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Transmit Optimization with Improper Gaussian Signalingin Multiuser Interference Channels

Rui Zhang

National University of Singapore

MIIS 2013 at Xi’an, China

July 3, 2013

Rui Zhang (NUS) Improper Gaussian Signaling July 3, 2013 1 / 39

Outline

1 System Model

2 Related Work

3 Preliminaries

4 Achievable Rate with Improper Gaussian Signaling

5 Widely Linear Precoding

6 Transmit Optimization in SISO-IC

7 Transmit Optimization in MISO-IC

8 Conclusion

9 Reference


System Model

MIMO Interference Channel (MIMO-IC)

K -user MIMO-IC. Applications include, e.g.,

Coordinated MIMO downlink transmission by multiple BSs (Multi-cell)Multiuser MIMO downlink linear precoding (Single-cell)

yk = Hkkxk +∑j 6=k

Hkjxj + nk , k = 1, · · · ,K .

yk ∈ CN×1: received signal vector at RX k.

xk ∈ CM×1: transmitted signal vector byTX k.

Hkk ∈ CN×M : direct channel for the kthuser TX and RX pair.

Hkj ∈ CN×M : cross channel from TX j toRX k, k 6= j .

nk ∈ CN×1: additive white Gaussian noise

(AWGN) at RX k. nk ∼ CN (0, σ2IN ).


Related Work

Related Work (1/3)

Information-theoretical

Capacity region unknown in generalStrong interference: interference decoding [Carleial75] [Sato81]

[ShangPoor12]

Inner bound: Han-Kobayashi rate splitting [HanKobayashi81], capacitywithin 1 bit [EtkinTseWang08]Degrees of freedom (DoF): interference alignment [CadambeJafar08]

Interference treated as noise

Sum capacity optimal for noisy interference channel[ShangKramerChen09]

Transmit beamforming with SINR constraint: uplink-downlink duality[DahroujYu10]

Minimum mean square error (MMSE) [ShenLiTaoWang10]

(Weighted) sum-rate maximization (WSRMax)Achievable rate region


Related Work

Related Work (2/3)

(Weighted) sum-rate maximization (WSRMax)

NP-hard in general [LiuDaiLuo11]

Optimal binary power allocation for two-user SISO-IC[GjendemsjøGesbertØienKiani08]

Gradient descent for precoding or transmit covariance matrices forMIMO-IC [YeBlum03][SungLeeParkLee10]

Iterative weighted-MMSE approach [ShiRazaviyaynLuoHe11]

[RazaviyaynSanjabiLuo12]

Interference pricing [HuangBerryHonig06], virtual SINR framework[ZakhourGesbert09]

Global optimal via monotonic optimization [QianZhangHuang09]

[JorswieckLarsson10] [LiuZhangChua12]

[BjornsonZhengBengtssonOttersten12]


Related Work

Related Work (3/3)

Achievable rate region

SISO-IC [CharafeddineSezginPaulraj07]MISO-IC

More transmit antennas than number of users (M > K)[JorswieckLarssonDanev08]WSRMax approach [ShangChenPoor11]Rate-profile approach, interference-temperature approach [ZhangCui10]Optimality of transmit beamforming (rank-1 transmit covariancematrix) [ZhangCui10] [ShangChenPoor11] [MochaourabJorswi11]

MIMO-IC [BjornsonBengtssonOttersten12] [CaoJorswieckShi] [ParkSung]


Related Work

Achievable Rate: Prior Result

Achievable rate expression with interference treated as noise:

Rk = log

∣∣σ2I +∑K

j=1 HkjCxj HHkj

∣∣∣∣σ2I +∑

j 6=k HkjCxj HHkj

∣∣ , k = 1, · · · ,K

xk ∼ CN (0,Cxk ),∀k : Circularly Symmetric Complex Gaussian(CSCG)

Cxk : covariance matrix, i.e., Cxk = E(xkxHk )

MISO-IC: Rk = log(

1 +hkkCxk

hHkk

σ2+∑

j 6=k hkjCxj hHkj

)Beamforming (Cxk = tktHk ): Rk = log

(1 + |hkk tk |2

σ2+∑

j 6=k |hkj tj |2

)SISO-IC: Rk = log

(1 +

|hkk |2Cxkσ2+

∑j 6=k |hkj |2Cxj

). Cxk : power of user k


Related Work

Achievable Rate: Revisit

Rk = log

∣∣σ2I +∑K

j=1 HkjCxj HHkj

∣∣∣∣σ2I +∑

j 6=k HkjCxj HHkj

∣∣ (1)

Question: Is (1) the best achievable rate with single-user decoding?

