applications of robust optimization in signal processing...

Applications of Robust Optimization in Signal

Processing: Beamforming and Power Control

Fall 2012

Instructor: Farid AlizadehScribe: Shunqiao Sun

12/09/2012

1 Overview

In this presentation, we study the applications of robust optimization in signalprocessing: robust beamforming in array and power control in wireless networks.First, the background of beamforming and the traditional diagonal loading (DL)method are introduced. Then, the robust beamforming designs are proposedby considering the worst-case distortionless and probabilistic constraints of thesteering vector, respectively. It is shown that the robust beamforming designcan be formulated as a seconder order cone programming (SOCP) problem ora semidefinite programming (SDP) problem, and then can be solved efficiently.In the last, the robust power control problem in cognitive radio networks canbe formulated as a SOCP problem by considering the worst-case uncertaintiesin the wireless channel gains.

2 Background of Beamforming

Beamforming has been widely used in wireless communications and radar sys-tems. In practice, the performance of beamforming may undergo degradationdue to the mismatch in the steering vectors. The mismatch could be a resultof imperfect array calibration, look direction and signal pointing errors. Ar-ray signal processing is very sensitive to these mismatches and robust design isrequired [1-5].

Consider a narrow-band beamformer:

y (k) = wHx (k) ,

where k is the time index, x (k) = [x1 (k) , ..., xM (k)]T ∈ CM×1 is the complex

output of the array. w = [w1, ..., wM]T ∈ CM×1 is the complex beamforming

1

Shunqiao Sun Date: 12/09/2012

v

rdrd

Fusion center

Receivers

Target

sinrd 2 sinrd

1x k Mx k

1 sinrM d

1wMw

Figure 1: Uniform Linear Array Model.

weights. M is the number of sensors in the array (see Figure 1). The observationvector is

x (k) = s (k)a+ i (k) + n (k) , (1)

where s (k) is the signal waveform; i (k) and n (k) are the interference and noise,respectively. Here, a the steering vector and is defined as

a =[1, ej

2πλdr sinθ, ..., ej

2πλ

(M−1)dr sinθ]T.

The signal-to-interference-plus-noise ratio (SINR) is given as

SINR =σ2s∣∣wHa∣∣2

wHRi+nw, (2)

where Ri+n = E{(i (k) + n (k)) (i (k) + n (k))

H}

is the interference-plus-noise

covariance matrix, and σ2s is the signal power. The beamforming vector w ischosen to maximize the SINR. The beamforming vector design problem can beformulated as

minw

wHRi+nw s.t. wHa = 1. (3)

It is well known that the solution of problem (3) is

wopt = αR−1i+na, (4)

2


where α =(aHR−1

i+na)−1

. The solution is referred as the minimum variancedistortionless response (MVDR) beamformer.

In practice, it is difficult to know the interference-plus-noise covariance ma-trix Ri+n. So, the sample covariance matrix

R =1

N

N∑n=1

x (n) xH (n) (5)

is used instead of Ri+n. And the sample version beamforming design problemis written as

minw

wHRw s.t. wHa = 1. (6)

The solution of (6) is referred as the sample matrix inversion (SMI)-based min-imum variance (MV) beamforming and is written as

wSMI = αR−1a, (7)

where α =(aHR−1a

)−1. For the signal free case, the optimal SINR is

SINRopt = σ2saHR−1

i+na. (8)

The fact that sample covariance matrix is used instead of the interference-plus noise covariance matrix would dramatically affect the beamforming perfor-mance when there is mismatch in the steering vector. The reason is as follows.

In the mismatch case, the actual steering vector a that characterizes thespatial signature of the signal is

a = a+ ∆ 6= a, (9)

where ∆ is unknown complex vector that describes the mismatch of steering vec-tor. Therefore, the SMI beamformer would view the desired signal componentsin the samples as interference and try to suppress them. This suppression iscalled “signal cancellation” phenomenon which will degrade the SINR. A littlemismatch in the steering vector would lead to severe degradation in the SINR.

