Coordinated Multiantenna Interference Management in …atolli/IM_tutorial_2015_full.pdf · I Resource allocation and linear transceiver design ... Coordinated Multi-cell Transmission

8 September, 2015 Coordinated Multiantenna Interference Management in 5G Networks 1

Coordinated Multiantenna Interference Managementin 5G Networks

Antti [email protected]

Department of Communications Engineering (DCE)University of Oulu, Finland

8 September, 2015

c©Antti Tolli, Department of Comm. Engineering


OutlineEvolution of multiantenna systemsMIMO with large antenna arraysLinear transceiver design and resource allocation

I Introduction to convex optimisationI Resource allocation and linear transceiver designI Coordinated transceiver optimisationI Coherent vs. coordinated beamforming

Minimum power multicell beamforming with QoS constraintsI Centralised solutionI Decentralised solution via optimisation decompositionI Large system approximation

Throughput optimal linear TX-RX designI Weighted sum rate maximisation (WSRM) via MSE minimisationI WSRM with rate constraintsI Decentralised solution via precoded UL pilotI Bidirectional signalling strategies for dynamic TDDI Mode selection and transceiver design in underlay D2D MIMO

systemsc©Antti Tolli, Department of Comm. Engineering


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-node 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



General ObjectiveGoal: Design dynamic multi-dimensional radio resource managementacross time, frequency, and space (location)Assumption: Heterogeneous network composed of

I 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



Evolution of Multiantenna Systems

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

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

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

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

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

+-.+!+*(



Coordinated Multi-cell Transmission/Reception

Coherent (joint) 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/ReceptionCoordinated beamforming

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



Interference AlignmentConsider a symmetric constant MIMO interference channel with KTX-RX pairs each node equipped with M antennas

TX1

TX2

TX3

RX1

RX2

RX3



Interference AlignmentExact capacity characterization of the K-user interference channel isunknownDegrees of freedom (or multiplexing gain)DoF = lim

SNR→∞sum ratelog2 SNR

Achievable via interference alignment (IA) 1

Feasibility conditions for IA (for constant MIMO channel, K ≥ 3)234

(K + 1)d ≤ 2M =⇒ Total DoF ≤ 2MK

K + 1≤ 2M (1)

1V.R. Cadambe and S.A. Jafar. ”Interference Alignment and Degrees of Freedom of the K-User Interference Channel.”IEEE Trans. Inform. Theory, August 2008.

2C. Yetis, T. Gou, S. Jafar, and A. Kayran, ”On feasibility of interference alignment in MIMO interference networks,” IEEETrans. Signal Process., 2010.

3M. Razaviyayn, G. Lyubeznik, and Z.Q. Luo, ”On the Degrees of Freedom Achievable Through Interference Alignment ina MIMO Interference Channel,” IEEE Trans. Signal Process., 2012

4G. Bresler, D. Cartwright, and D. Tse, Feasibility of Interference Alignment for the MIMO Interference Channel,” in IEEETrans. Info. Theory, 2014









MIMO with Large Antenna Arrays

D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge,England: Cambridge University Press, 2005.

A. M. Tulino and Sergio Verd, Random Matrix Theory and Wireless Communications,Foundations and Trends in Communications and Information Theory, vol. 1, no. 1, pp1-182.

Rusek, F.; Persson, D.; Buon Kiong Lau; Larsson, E.G.; Marzetta, T.L.; Edfors, O.;Tufvesson, F., ”Scaling Up MIMO: Opportunities and Challenges with Very LargeArrays,” Signal Processing Magazine, IEEE , vol.30, no.1, pp.40–60, Jan. 2013

Foundations and Trends in Communications and Information Theory, Vol. 1, Issue 1,”Random Matrix Theory and Wireless Communications” by A. Tulino and S Verdu

R. Couillet and M. Debbah, Random Matrix Methods for Wireless Communications, 1sted. Cambridge University Press, 2011.



Introduction

Point-to-point MIMO channel capacity with large antenna arraysI Capacity formulationI Approximate capacity from deterministic eigenvalue distribution

(nr, nt →∞)I Asymptotic capacity when nr (nt) is fixed while nt (nr) →∞

Multiuser MIMO sum rate in UL/DL with large antenna arraysI Capacity formulationI Asymptotic capacity when number of users K is fixed while nt (nr)→∞

Multi-user (Multi-cell) MIMO Uplink with linear MMSE receivers –Large System Approximation

I MMSE receiver derivationI MMSE SINR – large system approximation



Point-to-point MIMO Architecture334 MIMO II: capacity and multiplexing architectures

Figure 8.1 The V-BLASTarchitecture for communicatingover the MIMO channel.

+

Pnt

P1

Qx[m]

H[m]

w[m]

y[m] Jointdecoder

AWGN coderrate R1

AWGN coderrate Rnt

····

········

coordinate system given by a unitary matrix Q, not necessarily dependent onthe channel matrix H. This is the V-BLAST architecture. The data streamsare decoded jointly. The kth data stream is allocated a power Pk (such thatthe sum of the powers, P1+· · ·+Pnt

, is equal to P, the total transmit powerconstraint) and is encoded using a capacity-achieving Gaussian code with rateRk. The total rate is R=!nt

k=1Rk.As special cases:

• If Q=V and the powers are given by the waterfilling allocations, then wehave the capacity-achieving architecture in Figure 7.2.

• If Q= Inr , then independent data streams are sent on the different transmitantennas.

Using a sphere-packing argument analogous to the ones used in Chapter 5,we will argue an upper bound on the highest reliable rate of communication:

R < logdet"Inr +

1N0

HKxH!#bits/s/Hz! (8.2)

Here Kx is the covariance matrix of the transmitted signal x and is a functionof the multiplexing coordinate system and the power allocations:

Kx "=Q diag#P1$ % % % $Pnt&Q!! (8.3)

Considering communication over a block of time symbols of length N , thereceived vector, of length nrN , lies with high probability in an ellipsoid ofvolume proportional to

det'N0Inr +HKxH!(N ! (8.4)

This formula is a direct generalization of the corresponding volume for-mula (5.50) for the parallel channel, and is justified in Exercise 8.2. Sincewe have to allow for non-overlapping noise spheres (of radius

"N0 and,

hence, volume proportional to NnrN0 ) around each codeword to ensure reliable

[D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.]

