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PIMRC 2013 TutorialOptimal Resource Allocation in Coordinated

Multi-Cell Systems

Emil Bjornson1 and Eduard Jorswieck2

1 Signal Processing Lab., KTH Royal Institute of Technology, Sweden andAlcatel-Lucent Chair on Flexible Radio, Supelec, France

2 Dresden University of TechnologyDepartment of Electrical Engineering and Information Technology, Germany

8 September 2013

Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 1 / 64

Introduction: Part II Communications

Theory

• The multi-cell system model analyzed in Part I was based on threesimplifying assumptions: perfect CSI, unlimited backhaulcapacity, and ideal transceiver hardware.

• These assumptions are generalized in Part II to more realisticconditions. We show that many of the previous results are readilygeneralizable.

• Furthermore, we describe how the system model can be extended toalso incorporate multi-cast transmission, multi-carrier systems,and multi-antenna receivers.

• Finally, we discuss some recent work on the design of dynamiccooperation clusters and show that our framework is applicable incognitive radio systems and for physical layer security.

• There are many open problems in each topic and in thecombination of multiple topics.


Outline Communications

Theory

1. Robustness to Channel Uncertainty

2. Distributed Resource Allocation

3. Transceiver Hardware Impairments

4. Multi-Cast Transmission

5. Multi-Carrier Systems

6. Multi-Antenna Users

7. Design of Dynamic Cooperation Clusters

8. Cognitive Radio Systems

9. Physical Layer Security


Channel Uncertainty Model Communications

Theory

• The channel uncertainty originates from a variety of sources; forexample, imperfect channel estimation, feedback quantization,inadequate channel reciprocity, and delays in CSI acquisition onfading channels.

• The additive error model is given by

hk = hk + εk ∀k (1)

where hk = [hT1k . . . hTKtk]T ∈ CN×1 is the CSI available at the base

stations and εk ∈ CN×1 is the combined error vector.

• The system cannot account for any error; the stochastic errorvector εk could potentially cancel out the nominal vector asεk = −hk or be very large (unbounded for Rayleigh fadingchannels).

• Therefore, we consider a subset of error vectors, the uncertaintyset, that has high probability of containing the error


Ellipsoidal Uncertainty Set Communications

Theory

• We define jointly for all base stations to a certain user, theellipsoidal uncertainty set

Uk(hk,Bk) ={

hk : hk = hk + Bkεk, ‖εk‖2 ≤ 1}∀k. (2)

Uk(hk,Bk) ={hk : hk = hk + Bkεk, ‖εk‖2 ≤ 1

}

0

{εk : ‖εk‖2 ≤ 1}

hk

Unit Sphere Channel Uncertainty Set

Transformation

Eigenvectors of Bk

Illustration of the ellipsoidal uncertainty set Uk(hk,Bk) in (2) andthe unit sphere {εk : ‖εk‖2 ≤ 1} that it is created from.


Example: CSI Estimation Uncertainty Communications

Theory

• Modeling the channels by hk ∼ CN (0,Rk), and performingchannel estimation results in estimation errors εk ∼ CN (0,Ek).

• The error covariance matrix Ek depends on Rk and on the typeof channel estimation. It is block-diagonal asEk = diag(E1k, . . . ,EKtk) if the channel from each base station toeach user is estimated separately.

• The estimation error εk belongs with probability pk to the ellipsoidalset {εk : 2εHk E−1

k εk ≤ χ2pk

(2N)}, where χ2pk

(n) denotes thepk-percentile of the chi-squared distribution with n degrees offreedom.

• Alternatively, the estimation error εjk of the channel component hjkbelongs with probability pjk to the ellipsoidal set{εjk : 2εHjkE

−1jk εjk ≤ χ2

pjk(2Nj)}.


Worst-Case Robust Optimization Problem Communications

Theory

• We formulate the feasibility problem with SINR requirementsSINRk ≥ γk as

find S1 . . . ,SKr (3)

s. t.hHk CkDkSkD

Hk CH

k hkσ2k+hHk Ck(

∑i6=k DiSiDH

i )CHk hk≥ γk ∀hk ∈ Uk ∀k, (4)

∑Krk=1 tr(QlkSk) ≤ ql ∀l (5)

where the beamforming vectors vkvHk have been replaced by signal

correlation matrices Sk.

• This is a generalization of the “easy” problem.

• Is single-stream beamforming (rank one Sk) still optimal?


S-Procedure Communications

Theory

Lemma (S-Procedure)

Let θm(ε) = εHZmε+ zHmε+ εHzm + zm, m = 1, 2, be two quadraticfunctions in ε and let Zm be Hermitian. Suppose it exists ε such thatθ1(ε) > 0, then the implication

θ1(ε) ≥ 0⇒ θ2(ε) ≥ 0 (6)

holds true if and only if it exists λ ≥ 0 such that

Z2 z2

zH2 z2

− λ

Z1 z1

zH1 z1

� 0. (7)


Feasibility of Robust SINR Constraints Communications

Theory

Theorem

Let S1, . . . ,SKr be a transmit strategy and let γk ≥ 0 be given, then therobust SINR constraint

hHk CkDkSkDHk CH

k hk

σ2k + hHk Ck(

∑i 6=k DiSiDH

i )CHk hk

≥ γk ∀hk ∈ Uk(hk,Bk) (8)

is fulfilled if and only if it exists λk ≥ 0 such that

BH

k AkBk BHk Akhk

hHk AkBk hHk Akhk − σ2k

+

λkIN 0N×1

01×N −λk

� 0N+1 (9)

where Ak = 1γk

CkDkSkDHk CH

k −∑

i 6=k CkDiSiDHi CH

k .


Observations on Robust SINR Opt. Communications

Theory

• This theorem converts the infinite number of SINR constraints intojust one (linear) semi-definite constraint per user—at the costof adding Kr extra variables {λk}Krk=1 that indirectly represents theworst channel conditions in the uncertainty set.

• Channel uncertainty naturally increases the computationalcomplexity, but Theorem 2 shows that the robust problem is alsoconvex. The complexity is thus still polynomial in the number ofantennas N , users Kr, and power constraints L.