One popular method to deal with the mismatch issue is diagonal loading.The basic idea of this method is to replace the sample covariance matrix R withthe diagonal loading covariance matrix

Rdl = ξI+ R (10)

in the SMI expression (7). Here, ξ is a diagonal loading factor and I is the iden-tity matrix. The diagonal loading method increase the variance of the artificialwhite noise by the amount of ξ. This modification would force the beamformerto put more effort to minimize the white noise rather than the interference.As we know, when there is mismatch in the steering vector, the desired signalcomponents are suppressed as interference. As the beamformer puts more effortin suppressing the white noise, the signal cancellation is reduced. But if ξ istoo large, the beamformer fails to suppress strong interference since most effortis used to suppress the white noise. How to choose a proper diagonal loadingfactor ξ is not clear.

3


3 Uncertainty in Steering Vector: SOCP For-mulation

3.1 Problem Formulation

In [1], the norm of the steering vector distortion ∆ is assumed to be bounded

‖∆‖ ≤ ε, (11)

where ε > 0 is some constant. The steering vector belongs to the set

A (ε) = {c |c = a+ e, ‖e‖ ≤ ε} . (12)

In [3], the steering vector is assumed to be lie in an n-dimensional ellipsoid

ζ = {Pµ+ c |‖µ‖ ≤ 1} . (13)

The set ζ describes an ellipsoid whose center is c and whose principal semiaxesare the unit-norm left singular vectors of P scaled by the corresponding singularvalues.

For all vectors belonging to A (ε), the absolute value of the array responseshould not be smaller than one∣∣wHc∣∣ ≥ 1 for all c ∈ A (ε) . (14)

With (14), the robust beamforming can be formulated as

minw

wHRw s.t.∣∣wHc∣∣ ≥ 1 for all c ∈ A (ε) . (15)

For each c ∈ A (ε), the constraint∣∣wHc∣∣ ≥ 1 is nonconvex. Since there is an

infinite number of vectors c in A (ε), there is an infinite number of constraints.Therefore, (15) is a semi-infinite nonconvex quadratic problem. We consider theworst-case constraint in (15) and rewrite the problem as

minw

wHRw s.t. minc∈A(ε)

∣∣wHc∣∣ ≥ 1. (16)

We rewrite the constraint in (16) as

mine∈D(ε)

∣∣wHa+wHe∣∣ ≥ 1, (17)

where D (ε) = {e |‖e‖ ≤ ε }. Applying the triangle and Cauchy-Schwarz inequal-ities, we have∣∣wHa+wHe

∣∣ ≥ ∣∣wHa∣∣− ∣∣wHe∣∣ ≥ ∣∣wHa∣∣− ε ‖w‖ . (18)

The equation holds with∣∣wHa∣∣ > ε ‖w‖ and e = − w

‖w‖εejφ, where φ =

angle{wHa

}. So, we have the worst-case constraint as

minc∈D(ε)

∣∣wHc∣∣ = ∣∣wHa∣∣− ε ‖w‖ . (19)

4


The semi-infinite programming (SIP) problem (16) can be formulated as

minw

wHRw s.t.∣∣wHa∣∣− ε ‖w‖ ≥ 1. (20)

In problem (20), the constraint is nonconvex due to the absolute value operation.An important observation is that the cost function in (20) is unchanged whenthere is an arbitrary phase rotation in w. So if w0 is the optimal solutionto (20), we can rotate rotate the phase of w0 without affecting the objectivefunction, so that wHa is real. Thus, we can choose w such that

Re{wHa

}≥ 0, Im

{wHa

}= 0. (21)

So the robust beamforming problem can be rewritten as

minw

wHRw s.t. wHa ≥ ε ‖w‖+ 1, Im{wHa

}= 0. (22)

3.2 Relationship with SMI Beamformer

It is easy to see the constraint in (16) is equivalent to

minc∈A(ε)

∣∣wHc∣∣ = 1. (23)

So the constraint in (22) is equivalent to equality constraint wHa = ε ‖w‖+ 1.We can rewrite the problem

minw

wHRw s.t.∣∣wHa− 1

∣∣2 = ε2wHw. (24)

The Lagrange function is

H (w, λ) = wHRw+ λ(ε2wHw−wHaaHw+wHa+ aHw− 1

), (25)

where λ is a Lagrange multipler. Taking the gradient of H (w, λ) and equatingit to zero yields

w = −λ(R+ λε2I− λaaH

)−1a. (26)

With matrix inversion lemma, we get

w =λ

λaH(R+ λε2I

)−1a− 1

(R+ λε2I

)−1a. (27)

This indicates that the proposed robust beamforming belongs to diagonal load-ing methods.