Answer: No

One important assumption of (1): xk ∼ CN (0,Cxk ), CircularlySymmetric Complex Gaussian (CSCG), or proper Gaussian

Y. Zeng, C. M. Yetis, E. Gunawan, Y. L. Guan, and R. Zhang, “Transmit optimization with improper Gaussian signaling forinterference channels,” IEEE Trans. Signal Process., vol. 61, no. 11, pp. 2899-2913, Jun. 2013.


Preliminaries

Proper and Improper Random Vectors

Zero-mean complex-valued random vector (RV): z = u + jv

Covariance matrix: Cz , E(zzH)

Pseudo-covariance matrix: Cz , E(zzT )

In general, both Cz and Cz are required for complete second-ordercharacterization [1]. Let Rxy = E(xyT ), then

Ruu =1

2<{Cz + Cz}, Rvv =

1

2<{Cz − Cz},

Rvu =1

2={Cz + Cz}, Ruv = −1

2={Cz − Cz}.

Definition

[1]: Proper: A complex RV z is called proper if Cz = 0; otherwise, it iscalled improper.

Proper =⇒ Ruu = Rvv, Ruv = −RTuv


Preliminaries

Proper Versus Circularly Symmetric

Definition

[2]: Circularly Symmetric: A complex RV z is circularly symmetric if zand z′ = eαz have the same distribution for any real value α.

For circularly symmetric RV z, we have

Cz = Cz′ = E(z′z′T ) = e2αCz, ∀α,=⇒ Cz = 0.

So circularity =⇒ properness

But properness 6=⇒ circularity

For Zero-Mean Gaussian: Circularity ⇐⇒ Properness

xk ∼ CN (0,Cxk ) (CSCG) ⇐⇒ xk zero-mean proper Gaussian

(Cxk = 0)


Preliminaries

Entropy of Complex Gaussian RV

For an arbitrary complex RV z, define the augmented covariancematrix

Cz , E([

zz∗

][zz∗

]H)=

[Cz Cz

C∗z C∗z

]The entropy of complex Gaussian RV z ∈ Cn is [2]

h(z) =1

2log((πe)2n|Cz|

)For proper Gaussian RV (Cz = 0), entropy reduces to

h(z) = log((πe)n|Cz|

)Rui Zhang (NUS) Improper Gaussian Signaling July 3, 2013 11 / 39

Preliminaries

Why Proper Gaussian Signaling?

Why is proper complex Gaussian (or CSCG) signaling usually assumedin the literature?

Noise is modeled as proper Gaussian, n ∼ CN (0,Cn)Maximum-entropy theorem [1], z ∈ Cn, given E(zzH) = Cz,

h(z) ≤ log [(πe)n|Cz|] , (= iff z ∼ CN (0,Cz))

Hence CSCG signaling is optimal for Gaussian pt-to-pt [3], MAC [4]and BC [5] (nonlinear decoding/precoding)


Achievable Rate with Improper Gaussian Signaling

Improper Gaussian Signaling for IC

For ICs, improper complex Gaussian signaling achieves

higher DoF for 3-user time-invariant SISO-IC [6]higher rate for 2-user SISO-IC [7, 8]

Existing approach:

complex-valued SISO-IC −→ real-valued MIMO-IC

Our approach: complex-valued covariance and pseudo-covariancematrices optimization

More insightsGuaranteed performance gain over proper Gaussian signaling



Achievable Rate: Improper Gaussian Signaling

yk =Hkkxk +∑j 6=k

Hkjxj + nk︸︷︷︸sk

, E(xkxHk ) = Cxk , E(xkxTk ) = Cxk , ∀k.