• The transmit strategy S∗1, . . . ,S∗Kr

that solves might in general haverank(S∗k) > 1 for some users. As single-stream beamforming isalways sufficient under perfect CSI, we can however expect it to alsowork well when the uncertainty is small.


Illustrations: Uncertainty I Communications

Theory

0 1 2 3 4 5 6 70

1

2

3

4

5

6

Max SumPerformance

Max Prop. FairnessMax−Min Fairness

ξ ∈ {0, 0.01, 0.05, }0.1

log2(1+SINR1) [bits/channel use]

log 2

(1+

SIN

R2)

[bi

ts/c

hann

el u

se]


Illustrations: Uncertainty II Communications

Theory

0 1 2 3 4 5 60

1

2

3

4

5

Max Sum Performance

Max Prop. FairnessMax−Min Fairness

ξ ∈ {0, 0.01, 0.05, }0.1


log 2

(1+

SIN

R2)

[bi

ts/c

hann

el u

se]


Further Remarks on Channel Uncertainty Communications

Theory

• We can solve any robust multi-cell resource allocation problemas a sequence of the convex problems described above.

• We can expect a single-stream beamforming solution.• Worst-case robustness under separate uncertainty

Theorem

For a MISO interference channel with per-transmitter constraints of qj ,all Pareto optimal points of the robust performance region are achievedby beamforming vectors vjkk(λk) for λk ∈ [0, 1]Kr−1 for k = 1, . . . ,Kr:

vjkk(λk) = arg maxvjkk

<(hHjkkCjkkvjkk)− ‖BHjkk

Cjkkvjkk‖

subject to =(|hHjkkCjkkvjkk

)= 0, ‖vjkk‖ ≤ qjk ,

|hHjkiCjkivjkk|+ ‖BjkiCjkivjkk‖2 ≤√λkiΓki ∀i 6= k,

for fixed values of Γki. The elements in λk are denoted λki for all i 6= k.


Distributed Implementation Communications

Theory296 Extensions and Generalizations

(a) Centralized Optimization and Resource Allocation

(b) Distributed Allocation with Control Signaling (Subsection 4.2.1)

(c) Truly Distributed Allocation (Subsection 4.2.2)

Fig. 4.4 Different implementations of resource allocation in multi-cell systems: (a) Global

CSI is gathered at a central station that allocates resources; (b) Base stations perform dis-

tributed resource allocation by iteratively exchanging control variables (but not CSI); and

(c) Base stations perform distributed resource allocation without exchanging any informa-

tion (but the problem formulation and local CSI is available).

The very essence of resource al-location problems is the couplingbetween the users, in terms ofinter-user interference and powerconstraints.

Different implementations of resource al-

location in multi-cell systems: (a) Global

CSI is gathered at a central station that

allocates resources; (b) Base stations per-

form distributed resource allocation by it-

eratively exchanging control variables (but

not CSI); and (c) Base stations perform

distributed resource allocation without ex-

changing any information (but the problem

formulation and local CSI is available).


Example Decomposition Communications

Theory

As a first step, the feasibility problem with SINR requirements (withSINRk ≥ γk ∀k) is rewritten as

4.2 Distributed Resource Allocation 297

procedure in Section 2.2 can be solved to global optimality in a

distributed manner. The former problem is considered in [204, 257, 265]

under total power minimization, while the latter is considered in

[14, 239, 257].

As a first step, the feasibility problem with QoS requirements

in (2.29) (with SINRk ≥ γk ∀k) is rewritten as

find vk,Θik,Θik, qlk ∀k,i, l, i �= k

subject to|hH

k CkDkvk|2σ2

k +∑i�=k

Θ2ik

≥ γk ∀k,

|hHk CkDivi|2 ≤ Θ2

ik, Θik ≤ Θik ∀k,i, i �= k,

vHk Qlkvk ≤ qlk,

Kr∑

i=1

qli ≤ ql ∀l,k

(4.29)

by adding nonnegative auxiliary variables Θik,Θik, qlk. The squared

variable Θ2ik is the actual interference generated at MSk by signals

intended for MSi, while Θ2ik is its believed value in the beam-

forming optimization for MSk. The reason for defining these vari-

ables as the square roots of the interference is to enable the SINR

constraints to be expressed as second-order cones �(hHk CkDkvk) ≥

√γk

√σ2

k +∑

i�=k Θ2ik. Similarly, the power constraints are separated

into per-user constraints using the variables qlk.

Looking at (4.29), we observe that the transmission to different

users is only coupled by the so-called consistency constraints Θik ≤ Θik

and∑Kr

i=1 qli ≤ ql. If the coupling variables Θik,Θik, qlk are fixed, the

beamforming optimization for the different users would decouple. The

classic decomposition methods basically pretend that these variables

are constants and update them iteratively [194, 204, 265].

We will take a dual decomposition approach where the coupling

is relaxed by forming a partial Lagrangian for the consistency con-

straints. If the Lagrange multipliers are denoted yik and zl for the

interference and power consistency constraints, respectively, (4.29) can

by adding non-negative auxiliary variables Θik, Θik, qlk. The squaredvariable Θ2

ik is the actual interference generated at User k by signals

intended for User i, while Θ2ik is its believed value in the beamforming

optimization for User k.


• Dual decomposition approach.

• Lagrange multipliers are denoted yik and zl for the interference andpower consistency constraints, respectively, decompose the probleminto Kr

298 Extensions and Generalizations

be decomposed into Kr subproblems where the problem for MSk is

minimizevk,{Θki}∀i,{Θik}∀i,{qlk}∀l

∑

i�=k

(ykiΘki − yikΘik

)+

L∑

l=1

zlqlk

subject to|hH

k CkDkvk|2σ2

k +∑i�=k

Θ2ik

≥ γk ∀k,

vHk Qlkvk ≤ qlk, qlk ≤ ql ∀l,

|hHi CiDkvk|2 ≤ Θ2

ki i �= k

(4.30)

and a master dual problem

maximize{yik}∀k,i,{zl}∀l

Kr∑

k=1

∑

i�=k

yik

(Θ∗

ik − Θ∗ik

)+

L∑

l=1

zl

(Kr∑

k=1

q∗lk − ql

), (4.31)

where Θ∗ik,Θ

∗ik, q

∗lk,v

∗k are the subproblem solutions.

