Robust Two-Way Cognitive Relaying: Precoder Designs ... - IEEE Xplore

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Robust Two-Way Cognitive Relaying: Precoder Designs under Interference Constraints and Imperfect CSI P. Ubaidulla, Member, IEEE, and Sonia A¨ıssa, Senior Member, IEEE

Abstract—We present various robust precoder designs for twoway relaying in a cognitive radio network, where a pair of cognitive (or secondary) transceiver nodes communicate with each other assisted by a set of cognitive two-way relays. The secondary nodes share the spectrum with a licensed primary user (PU) node while keeping the interference to the PU below a specified threshold. The PU node and the cognitive transceivers employ single transmit/receive antennas whereas the secondary relay nodes employ multiple transmit/receive antennas. The proposed precoder designs ensure robust performance in the presence of errors in the channel state information (CSI). Such robust designs are of significant interest since in practice it is very difficult to obtain perfect CSI. We consider CSI errors with two different types of characterization and corresponding robust designs. First, we consider robust relay precoder designs that are applicable when CSI errors have known first and second moments. Next, we consider robust designs that are applicable when the CSI error can be characterized in terms of spherical uncertainty region. We show that the proposed designs can be reformulated as convex optimization problems that can be solved efficiently. Through numerical simulations and comparisons we illustrate the performance of the proposed designs. Index Terms—Cooperative networks, cognitive radio networks, robust precoder design, two-way relaying.

I. I NTRODUCTION

C

OGNITIVE radio (CR), as a promising technology to enhance the spectrum utilization, has recently been receiving significant attention from the research community [1]–[4]. Different schemes for spectrum sharing among the cognitive or secondary users (SUs) and primary users (PUs) have been proposed. Notable among these are schemes which involve the concept of cooperation to increase the transmission opportunities among SUs and enhance the performance of the SUs’ communication process. Indeed, the SUs in a cognitive radio network often operating with low transmit power in order to limit the interference to the PUs, collaborative and relayassisted transmission techniques are not only beneficial but can also be necessary for improving the rate and range of communication among the SUs while keeping the interference to the PUs minimal [5]–[8]. Relay-assisted communication Manuscript received January 14, 2013; revised July 4 and November 7, 2013; accepted January 13, 2014. The associate editor coordinating the review of this paper and approving it for publication was H. Yousefi’zadeh. P. Ubaidulla is with the Signal Processing and Communications Research Center (SPCRC), International Institute of Information Technology (IIIT), Hyderabad, India (e-mail: [email protected]). S. A¨ıssa is with the Institut National de la Recherche Scientifique (INRS-EMT), University of Quebec, Montreal, QC, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2014.031714.130082

techniques can provide benefits like improvement in link quality and transmission reliability, as well as increase in coverage [9]–[12]. Various relaying schemes have been proposed in the literature. Among them, regenerative and non-regenerative schemes have been studied widely [9], [10], [13], [14]. On the other hand, two-way relaying schemes that provide better spectral efficiency compared to the one-way relaying schemes described above have started to attract significant attention [15]. Various protocols have been proposed for two-way relay networks wherein the information exchange between two transceiver nodes is aided by a relay. Three-phase protocols wherein information exchange takes place in three phases or time-slots have been proposed in [16]–[18]. In this protocol, the transceiver nodes transmit sequentially their messages in first and second phases to a relay node. In the third phase, the relay node broadcasts a processed version of the signals received from the transceiver nodes. Two-phase protocols have been proposed in [15], [18], [19]. In two-phase two-way relaying, the information exchange between the transceiver pair takes place in two time slots. In the first slot, the transceiver nodes simultaneously transmit the messages to the relay(s). In the second time slot, the relay(s) process the combined message received in the previous time slot and transmit to the transceivers. Different techniques for processing the combined signal at the relays, also known as relay precoding or beamforming, have been reported [20]–[22]. Optimal relay precoding for the sum-rate maximization is studied in [20]. In [21], the authors propose a locally optimal multiple-input multiple-output (MIMO) relay precoder based on sum-rate maximization. Relay precoder designs for a network with single antenna nodes and perfect or imperfect CSI have been proposed in [22]. In this paper, we propose relay precoder designs for a set of cognitive (secondary) two-way relay nodes that assist a pair of secondary transceivers communicating with each other while sharing the spectrum with a PU. The SU network employs two-phase two-way relaying protocol. The proposed precoder designs are based on three criteria: i) minimization of the sum mean-square error (MSE) with a constraint on the total relay transmit power, ii) balancing of MSEs at the secondary transceivers with a constraint on the total relay transmit power, and iii) minimization of total relay transmit power with a constraint on the MSEs at the secondary transceiver nodes. Different optimality criteria adopted in the proposed designs result in precoders with different properties. For instance, the

