈Ub

∣∣hb,kmi

∣∣2 ≤ ζ(b)2b,k , ∀ k 6∈ Ub∀ b

ζ(b)b,k = ζ

(bk)b,k , ∀ k, b ∈ Bk

(29)

where the variables mk, and ζ(b)b,k ∀ k, b ∈ Bk are local for each BS b

8S. Boyd, L. Xiao, A. Mutapcic, and J. Mattingley, ”Notes on decomposition methods: course reader for convexoptimization II, Stanford,” 2008. Available online: http://www.stanford.edu/class/ee364b/

9D. P. Palomar and M. Chiang, ”A tutorial on decomposition methods for network utility maximization,” IEEE J. Sel.Areas Commun., vol. 24, no. 8, pp. 1439-1451, Aug. 2006.


Decentralised Solution via Dual DecompositionDual decomposition approach: the consistency constraints in (29) arerelaxed by forming the partial Lagrangian as

L(M1, . . . ,MB , ζ

(1), . . . , ζ(B),ν1, . . . ,νB

)=

B∑b=1

∑k∈Ub

∥∥mk

∥∥2

2+

K∑k=1

∑b∈Bk

νb,k(ζ(b)b,k − ζ

(bk)b,k )

=B∑b=1

∑k∈Ub

∥∥mk

∥∥2

2+

B∑b=1

νTb ζ(b) (30)

where νb,k ∀ k, b ∈ Bk are real valued Lagrange multipliers associated withthe consistency constraints

The dual function can now be written as

g(ν1, . . . ,νB) =∑B

b=1gb(νb) (31)

where gb(νb) is the minimum value of the partial Lagrangian solved for agiven νb

gb(νb) = infmk,ζ(b)

∑

k∈Ub

∥∥mk

∥∥22

+ νTb ζ

(b). (32)


Decentralised Solution via Dual Decomposition

Finally, the local problem of BS b can be formulated as

min.∑

k∈Ub

∥∥mk

∥∥22

+ νTb ζ

(b)

s. t. Γ(b)k ≥ γk,∀ k ∈ Ub∑

i∈Ub

∣∣hb,kmi

∣∣2 ≤ ζ(b)2b,k , ∀ k 6∈ Ub(33)

where the variables are mk ∀ k ∈ Ub, and ζ(b)

I Locally solved as SOCPs in each BS b with knowledge of νb

The master problem: max. g(ν1, . . . ,νB) with variables νb ∀ bI Solved iteratively with the following updates:

νb,k(t+ 1) = νb,k(t) + µ(ζ(b)b,k(t)− ζ(bk)b,k (t)

),∀ b, k (34)

I t is the iteration/time index, µ is a positive step-size

Feasible solution by using ζb,k(t) = 12(ζ

(bk)b,k (t) + ζ

(b)b,k(t)) in (28)


Distributed Algorithm

xx

xx

xx

xx

xx

xx

xx

xx

xxxx

xx

xx

xx

xx

1=b

3=k

24,1

ceinterferen !"

23,1

ceinterferen !"

21,2

ceinterferen !"

22,2

ceinterferen !"

4=k

2=k

1=k 2=b

!"#$%&'()*+),-.

/0(#1+1#)1&2(3.#(44)

1&2(3+(3(&#()

1&+*35%21*&)

( )21,2

!( )22,2

!( )13,1

! ( )14,1

! ( )11,2

! ( )12,2

! ( )24,1

! ( )23,1

!

Information exchange between adjacent BSs

Real-valued inter-cell interference terms


Distributed Algorithm

Always feasible solution: use the average vector ζ(t) in (33)

Feasible γk, ∀ k can be guaranteed even if the update rate of ζ(b)(t)between BSs is slower than the channel coherence time

Special cases with reduced backhaul information exchange

1 Group-specific inter-cell interference constraint,ζb,k = ζb,i ∀ k, i ∈ C, k, i 6∈ Ub

2 BS-specific inter-cell interference constraint, ζb,k = ζb ∀ k 6∈ Ub.3 One common constraint for all BSs (within a given joint processing

area), ζb,k = ζ ∀ k, b.Cases that do not require exchange of ζb,k

1 The constraints ζb,k can be fixed to some values that may depend forexample on the BS- and user-specific operating environment.