Generalized architecture to multiplex nt independent data streams

The choice of Q depends on the CSITI Q = V requires full CSIT → capacity achieving scheme with WF

power allocationI Q = I requires no CSIT → independent streams sent on each TX

antenna



Capacity in Fast Fading MIMO Channel

Capacity of fast fading channel H ∈Cnr×nt with CSIR

C = maxE[‖x‖2]≤P

I(x; y,H) = maxE[‖x‖2]≤P

I(x; y|H) (2)

For fixed MIMO channel realization H = H

I(x; y|H = H) = h(y)− h(y|x)

= h(y)− h(n) = h(y)− log(πeN0)nr

≤ log((πe)nr det(N0Inr + HKxHH))− log(πeN0)

nr

= log det

(Inr +

1

N0HKxH

H

)

where Kx = QPQH is the covariance matrix of x, x ∼ CN (0,Kx),and P = diag (p1, . . . , pnt)



Capacity in Fast Fading MIMO ChannelCSI at receiver only

In general, Kx depends on the stationary distribution of the fadingprocess {H[m]}When the elements of H are i.i.d. CN (0, 1), the optimal Kx is

Kx =P

ntInt (3)

The ergodic capacity of MIMO channel without CSIT is simplified to

C = E[log det

(Inr +

P

ntN0HHH

)]

=

nmin∑

i=1

E[log

(1 +

P

ntN0λ2i

)](4)

where nmin = min(nt, nr)



Large Antenna Array RegimeFocus on the square channel n = nt = nr

Cnn(SNR) = E

[n∑

i=1

log

(1 + SNR

λ2in

)](5)

where λi/√n are the singular values of H/

√n

When n→∞, thedistribution of λi/

√n

becomes deterministic

f∗(x) =

{1π

√4− x2 0 ≤ x ≤ 2,

0 else

342 MIMO II: capacity and multiplexing architectures

due to Marcenko and Pastur [78], the empirical distribution of the singularvalues of H/

!n converges to a deterministic limiting distribution for almost

all realizations of H. Figure 8.4 demonstrates the convergence. The limitingdistribution is the so-called quarter circle law.3 The corresponding limitingdensity of the squared singular values is given by

f "!x"=

!"

#

1#

$1x# 1

4 0 $ x $ 4$

0 else%(8.23)

Hence, we can conclude that, for increasing n,

1n

n%

i=1

log&1+ SNR

&2i

n

'%

( 4

0log!1+ SNRx"f "!x"dx% (8.24)

If we denote

c"!SNR" '=( 4

0log!1+ SNRx"f "!x"dx$ (8.25)

Figure 8.4 Convergence of theempirical singular valuedistribution of H/

!n. For

each n, a single randomrealization of H/

!n is

generated and the empiricaldistribution (histogram) of thesingular values is plotted. Wesee that as n grows, thehistogram converges to thequarter circle law.

0 0.5 1 1.5 20

1

2

3

4n = 32

0 0.5 1 1.5 20

2

4

6

8

10n = 64

0 0.5 1 1.5 20

5

10

15

20n = 128

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7Quarter circle law

3 Note that although the singular values are unbounded, in the limit they lie in the interval(0$2) with probability 1.

[D. Tse and P. Viswanath, Fundamentals of Wireless Communication,Cambridge University Press, 2005]



Large Antenna Array RegimeFor increasing n 5

1

n

n∑i=1

log

(1 + SNR

λ2i

n

)→∫ 4

0log(1 + SNRx)f∗(x)dx = c∗(SNR) (6)

and the closed form solution to the integral is

c∗(SNR) = 2 log

(1 +√4SNR + 1

2

)−

log e

4SNR

(√4SNR + 1− 1

)2(7)

Furthermore

limn→∞

Cnn(SNR)

n= c∗(SNR)→ Cnn(SNR) ≈ nc∗(SNR) (8)

Capacity grows linearly in n at any SNR5More in: Foundations and Trends in Communications and Information Theory, Vol. 1, Issue 1, ”Random Matrix Theory

and Wireless Communications” by A. Tulino and S Verdu



Example: Approximate vs. Exact Capacity

343 8.2 Fast fading MIMO channel

we can solve the integral for the density in (8.23) to arrive at (see Exer-cise 8.17)

c!!SNR"= 2 log!1+ SNR" 1

4F!SNR"

"" log e

4SNRF!SNR"# (8.26)

where

F!SNR" $=##

4SNR+1"1$2

% (8.27)

The significance of c!!SNR" is that

limn$%

Cnn!SNR"n

= c!!SNR"% (8.28)

So capacity grows linearly in n at any SNR and the constant c!!SNR" is therate of the growth.We compare the large-n approximation

Cnn!SNR"& nc!!SNR"# (8.29)

with the actual value of the capacity for n = 2#4 in Figure 8.5. We see theapproximation is very good, even for such small values of n. In Exercise 8.7,we see statistical models other than i.i.d. Rayleigh, which also have a linearincrease in capacity with an increase in n.

Linear scaling: a more in-depth lookTo better understand why the capacity scales linearly with the number ofantennas, it is useful to contrast the MIMO scenario here with three otherscenarios:

Figure 8.5 Comparisonbetween the large-napproximation and the actualcapacity for n= 2! 4.

–5 0 10 15 20SNR (dB)

25 30

Approximate capacity c!

–10

9

8

7

6

5

4

3

2

1

0

Rat

e(b

its /s

/ Hz)

5

Exact capacity 14 C44

Exact capacity 12 C22

Figure: Comparison between the large-n approximation and the actual capacityfor n = 2, 4. [D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005]



Large Antenna Array Regime, nr >> nt

Assume nr >> nt, and the elements of H are i.i.d. CN (0, 1)

HHH

nr≈ Int (9)

then

log

∣∣∣∣Inr +P

ntN0HHH

∣∣∣∣ = log

∣∣∣∣Int +P

ntN0HHH

∣∣∣∣

≈ nt log(1 +PnrntN0

) (10)

Match filter receiver is asymptotically optimal



Large Antenna Array Regime, nt >> nr

Assume nt >> nr, and the elements of H are i.i.d. CN (0, 1)

HHH

nt≈ Inr (11)

then the rate expression without CSIT is simplified to

log

∣∣∣∣Inr +P

ntN0HHH

∣∣∣∣ ≈ nr log(1 +P

N0) (12)

The rate expression with full CSIT (at high SNR) is simplified to

log

∣∣∣∣Inr +1

N0HKxH

H

∣∣∣∣ ≈ nr log(1 +ntnr

P

N0) (13)

The rows of H are asymptotically orthogonal:Kx = VPVH ≈ P

nrntHHH



Uplink System ModelAssume time-invariant uplink channel with K single-antenna usersand a single BS with nr receive antennas.

426 MIMO IV: multiuser communication

the downlink. We conclude in Section 10.5 with a discussion of the systemimplications of using MIMO in cellular networks; this will link up the newinsights obtained here with those in Chapters 4 and 6.

10.1 Uplink with multiple receive antennas

We begin with the narrowband time-invariant uplink with each user havinga single transmit antenna and the base-station equipped with an array ofantennas (Figure 10.1). The channels from the users to the base-station aretime-invariant. The baseband model is

y!m"=K!

k=1

hkxk!m"+w!m"# (10.1)

with y!m" being the received vector (of dimension nr , the number of receiveantennas) at time m, and hk the spatial signature of user k impinged on thereceive antenna array at the base-station. User k’s scalar transmit symbol attime m is denoted by xk!m" and w!m" is i.i.d. !" $0#N0Inr% noise.