c 2014 IEEE 1536-1276/14$31.00 

UBAIDULLA et al.: ROBUST TWO-WAY COGNITIVE RELAYING: PRECODER DESIGNS UNDER INTERFERENCE CONSTRAINTS AND IMPERFECT CSI

minimum sum MSE precoder design has been studied in the context of multiuser downlink and it is related to the sumrate maximization [23]. On the other hand, MSE-balancing precoder design leads to a fair mini-max distribution of the MSEs at the transceiver. In certain scenarios, it is required to maintain a certain quality-of-service (QoS) at the transceivers during the communication process. The MSE-constrained precoder design aims at ensuring the specified QoS, i.e., MSE in the present case, while minimizing the total transmit power so as to optimize the power budget usage. Herein, the proposed designs ensure that the interference to the PU receiver resulting from the transmission by the relay nodes are below a specified threshold. When the available CSI is imperfect, as is usually the case in practice, the precoder designs that assume perfect CSI result in performance degradation. Moreover, in a cognitive radio network, such non-robust designs can lead to interference to the PUs in excess of the specified limit [24]. Hence, to ensure acceptable performance in practical operating environments, it is necessary to develop precoders that are robust to imperfections in the CSI. We consider CSI errors with two different types of characterization and propose corresponding robust designs. First, we consider robust relay precoder designs that are applicable when CSI errors have known first and second moments. Then, we consider robust designs that are applicable when the CSI error can be characterized in terms of a spherical uncertainty region. The robust precoder designs proposed in this paper take into account the known properties of errors in the CSI and ensure that the interference to the PU does not exceed the limit even in the presence of imperfect CSI. We show that the relay precoder designs can be reformulated as convex optimization problems that can be solved efficiently. Specifically, for the minimum sum-MSE precoder design, the problem can be tackled by iteratively solving a pair of sub-problems in the case of CSI error with known moments. The first sub-problem is formulated as a convex optimization problem and it can be solved efficiently using interior-point methods [25]. The second sub-problem can be solved analytically. The MSEconstrained design, both for the cases of perfect and imperfect CSI, is reformulated as a convex optimization problem consisting of second-order cone (SOC) constraints. The MSEbalancing design, with perfect and imperfect CSI, is solved via an iterative algorithm involving a bisection search and the solution to a convex feasibility problem. Next, we consider an MSE-constrained robust design applicable when the CSI errors belong to an uncertainty region of known size [26] and propose tractable iterative solutions to this design problem. For this type of CSI error, it is also possible to develop robust precoder designs based on minimum sum-MSE and MSEbalancing criteria following similar techniques as in the case of MSE-constrained design. But, we do not address these problems in this paper due to their excessive computational complexity. In detailing the contributions highlighted above, the rest of the paper is organized as follows. The system model is detailed in Section II. The proposed robust designs are presented in Sections III and IV. Section V provides simulation results and comparisons. Conclusions are presented in Section VI. Notations: Vectors are denoted by boldface lowercase let-

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ters, and matrices are denoted by boldface uppercase letters. [·]T and [·]H denote transpose and Hermitian operations, respectively. [·]∗ denotes complex conjugation. [A]ij denotes the element on the ith row and jth column of matrix A. vec(·) operator stacks the columns of the input matrix into one column-vector. E{·} denotes the expectation operator. II. S YSTEM M ODEL We consider a cognitive radio network with a pair of secondary transceiver nodes, T1 and T2 , communicating with each other aided by M cognitive relay nodes that share the spectrum with a PU. Each node of the secondary transceiver pair and the PU receiver employ a single transmit/receive antenna. The ith SU relay node is equipped with Ni antennas. The SUs employ the two-way relaying scheme. As such, the communication between the two transceiver nodes takes place in two time slots. In the first slot (or phase), each of the transceiver nodes transmits the signal meant for the other, and the relays receive the signals. Let s1 ∈ C and s2 ∈ C be the symbols transmitted by the transceiver nodes T1 and T2 , respectively. We assume that E{|s1 |2 } = E{|s2 |2 } = 1. Then, the signal received by the ith relay, denoted by ri , can be written as √ √ (1) ri = p1 gi s1 + p2 hi s2 + μi , 1 ≤ i ≤ M, where p1 and p2 are the transmit powers of T1 and T2 , respectively, gi denotes the vector of channel gains from T1 to the ith relay, hi denotes the vector of channel gains from T2 to the ith relay node, and μi is an independent and identically distributed (i.i.d.) complex Gaussian random vector 2 with zero mean and E{μi μH i } = Σi = σμi I representing the additive noise at the ith relay. In the second phase, each of the relay nodes transmits the signal it received in the first phase after multiplying it by a precoding matrix, and each of the transceiver nodes receives the signals from the relays. The signal received by T1 can be written as z1 =