Rk = log

∣∣σ2I +∑K

j=1 HkjCxj HHkj

∣∣∣∣σ2I +∑

j 6=k HkjCxj HHkj

∣∣︸︷︷︸,Rk,proper({Cxj })

+1

2log

∣∣I− C−1yk Cyk C−Tyk CHyk

∣∣∣∣I− C−1sk Csk C−Tsk CHsk

∣∣

Cyk = E(ykyHk ) =

K∑j=1

HkjCxj HHkj + σ2I, Cyk = E(ykyT

k ) =K∑j=1

Hkj Cxj HTkj

Csk = E(sksHk ) =∑j 6=k

HkjCxj HHkj + σ2I, Csk = E(sksTk ) =

∑j 6=k

Hkj Cxj HTkj



Achievable Rate: Improper Gaussian Signaling

Rk = log

∣∣σ2I +∑K

j=1 HkjCxj HHkj

∣∣∣∣σ2I +∑

j 6=k HkjCxj HHkj

∣∣︸︷︷︸,Rk,proper({Cxj })

+1

2log

∣∣I− C−1yk Cyk C−Tyk CHyk

∣∣∣∣I− C−1sk Csk C−Tsk CHsk

∣∣

Additional term (in red) not present in proper Gaussian signaling

Special case: Cxk = 0, ∀k , i.e., proper Gaussian signaling

Rate improvement over proper Gaussian signaling by choosing {Cxj}to make the additional term positive



Achievable Rate Region: Improper Gaussian Signaling

Cxk and Cxk is a valid pair of covariance and pseudo-covariancematrices if and only if (iff) [2]

Cxk ,

[Cxk Cxk

C∗xk C∗xk

]� 0

Conditions that Cxk � 0 and Cxk symmetric are implied by Cxk � 0

Power constraint: Tr{Cxk} ≤ Pk , ∀kAchievable rate region:

R ,⋃

Tr{Cxk}≤Pk ,

Cxk�0,∀k

{(r1, · · · , rK ) : 0 ≤ rk ≤ Rk , ∀k

}



Problem Formulation: Rate-Profile Approach toCharacterize Rate-Region Pareto Boundary

Given a rate profile α= (α1 · · ·αK ) � 0,∑K

k=1 αk = 1, thecorresponding Pareto-optimal rate tuple can be found by solving

max .{Cxk

},{Cxk},R

R

s.t. Rk ≥ αkR, ∀k,Tr(Cxk ) ≤ Pk , ∀k ,[

Cxk Cxk

C∗xk C∗xk

]� 0, ∀k,

R?· α is on Pareto boundary of rate region [9]

Circumvent non-convexity of traditional weighted sum-ratemaximization (WSRMax)


Widely Linear Precoding


Conventional proper Gaussian signaling: linear precoding withoptimized covariance matrix Cxk

Cxk = C12xk (C

12xk )H , via Eigenvalue Decomposition (EVD):

Cxk = UDUH =⇒ C12xk = UD

12

Improper Gaussian signaling: with optimized Cxk and Cxk , how toobtain zk such that




Linear precoding:

zk = Ukdk =⇒ E(zkzTk ) = Uk Cdk UTk = 0

So conventional linear precoding is not sufficient

With augmented covariance matrix, we need to have

Czk= E

([zkz∗k

][zkz∗k

]H)=

[Cxk Cxk

C∗xk C∗xk

]= Cxk

(2)

(2) is satisfied with mapping[zkz∗k

]= C

12xk

[dk

d∗k

]Rui Zhang (NUS) Improper Gaussian Signaling July 3, 2013 19 / 39



[zkz∗k

]= C

12xk

[dk

d∗k

](3)

Finding C12xk via conventional EVD cannot satisfy (3) in general due to

the complex-conjugate relation between the top and bottom blocks.

Instead, need to find C12xk with the following structure:

C12xk =

[B1 B2

B∗2 B∗1

]So that (3) can be satisfied since it is equivalent to the following twoconsistent equations:

zk = B1dk + B2d∗k ,

z∗k = B∗2dk + B∗1d∗k




Theorem

[10] There exists one form of EVD for the augmented covariance matrixCxk ∈ C2M×2M such that

Cxk = (TV)Λ(TV)H ,

where T , 1√2

[IM iIMIM −iIM

], V ∈ R2M×2M : real-valued orthogonal matrix,

Λ: eigenvalues.

Then we have C12xk = T(VΛ1/2)TH =

[B1 B2

B∗2 B∗1

]The specific structure of C

12xk is satisifed due to the special unitary

matrix T, and the fact that VΛ1/2 is real-valued




[zkz∗k

]= C

12xk

[dk

d∗k

]

C12xk = T(VΛ1/2)TH =

[B1 B2

B∗2 B∗1

]Widely linear precoding: zk = B1dk + B2d∗k


Transmit Optimization in SISO-IC

Case Study I: K -User SISO-IC

K -user SISO-IC:

yk = hkkxk +∑j 6=k

hkjxj + nk

Cxk = E(xkx∗k ), Cxk = E(xkxk)

Rate expression reduces to

Rk =1

2log

C 2yk− |Cyk |2

C 2sk− |Csk |2

= log(

1 +|hkk |2Cxk

σ2 +∑

j 6=k |hkj |2Cxj

)︸︷︷︸

,Rproperk ({Cxj

})

+1

2log

1− C−2yk|Cyk |2

1− C−2sk |Csk |2.