2 Zero-forcing for the inter-cell interference, ζb,k = 0 ∀ k, b.


Simulation ScenarioA flat fading multiuser MIMO system

K = 4 users served simultaneously by 2 BSs

{NT, NRk} = {4, 1}a1,1 = a1,2 = a2,3 = a2,4 = a

Path gain to noise ratio is normalized to a2/N0 = 1

1=k

1=b 2=b

22,1

21,1 aa = 2

4,223,2 aa =

3=k

23,1a

23,1

21,1

a

a=α2=k 4=k


Numerical Results – Convergence behaviour

10 20 30 40 50 60 70 80

10−6

10−4

10−2

100

102

Iterations [t]

Pd

eco

mp(t

) −

Po

pt

0 dB SINR target

10 dB SINR target

Figure : Suboptimality of the distributed algorithm versus the iteration numbert for 0 dB and 10 dB SINR targets.


Numerical Results – Block Fading

0 3 6 10 20 50−2

0

2

4

6

8

10

12

14

16


Tra

nsm

it p

ow

er

[dB

]

ZF for all interference

ZF for inter−cell interference

coordinated, one constr.

coordinated, per BS constr.

coordinated, per user constr.

coherent

Figure : Sum power of {K,B,NT} = {4, 2, 4} system with 0 dB SINR target.CrownCom’14 59 Antti Tolli

Numerical Results – Block Fading

0 3 6 10 20 5018

20

22

24

26

28

30

32

34

36


Tra

nsm

it p

ow

er

[dB

]

ZF for all interference

ZF for inter−cell interference

coordinated, one constr.

coordinated, per BS constr.

coordinated, per user constr.

coherent

Figure : Sum power of {K,B,NT} = {4, 2, 4} system with 20 dB SINR target.CrownCom’14 60 Antti Tolli

Time correlated fading - Example

1.12 1.122 1.124 1.126 1.128 1.13 1.132 1.134

x 104

100

101

102

time

Tra

nsm

it po

wer

coordinated, per user constr.coordinated (ideal), per user constr.ZF for inter−cell interference

Figure : Time evolution of the distributed algorithm with 0 dB SINR target,TSfd = 0.1 (e.g., 30 km/h with 2 ms reporting period).

TS is the signalling period and fd is the maximum Doppler shift.CrownCom’14 61 Antti Tolli

Extensions

Cognitive underlay cellular network10 - a sum interference constraintis imposed to every primary user k ∈ UP from the secondary BSs BS

∑i∈US

b

∣∣hb,kmi

∣∣2 ≤ φb,k,∀b ∈ BS,∀k ∈ UP

∑b∈BS

φb,k ≤ Φk, ∀k ∈ UP

(35)

(36)

Worst case beamformer design with ellipsoid CSIT uncertainty11,12

hb,k = hb,k + ub,k ∀ b ∈ B, k ∈ UEb,k = {ub,k : ub,kEb,ku

Hb,k ≤ 1} ∀ b ∈ B, k ∈ U

(37)

(38)

where hb,k and ub,k are the estimated channel at the BS and the CSIerror, respectively, and PSD matrix Eb,k defines the CSI accuracy.

10H. Pennanen, A. Tolli and M. Latva-aho, ”Multi-Cell Beamforming with Decentralized Coordination in Cognitive andCellular Networks”, IEEE Transactions on Signal Processing, vol. 62, no. 2, pp. 295 - 308, January 2014

11C. Shen, T. H. Chang, K. Y. Wang, Z. Qiu, and C. Y. Chi, ”Distributed robust multi-cell coordinated beamforming withimperfect CSI: An ADMM approach,” IEEE Trans. Signal Processing, vol. 60, no. 6, pp. 2988 - 3003, Jun. 2012.

12H. Pennanen, A. Tolli and M. Latva-aho, ”Decentralized Robust Beamforming for Coordinated Multi-Cell MISONetworks”, IEEE Signal Processing letters, vol. 21, no. 3, pp. 334 - 338, March 2014


Decentralising the Optimal Multi-cell Beamforming viaLarge System Analysis

H. Asgharimoghaddam, A. Tolli & N. Rajatheva, Decentralizing theOptimal Multi-cell Beamforming via Large System Analysis, in Proc. IEEEICC 2014, Sydney, Australia, June, 2014

H. Asgharimoghaddam, A. Tolli & N. Rajatheva, ”Decentralized Multi-cellBeamforming Via Large System Analysis in Correlated Channels”, in Proc.EUSIPCO 2014, Lisbon, Portugal, September, 2014


Very Large Antenna Array

Assume a large antenna array scenario where

1 the number of antennas NT at the serving node is large while thenumber of users |Ub| in the cell b is fixed, NT >> |Ub|

2 both NT and |Ub| are large, NT, |Ub| → ∞ while NT/|Ub| > 1

The main research problem is to study how the increased degrees offreedom can be utilised both

I to simplify the transmitter/receiver processing and

I to reduce the backhaul signalling.