M  i=1

√ √ gi ( p1 Wi gi s1 + p2 Wi hi s2 + Wi μi ) + ν1 , (2)

where gi is a row vector of channel gains from the ith relay to T1 , Wi is the precoding matrix at the ith relay, and ν1 is a zero-mean complex Gaussian random variable of variance σν21 modeling the additive noise at node T1 . Similarly, the signal received by T2 can be written as z2 =

M 

√ √ hi ( p1 Wi gi s1 + p2 Wi hi s2 + Wi μi ) + ν2 , (3)

i=1

where hi is a row vector of channel gains from the ith relay to T2 and ν2 is a zero-mean complex Gaussian random variable of variance σν22 representing the additive noise at node T2 . In two-way relaying, once the second transmission phase is accomplished, the signal transmitted by the transceiver nodes reappear as self-interference. The first term in (2) and the second term in (3) represent the self-interference. As T1 and T2 know the CSI and their transmitted signal, the self-interference can be perfectly canceled before further processing the received signals. We assume that the global CSI is available with all the nodes [27]. The channel gain from

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ith relay to Tj can be estimated by Tj and the gain of the reverse channel can be estimated by the ith relay. The nodes exchange the acquired CSI with each other. After canceling the self-interference, T1 estimates the symbol meant for it as M

M

i=1

i=1

 √  s2 = a1 p2 g i W i hi s 2 + a 1 gi Wi μi + a1 ν1 , (4) where a1 ∈ C is the receive scaling factor at T1 . Similarly, the estimate at T2 is given by M

M

i=1

i=1

 √  hi W i g i s 1 + a 2 hi Wi μi + a2 ν2 , (5) s1 = a2 p1 where a2 ∈ C is the receive scaling factor at T2 . Thus, the MSE at transceiver node T1 can be expressed as M

√  1 =E{| s2 − s2 |2 } = |a1 p2 gi Wi hi − 1|2 i=1

+ |a1 |2

M  i=1

E{|gi Wi μi |2 } + |a1 |2 σν21 ,

(6)

and that at T2 is given by M

√  s1 − s1 |2 } = |a2 p1 hi Wi gi − 1|2 2 =E{| i=1

+ |a2 |2

M  i=1

E{|hi Wi μi |2 } + |a2 |2 σν22 .

(7)

p1 |

M

∗ ∗ ∗ i=1 hi Wi gi ∗ ∗ ∗ 2 H H i=1 hi Wi gi | + hi Wi Σi Wi hi

M

p1

+ σν22

. (9)

During the two-way relaying process explained above, the transmission from the SU nodes causes interference to the PU receiver. In the second phase, the signal transmitted from the relay nodes appear as the interference signal at the PU receiver: u=

M 

√ √ fiH ( p1 Wi gi s1 + p2 Wi hi s2 + Wi μi ) ,

where fi is the vector of channel gains from the ith secondary relay to the PU receiver. The power of this interference signal, I = E{|u|2 }, should not exceed a specified limit θ. We note that the transmission from the transceiver nodes T1 and T2 during the first phase causes interference to the PU. This interference depends on the transmit powers p1 , p2 and on the channel gains from T1 , T2 to the PU. Herein, we assume that the values of p1 and p2 are appropriately selected so as to keep the total interference to the PU from the nodes T1 and T2 below the specified interference threshold θ [4]. Specifically, p1 and p2 should satisfy the following condition: 2