This decomposition enables an iterative procedure where the

subproblems in (4.30) are solved for constant values on the Lagrange

multipliers yki,yik ∀i �= k and zl ∀l. This is called dual decomposition

because the Lagrange multipliers are viewed as constants in the sub-

problems, and not the coupling variables themselves. The Lagrange

multipliers can be viewed as prices for causing interference and for

consuming transmit power, and these prices are iteratively adjusted

by the master problem (4.31) until convergence. The update procedure

requires backhaul signaling, but we will see that it can be implemented

by distributed message passing between the involved transmitters. In

other words, the heavy CSI signaling required to solve the resource allo-

cation problem centrally is replaced by iterative interference and power

control signaling. This confirms the observation in Section 3.2 that coor-

dinated decision making is the limiting factor in multi-cell resource

allocation, and not the localness of the CSI at the base stations.

The subproblems in (4.30) resembles the interference-constrained

beamforming in Subsection 2.2.1 (with interference limits Γik = Θ2ik),

with the difference that also the power constraints are decoupled. The

problem (4.30) is convex and can be solved to global optimality using

standard techniques. The master problem has a more complicated

• with a master dual problem

298 Extensions and Generalizations

be decomposed into Kr subproblems where the problem for MSk is

minimizevk,{Θki}∀i,{Θik}∀i,{qlk}∀l

∑

i�=k

(ykiΘki − yikΘik

)+

L∑

l=1

zlqlk

subject to|hH

k CkDkvk|2σ2

k +∑i�=k

Θ2ik

≥ γk ∀k,

vHk Qlkvk ≤ qlk, qlk ≤ ql ∀l,

|hHi CiDkvk|2 ≤ Θ2

ki i �= k

(4.30)

and a master dual problem

maximize{yik}∀k,i,{zl}∀l

Kr∑

k=1

∑

i�=k

yik

(Θ∗

ik − Θ∗ik

)+

L∑

l=1

zl

(Kr∑

k=1

q∗lk − ql

), (4.31)

where Θ∗ik,Θ

∗ik, q

∗lk,v


This decomposition enables an iterative procedure where the

subproblems in (4.30) are solved for constant values on the Lagrange

multipliers yki,yik ∀i �= k and zl ∀l. This is called dual decomposition

because the Lagrange multipliers are viewed as constants in the sub-

problems, and not the coupling variables themselves. The Lagrange

multipliers can be viewed as prices for causing interference and for

consuming transmit power, and these prices are iteratively adjusted

by the master problem (4.31) until convergence. The update procedure

requires backhaul signaling, but we will see that it can be implemented

by distributed message passing between the involved transmitters. In

other words, the heavy CSI signaling required to solve the resource allo-

cation problem centrally is replaced by iterative interference and power

control signaling. This confirms the observation in Section 3.2 that coor-

dinated decision making is the limiting factor in multi-cell resource

allocation, and not the localness of the CSI at the base stations.

The subproblems in (4.30) resembles the interference-constrained

beamforming in Subsection 2.2.1 (with interference limits Γik = Θ2ik),

with the difference that also the power constraints are decoupled. The

problem (4.30) is convex and can be solved to global optimality using

standard techniques. The master problem has a more complicated

where Θ∗ik, Θ∗ik, q

∗lk,v



• This decomposition enables an iterative procedure where thesubproblems are solved for constant values on the Lagrangemultipliers yki, yik ∀i 6= k and zl ∀l.

• This is called dual decomposition because the Lagrange multipliersare viewed as constants in the subproblems, and not the couplingvariables themselves.

• The master problem has a more complicated structure and istypically solved by subgradient methods.

• This implies that the Lagrange multipliers in iteration n+ 1 areachieved as

y(n+1)ik =

[y

(n)ik − ξ(Θ

(n)ik −Θ

(n)ik )]

+(10)

z(n+1)l =

[z

(n)l − ξ

(ql −

Kr∑

k=1

q(n)lk

)]+

(11)

where ξ > 0 is the step size.


Decomposition Remarks Communications

Theory

• Since the approach described above provides a distributed solutionto resource allocation with fixed SINR requirements, it can also beused as a subproblem in algorithms that successively increasethe SINR requirements for the purpose of obtaining a Paretooptimal point.

• Therefore, it can be generalized to find certain operating pointson the Pareto boundary as, e.g., the Max-min fairness point.

• To summarize, the dual decomposition approach should be seen asproof-of-concept: convex and quasi-convex resource allocationproblems can be implemented in a distributed fashion byexchanging control variables rather than CSI.


Decomposition Example Illustration Communications

Theory

0 1 2 3 4 5 6 70

1

2

3

4

5

6


log 2

(1+

SIN

R2)

[bits

/cha

nnel

use

]

Iteration 1

Iteration 2

Iteration 6

Line of Equal Performance

Max SumPerformance

Max−Min Fairness

Illustration of the convergence of the distributed resource allocation.Max-min fairness is the system utility function and 98% of the optimalperformance is achieved after 6 iterations.


Hardware Distortion Model Communications

Theory

• Practical transceiver hardware suffer from impairments:phase-noise, I/Q imbalance, nonlinearities, etc.

• These impairments distort the transmitted and received signals.

• Residual distortion noise remain after calibration algorithms.

• A generalization of the multi-cell system model is given by

yk = hHk Ck

(Kr∑

i=1

Divisi + ξ(t)

)+ nk + ξ

(r)k (12)

where visi is the transmitted signal to User i.

• The new terms ξ(t) ∈ CN and ξ(r)k ∈ C ∀k are the additive

transmitter-distortion and receiver-distortion, respectively.

• These are modeled as zero-mean complex Gaussian and arestatistically independent of the data signals, but the covariancematrices depend on the power of the transmitted and receivedsignals, respectively.


Additive Impairment Mismatch Communications

Theory

• The transmitter-distortion ξ(t) describes the mismatch betweenthe aggregate data signal

∑Kri=1 Divisi designed by the resource

allocation and what is actually transmitted by the RF hardware.The mismatch is due to transceiver hardware impairments in thetransmitter.