The impact of non-idealities is assessedI Non-zero antenna correlation

Tools from random matrix theory can be utilised


System assumptions and research problem

Assumptions:

Local CSI available at eachBS

Linear TX-RX processing

TDD mode – channelreciprocity

Nb >> Kb

1

2

1 BK-1

K

Problem: Minimum power beamforming

Sum power minimization over BSs with user specificminimum rate/SINR targets


Simplified solutions

Matched filter: mk = cbk,khHbk,k

/‖hbk,k‖2, where cbk,k is scaled such

that γk = |hbk,kmk|2/N0

Channel inversion (ZF): the required sum power for a given userallocation Ub is obtained from

∑Bb=1

∑k∈Ub γk‖zb,k‖

22N0 (39)

where the ZF precoders are Zb = [zb,1, . . . , zb,K ] = HHb (HbH

Hb )−1

and where Hb = [hTb,1, . . . ,h

Tb,K ]T, ∀ b.

Fixed ICI thresholds {ζb,k}, with independent optimisation per BS

min.{mk}

∑k∈Ub

∥∥mk

∥∥22

s. t. Γk ≥ γk,∀ k ∈ Ub∑i∈Ub

∣∣hb,kmi

∣∣2 ≤ ζ2b,k, ∀ k 6∈ Ub(40)


UL-DL Duality RevisitedRecall the dual uplink presentation of (9) and (28)

minimizeuk,qk

∑

b∈B

∑

k∈Ub

qk

subject toqk|uH

khHbk,k|2

∑l 6=k ql|uH

khHbk,l|2 +N0‖uk‖2

≥γk ∀k∈U(41)

The dual uplink power of each user is found by fixed point iteration13

qk =1

(1 + 1γk

)hbk,k(∑

l∈U qlhHbk,l

hbk,l + I)−1hHbk,k

(42)

The dual uplink detection vector ub,k is given by the MMSE receiver

uk = (∑

l∈Uqlh

Hbk,l

hbk,l +N0I)−1hHbk,k

(43)

13Note that (42) is equivalent to (23)


UL-DL Duality Revisited

A link between the DL and UL beamformers is provided by

mk =√δkuk (44)

where δk can be found by

δ = G−11K (45)

and where δ is a vector that contains all δk values14. The elementsof G are given as:

Gi,j =

{ 1γi|hbi,iui|2 i = j

−|hbj ,iuj |2 i 6= j.(46)

14δk = pk and G = D−1a −G in (16) if ‖uk‖2 = 1


Large Dimension Approximation of (42) for the i.i.d. caseLarge system approximation for the optimal UL power is given as15

qk = ((1 +1

γk)(

a2bk,kmΣbk(−1)

1 + qka2bk,k

mΣbk(−1)

))−1 (47)

where a2bk,k is the pathloss from user k to BS bk, mΣbk(−1) is the

Stieltjes transform of Σbk at z = −1 and Σbk =∑

l∈U ,l 6=kqlh

Hbk,l

hbk,l

Similarly, the entries of matrix G can be approximated by

Gi,j =

1γi

(a2bi,i

mΣbi(−1)

ηbi,i)2 i = j

−1Na

a2bj ,i

a2bj ,j

m′Σbj(−1)

η2bj ,i

η2bj ,j

i 6= j(48)

where ηbj ,i = 1 + qia2bj ,imΣbj

(−1) and m′Σb(−1) is the differential

of mΣb(z) with respect to z at point z = −1

15S. Lakshminaryana, J. Hoydis, M. Debbah and M.Assaad, ”Asymptotic analysis of distributed multi-cell beamforming”, inPIMRC 2010, 2010, pp. 2105–2110.


Approximation of Intercell Interference TermsFrom (28) and (26), the total ICI from all BSs towards user k is

∑

b 6=bk

ζ2b,k =∑

b6=bk

∑

l∈Ub

|hb,kml|2 (49)

Considering (44), the ICI term ζ2b,k in (49) can be written as follows,

ζ2b,k =∑

l∈Ub

√δl|hb,kul|2 ≈

∑

l∈Ub

√δlGl,k (50)

where δb,l is found from (45) and |hb,kub,l|2 ≈ Gl,k is from (48),

Therefore, the ICI from the bth BS to the kth user can be written as,

ζ2b,k =∑

l∈Ub

√δl

1

Na

a2bl,ka2bl,lm′Σbl

(−1)

η2bl,kη2bl,l

(51)

Approximately optimal ICI based on only large scale characteristicsof the user channels.