Pareto Boundary for SISO-IC with Improper GaussianSignaling

The Pareto boundary problem for SISO-IC reduces to

max{Cxk

},{Cxk},R

R

s.t. Rk ≥ αkR, ∀k0 ≤ Cxk ≤ Pk , ∀k

|Cxk |2 ≤ C 2

xk, ∀k

Rk = 12 log

C2yk−|Cyk

|2

C2sk−|Csk

|2

⇓

max{Cxk

},{Cxk}

mink=1,··· ,K

1

2αklog

C 2yk− |Cyk |2

C 2sk− |Csk |2

s.t. 0 ≤ Cxk ≤ Pk , ∀k

|Cxk |2 ≤ C 2

xk, ∀k .

Approximatesolution bysemidefiniterelaxation (SDR)



SDR: the Main Concept

Consider a general quadratically constrained quadratic program(QCQP)

minx∈Cn

xHCx

s.t. xHAix ≤ bi , i = 1, · · · ,m,

where C, Ai ∈ Hn, but not necessarily positive semidefinite.

With the identity xHCx = Tr(CxxH), xHAix = Tr(AixxH), theQCQP problem can be written as [11]

minX∈Hn

Tr(CX)

s.t. X � 0, Tr(AiX) ≤ bi , i = 1, · · · ,m,rank(X) = 1.



SDR: the Main Concept

Drop the rank-one constraint to obtain a relaxed problem

(SDR): minX∈Hn

Tr(CX)

s.t. X � 0, Tr(AiX) ≤ bi , i = 1, · · · ,m,

which is convex.

If an SDR solution X? is of rank one, i.e., X? = x?x?H , then x? is thesolution to the original QCQP.

Otherwise, an approximate solution x to QCQP can be generatedfrom X? with Gaussian randomization [11].



Joint Covariance and Pseudo-Covariance Optimization forSISO-IC

With some manipulations, the problem for SISO-IC can be formulatedas

maxc,q,t

mink

1

2αklog

(σ2t + aTk c)2 − qHFkq

(σ2t + bTk c)2 − qHGkq

s.t. cTEkc ≤ P2k , eT

k ct ≥ 0, ∀k

qHEkq ≤ cTEkc, ∀k, t2 = 1.

c ∈ RK : covariances, q ∈ CK : pseudo-covariances, t: slack variable

The SDR problem:

maxC∈SK+1,Q∈HK

mink

1

2αklog

Tr(AkC)− Tr(FkQ)

Tr(BkC)− Tr(GkQ)

s.t. Tr(EkC) ≤ P2k , Tr(KkC) ≥ 0, ∀k

Tr(EkQ) ≤ Tr(EkC), ∀k, C11 = 1

C � 0, Q � 0.




With some manipulations, the problem for SISO-IC can be formulatedas

maxc,q,t

mink

1

2αklog

(σ2t + aTk c)2 − qHFkq

(σ2t + bTk c)2 − qHGkq

s.t. cTEkc ≤ P2k , eT

k ct ≥ 0, ∀k

qHEkq ≤ cTEkc, ∀k, t2 = 1.

c ∈ RK : covariances, q ∈ CK : pseudo-covariances, t: slack variableThe SDR problem:

maxC∈SK+1,Q∈HK

mink

1

2αklog

Tr(AkC)− Tr(FkQ)

Tr(BkC)− Tr(GkQ)

s.t. Tr(EkC) ≤ P2k , Tr(KkC) ≥ 0, ∀k

Tr(EkQ) ≤ Tr(EkC), ∀k, C11 = 1

C � 0, Q � 0.




The SDR problem is a quasi-convex, and hence can be optimallysolved with bisection method.

An approximate solution to the original covariance andpseudo-covariance optimization problem for the SISO-IC is thenobtained with Gaussian randomization [11].



Numerical Results

Achievable rate region for two-user SISO-IC, SNR = 10 dB.



Numerical Results

Average max-min rates for two-user SISO-IC.