Distribution ofMismatch ξ(t)

∑Kri=1 Divisi

Intended Signal

Signal Dimension 1

Sign

al D

imen

sion

2


Additive Mismatch Model Communications

Theory

• We assume that the elements of ξ(t) are uncorrelated, which canbe confirmed by measurements on decoupled antenna branches, i.e.,ξ(t) ∼ CN (0,Ξ) where Ξ ∈ CN×N is a diagonal covariance matrix.

• The signal power allocated to the nth transmit antenna can becomputed as ‖TnVtot‖2F , where Tn ∈ CN×N is zero except at thenth diagonal element and Vtot = [D1v1 . . . DKrvKr ].

• The distortion at the n-th antenna increases with ‖TnVtot‖2F , thus

Ξ =

(η1(‖T1Vtot‖F )

)2. . .

(ηN (‖TNVtot‖F )

)2

(13)

where ηn(·) is a continuous and monotonically increasing function.


Error Vector Magnitude (EVM) Communications

Theory

• The error vector magnitude (EVM) is the ratio between theaverage distortion magnitude and the average transmit magnitude,defined as

EVM(t)n =

√√√√ E{∣∣[ξ(t)]n

∣∣2}

E{∣∣[∑Kr

k=1 Dkvksk]n

∣∣2}=ηn (‖TnVtot‖F )

‖TnVtot‖F(14)

and is often reported as a percentage.

• The EVM requirements in 3GPP Long Term Evolution (LTE) are8-17.5 % at the transmitter, depending on the anticipated spectralefficiency.


Power Constraints with Distortion Communications

Theory

• Due to the statistical independence of all terms, the powerconstraints are generalized as

Kr∑

k=1

vHk Qlkvk + tr(QlξΞ) ≤ ql l = 1, . . . , L. (15)

• The weighting matrix Qlξ ∈ CN×N is positive semi-definite andshould typically have the same structure as Qlk, but we might havetr(Qlξ) < tr(Qlk).

• An alternative model is to keep the original power constraints (i.e.,set Qlξ = 0N ) and simply reduce each limit ql to account for thedistortions.


Receiver Impairments Communications

Theory

• The receiver-distortion ξ(r)k of User k models the mismatch

between ideal and practical reception:

ξ(r)k ∼ CN (0, σ2

k,ξ) where σk,ξ = νk(‖hHk CkVtot‖F

)(16)

and νk(·) is a continuous and monotonically increasing function.

• The main error sources are phase noise and I/Q-imbalance, thus

we can expect νk(·) to behave as νk(x) = EVM(r)k x, where EVM

(r)k

is a constant EVM-term.


Res. Alloc. with Transceiver Impairments Communications

Theory

• The generalized system model in (12) results in generalized powerconstraints (15) and a generalized SINR expression for User k:

SINRk =|hHk CkDkvk|2

σ2k + σ2

k,ξ +∑

i 6=k |hHk CkDivi|2 + hHk CkΞCHk hk

. (17)

• Otherwise the multi-objective and single-objective resourceallocation problems are the same.

• The feasibility problem is generalized (where {γk}Krk=1 fixed)

find v1 . . . ,vKr (18)

subject to |hHk CkDkvk|2 ≥ γk(σ2k + σ2

k,ξ

+∑

i 6=k|hHk CkDivi|2 + hHk CkΞCH

k hk

)∀k,

Kr∑

k=1

vHk Qlkvk + tr(QlξΞ) ≤ ql ∀l.


Convex Reformulation under Impairments Communications

Theory

Corollary

The feasibility problem in (18) can be reformulated as

find vk, tn, ρk ∀k, n (19)

subject to

Kr∑

k=1

vHk Qlkvk +N∑

n=1

tr(QlξTn)t2n ≤ ql ∀l, (20)

√√√√σ2k + ρk +

Kr∑

i=1

|hHk CkDivi|2 +N∑

n=1

t2nhHk CkTnCHk hk

≤√

1 + γk

γk<(hHk CkDkvk) ∀k, (21)

=(hHk CkDkvk) = 0 ∀k, (22)

ηn(‖TnVtot‖F ) ≤ tn ∀n, (23)

νk

(‖hHk CkVtot‖F

)≤ ρk ∀k (24)

and is jointly convex in the beamforming vectors and in the auxiliary optimization variables{tn}Nn=1, {ρk}

Krk=1 if ηn(·) and νk(·) are monotonically increasing convex functions.


Rate Region under Hardware Impairments Communications

Theory

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

3.5

4

Max SumPerformance

Max Prop. Fairness

Max−Min Fairness

EVM(t)n = EVM

(r)k ∈ {0, 0.05, 0.1, }0.15


log 2

(1+

SIN

R2)

[bi

ts/c

hann

el u

se]

Performance regions for two different channel realizations under globaljoint transmission with different levels of (linear) transceiverhardware impairments; the EVM at the transmitters and receivers is 0,5, 10, or 15 percent.


Problem Statement: Multicast Communications

Theory

• The maximization of the minimum achievable SNR leads to

maxv:‖v‖≤q

mink∈Kj

|hHk CkDkv|2σ2k

. (25)

• Alternatively, the power minimization under SNR requirement γ

minv‖v‖2 subject to

|hHk CkDkv|2σ2k

≥ γ ∀k ∈ Kj . (26)

Corollary

For a fixed total transmit power q, the multi-cast beamforming vectorwhich solves (25) is given by

v∗k(λ) = q vmax

∑

k∈KjλkD

Hk CH

k hkhHk CkDk

(27)

for some set of |Kj | parameters that satisfies λk ≥ 0 and∑

k∈Kj λk = 1.


Multicast Illustration Communications

Theory

(a) Almost Orthogonal Channels

(c) Unequal Path Losses

SNR 1(v)

SNR 2(v)

SNR 1(v)

SNR 2(v)

SNR 1(v)

x2(v)

(d) Equal Path Losses

(b) Almost Equal Channels

SNR 1(v)

SNR 2(v)Max−MinSNR

Max−MinSNR

Max−MinSNR

Max−MinSNR

Possible Operating Pointsfor Power Minimization


Multi-Carrier System Model Communications

Theory

• The received signal at User k on the cth subcarrier is

ykc = hHkcCk

Kr∑

i=1

Divicsic + nkc c = 1, . . . ,Kc. (28)

• Assuming that all signal and noise variables are independent, theSINR at User k on the cth subcarrier is

SINRkc(v1c, . . . ,vKrc) =|hHkcCkDkvkc|2

σ2kc +

∑i 6=k|hHkcCkDivic|2 (29)

with vkc is the beamforming vector and the power constraints are

Kr∑

k=1

Kc∑

c=1

vHkcQlkcvkc ≤ ql l = 1, . . . , L (30)

where Qlkc � 0N might model subcarrier-specific characteristics.