10.1.1 Space-division multiple access

In the literature, the use of multiple receive antennas in the uplink is oftencalled space-division multiple access (SDMA): we can discriminate amongstthe users by exploiting the fact that different users impinge different spatialsignatures on the receive antenna array.An easy observation we can make is that this uplink is very similar to

the MIMO point-to-point channel in Chapter 5 except that the signals sent

Figure 10.1 The uplink withsingle transmit antenna at eachuser and multiple receiveantennas at the base-station.

out on the transmit antennas cannot be coordinated. We studied preciselysuch a signaling scheme using separate data streams on each of the transmitantennas in Section 8.3. We can form an analogy between users and transmitantennas (so nt , the number of transmit antennas in the MIMO point-to-pointchannel in Section 8.3, is equal to the number of users K). Further, theequivalent MIMO point-to-point channel H is !h1# & & & #hK", constructed fromthe SIMO channels of the users.Thus, the transceiver architecture in Figure 8.1 in conjunction with the

receiver structures in Section 8.3 can be used as an SDMA strategy. Forexample, each of the user’s signal can be demodulated using a linear decorre-lator or an MMSE receiver. The MMSE receiver is the optimal compromisebetween maximizing the signal strength from the user of interest and sup-pressing the interference from the other users. To get better performance, onecan also augment the linear receiver structure with successive cancellationto yield the MMSE–SIC receiver (Figure 10.2). With successive cancella-tion, there is also a further choice of cancellation ordering. By choosing a

[D. Tse and P.Viswanath,

Fundamentals ofWireless Communication,

Cambridge UniversityPress, 2005]

The received signal vector at symbol time m isdescribed by

y[m] =

K∑

k=1

hkxk[m] + n[m]

= Hx[m] + n[m]

(14)

whereI xk is the TX symbol of user k, subject to

E[|x|2] ≤ Pk,I y ∈Cnr is the RX signal,I n ∼ CN (0, N0Inr

) complex white Gaussian noise,I hk =

√akhk ∈Cnr is the channel vector of user k,

where ak is the large scale fading factor and hk is thenormalized channel



Sum Capacity for Multiuser Uplink

Sum capacity expression for SDMA is equal to SU-MIMO withoutCSIT

log

∣∣∣∣∣Inr +

K∑

k=1

PkN0

hkhHk

∣∣∣∣∣ = log

∣∣∣∣Inr +1

N0HKxH

H

∣∣∣∣

=

K∑

i=1

log(1 + γmmse−sick ) (15)

where H = [h1, . . . ,hK ], Kx = diag(P1, . . . , PK)427 10.1 Uplink with multiple receive antennas

MMSE Receiver 2

MMSE Receiver 1

y[m]

User 2Decode User 2

Subtract User 1

User 1Decode User 1

different order, users are prioritized differently in the sharing of the commonFigure 10.2 The MMSE–SICreceiver: user 1’s data is firstdecoded and then thecorresponding transmit signalis subtracted off before the nextstage. This receiver structure,by changing the ordering ofcancellation, achieves the twocorner points in the capacityregion.

resource of the uplink channel, in the sense that users canceled later are treatedbetter.

Provided that the overall channel matrix H is well-conditioned, all ofthese SDMA schemes can fully exploit the total number of degrees of free-dom min!K"nr# of the uplink channel (although, as we have seen, differentschemes have different power gains). This translates to being able to simul-taneously support multiple users, each with a data rate that is not limitedby interference. Since the users are geographically separated, their trans-mit signals arrive in different directions at the receive array even whenthere is limited scattering in the environment, and the assumption of a well-conditionedH is usually valid. (Recall Example 7.4 in Section 7.2.4.) Contrastthis to the point-to-point case when the transmit antennas are co-located, anda rich scattering environment is needed to provide a well-conditioned channelmatrix H.

Given the power levels of the users, the achieved SINR of each user canbe computed for the different SDMA schemes using the formulas derived inSection 8.3 (Exercise 10.1). Within the class of linear receiver architecture,we can also formulate a power control problem: given target SINR require-ments for the users, how does one optimally choose the powers and linearfilters to meet the requirements? This is similar to the uplink CDMA powercontrol problem described in Section 4.3.1, except that there is a furtherflexibility in the choice of the receive filters as well as the transmit powers.The first observation is that for any choice of transmit powers, one alwayswants to use the MMSE filter for each user, since that choice maximizes theSINR for every user. Second, the power control problem shares the basicmonotonicity property of the CDMA problem: when a user lowers its transmitpower, it creates less interference and benefits all other users in the system.As a consequence, there is a component-wise optimal solution for the pow-ers, where every user is using the minimum possible power to support theSINR requirements. (See Exercise 10.2.) A simple distributed power controlalgorithm will converge to the optimal solution: at each step, each user firstupdates its MMSE filter as a function of the current power levels of the otherusers, and then updates its own transmit power so that its SINR requirementis just met. (See Exercise 10.3.)

[D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005]



Large Antenna Array Regime, nr >> K

Assume nr >> K, and the elements of hk are i.i.d. CN (0, 1)

HHH

nr≈ AK (16)

where AK = diag(a1, . . . , aK). Then,

∣∣∣∣Inr +1

N0HKxH

H

∣∣∣∣ = log

∣∣∣∣IK +1

N0KxH

HH

∣∣∣∣

≈K∑

k=1

log(1 +nrPkakN0

) (17)

Match filter receiver is asymptotically optimal



Downlink System ModelAssume time-invariant downlink channel with K single-antennausers and a single BS with nt transmit antennas.

448 MIMO IV: multiuser communication

10.3 Downlink with multiple transmit antennas

We now turn to the downlink channel, from the base-station to the multiple

Figure 10.15 The downlinkwith multiple transmit antennasat the base-station and singlereceive antenna at each user.

users. This time the base-station has an array of transmit antennas but eachuser has a single receive antenna (Figure 10.15). It is often a practicallyinteresting situation since it is easier to put multiple antennas at the base-station than at the mobile users. As in the uplink case we first consider thetime-invariant scenario where the channel is fixed. The baseband model of thenarrowband downlink with the base-station having nt antennas and K userswith each user having a single receive antenna is

yk!m"= h!kx!m"+wk!m"# k= 1# $ $ $ #K# (10.31)

where yk!m" is the received vector for user k at time m, h!k is an nt dimen-

sional row vector representing the channel from the base-station to user k.Geometrically, user k observes the projection of the transmit signal in thespatial direction hk in additive Gaussian noise. The noise wk!m"" !" %0#N0&and is i.i.d. in time m. An important assumption we are implicitly makinghere is that the channel’s hk are known to the base-station as well as to theusers.