Numerical Examples – Multi-cell case

7 cell wrap-around, K = 28 users dropped randomly

Number of TX antennas per BS, NT = 4− 120

Frequency flat Rayleigh fading

Correlation between adjacent antennas, 0− 0.99


Numerical Examples – Multicell case

Sum power of {K,B,NT} = {28, 7, 28− 120} system with i.i.d.antennas

Figure : 0 dB SINR target Figure : 10 dB SINR target



Figure : Comparison of required transmit power of{K,B,NT} = {28, 7, 28− 120} system with i.i.d. antennas with 0 dB SINRtarget



Figure : Comparison of required transmit power of{K,B,NT} = {28, 7, 28− 120} system with i.i.d. antennas with 10 dB SINRtarget



Figure : Comparison of required transmit power of{K,B,NT} = {28, 7, 28− 120} system with 0.9 correlation between adjacentantennas with 0 dB SINR target


Throughput Optimal Linear Transmitter-Receiver Design

S. S. Christensen, R. Agarwal, E. Carvalho, and J. Cioffi, ”Weighted sum-rate maximization using weighted MMSEfor MIMO-BC beamforming design,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 47924799, Dec. 2008.

Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, An iteratively weighted MMSE approach to distributed sum-utilitymaximization for a MIMO interfering broadcast channel, IEEE Trans. Signal Processing, vol. 59, no. 9, pp. 4331 –4340, Sep. 2011

Kaleva, J.; Tolli, A.; Juntti, M.; , ”Weighted Sum Rate Maximization for Interfering Broadcast Channel viaSuccessive Convex Approximation”, Global Communications Conference, 2012. GLOBECOM 2012. IEEE , Dec. 2012

P. Komulainen, A. Tolli & M. Juntti, Effective CSI Signaling and Decentralized Beam Coordination in TDDMulti-Cell MIMO Systems, IEEE Transactions on Signal Processing, vol. 61, no. 9, pp. 2204 – 2218, May 2013

J. Kaleva, A. Tolli & M. Juntti, ”Primal Decomposition based Decentralized Weighted Sum Rate Maximization withQoS Constraints for Interfering Broadcast Channel”, in Proc. IEEE SPAWC 2013, Darmstadt, Germany, June, 2013

J. Kaleva, A. Tolli & M. Juntti, ”Decentralized Beamforming for Weighted Sum Rate Maximization with RateConstraints”, in Proc. IEEE PIMRC 2013 - Workshop on Cooperative and Heterogeneous Cellular Networks, London,UK, Sep. 2013

J. Kaleva, A. Tolli & M. Juntti, ”Decentralized Sum Rate Maximization with QoS Constraints for InterferingBroadcast Channel via Successive Convex Approximation”, IEEE Transactions on Signal Processing, submitted Feb2014, major revision Jun 2014


System Assumptions and Research ProblemAssumptions:

Cellular multi-userMIMO system.

Each user is associatedto single BS(non-cooperative)

TDD and perfect CSIquantization.

User 3User 1

User 2 User 4

BS2BS1

Decentralised beamformer design

Objective: WSRM problem (with user specific rate constraints)I NP-hard problemI Efficient relaxation methods are required for tractability

Focus: low computational complexity and CSI acquisition(pilot/backhaul signalling)


WSRM Problem Formulation

Maximise the WSRM over a set of transmit covariance matricesKxk = MkM

Hk

max .Kxk

B∑

b=1

∑

k∈Ub

µk log det(I + R−1k Hbk,kKxkH

Hbk,k

)

s. t.∑

k∈Ub

Tr(Kxk) ≤ Pb, b = 1, . . . , B,(52)

where the interference+noise covariance matrix for user k is

Rk =

K∑

i=1,i 6=kHbi,kKxiH

Hbi,k

+N0I (53)

Difficult non-convex optimisation problem in general (except whenB = 1 or Bk = B ∀ k)


MSE Reformulation

The MSE of the received data vector dk = UHk yk for user k is

Ek , E[(dk − dk)(dk − dk)H]