Multi-Carrier Resource Allocation Communications

Theory

• The multi-objective resource allocation problem undermulti-carrier transmission can be formulated as

maxvkc�0N ∀k,c

{g1, . . . , gKr} (31)

subject to gk = gk

({SINRkc(v1c, . . . ,vKrc)}Kcc=1

)∀k, (32)

Kr∑k=1

Kc∑c=1

vHkcQlkcvkc ≤ ql ∀l. (33)

• The overwhelming multi-carrier complexity can be handled bydividing the subcarriers into subsets of manageable size and solvethese separately (cf. physical resource blocks in 3GPP LTE).


Multi-Carrier Parametrization Communications

Theory

Corollary

Every feasible point g ∈ R is achieved by beamforming vectors vkc =√pkcvkc for all k, c,

where

vkc =Ψ†kcD

Hk hkc

‖Ψ†kcDHk hkc‖

, (34)

[p1c . . . pKrc

]=[γ1cσ2

1c . . . γKrcσ2Krc

]M†c, (35)

Ψkc =

( L∑

l=1

µl

qlQlkc +

Kr∑

i=1

λic

σ2ic

DHk CH

i hichHicCiDk

), (36)

γkc =λkc

σ2kc

hHkcDk

(Ψkc −

λkc

σ2kc

DHk CH

k hkchHkcCkDk

)†DHk hk, (37)

[Mc]ik =

{|hHicCiDivic|2, i = k,

−γkc|hHkcCkDivic|2, i 6= k,(38)

for some non-negative parameters {λkc}Kr,Kck=1,c=1 and {µl}Ll=1.


System and Channel Model Communications

Theory

• Resource allocation is harder to optimize with multi-antenna users.

• Consider a two-user MIMO IC denoted by TXi 7→ RXi, i = 1, 2,where each transmitter TXi and each receiver RXi are equippedwith NT ≥ 2 and NR ≥ 2 antennas and a single data stream istransmitted to each user.

• In this IC, the received signal at RXi is modelled as

yi = gHi (Hiivisi + Hkivksk + ni) i, k ∈ {1, 2}, k 6= i

where si ∼ CN (0, 1) is the transmitted symbol of TXi by thetransmit beamforming vector vi ∈ CNT×1.

• The matrices Hii,Hki ∈ CNR×NT are channel matrices of directlink TXi 7→ RXi and cross-talk link TXk 7→ RXi, respectively.

• Each transmitter has a power constraint set to 1 and define the setof feasible transmit beamforming vectors as

V ∆={

v ∈ CNT×1 : ‖v‖2 ≤ 1}

.


Achievable Rates Communications

Theory

• The achievable rate of the link TXi 7→ RXi is given by

Ri(v1,v2,gi) = log2

(1 + SINRi(v1,v2,gi)

)(39)

where SINRi(v1,v2,gi) =|gHi Hiivi|2

σ2i+|gHi Hkivk|2 .

• With linear MMSE receive filter

SINRi(v1,v2) = vHi HHii

(σ2i I + Hkivkvk

HHkiH)−1

Hii︸︷︷︸∆=Ai(vk)

vi.

Proposition

For the two-user single-beam MIMO IC, the SINR can be reformulated as

SINRi(v1,v2) = sin2(θH,i) ·‖Hiivi‖2

σ2i

+ cos2(θH,i) ·‖Hiivi‖2

σ2i + ‖Hkivk‖2

where cos(θH,i) =∣∣−−−→Hiivi

H · −−−−→Hkivk∣∣ and θH,i ∈ [0, π/2].


Pareto Boundary Points Communications

Theory

Proposition

For the two-user single-stream MIMO IC, all the operating points on thestrict Pareto boundary can be achieved only when both the transmittersspend the full power, i.e., ‖v1‖2= ‖v2‖2= 1.

• A single-user point can be easily achieved when only TXi worksand simultaneously operates “egoistically” to maximize its own rate.

• The maximum achievable rate Ri of the link TXi 7→ RXi and itsassociated “egoistic” strategy vEgo

i are

Ri = log2

(1 +

λ1(HHii Hii)

σ2i

), vEgo

i = u1(HHii Hii) ∀i. (40)


Ending Points of Strict PB Communications

Theory

• Each ending point of the strict Pareto boundary can be achievedwhen one transmitter employs an “altruistic” strategy to create nointerference to the other receiver and simultaneously to maximize itsown rate and the other transmitter operates “egoistically”.

Proposition

E1(R1, R2) can be achieved by (vEgo1 ,vAlt

2 ), where R1 and vEgo1 are as

above and

R2 = log2

(1 + vAlt,H

2 A2(vEgo1 )vAlt

2

),

vAlt2 =

−−−−−−−−−−−−−−−−−−−−−−−−−→Π⊥

HH21H11vEgo

1

u1

(B1,Π

⊥HH

21H11vEgo1

), (41)

with B1∆= Π⊥

HH21H11vEgo

1

A2(vEgo1 )Π⊥

HH21H11vEgo

1

.


Non-strict PB and ZF Points Communications

Theory

• An arbitrary point (R?1, R?2) on the non-strict Pareto boundary

can be computed as

R?i = γ ·Ri and R?k = Rk, ∀i, k ∈ {1, 2}, k 6= i

where i = 1 and i = 2 correspond to the horizontal part and thevertical part, respectively.

• The ZF transmit strategies

vZF1 = u`(H

H22H12,H

H21H11),∀` ∈ {1, 2, ..., NT },

vZF2 =

NT−1∑

`=1

c`u`(Π⊥HH

21H11vZF1

), (42)

where {c`}NT−1`=1 are complex-valued numbers with

∑NT−1`=1 |c`|2 = 1.