10.3.1 Degrees of freedom in the downlink

If the users could cooperate, then the resulting MIMO point-to- point channelwould have min%nt#K& spatial degrees of freedom, assuming that the rank ofthe matrix H= !h1# $ $ $ #hK" is full. Can we attain this full spatial degrees offreedom even when users cannot cooperate?Let us look at a special case. Suppose h1# $ $ $ #hK are orthogonal (which is

only possible if K # nt). In this case, we can transmit independent streams ofdata to each user, such that the stream for the kth user 'xk!m"( is along thetransmit spatial signature hk, i.e.,

x!m"=K!

k=1

xk!m"hk) (10.32)

The overall channel decomposes into a set of parallel channels; user k receives

yk!m"= $hk$2xk!m"+wk!m") (10.33)

Hence, one can transmit K parallel non-interfering streams of data to theusers, and attain the full number of spatial degrees of freedom in the channel.What happens in general, when the channels of the users are not orthogonal?

Observe that to obtain non-interfering channels for the users in the exampleabove, the key property of the transmit signature hk is that hk is orthogonal

[D. Tse and P.Viswanath,

Fundamentals ofWireless Communication,

Cambridge UniversityPress, 2005]

The received signal vector at symbol time m is

yk[m] = hHk x[m] + wk[m]

= hHkuk√pkdk[m] +

K∑

i=1,i 6=khHkui√pidi[m] + wk[m]

(18)

I x ∈Cnt is the TX signal vector, subject to powerconstraint E[Tr(xxH)] =

∑Kk=1 pk ≤ P ,

I uk ∈Cnt is the normalised beamformer, ‖uk‖ = 1I dk ∈C is the normalised data symbol, E

[|dk|2

]= 1

I yk ∈C is the RX signal,I wk ∼ CN (0, N0) complex white Gaussian noise,I hk =

√akhk ∈Cnt is the channel vector of user k

assumed to be ideally known at the transmitter



Sum Capacity for the Multiuser DownlinkMaximisation of the DL sum rate via dual uplink reformulationOptimal solution from constrained optimisation problem

maxqk

log2

∣∣∣∣∣Int +1

N0

K∑

k=1

qkhkhHk

∣∣∣∣∣

subject to

K∑

k=1

qk ≤ P, qk ≥ 0, k = 1, . . . ,K (19)

where qk is the dual UL power such that∑K

k=1 qk =∑K

k=1 pk = PWhen nt >> K, the objective of (19) is simplified to

maxqk

log

∣∣∣∣IK +1

N0KxH

HH

∣∣∣∣ ≈ maxqk

K∑

k=1

log(1 +qkntakN0

) (20)

where Kx = diag(q1, . . . , qK) and HHHnt≈ diag(a1, . . . , aK)



Linear MMSE Receiver – DerivationConsider the uplink system model y[m] = Hx[m] + n[m] (41),where the decision variables are generated as x = WHyAssume n ∼ CN (0,Kn) and x ∼ CN (0,Kx) are independent, theoptimal linear MMSE receiver is found by minimizing

W = arg minW

E[‖x−WHy‖2

]

︸︷︷︸MSE

(21)

After differentiation with respect to W and setting the gradient tozero, the optimal MMSE filter is

W =(HKxHH + Kn

)−1HKx (22)

If n ∼ CN (0, N0Inr) and x ∼ CN (0, PntInt)

W =

(P

ntHHH +N0I

)−1HP

nt(23)



Linear MMSE Receiver – SINR Derivation

Low SNR: P << N0 → W ≈ H Pnt

= Matched filter

High SNR: P >> N0 → W ≈(HHH

)−1H = ZF receiver

Output SINR for the kth stream with input power pk (P/nt) is

γmmsek =

E[|wH

k hkxk|2]

E[|wH

k (∑

i 6=k hixi + nk)|2] =

pkwHk hkh

Hkwk

wHkRkwk

= . . .

= pkhHkR−1k hk (24)

where Rk =∑

i 6=k pihihHi +N0Inr and wk is kth column of W

Achievable rate:

Rmmse = E

[nt∑

k=1

log(1 + γmmsek )

](25)



Large System Approximation of γmmsek

Some important lemmas

Lemma 1: Assume a vector x ∈ CN with i.i.d elements which havezero mean and variance equal to 1

N . Also consider a Hermitianmatrix A ∈ CN×N with elements independent of x, then

xHAx− 1

Ntr(A)

N→∞−−−−→ 0 (26)

Lemma 2:Let AN be a complex N ×N matrix with uniformlybounded spectral norm. Also, consider random Hermitian matrixCN such that for smallest eigenvalue of CN there exist an ε withprobability one such that λmin > ε for all large N , then

1

Ntr[ANC−1N ]− 1

Ntr[AN (CN + uuH)−1]

N→∞−−−−→ 0 (27)

where u is a complex vector.




Theorem 3: Stieltjes transform of Gram matrix YYH. Consider aN × n random matrix denoted by Y, such that elements of Y areindependent and zero mean. The variance of entries is given byE[|yi,j |2] = α2

i,j

mYYH(z) =

1

Ntr(YYH − zIN)−1 − 1

Ntr(Θ(z))

n→∞, Nn→ c−−−−−−−−→ 0

(28)where Θ(z) = diag(θ1(z), ..., θN (z)). The entries θi can be foundby fixed point iteration

θi(z) =−1

z(1 + 1n

n∑j=1

α2i,j θj(z))

∀1 ≤ i ≤ N (29)

θj(z) =−1

z(1 + 1n

N∑i=1

α2i,jθi(z))

∀1 ≤ j ≤ n (30)




The instantaneous random variable γmmsek (p) can be approximated

by a deterministic quantity γdetk (p) such that

γmmsek (p)− γdetk (p)

nr →∞, nrK→ c−−−−−−−−−→ 0 (31)

Sketch of proofI Define Σk =

∑i 6=k pihih

Hi and Σ = Hdiag(p1, . . . , pK)HH

I Applying Lemma 1, γmmsek can be approximated as

pkakhHk (Σk +N0Inr

)−1

hk →pkaknr

Tr(

(Σk +N0Inr)−1)

(32)

I Applying Lemma 2, we have

pkaknr

Tr(

(Σk +N0Inr)−1)≈ pkak

nrTr(

(Σ +N0Inr)−1)

= pkakmΣ(−N0) (33)

where mΣ(−N0) is the Stieltjes transform of Σ and it only dependson p and a1, . . . , ak




Uplink with K single-antenna users and a single BS with nr receiveantennas.