Transmit Optimization in MISO-IC

Case Study II: K -User MISO-IC

K -user MISO-IC:

yk = hkkxk +∑j 6=k

hkjxj + nk︸︷︷︸sk

E(xkxHk ) = Cxk , E(xkxTk ) = Cxk

Rate expression reduces to

Rk = log

(1 +

hkkCxk hHkk

σ2 +∑

j 6=k hkjCxj hHkj

)︸︷︷︸

,Rproperk ({Cxj

})

+1

2log

1− C−2yk |Cyk |2

1− C−2sk |Csk |2

Y. Zeng, R. Zhang, E. Gunawan, and Y. L. Guan, “Optimized Transmission with Improper Gaussian Signaling inthe K-User MISO Interference Channel,” submitted to IEEE Trans. on Wireless Communications. (available athttp://arxiv.org/abs/1303.2766)



Pareto Boundary of MISO-IC with Improper GaussianSignaling

The Pareto boundary problem for MISO-IC reduces to

max{Cxk},{Cxk

},RR

s.t. Rk ≥ αkR, ∀kTr{Cxk} ≤ Pk , ∀k[

Cxk Cxk

C∗xk C∗xk

]� 0, ∀k

Rk = log

(1 +

hkkCxk hHkk

σ2 +∑

j 6=k hkjCxj hHkj

)︸︷︷︸

,Rproperk

({Cxj})

+1

2log

1− C−2yk |Cyk |

2

1− C−2sk |Csk |2

.

Separate covariance and pseudo-covariance optimization1 Covariance optimization: with Cxk = 0, ∀k, optimize covariance

matrices {C?xk} with proper Gaussian signaling.2 Pseudo-covariance optimization: fixing {C?xk}, optimize

pseudo-covariance matrices {Cxk} to improve rate over (optimal)proper Gaussian signaling.



Pareto Boundary of MISO-IC with Improper GaussianSignaling

The Pareto boundary problem for MISO-IC reduces to

max{Cxk},{Cxk

},RR

s.t. Rk ≥ αkR, ∀kTr{Cxk} ≤ Pk , ∀k[

Cxk Cxk

C∗xk C∗xk

]� 0, ∀k

Rk = log

(1 +

hkkCxk hHkk

σ2 +∑

j 6=k hkjCxj hHkj

)︸︷︷︸

,Rproperk

({Cxj})

+1

2log

1− C−2yk |Cyk |

2

1− C−2sk |Csk |2

.

Separate covariance and pseudo-covariance optimization1 Covariance optimization: with Cxk = 0, ∀k , optimize covariance

matrices {C?xk} with proper Gaussian signaling.2 Pseudo-covariance optimization: fixing {C?xk}, optimize

pseudo-covariance matrices {Cxk} to improve rate over (optimal)proper Gaussian signaling.



Covariance Optimization

With Cxk = 0, ∀k , the problem reduces to

maxr ,{Cxk

}r

s.t. log

(1 +

hkkCxk hHkk

σ2 +∑

j 6=k hkjCxj hHkj

)≥ αk r , ∀k,

Tr{Cxk} ≤ Pk , Cxk � 0, ∀k .

Since beamforming is optimal, the problem can be solved via thefollowing SOCP feasibility problem together with bisection:

Find {tk}

s.t. σ2 +K∑j=1

|hkjtj |2 ≤(

1 +1

eαk r − 1

)(hkktk)2, ∀k,

={hkktk} = 0, ‖tk‖2 ≤ Pk , ∀k .



Pseudo-Covariance Optimization

With {r?,C?xk = tktHk } the solution to the covariance optimization problem, thepseudo-covariance optimization problem is

maxR,{Cxk

}R

s.t. αk r? +

1

2log

1− C−2yk |Cyk |2

1− C−2sk |Csk |2≥ αkR, ∀k[

tktHk Cxk

C∗xk (tktHk )∗

]� 0, ∀k

Lemma

The positive semidefinite constraint in the above problem is satisfied if and only if

Cxk = Zk tk tTk , k = 1, · · · ,K ,

where Zk is a complex scalar variable with constraint |Zk | ≤ ‖tk‖2, and tk = tk/‖tk‖ isthe normalized (proper) beamforming vector.



Pseudo-Covariance Optimization

Cxk = Zk tk tTk , k = 1, · · · ,K

Rank-1 pseudo-covariance matrices are optimal.

Cxk parameterized by a single scalar variable Zk only.