• ZF achievable rates are

RZFi (vZF

i ,vZF2 ) = log2

(1 +‖Hiiv

ZFi ‖

2

σ2i

)∀i. (43)


Strict Pareto Boundary Communications

Theory

• We propose the following optimization problem

(P0)

maximizev1,v2∈WFP

SINR1(v1,v2)

subject to SINR2(v1,v2) = SINR?2.

where SINR?2 ∈ (2R2 − 1, 2R2 − 1) is an SINR constraint, and

v1,v2 should be in VFP ∆={

v ∈ CNT×1 : ‖v‖2 = 1}

.

• Then,(R?1, R

?2) = (log2 (1 + SINR1(v?1,v

?2)) , log2 (1 + SINR?

2(v?1,v?2))) is

achieved by the optimal solution (v?1,v?2) to (P0).

• Direct joint optimization of v1 and v2 is analytically intractable.

• To solve (P0), an alternating optimization algorithm is applied tooptimize v1 and v2 alternatively.


Rate Region Boundary Communications

Theory

4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8

5.6

5.8

6

6.2

6.4

6.6

6.8

R1 [bits/channel use]

R2

[b

its/c

ha

nn

el u

se

]

WMMSE_10

Eq.(16)

Eq.(15)

RandBeam_10mil

SimpleReceiver

Proposed_1+9

Ending point E1

Ending point E2

ZF Points

Proposed_1

WMMSE_1

P2

P1

Achievable boundary with SNR=10dB and NT = 3, NR = 2 for arandom Gaussian channel data.


Multi-User MIMO IC Extension Communications

Theory

Achievable rate of the link TXk 7→ RXk

Rk({vk}K) = log2

(1 + SINRk({vk}K)

)∀k ∈ K = {1, ...,K}, where

SINRk({vi}K) = vHk HHkk

∑

i 6=kHikvivi

HHikH + σ2

kI

−1

Hkk

︸︷︷︸Ak(v−k)

vk (44)

is the SINR expression of the kth user. v−k denotes {vi}K\{k}.

• Fix all but one SINR values

(Q0)

maximize{vi}K

SINR1({vi}K)

subject to SINRk({vi}K) = SINR?k, ∀k ∈ K\{1}.

vHi vi ≤ 1, ∀i ∈ K.(Q0) is a non-convex problem w.r.t. K beamformers, i.e., {vi}K.


Algorithm for Multi-User MIMO IC Communications

Theory

22

Algorithm 3 The Proposed Alternating Optimization Algorithm for K-user Case

Input: Rate requirements {R⋆k}K\{1} where R⋆

k = log2 (1 + SINR⋆k) and SINR⋆

k ∈(0, 1

σ2kλ1(HkkH

Hkk)].

Output: A convergent operating point (R(ℓ)1 , R⋆

2, ..., R⋆K) with {w

(ℓ)i }K.

beginInitialization: Set a feasible w

(0)−1, ℓ = 0.

while some convergence criterion is not satisfied doℓ + +.for k = 1 → K do

Given w(ℓ−1)−k , obtain an optimal W k to (Qk′);

Extract a tight/approximate wk from W k to (Qk).if K ≥ 5 and SINR1

({w

(ℓ)i }i≤k, {w

(ℓ−1)i }i>k

)< SINR1

({w

(ℓ)i }i<k, {w

(ℓ−1)i }i≥k

)8 then

w(ℓ)k = w

(ℓ−1)k ;

Compute R(ℓ)1 = log2

(1 + SINR1

({w

(ℓ)k }K

)).

Following the same line of the proof of the Algorithm 1’s convergence in Section IV-C-2, the convergence

of the Algorithm 3 is also guaranteed. �

V. ILLUSTRATIONS AND DISCUSSIONS

To illustrate the achievable rate region by the proposed algorithm, we consider a two-user Gaussian

MIMO IC, where NT = 3 and NR = 2. The transmit power budget is set to 1 for the two users, and

noise power σ21 = σ2

2 = 10− SNR

10 where SNR=10dB. The channels H11, H12,H21 and H22 are

(−0.3034 + 1.9096i −0.3790 + 0.4201i 0.0357 + 0.7337i

−0.6358 − 0.8030i −0.7881 − 0.1273i 0.7534 + 0.8348i

),(

−0.6758 + 0.1040i −0.5949 − 0.0344i 0.4311 + 0.9658i

−2.1621 + 0.5404i −0.0037 + 0.6627i 0.8611 + 1.2318i

),

(0.3999 + 0.1567i 0.3798 − 0.5619i −0.1005 + 0.2836i

−0.5494 − 0.4648i 1.1971 − 0.5297i −0.7271 + 0.2114i

),(

−0.0308 − 0.1133i 0.0433 − 0.3313i 0.3047 − 1.2157i

−1.4947 − 1.8676i −0.9430 + 0.5704i −1.3328 + 1.4638i

).

A. Convergence and Performance of Initialization by Eq.(16)

To study the convergence rate of the alternating optimization algorithm and evaluate the effectiveness

of the initialization in (16), we respectively use (16) and 200 randomly generated feasible normalized

vectors as initial w2. Then, we run Algorithm 1 until |R(ℓ)1 − R

(ℓ−1)1 | ≤ 10−3.

Fig. 2 and Fig. 3 show the performance of the alternating optimization algorithm by different initializa-

tions. More precisely, Fig. 2 is for a given R⋆2 = R2 + 2

19 · (R2 −R2) = 5.6398 (close to the ending point

(R1, R2)). Fig. 2(a) shows that the achieved R1 with initialization by (16) nearly always outperforms

that with 200 random initializations. Fig. 2(b) implies the convergence rate of the algorithm. Fig. 3 is

May 6, 2013 DRAFT


3D Achievable Rate Region Communications

Theory

Achievable rate region for three-user MIMO IC with SNR=0dB andNT = 3, NR = 2. The color bar shows the sum rate.


Dynamic Cooperation Cluster Model Communications

Theory

C2C1

D1 D2 C3

D3

BS3BS1

BS2

Illustration of the overlapping nature of dynamic cooperation clusters.BSj forms different cooperation constellations when serving differentusers. Dj are the users that it serves with data and Cj are the usersconsidered in the spatial interference coordination.