Number of users

48 16 32 64 128 256

Va

r(γ

mm

se

- γ

de

t)

0

1

2

3

4

5

6

7

8

9

Figure: Variance of γmmsek − γdetk , K = nr and pkak/N0 = 10









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(34)

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(35)

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

x1

x2

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



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



Resource Allocation with MIMO

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



Multi-cell MIMO System ModelB BSs, NT TX antennas per BS and NRk RX antennas per user kA user k is served by Mk = |Bk| BSs from the joint processing setBk, Bk ⊆ B = {1, . . . , B}6

yk =∑

b∈Bab,kHb,kx

′b + nk (36)

=∑

b∈Bk

ab,kHb,kxb,k +∑

b∈Bk

ab,kHb,k

∑

i6=kxb,i

+∑

b∈B\Bk

ab,kHb,kx′b + nk

whereI 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 k6Extension 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 optimal linear 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 (37)



Linear Transceiver DesignEntire capacity region of multiuser MIMO DL has been recentlydiscovered

I Also with individual peak power constraint per BS antenna78

I Require complex nonlinear precoding based on dirty paper coding

I Sub-optimal but less complex transmission methods are neededLinear beamforming is usually remarkably simpler in practice

I Dimensionality contraint per BS:

0 ≤∑

k∈Ubmk ≤ NT, 0 ≤ mk ≤ NRk

. (38)

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)

7W. 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.

8H. 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.



Linear Transceiver DesignA generalised method for joint design of linear transceivers with

I 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



Linear Transceiver Design

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(39)

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

(40)

Additional Quality of Service constraints (QoS) can be alsoincorporated in (40), e.g., minimum/maximum SINR or rateconstraints per stream or per user




Optimisation criteria, e.g.,

1. Sum power minimisation with fixed per stream SINR constraints:f(γ1, . . . , γS) = −∑B

b=1 Pb

2. Weighted sum MSE minimisation:f(γ1, . . . , γS) = −∑S

s=1 βsMSEs = −∑Ss=1

βs

(1+γs)

3. Weighted sum rate maximisation:f(γ1, . . . , γS) =

∑Ss=1 βs log2(1 + γs) = log2

∏Ss=1(1 + γs)

βs

4. Max min weighted SINR per data stream, i.e., SINR balancing :f(γ1, . . . , γS) = max mins=1,...,S β

−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 k




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

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 beamformingEach 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. [A. Tolli, H. Pennanen and P. Komulainen, ”On the Value of Coherent and CoordinatedMulti-cell Transmission”, IEEE ICC’09, Dresden, Germany, June 2009]



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. [A. Tolli, H. Pennanen and P. Komulainen, ”On the Value of Coherent and CoordinatedMulti-cell Transmission”, IEEE ICC’09, Dresden, Germany, June 2009]



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. [A. Tolli, H. Pennanen and P. Komulainen, ”On the Value of Coherent and Coordinated Multi-cellTransmission”, IEEE ICC’09, Dresden, Germany, June 2009]



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. [A. Tolli, H. Pennanen and P. Komulainen, ”On the Value of Coherent and Coordinated Multi-cellTransmission”, IEEE ICC’09, Dresden, Germany, June 2009]









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 (36) is simplified to

yk = hbk,kmkdk +

K∑

i=1,i 6=khbi,kmidi + nk (41)

whereI 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

(42)

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



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 (43)

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

ak =γk

(1 + γk) |hbk,kuk|2(44)

Rewrite (43) as

(IK −DaG) p = N0a (45)

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 (46)




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

bk =γulk

(1 + γulk ) |hbk,kuk|2(47)

Rewrite (46) as (IK −DbG

T)

q = N0b (48)

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

Solve p from (45) and q from (48)

p = N0 (IK −DaG)−1 a = N0

(D−1a −G

)−11

q = N0

(IK −DbG

T)−1

b = N0

(D−1b −GT

)−11

(49)

(50)




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 (51)

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



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

minimise∑

k pksubject to γk(p) ≤ γtargetk , ∀ k (52)

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

γk(p) =Gk,kpk∑

i 6=kGk,ipi + wk(53)

Equivalent to a linear program (LP)

minimise 1Tp

subject to(IK −Dtarget

a G)

p � N0a(54)

For a feasible γtarget, the closed form solution is given by (49)-(50).



Iterative Solution

Transmit beamforming and power loading can be solved optimallyvia dual uplink formulation9

min.qk,uk ∀ k

∑k qk

subject to γk(q) ≥ γtargetk , ∀ k(55)

Iterative solution – alternate until convergence

1. 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. (50) is optimal for fixed receivers uk ∀ kJoint update method

q[t+1]k =

γtargetk

hHk

(∑i 6=k q

[t]i hH

bk,ihHbk,i

+N0I)−1

hbk,k

=γtargetk

γk[t]q[t]k (56)

9More 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,

(57)

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



SOCP ReformulationBy rearranging the constraint in (42) as

N0 +K∑i=1

∣∣hbi,kmi

∣∣2 ≤(

1 + 1γtargetk

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

Eq. (42) 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(58)

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

Standard form SOCPc©Antti Tolli, Department of Comm. Engineering


Decentralised Solution via Optimisation Decomposition

TDD assumption:

each BS is able to measure atleast the channels of all celledge 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



Decentralised Solution via Optimisation DecompositionProposed distributed solution

Beamformers are designed locally relying on limited informationexchanged between adjacent BSs

The coupled terms are decoupled by a dual decomposition10,Alternating direction method of multipliers (ADMM)11, or primaldecomposition12 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

10A. 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

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 Coordinated Downlink Beamforming via Primal Decomposition”,IEEE Signal Processing Letters, vol. 8, no.11, pp. 647 - 650, November 2011



Decentralised Solution via Optimisation DecompositionFor 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 (59)

where the inter-cell interference term is

ζ2b,k =∑

i∈Ub

∣∣hb,kmi

∣∣2 (60)



Decentralised Solution via Optimisation Decomposition

Now, (42) 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(61)

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 Decomposition13

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

(62)

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

13S. 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/



Decentralised Solution via Dual DecompositionDual decomposition approach: the consistency constraints in (62) arerelaxed by forming the partial Lagrangian as

L(M1, . . . ,MB , ζ

(1), . . . , ζ(B),ν1, . . . ,νB

)(63)

=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)

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) (64)

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). (65)



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(66)

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 (67)

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 (61)



Distributed Algorithm

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

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 (66)

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]

Pdecom

p(t

) −

Popt

0 dB SINR target

10 dB SINR target

Figure: Suboptimality of the distributed algorithm versus the iteration number tfor 0 dB and 10 dB SINR targets. [Tolli, H. Pennanen, and P. Komulainen, ”Decentralized MinimumPower Multi-cell Beamforming with Limited Backhaul Signalling”, IEEE Trans. on Wireless Comm., vol. 10, no. 2, pp. 570 -580, February 2011]



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.[Tolli, H. Pennanen, and P. Komulainen, ”Decentralized Minimum Power Multi-cell Beamforming with Limited BackhaulSignalling”, IEEE Trans. on Wireless Comm., vol. 10, no. 2, pp. 570 - 580, February 2011]



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.[Tolli, H. Pennanen, and P. Komulainen, ”Decentralized Minimum Power Multi-cell Beamforming with Limited BackhaulSignalling”, IEEE Trans. on Wireless Comm., vol. 10, no. 2, pp. 570 - 580, February 2011]



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). [Tolli, H. Pennanen, and P.Komulainen, ”Decentralized Minimum Power Multi-cell Beamforming with Limited Backhaul Signalling”, IEEE Trans. onWireless Comm., vol. 10, no. 2, pp. 570 - 580, February 2011]

TS is the signalling period and fd is the maximum Doppler shift.