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3.3 SOCP Problem Implementation

First, we convert the quadratic objective function in (22) to a linear one. As-

sume the Cholesky factorization of R as R = UHU. Then, wHRw = ‖Uw‖2.Therefore, the problem (22) can be converted into the following problem

minτ,w

τ s.t. ‖Uw‖ ≤ τ, ε ‖w‖ ≤ wHa− 1, Im{wHa

}= 0. (28)

By introducing

^w =

[Re {w}

T, Im {w}

T]T

^a =

[Re {a}

T, Im {a}

T]T

a =[Im {a}

T,−Re {a}

T]T

^

U =

[Re {U} − Im {U}

Im {U} Re {U}

],

the problem (28) can be written as

minτ,

^w

τ s.t.

∥∥∥∥^

U^w

∥∥∥∥ ≤ τ, ε∥∥∥^w∥∥∥ ≤ ^

wT

^a − 1,

^wT

a = 0. (29)

The problem (29) is a SOCP problem which can be efficiently solved.

4 Uncertainties in Steering Vector and Covari-ance Matrix: SDP Formulation

In this section, we consider the scenario that there are uncertainties in both thesteering vector and covariance matrix and the uncertainties are known to belongto a convex set U of CM × HM++. Here, HM++ denotes the set of all Hermitianpositive definite matrices of size M ×M. The convexity means that, for anytwo pairs (a1, Σ1) and (a2, Σ2) in U,

η (a1, Σ1) + (1− η) (a2, Σ2) ∈ U, ∀η ∈ (0, 1) . (30)

Here, we use Σ to denote the covariance. Given weight vector w, the worst-caseSINR analysis problem of a finding a steering vector and a covariance matrixthat minimize the SINR is

min SINR (w,a, Σ) s.t. (a, Σ) ∈ U, (31)

where the variables are a and Σ. The optimal value of the this problem is theworst-case SINR, denoted as SINRwc (w),

SINRwc (w) = inf(a,Σ)∈U

SINR (w,a, Σ) .

6


The problem (31) is a convex optimization problem as the SINR (w,a, Σ) is aconvex function of a and Σ for a given weight vector w.

The robust beamforming considering the worst-case SINR maximization isformulated as

max SINRwc (w) s.t. w 6= 0 (32)

with variable w. The problem (32) is nonconvex. In [2], a proposition is givenas below

Proposition: If (a∗, Σ∗) solve the problem

min supw6=0

SINR (w,a, Σ) s.t. (a, Σ) ∈ U (33)

with variables a ∈ CM and Σ ∈ HM. Then (w∗,a∗, Σ∗) with w∗ = Σ∗−1a∗

satisfies the saddle-point property

SINR (w,a∗, Σ∗) ≤ SINR (w∗,a∗, Σ∗) ≤ SINR (w∗,a, Σ) ,∀w ∈ CM\ {0} ,∀ (a, Σ) ∈ U. (34)

From the saddle-point property, we can show that

SINR (w∗,a∗, Σ∗) = inf(a,Σ)∈U

SINR (w∗,a, Σ) = supw6=0

SINR (w,a∗, Σ∗)

= inf(a,Σ)∈U

supw6=0

SINR (w,a, Σ) = supw6=0

inf(a,Σ)∈U

SINR (w,a, Σ) .

We conclude that w∗ = Σ−1a solves the worst-case SINR maximization problem(32).

According to (8), the SINR maximization problem (33) is equivalent to

min aHΣ−1a s.t. (a, Σ) ∈ U. (35)

By expanding the real and imaginary parts of a and Σ, we have

aHΣ−1a = zTR−1z, (36)

where

z =

[Re {a}Im {a}

]∈ R2M,

R =

[Re {Σ} − Im {Σ}Im {Σ} Re {Σ}

]∈ R2M×2M. (37)

Therefore, problem (35) is equivalent to

min zTR−1z s.t. (z,R) ∈ V, (38)

where, V is a subset of R2M × S2M++

V =

{([Re {a}Im {a}

],

[Re {Σ} − Im {Σ}Im {Σ} Re {Σ}

])|(a, Σ) ∈ U

}. (39)

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Here, S2M++ denote the set of all 2M× 2M symmetric positive definite matrices.The objective function in (38) is a matrix fractional function and is convex. V

is convex since U is. Using the Schur complement, the convex problem (38) canbe re-written as the following SDP problem

min t s.t.