Resource Allocation and DCC Communications

Theory

• While this tutorial provides a thorough framework for resourceallocation for given DCCs, the practical design of DCCs is arelatively new and unexplored research area.

• The sets {Dj}Krj=1 of users served by each base station are selectedunder the following conditions:

• Each active user should have a master base station (MBS) thatguarantees its data services.

• The backhaul infrastructure should support the joint transmissionto a user, i.e., joint transmission is used only when the increase inthroughput outweighs the increased demands on the backhaul.

• Proximity is not measured geographically but by the average channelgain, taking possible differences in transmit power between basestations into account.

• The base stations and many objects in the propagation environmentare static.


RA and DCC - user sets Communications

Theory

• The sets {Cj}Krj=1 of users that are considered in the beamforming ateach base station are selected under the following conditions:

• The channels between the base station and all the users that itincludes in its interference coordination need to be estimated.

• All neighboring base stations in TDD can estimate their ownindividual channel components by listening to the same uplinktraining signal from a user.

• All neighboring base stations in FDD can overhear the feedbackintended for their neighbors and thus obtain channels from a user tomultiple base stations.

• Both CSI estimation errors and tolerance to CSI errors increasewith distance.


Simple Clustering Algorithm Communications

Theory

• A simple clustering algorithm would be to include User k in Dj ifthe average channel gain E{‖hjk‖22} (over some suitabletime-window) is above a certain threshold value. The fulfillment ofthis condition is rechecked at the same time-scale as the estimateof E{‖hjk‖22} is updated. User k appoints base station

m = arg max{j: k∈Cj}

E{‖hjk‖22} (45)

as its MBS, which is the one with the strongest channel.

• The user then computes the ratio between the average channel gainof the MBS and the gain of the second strongest base station,

E{‖hmk‖22}max{j:k∈Cj}\{m} E{‖hjk‖22}

. (46)

Joint transmission occurs if this is above a threshold.


Cognitive Radio Introduction Communications

Theory

• In a cognitive radio scenario, the systems are capable of detectingtheir environment and reconfiguring their operations accordingly.

• Consider a network composed of licensed primary users. Additionalusers, called secondary users, having cognitive radio capabilities,can be supported without degrading the QoS of the primary users.

• CR systems can be categorized into three models:• An interweave cognitive radio is an intelligent wireless

communication system that periodically monitors the radio spectrum,intelligently detects occupancy in the different parts of the spectrum,and then opportunistically communicates over spectrum holes withminimal interference to the active primary users.

• The underlay paradigm mandates that concurrent non-cognitive andcognitive transmissions may occur only if the interference generatedby the cognitive devices at the non-cognitive receivers is below someacceptable threshold.

• Cognitive radio systems that have cooperation with the primarysystem as key feature are denoted as overlay cognitive radio systems.


Underlay Cognitive Radio Model Communications

Theory

BS1

BS2

BS3

C1=C2=C3

D1=D2=D3

Primary system Secondary system

• We focus on theunderlay cognitive radiosystem and begin withnull-shaping for allk ∈ Kprimary

vHk Qklvk = 0 (47)

or general interferencetemperature constraints(ITC)

vHk Qklvk ≤ ql. (48)


Null Shaping Constraints Communications

Theory

• Collect all null-shaping constraints for transmission to User k in amatrix

Zk =[Qk1 . . . QkKprimary

]. (49)

• To satisfy the null-shaping constraints, we can define a neweffective channel of User k as

hk = Π⊥Zkhk (50)

by projecting the original channel vector hk onto the null-space ofthe null-shaping matrix Zk.

• Based on the effective channels hk in (50), the completeframework developed in this tutorial can be applied.


Interference Temperature Constraints Communications

Theory

• Let us consider a simple two-link example: primary link one andsecondary link two.

• ITC: vH2 Q21v2 ≤ q1 with primary rate constraint

R1(load) = load · log2

(1 +

|hH1 C1D1v1|2σ2

1

).

• This results in q1 = |hH1 C1D1v1|2(2R1(load) − 1

)−1 − σ21.

• Resource allocation problem for secondary precoding is given by

maxv2: ‖v2‖≤1

g2(SINR2(v2)) subject to vH2 Q21v2 ≤ q1. (51)

• Again, the complete framework can be applied to handle theadditional ITC.


Pareto Boundary Characterization Communications

Theory

Theorem

The optimization problem (51) is solved by

v2(λ∗) =√λ∗

ΠDH2 CH

1 h1DH

2 CH2 h2

‖ΠDH2 CH

1 h1DH

2 CH2 h2‖

+√

1− λ∗Π⊥

DH2 CH

1 h1DH

2 CH2 h2

‖Π⊥DH

2 CH1 h1

DH2 CH

2 h2‖(52)

with

λ∗ =

λMRT, λMRT ≤ q1

‖DH2 CH

1 h1‖2 ,q1

‖DH2 CH

1 h1‖2 , otherwise,(53)

and λMRT =

∥∥ΠDH

2CH

1h1

DH2 CH

2 h2

∥∥2

∥∥ΠDH

2CH

1h1

DH2 CH

2 h2

∥∥2+∥∥Π⊥

DH2

CH1

h1DH

2 CH2 h2

∥∥2 .


Physical Layer Security: Motivation Communications

Theory

• The data processing, transmission, and encryption in moderncommunication systems are carried out separately.

• The typical purpose of the physical layer is to guarantee error-freetransmission, whereas encryption is performed at a higher layer inthe protocol stack.

• State-of-the-art encryption algorithms rely on mathematicaloperations assumed to be hard to compute, however, the classicalapproach to security becomes increasingly difficult to justify, inparticular if we consider that:

(a) the underlying intractability assumptions may be wrong;

(b) efficient attacks could be developed;

(c) the advent of quantum computers is likely to compromisethis type of encryption; and

(d) fast and reliable communications over ad hoc wirelessnetworks require light and effective security architectures.


Physical Layer Secrecy Observations Communications

Theory

• Physical layer security provides perfect secrecy.

• By clever coding and transmit strategies the data can be reliablyand securely sent at secrecy rates.