ExtensionsCognitive underlay cellular network14 - 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

(68)

(69)

Worst case beamformer design with ellipsoid CSIT uncertainty15,16

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

Hb,k ≤ 1} ∀ b ∈ B, k ∈ U

(70)

(71)

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.

14H. 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

15C. 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.

16H. 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 the Optimal Multi-cellBeamforming via Large System Analysis, in Proc. IEEE ICC 2014, Sydney, Australia,June, 2014

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

H. Asgharimoghaddam, A. Tolli & N. Rajatheva, ”Decentralizing the Optimal Multi-cellBeamforming in Correlated Channels via Large System Analysis”, submitted to IEEETrans on Signal Prog., July 2015



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

NT > Kb

1

2

1 BK-1

K

Problem: Minimum power beamforming

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



Simplified solutionsMatched filter: mk = cbk,kh

Hbk,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 (72)

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(73)



UL-DL Duality RevisitedRecall the dual uplink presentation of (42) and (61)

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(74)

The dual uplink power of each user is found by fixed point iteration17

qk =1

(1 + 1γk

)hbk,k(∑

l∈U qlhHbk,l

hbk,l + I)−1hHbk,k

(75)

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

uk = (∑

l∈Uqlh

Hbk,l

hbk,l +N0I)−1hHbk,k

(76)

17Note that (75) is equivalent to (56)



UL-DL Duality Revisited

A link between the DL and UL beamformers is provided by

mk =√δkuk (77)

where δk can be found by

δ = G−11K (78)

and where δ is a vector that contains all δk values18. The elementsof G are given as:

Gi,j =

{ 1γi|hbi,iui|2 i = j

−|hbj ,iuj |2 i 6= j.(79)

18δk = pk and G = D−1a −G in (49) if ‖uk‖2 = 1



Large Dimension Approximation of (75)As N,K →∞ with K/N ∈ (0, B)

qk − qk −→ 0 (80)

almost surely, where the approximated power qk is given as theresult of the convergence of the following iterative formula

q(t+1)k = q

(t)k

γk

e(t)bk,k

. (81)

e(t)bk,k

is the large system approximation of kth user’s SINR atiteration t. The functions ebk,1, ..., ebk,n are given as the uniquenonnegative solution of the following system of equation,

e(t+1)bk,i

=1

NTr{q(t)i θbk,i(

1

N

∑

l∈U

q(t)l θbk,l

1 + e(t)bk,l

+ µbkIN )−1} ∀i. (82)

where θbk,k is the correlation matrix from user k to BS bk,



Large Dimension Approximation of (75)

Similarly, the entries of matrix G can be approximated by ,19

Gl,k =

γk(λk)2

l = k

− 1N

e′bk,l

λk(1+e∗bk,l

)2l 6= k

(83)

where, e′bk,l(−1) is the differential of ebk,l(z) with respect to z atpoint z = −1

19H. Asgharimoghaddam, A. Tolli & N. Rajatheva, ”Decentralizing the Optimal Multi-cell Beamforming in CorrelatedChannels via Large System Analysis”, submitted to IEEE Trans on Signal Processing, July 2015



Approximation of Intercell Interference TermsFrom (61) and (59), the total ICI from all BSs towards user k is

∑

b 6=bk

ζ2b,k =∑

b6=bk

∑

l∈Ub

|hb,kml|2 (84)

Considering (77), the ICI term ζ2b,k in (84) can be written as follows,

ζ2b,k =∑

l∈Ub

√δl|hb,kul|2 ≈

∑

l∈Ub

√δlGl,k (85)

where δb,l is found from (78) and |hb,kub,l|2 ≈ Gl,k is from (83),

In i.i.d case the ICI can be expressed in a simple form as follows,

ζ2b,i ≈α2b,i

N(1 + e∗b,i)2

∑

j∈Ub

pb,j . (86)

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

Algorithm 1: Approximation ofuplink/downlink powers anddetection/beamformering vectors.

Algorithm 2: Decentralized beamformingwith approximated ICI values.



Numerical Examples

0 10 20 30 40 5072

74

76

78

80

Trial index

Tra

nsm

it S

NR

[dB

]

Transmit SNR Alg 1Transmit SNR Alg 2

0 10 20 30 40 506

8

10

12

Trial index

Res

ulte

d S

INR

[dB

]

Resulted SINR Alg 1Resulted SINR Alg 2

Figure: Transmit powers and SINR variation of Algorithms 1 and 2,N = K = 56, ρ = 0.



Numerical Examples

14 28 42 8459

60

61

62

63

64

65

66

67

Number of antennas

Tra

nsm

it S

NR

[dB

]

14 28 42 84−4

−3.5

−3

−2.5

−2

−1.5

MR

T S

INR

(dB

)

MRT SINR

MRTZFCentralizedAlgorithm 2

(a) 0 dB SINR target

14 28 42 8469

70

71

72

73

74

75

76

77

Number of antennas

Tra

nsm

it S

NR

[dB

]

14 28 42 84−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

MR

T S

INR

(dB

)

MRT SINR


(b) 10 dB SINR target

Figure: Transmit SNR versus the number of antennas, K = N/2, ρ = 0.8.



Numerical Examples

16 28 58 88 12055

60

65

70

75

80

85

90

95

Number of antennas

Tra

nsm

it S

NR

[dB

]

16 28 58 88 120−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

MR

T S

INR

(dB

)

MRT SINR


Figure: Transmit SNR vs. the number of antennas, γk = 0 dB, ρ = 0.8,K = 28.









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 May 2015



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,(87)

where the interference+noise covariance matrix for user k is

Rk =

K∑

i=1,i 6=kHbi,kKxiH

Hbi,k

+N0I (88)

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,(89)

where the received signal covariance Rk for user k is

Rk , E[ykyHk ] =

K∑

i=1

Hbi,kMiMHi HH

bi,k+ σ2kI. (90)

When the MMSE receiver (37) is employed in (89), the MSE matrixbecomes

EMMSEk = I−MH

kHHbk,k

R−1k Hbk,kMk (91)

Furthermore

EMMSEk =

(I + MH

kHHbk,k

R−1b,kHbk,kMk

)−1(92)



MSE Reformulation

Applying (92) to (87), 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,

(93)

where Pb, b = 1, . . . , B are separable convex per-BS powerconstraints, and the relaxation Ek 4 Ek, k = 1, . . . ,K bounds theachieved MSE20.