[R zzT t

]�= 0, (z,R) ∈ V. (40)

5 Uncertainty in Steering Vector: ProbabilisticConstraints

The idea of probability constraint in the beamforming is to model the steeringvector statistically rather than in a deterministic way. And the beamformingmaintains the distortionless response in a high probability rather than for allmismatch conditions in the uncertainty set.

The probability-constrained robust beamforming problem can be written as

minw

wHRw s.t. Pr{∣∣wHa∣∣ ≥ 1} ≥ p, (41)

where p is a certain probability value. According to (9), with triangle inequality,∣∣wH (a+ ∆)∣∣ ≥ ∣∣wHa∣∣− ∣∣wH∆∣∣ . (42)

Therefore, problem (41) can be written as

minw

wHRw s.t. Pr{∣∣wH∆∣∣ ≤ ∣∣wHa∣∣− 1} ≥ p. (43)

5.1 Gaussian Mismatch Case

Let ∆ follow a complex circularly symmetric Gaussian distribution with zeromean and covariance matrix C∆, i.e., ∆ ∼ NC (0M,C∆). The covariance ma-trix C∆ captures the second order statistics of the uncertainties in the steeringvector. It is easy to show that the random variable wH (a+ ∆) has the follow-

ing complex Gaussian distribution: wH (a+ ∆) ∼ NC

(wHa,

∥∥∥C1/2∆ w∥∥∥2). As

wH∆ is circular zero mean complex Gaussian, its real and imaginary parts arereal independent identically distributed Gaussian

Re{wH∆

}∼ NR

(0,∥∥∥C1/2∆ w

∥∥∥2/2)Im{wH∆

}∼ NR

(0,∥∥∥C1/2∆ w

∥∥∥2/2) .If x and y are two real i.i.d. zero mean Gaussian random variables with vari-ance as δ2, then z =

√x2 + y2 is Rayleigh distributed with cumulative density

function (CDF) as

F (z) = 1− e−z2/2δ2 .

8


Thus, the constraint in (43) can be written as

Pr{∣∣wH∆∣∣ ≤ ∣∣wHa∣∣− 1} = 1− exp

−

(∣∣wHa∣∣− 1)2∥∥∥C1/2∆ w∥∥∥2

≥ p. (44)

The inequality in the right hand side of (44) can be equivalently written as∥∥∥C1/2∆ w∥∥∥ ≤ 1√

− ln (1− p)

(∣∣wHa∣∣− 1) . (45)

Since the cost function in (43) is unchanged when w undergoes an arbitraryphase rotation, we can choose w such that Re

{wHa

}≥ 0, Im

{wHa

}= 0.

The robust beamforming problem can be approximated by the following SOCPproblem

minw

wHRw s.t.∥∥∥C1/2∆ w

∥∥∥ ≤ 1√− ln (1− p)

(wHa− 1

). (46)

5.2 Worst Mismatch Distribution Case

Here, the case that the distribution of the steering vector mismatch ∆ is un-known is considered. Assume that ∆ is zero mean with given covariance. Theassumptions Re

{wHa

}≥ 0, Im

{wHa

}= 0 hold. Thus, the constraint in (43)

can be written as

Pr{∣∣wH∆∣∣ < wHa− 1

}≥ p. (47)

The Chebyshev inequality states that for any zero mean random variable ξ withvariance σ2ξ and any positive real number α,

Pr {|ξ| ≥ α} ≤σ2ξα2. (48)

The constraint (47) can be lower bounded as

Pr{∣∣wH∆∣∣ < wHa− 1

}≥ 1−

∥∥∥C1/2∆ w∥∥∥2

(wHa− 1)2

(49)

for any distribution of ∆. The above inequality holds if∥∥∥C1/2∆ w

∥∥∥2 < (∣∣wHa∣∣− 1)2.The deterministic constraint is obtain as below∥∥∥C1/2∆ w

∥∥∥ ≤√1− p (wHa− 1). (50)

So the robust beamforming problem can be approximated by the followingSOCP problem

minw

wHRw s.t.∥∥∥C1/2∆ w

∥∥∥ ≤√1− p (wHa− 1). (51)

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PU-Tx

CR-Tx

PU-Rx

CR-Rx

CR Link 1 … CR Link n

CR-Tx

CR-Rx

11G

1nG

nnG

0nG

nh

…

10G

1nG

1h

Figure 2: A cognitive radio network is composed of M cognitive links randomlyallocated in a cluster area. The dashed lines denote the interference channels.