Interference is Information Leakage

The amount of interference that is created by one link at anotherreceiver corresponds to the amount of information leakaged from thislink to the curious receiver.

• Additional amounts of spectrally, spatially, or temporally adjustedinterference/noise can improve the secrecy rates.


PhySec Example Scenario Communications

Theory

AliceBob

Eve

Hugovh

va hab

hae

hhbhhe

Noise

Noise

Illustration of a simple eavesdropping scenario with four nodes: Alice wants to communicate

privately with Bob. Eve is trying to overhear, while Hugo supports the private communication

by intentionally creating interference at Eve (while avoiding interference at Bob).


System Model Communications

Theory

• Two transmissions create interference to each other: from Alice toBob and from Hugo to Eve.

• The link from Alice to Bob is described by the vector channelhab = DH

1 CH1 h1, Alice to Eve hae = DH

1 CH2 h2, whereas the link

from Hugo to Bob is described by hhb = DH2 CH

1 h1 and Hugo toEve is hhe = DH

2 CH2 h2.

• The private communication uses the beamforming vector va andthe helper creates interference using vh.

• The achievable secrecy rate for reliable and secure datatransmission is given by

RS(va,vh) =

[log2

(1 +

|hHabva|21 + |hHhbvh|2

)

− log2

(1 +

|hHaeva|21 + |hHhevh|2

)]

+

.

(54)


Secrecy Rate Maximization Communications

Theory

• The optimization problem for maximizing the secrecy rate is

max0≤‖va‖2≤qa

max0≤‖vh‖2≤qh

RS(va,vh). (55)

Lemma

The beamforming vector v′a that solves (55) for fixed vh is given by

v′a(vh) = qaψ (56)

where ψ is the generalized eigenvector associated with the maximumgeneralized eigenvalue of the pencil (I + 1

1+z1habh

Hab, I + 1

1+z2haeh

Hae)

with z1 = |hHhbvh|2 and z2 = |hHhevh|2.


Optimal Beamforming at Helper Communications

Theory

Theorem

For fixed va, the beamforming vector v′h that solves (55) is given by

v′h(λ) =λΠhhbh

∗he + (1− λ)Π⊥hhbh

∗he

‖λΠhhbh∗he + (1− λ)Π⊥hhbh

∗he‖

(57)

for some λ ∈ [0, 1].

• An extension to multiple K helpers is possible and an iterativealgorithm can be derived.

• If the number of helpers approaches infinity, the average secrecyrate approaches the average rate of the peaceful system.


Physical Layer Security I Communications

Theory

−10 −5 0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

SNR [dB]

Ave

rage

Sec

recy

Rat

e

Upper Bound (without Eve) Optimal Alice & HelperOptimal Alice, No HelperAlice: MRT, Helper: ZFBFAlice: MRT, No Helper

Average secrecy rate with and without a helper, and using differenttransmit strategies. Alice and Hugo have two transmit antennas each.


Physical Layer Security Illustration II Communications

Theory

−10 −5 0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4

1 HelperUpper Bound 25102050100500

SNR [dB]

Ave

rage

Sec

recy

Rat

e

Instantaneous secrecy rate with helpers using optimal beamforming.Alice and all helpers are equipped with two transmit antennas.


Conclusions Communications

Theory

• Our ambition has been to provide a solid ground and understandingfor optimization of practical modern multi-cell systems.

• Modern multi-antenna techniques provide much higher datathroughput and flexibility but makes it harder to optimize.

• The same algorithms solve many “different” resource allocationproblems and handle many practical conditions and scenarios.

C2C1 D1 D2

BS1 BS2


Some Open Problems Communications

Theory

• How to combine several of the topics in the same model?

• Extension to Multi-Hop Networks including AF, DF, CF andvariants of physical layer network coding.

• Application of Game Theory to derive distributed non-cooperativeresource allocation and beamforming algorithms.

• Application of Pricing to achieve certain operating points on thePareto Boundary.

• Novel performance measures to optimize the Energy Efficiency inmulti-cell networks.

• Use of more Detailed Hardware Models, which will impact thepower constraints and hardware impairments.

• Proactive resource allocation that guarantee Long-Term UserFairness and exploits Prior Information user behavior/traffic.

• Coordinated interference management in Multi-Tier Networks.


References Communications

Theory

The main reference is our book:

Emil Bjornson, Eduard Jorswieck, “Optimal

Resource Allocation in Coordinated Multi-

Cell Systems,” Foundations and Trends in

Communications and Information Theory,

vol. 9, no. 2-3, pp. 113-381, 2013.

The book contains 330 referencesto research articles on these topics.

Download the e-book from ourhomepages or buy the printed bookat a special price (see Part I).


Acknowledgements Communications

Theory

• Emil Bjornson thanks his colleagues Bjorn Ottersten, Mats Bengtsson, David Gesbert,Gan Zheng, and Per Zetterberg for the inspiration and guidance they have provided overthe years. This tutorial is the final result of his doctoral journey at the Signal ProcessingLab at KTH Royal Institute of Technology. He would also like thank the fellow doctoralstudents David Hammarwall, Niklas Jalden, Simon Jarmyr, Randa Zakhour, XueyingHou, and Jinghong Yang for fruitful scientific collaborations.

• Eduard Jorswieck thanks his colleagues Erik G. Larsson, Eleftherios Karipidis, RamiMochaourab, Zuleita Ka Ming Ho, Jan Sykora, Leonardo Badia, Jian Luo, MartinSchubert, and all members of the SAPHYRE project for interesting and fruitfuldiscussions on the topic of resource sharing in wireless networks.

• We are indebted to Rasmus Brandt, Axel Muller, Serveh Shalmashi, Jiaheng Wang, andthe anonymous reviewers for their careful proofreading of our book.

• This work was supported in part by the AMIMOS project under the Advanced ResearchGrant 228044 from the European Research Council (ERC). It was also supported by theInternational Postdoc Grant 2012-228 from The Swedish Research Council.

• Part of this work was performed in the framework of the European research projectSAPHYRE, which was supported in part by the European Union under its FP7 ICTObjective 1.1 – The Network of the Future. This work was also supported in part by theDeutsche Forschungsgemeinschaft (DFG) under Grant Jo 801/4-1.


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