20The relaxation tightness follows from the matrix monotonicity of the determinant function [Boyd&Vandenbergh].



MSE ReformulationFor fixed Mk, the rate maximizing Uk are solved from the roots ofthe Lagrangian of (93) as Uk = R−1k Hbk,kMk, k = 1, . . . ,KFor fixed receive beamformers Uk, the concave objective function isiteratively linearised w.r.t Ek.21

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,

(94)

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 (93) on every iteration.

21This method in the context of weighted sum rate maximisation was established in [Shi et al, TSP’11], where it wasreferred to as (iteratively) weighted MMSE minimization.



TX precoder adaptation stepThe relaxation Ek 4 Ei

k is tight → Replace Eik with Ek (89) in the

objective of (94)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

(95)

Iterative solution from the KKT conditions

Mk =

(K∑

i=1

HHbk,i

UiWiUHi Hbk,i + νbkI

)−1HHbk,k

UkWk (96)

where the optimal νbk22 is found via bisection

22dual variable related to the power constraintc©Antti Tolli, Department of Comm. Engineering


Alternating Optimization, Global Algorithm

WSRM via WSMSE23

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

23Q. 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,

(97)

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.

(98)

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), (98) 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). (99)

Convex, ith, approximation of (98) 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.

(100)



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 (100).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.

(101)



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. (102)

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)

Su

m r

ate

(b

its/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. [Kaleva, J.; Tolli, A.; Juntti, M.; , ”Weighted Sum Rate Maximization forInterfering Broadcast Channel via Successive Convex Approximation”, Global Communications Conference, 2012.GLOBECOM 2012. IEEE , Dec. 2012]

[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 Constraints24

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

∑mkl=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

∑Ki=1

∑mij=1,

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

|uHk,lHbi,kmi,j |2 + σ2k‖uk,l‖2

.

24J. 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



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



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. [J. Kaleva, A. Tolli & M. Juntti, ”Decentralized Beamforming for WeightedSum Rate Maximization with Rate Constraints”, in Proc. IEEE PIMRC 2013 - Workshop on Cooperative and HeterogeneousCellular Networks, London, UK, Sep. 2013]



Numerical Results - ADMM extension

0 10 20 30 40 50 601

1.5

2

2.5

3

3.5

4

4.5

5

Iteration

Rat

e (b

its/H

z/s)

Alternating Direction Method of Multipliers (ADMM).

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Iteration

Rat

e (b

its/H

z/s)

Dual Decomposition.

k = 1k = 2k = 3k = 4

Figure: Convergence of the ADMM and dual decomposition methods withB = 2,Kb = 2, NT = 2, NR = 2, Rk = 3bits/Hz/sec, β = 2, ρ = 1.5, χ = 3dBand SNR = 15dB. [J. Kaleva, A. Tolli and M. Juntti, ”Rate Constrained Decentralized Beamforming for MIMOInterfering Broadcast Channel”, PIMRC’15, Hong Kong, June, 2015]



Decentralised Solution via Precoded UL Pilot

Assumptions:

Each user is associated tosingle BS (non-cooperative)

TDD and perfect CSI

Precoded UL pilot sequences

1

2

1 BK-1

K

Goal: Scheduling & TX – RX design to control interferencea

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 AlgorithmEach BS b calculates own weights Wk ∀ k ∈ Ub : distribute viabackhaulEach 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. [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]



Bidirectional Signalling Strategies for Dynamic TDD

P. Jayasinghe, A. Tolli, J. Kaleva and M. Latva-aho, ”Bi-directionalSignaling for Dynamic TDD with Decentralized Beamforming” ICCWorkshop on Small Cell and 5G Networks (SmallNets), London, UK, June,2015

P. Jayasinghe, A. Tolli, and M. Latva-aho, ”Bi-directional SignalingStrategies for Dynamic TDD Networks” IEEE SPAWC, Stockholm,Sweden, July, 2015

METIS Deliverable D3.3 on ”Final performance results and consolidatedview on the most promising multi-node/multi-antenna transmissiontechnologies” (A. Tolli & P. Jayasinghe), https://www.metis2020.com/


https://www.metis2020.com/


Background & Introduction

Figure: Dynamic TDD system

The load variation between adjacent small cells can be significant.

Fixed UL/DL switching would be highly suboptimal.

With the flexible UL/DL allocation provides large potential gains inspectral efficiency → dynamic traffic aware TDD transmission

Allowing such flexibility makes obviously the interferencemanagement considerably more challenging.



Interference scenarios

Figure: Interference at DL terminal Figure: Interference at UL BS

Additional interference associated with the dynamic TDDI UL-to-DL interference.I DL-to-UL interference.

Interference mitigated by coordinated beamforming.More measurements and info exchange also at the terminal side



Synchronous TDD - System Model

Assumptions:

Multi-cell multiuser MIMO

Each user served by one BS


TDD and perfect CSI

Coordination via pilot andbackhaul signaling

1

2

1 BK-1

K

Problem: System utility maximization

low computational complexity and CSI acquisition

Impact of forward-backward training on the system performance



Bi-directional training

Figure: Backward-Forward training structure [Changxin Shi; Berry, R.A.; Honig, M.L.,”Bi-Directional Training for Adaptive Beamforming and Power Control in Interference Networks,” Signal Processing, IEEETransactions on , vol.62, no.3, pp.607–618, Feb.1, 2014]

Bidirectional training(BIT) phase is used in the beginning of eachTDD frame to speed up the convergence of the iterative algorithms

A number of blocks of pilots are alternately transmitted in thedownlink and the uplink



Bi-directional training in Dynamic TDD

Figure: TDD frame structure with two bi-directional beamformer signalingiterations. [P. Jayasinghe, A. Tolli, and M. Latva-aho, ”Bi-directional Signaling Strategies for Dynamic TDD Networks”IEEE SPAWC, Stockholm, Sweden, July, 2015]



Bi-directional signalling: Simulation setup

4-antenna BSs, 4 2-antenna UEs per BSThe signaling overhead per one signaling iteration is γ and the totalsignaling overhead is ρ = BIT× γ. The actual throughput is(1− ρ)R, where R is the achieved WSR from the iterative algorithm.