6 Robust Power Control in Cognitive Radio Net-works

In this section, we study the robust power control problems in cognitive radionetworks considering the uncertainties in the wireless channel gains. In practice,the wireless channel gains are imperfectly known or time varying. Without con-sidering the uncertainty of these parameters, the solutions of nominal problemsare not robust.

Consider an ad hoc cognitive radio network, i.e., there is no central controlnode. Assume that there are M cognitive user links randomly allocated in acluster area, as shown in Figure 2. We assume that all the cognitive users canshare the same frequency resource with primary user under the constraint thatthe total interference generated by cognitive users does not exceed a thresholdthat PU-Rx can tolerate, i.e.,

M∑i=1

hipi ≤ I, (52)

where hi denotes the channel gain between the cognitive transmitter (CR-Tx)of link i to the PU-Rx. pi represents the transmitter power of the CR-Tx oflink i. I is the maximum interference level that the PU-Rx can tolerate.

The signal-to-interference-noise-ratio (SINR) at the cognitive receiver (CR-Rx) of link i is

SINRi =Giipi

M∑j6=iGijpj +Gi0p0 + ni

, (53)

where Gii represents the channel gain between the CR-Tx and the CR-Rx oflink i. Gij is the channel gain between the CR-Tx of link j and the CR-Rx of

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link i. Gi0 denotes the channel gain between the primary transmitter (PU-Tx)and the CR-Rx of link i. p0 is the transmit power of PU-Tx and ni is thebackground noise at the CR-Rx of link i.

To guarantee the QoS requirements of cognitive users, i.e., the packet delayconstraint or minimum transmit rate, the SINR at each CR-Rx should be largerthan a threshold, i.e., we have

SINRi ≥ γi, ∀i, (54)

where γi is the SINR requirement at the CR-Rx of link i.Constraint (54) can be equivalently written as

(D− F)p ≥m, (55)

where D is a M×M diagonal matrix with diagonal elements being 1γ1, ..., 1

γM.

Vector m =[

N1

G11, ..., NM

GMM

]T, where Ni = Gi0p0 + ni. F = [Fij] with

Fij =

{GijGii, i 6= j,

0, i = j.(56)

The optimization objective is to minimize the total transmit power of all CR-Txs under both the PU-Rx interference constraint and the SINR constraint ofeach CR-Rx. Mathematically, the problem is formulated as follows:Nominal power control problem (P1) :

min

M∑i=1

pi

s.t.

M∑i=1

hipi ≤ I

SINRi ≥ γi, ∀i, (57)

where the variables are 0 ≤ pi ≤ Pmax for all i, and Pmax is the maximumtransmit power for each CR-Tx.

We use ellipsoid sets to describe the uncertainty. Under the worst case, therobust power control problem can be formulated as a SOCP problem.

Let Fi denote the uncertainty set of the ith row of matrix F. Fi describesthe perturbation of interfering channel gains relevant to channel gain of link i.We use an ellipsoid to describe set Fi, for all i. The same method is also usedin [6]. Denote the normalized channel gain from the CR-Tx of link j to theCR-Rx of link i as Fij = Fij + ∆Fij, where Fij is the nominal value, and ∆Fijis the perturbation part. Denote the ith row of F as Fi, and the correspondingperturbation part as ∆Fi. Under ellipsoid approximation, the uncertainty set ofFi for Fi can be written as

Fi =

Fi + ∆Fi :∑j6=i

|∆Fij|2 ≤ ε2i

, (58)

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where εi ≥ 0 is the maximum deviation of each entry in Fi. Since the channelgain vector at each CR-Rx varies independently, the row-wise structure of Fi cancapture the uncertainty well. Besides, the row-wise uncertainty sets of channelgains have a decomposition structure which is important in implementing thedistributed algorithm.

