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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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 2 / 64
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
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 3 / 64
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
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 4 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 5 / 64
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)}.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 6 / 64
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?
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 7 / 64
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)
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 8 / 64
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 .
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 9 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 10 / 64
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]
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 11 / 64
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
log2(1+SINR1) [bits/channel use]
log 2
(1+
SIN
R2)
[bi
ts/c
hann
el u
se]
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 12 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 13 / 64
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
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).
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 14 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 15 / 64
• 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
∗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
where Θ∗ik, Θ∗ik, q
∗lk,v
∗k are the subproblem solutions.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 16 / 64
• 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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 17 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 18 / 64
Decomposition Example Illustration Communications
Theory
0 1 2 3 4 5 6 70
1
2
3
4
5
6
log2(1+SINR1) [bits/channel use]
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 19 / 64
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
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 20 / 64
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
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 21 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 22 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 23 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 24 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 25 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 26 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 27 / 64
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
log2(1+SINR1) [bits/channel use]
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 28 / 64
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
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 29 / 64
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
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 30 / 64
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
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 31 / 64
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).
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 32 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 33 / 64
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
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}
.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 34 / 64
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].
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 35 / 64
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)
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 36 / 64
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
.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 37 / 64
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)
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 38 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 39 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 40 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 41 / 64
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
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 42 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 43 / 64
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
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 44 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 45 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 46 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 47 / 64
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
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 48 / 64
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)
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 49 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 50 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 51 / 64
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 .
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 52 / 64
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
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 53 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 54 / 64
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).
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 55 / 64
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)
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 56 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 57 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 58 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 59 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 60 / 64
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
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 61 / 64
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.
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 62 / 64
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).
Bjornson & Jorswieck: Coordinated Multi-Cell Systems 8 September 2013 63 / 64
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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