= I−UHkHbk,kMk − (UH

kHbk,kMk)H + UH

kRkUk,(54)

where the received signal covariance Rk for user k is

Rk , E[ykyHk ] =

K∑

i=1

Hbi,kMiMHi HH

bi,k+ σ2kI. (55)

When the MMSE receiver (4) is employed in (54), the MSE matrixbecomes

EMMSEk = I−MH

kHHbk,k

R−1k Hbk,kMk (56)

Furthermore

EMMSEk =

(I + MH

kHHbk,k

R−1b,kHbk,kMk

)−1(57)


MSE Reformulation

Applying (57) to (52), we can reformulate the WSRMax objective asminMk

∑Kk=1 µk log det

(EMMSEk

)– still non-convex

Local solution: introduce new variables and split the problem intosolvable subproblems

min .Uk,Mk,Ek

K∑

k=1

µk log det(Ek

)

s. t. Ek 4 Ek, k = 1, . . . ,K,Mk ∈ Pbk , k = 1, . . . ,K,

(58)

where Pb, b = 1, . . . , B are separable convex per-BS powerconstraints, and the relaxation Ek 4 Ek, k = 1, . . . ,K bounds theachieved MSE16.

16The relaxation tightness follows from the matrix monotonicity of the determinant function [Boyd&Vandenbergh].


MSE Reformulation

For fixed Mk, the rate maximizing Uk are solved from the roots ofthe Lagrangian of (58) as Uk = R−1k Hbk,kMk, k = 1, . . . ,K

For fixed receive beamformers Uk, the concave objective function isiteratively linearised w.r.t Ek.17

The linearised convex subproblem in ith iteration is given as

min .Mk,E

ik

K∑

k=1

µkTr(Wi

kEik

)

s. t. Ek 4 Eik, k = 1, . . . ,K,

Mk ∈ Pbk , k = 1, . . . ,K,

(59)

where Wik = Gi

k and Gik = ∇Ei−1

k

(log det

(Ei−1k

))= [Ei−1

k ]−1 for

all k = 1, . . . ,K.

Monotonic improvement of the objective of (58) on every iteration.17This method in the context of weighted sum rate maximisation was established in [Shi et al, TSP’11], where it was

referred to as (iteratively) weighted MMSE minimization.


TX precoder adaptation stepThe relaxation Ek 4 Ei

k is tight → Replace Eik with Ek (54) in the

objective of (59)Local convex problem for each BS b

min∑

k∈Ub

(−2µkTr(WkU

HkHbk,kMk)

+

K∑

i=1

µiTr(MHkHH

bk,iUiWiU

Hi Hbk,iMk)

)

s. t.∑

k∈Ub

Tr(MkMHk ) ≤ Pb

(60)

Iterative solution from the KKT conditions

Mk =

(K∑

i=1

HHbk,i

UiWiUHi Hbk,i + νbkI

)−1HHbk,k

UkWk (61)

where the optimal νbk18 is found via bisection

18dual variable related to the power constraint


Alternating Optimization, Global Algorithm

WSRM via WSMSE19

1 Initialize TX beamformers Mk, i = 1, . . . ,K

2 Compute the optimal LMMSE receivers Uk ∀k, for given Mi ∀ i3 Compute the MSE weights Wk, for given Uk,Mk ∀b, k4 Compute Mk ∀k, for given Ui,Wi ∀i5 Repeat steps 2-4 until convergence

Every step can be calculated locally → decentralised design

Implementation challenges:I Uk∀ k needs to be conveyed to the BSs → precoded UL pilot

I Wk∀ k needs to be shared among BSs → backhaul exchange

19Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, An iteratively weighted MMSE approach to distributed sum-utilitymaximization for a MIMO interfering broadcast channel, IEEE Trans. Signal Processing, vol. 59, no. 9, pp. 4331 – 4340,Sep. 2011


Alternative WSRM Problem Formulation

Data stream specific processing

max .mk,l,uk,l

B∑

b=1

∑

k∈Ub

µk

(mk∑

l=1

log2 (1 + Γk,l)

)

s. t.∑

k∈Ub

mk∑

l=1

‖mk,l‖2 ≤ Pb, b = 1, . . . , B,

(62)

where SINR of data stream l of user k is

Γk,l =|uHk,lHbk,kmk,l|2

K∑

i=1

mi∑

j=1(i,j)6=(k,l)

|uHk,lHbi,kmi,j |2 + σ2k‖uk,l‖2

.