Denote the channel gain between the CR-Tx of link i and the PU-Rx ashi = hi + ∆hi, where hi is the nominal value, and ∆hi is the correspondingdeviation part. Let H be the uncertainty set of channel gain vector h, whereh = [h1, ...,hM]

T. Use an ellipsoid to describe the uncertainty set H as follows

H =

{h+ ∆h :

∑i

|∆hi|2≤ ε20

}, (59)

where ε0 is the maximum deviation of each entry in h.The robust power control problem that considers the channel uncertainty

can be written asRobust power control problem (P2) :

min∑i

pi

s.t.∑i

(hi + ∆hi

)pi ≤ I∑

j 6=i

(Fij + ∆Fij

)pj

pi+

Ni

Giipi≤ 1

γi, ∀i (60)∑

j6=i

|∆Fij|2 ≤ ε2i , ∀i∑

i

|∆hi|2 ≤ ε20.

Since problem P2 is subject to an infinite number of constraints with respect tothe sets Fi (εi), for all i and H (ε0), problem P2 is a SIP problem. A naturalmethod to solve the SIP problem is to transform it into an equivalent problemwith finite constraints.

We transform the SIP problem into an equivalent problem by considering theworst case in the constraints of problem P2. With Cauchy-Schwartz inequality,we have

maxFi∈Fi

∑j6=i

(Fij − Fij

)pj

= εi

√∑j6=i

p2j , (61)

maxh∈H

{∑i

(hi − hi

)pi

}= ε0

√∑i

p2i . (62)

Thus, under the worst case, the problem P2 is transformed as a SOCP problemwith the standard form as follows

12


0 50 1008

9

10

11

Iteration

SIN

R

CR1CR2CR3

(a)

0 50 1008

9

10

11

Iteration

SIN

R

CR1CR2CR3

(b)

Figure 3: Performance comparisons: (a) SINR under robust algorithm; (b)SINR under non-robust algorithms.

Equivalent robust problem (P3) :

min 1Tp

s.t. ε0‖p‖2 ≤ −hTp+ I (63)

εi

∥∥∥(I− eieTi

)1/2p∥∥∥2≤(1

γiei − FTi

)Tp−

Ni

Gii, ∀i,

where ei represents the ith standard basic vector, and the elements in p shouldsatisfy 0 ≤ pi ≤ Pmax, for all i.

The distributed algorithm is then developed to solve the SOCP problem ina distributed way. In addition, an asynchronous iterative algorithm is furtherproposed along with a proof of the convergence in [6]. Figure 3 gives an sim-ulation example. Assume the SINR requirement is 10dB. It can be found thatwhen there are channel uncertainties, under robust algorithm the SINR require-ment is guaranteed while the outage events happen frequently under non-robustalgorithms.

References

[1] S. A. Vorobyov, A. B. Gershman and Z. Q. Luo,“Robust adaptive beam-forming using worst-case performance optimization: A solution to the signalmismatch problem,” IEEE Trans. on Signal Processing, vol. 51, no.2, pp.313-324, 2003.

[2] S. J. Kim, A. Magnani, A. Mutapcic, S. Boyd and Z. Q. Luo, “Robustbeamforming via worst-case SINR maximization,” IEEE Trans. on SignalProcessing, vol. 56, no.4, pp. 1539-1547, 2008.

[3] R. G. Lorenz and S. Boyd, “Robust minimum variance beamforming,” IEEETrans. on Signal Processing, vol. 53, no.5, pp. 1684-1696, 2005.

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[4] A. B. Gershman, N. D. Sidiropoulos, S. Shahbazpanahi, M. Bengtsson andB. Ottersten, “Convex optimization-based beamforming,” IEEE Signal Pro-cessing Magazine, vol. 27, no. 3, pp. 62-74, 2010.

[5] S. A. Vorobyov, H. Chen and A. B. Gershman, “On the relationship betweenrobust minimum variance beamformers with probabilistic and worst-casedistortionless response constraints,” IEEE Trans. on Signal Processing, vol.56, no.11, pp. 5719-5724, 2008.

[6] S. Sun, W. Ni and Y. Zhu, “Robust power control in cognitive radio network-s: A distributed way,” in Proc. IEEE International Conference on Commu-nications (ICC), Kyoto, Japan, June, 2011.

[7] A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski, Robust Optimization. Prince-ton University Press, 2009.

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