Number of variables reduced: KM2 → K .

The problem can be reformulated as

maxz∈CK

mink=1,··· ,K

1

2αklog

1− zHMkz

1− zHWkz

s.t. |eHk z|2 ≤∥∥tk∥∥4, ∀k ,

where z ,[Z1 · · · ZK

]T.



SDR for Pseudo-Covariance Optimization

The SDR of the pseduo-covariance optimization problem:

(SDR): maxZ�0

mink=1,··· ,K

1

2αklog

1− Tr(MkZ)

1− Tr(WkZ)

s.t. Tr(EkZ) ≤∥∥tk∥∥4, ∀k .

The SDR problem is quasi-convex, which can be solved with bisection

An approximate solution to the original problem is obtained withGaussian randomization

Lemma

[12] For two-user MISO-IC (K = 2), SDR yields the optimalpseudo-covariance matrices.



Numerical Results

Achievable rate region for two-user MISO-IC, M = 2, SNR = 10 dB.



Numerical Results

Average max-min rate for three-user MISO-IC, M = 2.


Conclusion

Conclusion and Future Work

Conclusion

A new rate expression for K -user MIMO-IC applicable to the moregeneral improper complex Gaussian signalingWidely linear precoding to satisfy the optimized covariance andpseudo-covariance matricesJoint covariance and pseudo-covariance optimization for SISO-ICSeparate covariance and pseudo-covariance optimization for MISO-ICRank-1 optimality of the pseudo-covariance matrices for MISO-ICGuaranteed rate gain over proper complex Gaussian signaling

Future work

More general K -user MIMO-ICJoint covariance and pseudo-covariance optimization combined withtime/frequency symbol extension


Reference

[1] F. D. Neeser and J. L. Massey, “Proper complex random processeswith applications to information theory,” IEEE Trans. Inf. Theory,vol. 39, no. 4, pp. 1293 – 1302, Jul. 1993.

[2] P. J. Schreier and L. L. Scharf, Statistical Signal Processing ofComplex-Valued Data: The Theory of Improper and NoncircularSignals. Cambridge Univ. Press, 2010.

[3] E. Telatar, “Capacity of multi-antenna Gaussian channels,” EuropeanTrans. Tel., vol. 10, no. 6, pp. 585–596, Nov. 1999.

[4] D. N. C. Tse and S. V. Hanly, “Multiaccess fading channels. Part I:Polymatroid structure, optimal resource allocation and throughputcapacities,” IEEE Trans. Inf. Theory, vol. 44, no. 7, pp. 2796–2815,Nov. 1998.

[5] G. Caire and S. Shamai, “On the achievable throughput of amultiantenna gaussian broadcast channel,” IEEE Trans. Inf. Theory,vol. 49, no. 7, pp. 1691–1706, Jul. 2003.


Reference

[6] V. R. Cadambe, S. A. Jafar, and C. Wang, “Interference alignmentwith asymmetric complex signaling - settling theHost-Madsen-Nosratinia conjecture,” IEEE Trans. Inf. Theory, pp.4552 – 4565, Sep. 2010.

[7] Z. K. M. Ho and E. Jorswieck, “Improper Gaussian signaling on thetwo-user SISO interference channel,” IEEE Trans. Wireless Commun.,vol. 11, no. 9, pp. 3194 – 3203, Sep. 2012.

[8] S. H. Park, H. Park, and I. Lee, “Coordinated SINR balancingtechniques for multi-cell downlink transmission,” in Proc. VTC2010-fall, 2010.

[9] R. Zhang and S. Cui, “Cooperative interference management withMISO beamforming,” IEEE Trans. Signal Process., vol. 58, no. 10,pp. 5450–5458, Oct. 2010.

[10] P. J. Schreier and L. L. Scharf, “Second-order analysis of impropercomplex random vectors and processes,” IEEE Trans. Signal Process.,vol. 51, no. 3, pp. 714–725, Mar. 2003.


Reference

[11] Z.-Q. Luo, W. K. Ma, A. M. So, Y. Ye, and S. Zhang, “Semidefiniterelaxation of quadratic optimization problems,” IEEE SignalProcessing Mag, vol. 27, no. 3, pp. 20–34, May 2010.

[12] Y. Huang and D. P. Palomar, “Rank-constrained separablesemidefinite programming with applications to optimal beamforming,”IEEE Trans. Signal Process., vol. 58, no. 2, pp. 664–678, Feb. 2010.


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