Preliminaries

Rate maximising receive beamformers for fixed precoders are

uk,l =

K∑

j=1

mk∑

l=1

Hbj ,kmj,lmHj,lH

Hbj ,k

+ Iσ2k

−1

Hbk,lmk,l ∀ (k, l).

Mean-squared error (MSE) for data stream l of user k is defined as

εk,l , |1− uHk,lHbk,kmk,l|2+

K∑

i=1

mi∑

j=1(i,j)6=(k,l)

|uHk,lHbi,kmj,i|2 + σ2k‖uk,l‖2.

MSE and SINR have following relation (assuming MMSE receivebeamformers)

ε−1k,l = Γk,l + 1.


MSE formulation

max .tk,l,mk,l

B∑

b=1

∑

k∈Ub

µk

(mk∑

l=1

log2 (g(tk,l))

)

s. t. εk,l ≤ [g(tk,l)]−1 ∀ (k, l),

∑

k∈Ub

mk∑

l=1

‖mk,l‖2 ≤ Pb, b = 1, . . . , B.

(63)

MSE constraint can be formulated as a difference of convexfunctions program (DCP), by introducing upper boundary g(tk,l) foreach MSE term εk,l

εk,l ≤ [g(tk,l)]−1 ⇔ εk,l − [g(tk,l)]

−1 ≤ 0.

Note that the problem is still non-convex.

I g(tk,l) is monotonic and log-concave.I [g(tk,l)]

−1 is convex.


Successive convex approximation

At each point t(i)k,l ∀ (k, l), (63) is approximated as convex problem

by taking first-order Taylor series approximation of the MSE upperboundary.

f(x, x0) , f(x0) + (x− x0) ∂∂xf(x0). (64)

Convex, ith, approximation of (63) at point t(i)k,l ∀ (k, l) is

max .tk,l,mk,l

B∑

b=1

∑

k∈Ub

µk

(mk∑

l=1

log2 (g(tk,l))

)

s. t. εk,l ≤ g(tk,l, t(i)k,l) ∀ (k, l)

∑

k∈Ub

mk∑

l=1

‖mk,l‖2 ≤ Pb, b = 1, . . . , B.

(65)


SCA algorithm outline

1: Initialize t(1)k,l ∀ (k, l) in such a away that

[g(t

(1)k,l )]−1

= 1.

2: Initialize precoders mk,l ∀ (k, l) in such a way that sum powerconstraints are satisfied.

3: Set i = 1.4: repeat5: Generate MMSE receive beamformers uk,l ∀ (k, l).6: repeat7: Solve precoders mk,l ∀ (k, l) and tk,l ∀ (k, l) from (65).8: i = i+ 1.9: t

(i)k,l = tk,l ∀ (k, l).

10: until Desired level of convergence has been reached.11: until Desired level of convergence has been reached.


Exponential MSE boundary functions

An import class of functions for MSE boundary are exponentialfunctions g(tk,l) = αtk,l , where α > 0.

Allow efficient iterative solution and distributed design.

max .tk,l,mk,l

B∑

b=1

∑

k∈Ub

µk

(mk∑

l=1

log2(α)tk,l

)

s. t. εk,l ≤ α−t(i)k,l − log(α)(tk,l − t(i)k,l)α

−t(i)k,l ∀ (k, l)∑

k∈Ub

mk∑

l=1

‖mk,l‖2 ≤ Pb, b = 1, . . . , B.

(66)


Solution via KKT Conditions

Beamformers mk,l can be solved from the roots of the gradient ofthe Lagrangian as

mk,l =

∑

(i,j)

λi,jHHbk,i

ui,juHi,jHbk,i + νbkI

−1

λk,lHHbk,k

uk,l. (67)

The Lagrange multipliers are λk,l = µkαt(i)k,l

log(α) and optimal νb ∀ b arefound by bisection.

t(i+1)k,l can be found from the corresponding complementary

slackness constraint to be

t(i+1)k,l = t

(i)k,l +

1

log(α)(1− εk,lαt

(i)k,l) ∀ (k, l).

Here, term 1log(α) can be seen as step size for an exact line search.


Improved Rate of Convergence

To increase the rate of convergence it is beneficial to choose thenext point of approximation further than indicated by the exact linesearch.

I Choose larger step sizes.I Too large steps size causes oscillation in the objective.

Simple adaptive update of the step size can be, for example,formulated as

t(i+1)k,l = t

(i)k,l +

1

log(α− α−eiβ

)(1− εk,lαt

(i)k,l) ∀ (k, l).