Synchronous 2-cell Downlink, Strategy A

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.56

8

10

12

14

16

18

20

22

Overhead

ActualSum

Rate

(SNR

=10dB)

α = 0 dB & γ = 0.01

α = 0 dB & γ = 0.02

α = 0 dB , uncoordinated

α = 12 dB & γ = 0.01

α = 12 dB & γ = 0.02


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.55

10

15

20

25

30

Overhead

ActualSum

Rate

(SNR

=20dB)

α = 0 dB & γ = 0.01

α = 0 dB & γ = 0.02


α = 12 dB & γ = 0.01

α = 12 dB & γ = 0.02


Peak Rate

Peak Rate

Figure: Actual Sum rate vs overhead with different SNR (10, 20 dB) values. [P.Jayasinghe, A. Tolli, J. Kaleva and M. Latva-aho, ”Bi-directional Signaling for Dynamic TDD with DecentralizedBeamforming” ICC Workshop on Small Cell and 5G Networks (SmallNets), London, UK, June, 2015]



Dynamic TDD 3-cell, Strategy A

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

15

20

25

30

35

Overhead

ActualSum

Rate

α = 0dB ,β = 3dB, δ = 6dB, γ = 0.01

α = 0dB ,β = 3dB, δ = 6dB uncoordinated

α = 0dB ,β = 9dB, δ = 12dB, γ = 0.01


α = 6dB ,β = 3dB, δ = 6dB, γ = 0.01


α = 6dB ,β = 9dB, δ = 12dB, γ = 0.01


Peak Rate

Figure: Actual sum rate vs overhead at SNR = 20 dB, with diferent α, β, δ. [P.Jayasinghe, A. Tolli, J. Kaleva and M. Latva-aho, ”Bi-directional Signaling for Dynamic TDD with DecentralizedBeamforming” ICC Workshop on Small Cell and 5G Networks (SmallNets), London, UK, June, 2015]



Time-correlated Fading Scenario

50 55 60 65 70 75 80 85 90 95 10014

16

18

20

22

24

26

Channel Number

AverageActu

alSum

Rate

BIT =1, no reset

BIT =3, reset = 10

BIT =5, reset = 10

BIT =10, reset =10

Figure: Average sum rate over time-correlated channel at SNR = 20 dB withγ = 0.01. [P. Jayasinghe, A. Tolli, J. Kaleva and M. Latva-aho, ”Bi-directional Signaling for Dynamic TDD withDecentralized Beamforming” ICC Workshop on Small Cell and 5G Networks (SmallNets), London, UK, June, 2015]



Comparison of different signalling strategies

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

12

14

16

18

20

22

24

26

28

Overhead

Actu

alRate

atSNR

=20dB

Strategy A

Strategy B

Strategy C

Uncoordinated method

Figure: Actual sum rate at SNR = 20dB vs overhead for different bi-directionalsignaling strategies.



Mode Selection and Transceiver Design in Underlay D2DMIMO Systems

A. Tolli, J. Kaleva and P. Komulainen, ”Mode Selection and TransceiverDesign for Rate Maximization in Underlay D2D MIMO Systems”, IEEEICC, London, UK, June, 2015

A. Ghazanfari, A. Tolli and J. Kaleva, ”Joint Power Loading and ModeSelection for Network-assisted Device-to-Device Communication”, IEEEICC, London, UK, June, 2015



System assumptions and research problemAssumptions:

One multiantenna BS in charge of LD2D pairs

K = 2L multiantenna users in total


Centralized processing

Half-duplex nodes

Slotted transmission - UL(1) andDL(2) slots

Problem: Weighted sum ratemaximization (WSR)

Aggregate sum of end-to-end datarates carried via uplink, downlink,and D2D links



Optimization problemThe optimization objective is to maximize the aggregate sum of E2Edata rates carried via uplink(1), downlink(2), and D2D links

max .

L∑

l=1

µl

(R

(1)l +R

(2)l + min(R

(1)l , R

(2)l ))

(103a)

s. t. ‖M(1)l ‖2F + ‖Fl‖2F ≤ Pil∀ l (103b)

‖M(2)l ‖2F ≤ Pil∀ l (103c)

L∑

l=1

‖Bl‖2F ≤ PB. (103d)

Opt. variables: TX precoders for link l

M(1)l ,M

(2)l for direct D2D TX in time slots (1) and (2)

Fl for UL TX in time slot (1)

Bl for DL TX in time slot (2)



Reformulation via MSE minimization

By writing the function min(R(1)l , R

(2)l ) in its equivalent epigraph

form, (103) can be reformulated as an equivalent log-MSEminimization problem25

min .M

(1)l ,M

(2)l ,

Bl,Fl,tl

L∑

l=1

µl

(log det

(E

(1)l

)+ log det

(E

(2)l

)− tl

)

s. t. log det(E

(s)B,l

)−1≥ tl, s = 1, 2

(103b), (103c), (103d).

(104)

25R(1)l

= log det(E

(1)l

)−1, R

(2)l

= log det(E

(2)l

)−1, R

(1)l

= log det(E

(1)B,l

)−1, and

R(2)l

= log det(E

(2)B,l

)−1



Reformulation via MSE minimizationIntroducing new optimization variables for the RX filters as well asupper bounds for the MSE matrices, (104) can be written as

min .∑L

l=1µl

(log det

(E

(1)l

)+ log det

(E

(2)l

)− tl

)(105a)

s. t. log det(E

(s)B,l

)≤ −tl, s = 1, 2 (105b)

E(s)l 4 E

(s)l , E

(s)B,l 4 E

(s)B,l ∀ l, s = 1, 2 (105c)

(103b), (103c), (103d). (105d)

where the variables are, ∀ l = 1, . . . , L, s = 1, 2

1. MSE upper bounds E(s)l , E

(s)B,l and TX precoders M

(s)l ,Bl,Fl

2. Rx receivers A(s)l ,Gl,Ul.

Convex constraints for fixed RX → alternating TX-RX opt.(1 ↔ 2)

Successive linearization of the concave objective



Mode selection via overloadingInitial allocation

UL slot (1)

DL slot (2)

Final allocation after convergence:

UL slot (1)

DL slot (2)



Numerical ExamplesL = 4 D2D pairs, each with 2 antennasBS is equipped with 8 antennasFrequency flat Rician fading with factor κ

[A. Tolli, J. Kaleva and P. Komulainen, ”Mode Selection and Transceiver Design for Rate Maximization in Underlay D2DMIMO Systems”, IEEE ICC, London, UK, June, 2015]



Numerical Examples26

20 25 30 35 40 45 50 55 604

5

6

7

8

9

10

11

12

Ave

rage

sum

rat

e [b

its/s

ec/H

z].

Cell radius.

D2D + Cellular

Cellular Only

D2D Only

Figure: Average achieved sum rate peruser with NB = 8, Nk = 2 ∀ k, L = 4,κ = 10, SNR = 15dB and D2Ddistance of 20m.

20 25 30 35 40 45 50 55 607

8

9

10

11

12

13

14

15

Ave

rag

e s

um

ra

te [

bits/s

ec/H

z].

Cell radius.

Cellular Only

D2D + Cellular

D2D Only

Figure: Average achieved sum rate peruser with NB = 8, Nk = 2 ∀ k, L = 4,κ = 0, SNR = 15dB and D2D distanceof 20m.

26A. Tolli, J. Kaleva and P. Komulainen, ”Mode Selection and Transceiver Design for Rate Maximization in Underlay D2DMIMO Systems”, IEEE ICC, London, UK, June, 2015


Documents

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