I Use more aggressive search at first iterations. Converges to exact linesearch.


Numerical Results

5 10 15 20 25 30 354

6

8

10

12

14

16

18

20

22

SNR (dB)

Sum

rate

(bits/H

z/s

)

SCA (Imax

= 3)

[4] (Imax

= 3)

[3] (Imax

= 3)

SCA (Imax

= 6)

[4] (Imax

= 6)

[3] (Imax

= 6)

SCA (Imax

= 10)

[4] (Imax

= 10)

[3] (Imax

= 10)

10 Iterations

6 Iterations

3 Iterations

Figure : Impact of the limited number of iterations to the achievable sum ratewith NT = 4, NR = 2, K = 8.[3] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, An iteratively weighted MMSE approach to distributed sum-utility maximization for a MIMO

interfering broadcast channel, IEEE Trans. Signal Processing, Sep. 2011.

[4] T. Bogale and L. Vandendorpe, Weighted sum rate optimization for downlink multiuser MIMO coordinated base station systems: Centralized and

distributed algorithms, IEEE Trans. Signal Processing, Dec. 2011.


WSRmax with QoS Constraints20

Objective: WSRmax with per user QoS constraints

max .mk,l,uk,l

B∑

b=1

∑

k∈Ub

µk

(mk∑

l=1

log2 (1 + Γk,l)

)

s. t.

mk∑

l=1

log2 (1 + Γk,l) ≥ Rk ∀ k = 1, . . . ,K,

∑

k∈Ub

mk∑

l=1

‖mk,l‖2 ≤ Pb, b = 1, . . . , B,

where SINR of data stream l of user k is

Γk,l =|uHk,lHbk,kmk,l|2

K∑

i=1

mi∑

j=1,(i,j)6=(k,l)

|uHk,lHbi,kmi,j |2 + σ2k‖uk,l‖2

.

20J. Kaleva, A. Tolli & M. Juntti, ”Primal Decomposition based Decentralized Weighted Sum Rate Maximization with QoSConstraints for Interfering Broadcast Channel”, in Proc. IEEE SPAWC 2013, Darmstadt, Germany, June, 2013


MSE Reformulation

Introduce upper boundary 2tk,l for each MSE term εk,l

maxtk,l,mk,l

B∑

b=1

∑

k∈Ub

µk

(mk∑

l=1

tk,l

)

s. t. εk,l ≤ 2−tk,l ∀ (k, l),mk∑

l=1

tk,l ≥ Rk, k = 1, . . . ,K,

∑

k∈Ub

mk∑

l=1

‖mk,l‖2 ≤ Pb, b = 1, . . . , B,

Difference of convex functions program (DCP), εk,l − 2tk,l ≤ 0

Successive convex (linear) approximation is used to approximate theMSE bounds 2tk,l ∀ (k, l) iteratively.


Linear Approximation + Lagrangian Relaxation

maxtk,l,mk,l

B∑

b=1

∑

k∈Ub

µk

(mk∑

l=1

tk,l

)−

B∑

b=1

∑

k∈Ub

γk

(Rk −

mk∑

l=1

tk,l

)

s. t. εk,l ≤ a(i)k,ltk,l + b(i)k,l ∀ (k, l),

∑

k∈Ub

mk∑

l=1

‖mk,l‖2 ≤ Pb, b = 1, . . . , B.

Convex problem

Partial Lagrangian relaxation of the rate constraints→ WSRmax with new weights µk + γk per user

γk represent demand of rate for user k = 1, . . . ,K

Still coupled by the MSE constraints εk,l ≤ a(i)k,ltk,l + b(i)k,l ∀ (k, l)


Iterative SolutionTransmit beamformers are obtained directly from the KKT conditions as

mk,l = K†k,lHHbk,l

uk,lλk,l ∀ (k, l),

where

Kk,l =

K∑

i=1

mi∑

j=1

λi,jHHbk,i

ui,juHi,jHbk,i + Iνb.

MSE constraint related dual variables (MSE weights) can be written inclosed form as

λk,l = (µk + γk)/a(i)k,l ∀ (k, l).

I Depends only on the point of approximation and rate demand dual variablesγk = 1, . . . ,K.

The rate demand weight factors γk ∀ k = 1, . . . ,K are updated according tothe subgradients of the corresponding rate constraints

γ(i+1)k =

(γ(i)k + β

(i)k

(Rk +

mk∑

l=1

t(i)k,l)

))+

.

All steps can be executed locally at each BS / terminal


Features & Issues

Computational complexity

Receive beamformers uk,l can be solved in a closed form (LMMSE)

Transmit beamformers mk,l require simple bisection over νb

Dual variables λk,l are solved in a closed form

Feasibility

Each step is not required to be feasible (rate constraints)

Possibly strictly feasible only after convergence

Initialization

Does not require feasible initial pointI Simplifies distributed design


Algorithm Outline

1: Initialize precoders mk,l ∀ (k, l).2: Initialize γk,l = 0 ∀ (k, l).3: repeat4: Generate LMMSE receive beamformers uk,l ∀ (k, l).5: repeat6: Measure MSE εk,l ∀ (k, l).7: Update rate demand variables γk ∀ k = 1, . . . ,K.8: Assign weights λk,l ∀ (k, l).9: Exchange weights λk,l ∀ (k, l) between the adjacent BSs.

10: Solve transmit beamformers mk,l ∀ (k, l).11: until Desired level of convergence has been reached or i > Imax.12: until Desired level of convergence has been reached or j > Jmax.


Numerical Results (1/2)

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

Iteration

Rate

(b

its/H

z/s

)

k = 1, b = 1, R = 4 bits/s/Hz

k = 2, b = 1

k = 1, b = 2, R = 4 bits/s/Hz

k = 2, b = 2

Average sum rate per BS

Figure : Behaviour of the unconstrained users at SNR = 5dB with 3dB cellseparation, NT = 4, NR = 2, Kb = 2 and β = 10.


Numerical Results (2/2)

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

Iteration

Rate

(b

its/H

z/s

)

k = 1, b = 1, R = 4 bits/s/Hz

k = 2, b = 1, R = 2 bits/s/Hz

k = 3, b = 1, R = 2 bits/s/Hz

k = 1, b = 2, R = 4 bits/s/Hz

k = 2, b = 2, R = 2 bits/s/Hz

k = 3, b = 2, R = 2 bits/s/Hz

Figure : Convergence at SNR = 15dB with 3dB cell separation, NT = 4,NR = 2, Kb = 3 and β = 4.


Decentralised Solution via Precoded UL Pilot

Assumptions:

Each user is associated tosingle BS (non-cooperative)

TDD and perfect CSI

Precoded UL pilot sequencesavailable

1

2

1 BK-1

K

Goal: Beam coordination to avoid inter-cell interferencea

I Scheduling & TX – RX design

CSI acquisition: Pilot & Backhaul signalling

Decentralized, practical methods, based on locally available CSI

Support for independent user scheduling by BSs

aP. Komulainen, A. Tolli & M. Juntti, Effective CSI Signaling and Decentralized Beam Coordination in TDD Multi-CellMIMO Systems, IEEE Transactions on Signal Processing, vol. 61, no. 9, pp. 2204 – 2218, May 2013


Strategy A: Global Algorithm

Each BS b calculates own weights Wk ∀ k ∈ Ub : distribute viabackhaul

Each BS calculates own precoders Mk ∀ k ∈ Ub : use for datatransmission

Each UE calculates own receiver Uk : use for reception and ULsounding

Slow convergence


Strategy B: Separate Channel Sounding (CS) and BusyBurst (BB) Pilots

CS pilot Qk whitens the inter-cell interference at terminal k so thatQHkQk = R−1k , where Rk =

∑i 6∈Ubk

Hbi,kMiMHi HH

bi,k+N0I

The MSE weights calculated by the terminals can be incorporated to

the uplink BB signaling so that pilot precoder is W12k Uk

Allows local iterationsno backhaul required


Strategy C: Cell-specific Iterations with CS Only

Allows local iterationsfast convergence


Numerical Results: Setup

Two 4-antenna BSs, five 2-antenna UEs per BS

Cell separation defined as a1/a2

Uncorrelated Rayleigh (quasistatic) fading


Numerical Results: Convergence

2 4 6 8 10 12 14 16 18 204

6

8

10

12

14

16

18

Frame

Su

m r

ate

per

BS

[b

its/

Hz/

s]

M = 4, N = 2, K = 5, SNR = 25dB, cell sep. = 0dB

Alg. 1 (matrix weighted)

Strat. A: BB only

Strat. B: BB+CS

Strat. C: CS only

Strat. A (AP constraints)

Non−cooperative

Figure : Average convergence of the sum rate at 0dB cell separation, at 25dBSNR.


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