p1 |f 1 | + p2 |f 2 | ≤ θ,

i

i

i

i

i

i

 i are the estimated channel coefficients from i and h where g T1 and T2 respectively to the ith relay, and where αi and βi are the corresponding estimation error vectors. For the channel between the ith relay and the PU receiver, we have fi + ef . (14) fi =  The CSI error vectors are assumed to be Gaussian distributed 2 H 2 [29] with zero mean and E{αi αH i } = σα I, E{αi αi } = σα I, H 2 2 H E{β i β H i } = σβ I, E{β i β i } = σβ I and E{efi efi } = 2 σf I, 1 ≤ i ≤ M . In the presence of CSI errors modeled above, MSE, relay transmit power and PU interference power are random variables. Hence, in order to ensure robust performance, we adopt a stochastic approach to the robust design by minimizing the expected values of those objective and constraint functions that depend on the CSI error. A. Robust Minimum SMSE Relay Precoder Design When the CSI is imperfect, the self-interference at the SUs’ transceivers cannot be completely eliminated and this results in residual self-interference which adds to the MSE. In a twoway relaying scheme, the effect of the CSI error on the average MSE at the SUs’ transceivers can be decomposed into three factors. For instance, the average MSE at transceiver T1 can be written as

(10)

i=1

2

III. ROBUST C OGNITIVE R ELAY P RECODER D ESIGNS UNDER S TOCHASTIC CSI E RROR Now we turn our attention to operation under channel uncertainties and we present the robust designs of the cognitive relay precoder when the CSI is imperfect. The robust precoder designs in this section are based on a stochastic model for the error in CSI. In this model, the actual channel coefficients of the links between nodes T1 and T2 and the relays are represented as i + β , gi + αi , hi = h (12) gi = i  g = g + α , h = h + β , 1 ≤ i ≤ M, (13)

i

The optimal values of the scaling factors a1 and a2 that minimize the corresponding MSEs are given by M p2 i=1 g∗i Wi∗ h∗i , (8) a1 = M 2 p2 | i=1 g∗i Wi∗ h∗i |2 + gi Wi Σi WiH gH + σ ν 1 i a2 =

where f 1 and f 2 are the channel gains from T1 and T2 , respectively, to the PU receiver. We assume that the SU nodes possess an estimate, though not very accurate, of the channel gains from the SU nodes to the PU receiver. Such estimates can be obtained, for example, by periodically sensing the signals from the PU receiver in case the PU node employs time-division multiple access (TDMA) [28].

(11)

1 + 1 + 1 . ¯1 = Eα,β,α {1 } = 

(15)

The first term,  1 , corresponds to the MSE that would result if we neglect the CSI error. The second term, 1 , represents all the contributions to the MSE from the CSI errors except that corresponding to the imperfect cancellation of the selfinterference. Finally, the term 1 represents the contribution to the total MSE resulting from the failure to completely remove the self-interference in the presence of CSI errors. The residual self-interferenceMat node T1 is given by   √  i s1  i Wi g ζ1 = p1 a1 gi + αi )s1 − g ( gi + αi )Wi ( =

√ p 1 a1

i=1 M  i=1

i + αi Wi αi )s1 . (16) ( gi Wi αi + αi Wi g

UBAIDULLA et al.: ROBUST TWO-WAY COGNITIVE RELAYING: PRECODER DESIGNS UNDER INTERFERENCE CONSTRAINTS AND IMPERFECT CSI

Based on the above, the last term in (15) can be computed as 1

2

= E{|ζ1 | }

⎧ 2 ⎫ M ⎨  ⎬   2 i + αi Wi αi ) = |a1 | p1 E  ( gi Wi αi + αi Wi g ⎩  ⎭

(17)

i=1

= |a1 |2 p1

M  i=1

i 2 + σα2 σα2 Wi 2 ). (σα2  gi Wi 2 + σα2 Wi g

On the other hand, the sum of first two terms in (15) is given in (18). Then, combining (17) and (18), we obtain the following expression for ¯ 1 : ¯1

=

√  2 2  2 2 2  2 |a1 p2 A 1 z − 1| + |a1 | B1 z + |a1 | p1 σα Cz  2 + |a1 |2 Σ  1 z2 + |a1 |2 σν2 , +|a1 |2 p2 σα2 Dz (19) 1

where  z = vec(W1 )T  1 = h T ⊗ g A 1 1 1 = bdiag B



T T vec(W2 )T · · · vec(WM ) ,

(20)

 T ⊗ g h M , M

(21)

T ⊗ g h 2 2

1

i (p1 σα2 + p2 σβ2 + σμ2 i ) 2 I ⊗ g

= bdiag C = bdiag D and 1 = bdiag Σ

···



M i=1

,

M iT ⊗ I g ,

(23)

i=1



(22)

M T ⊗ I , h i

where   T T 2 = g  g  ··· g  A  1T ⊗ h  ⊗ h ⊗ h 1 2 M , 2 M M

 1 T 2 = bdiag (p1 σ 2 + p2 σ 2 + σ 2 ) 2 I ⊗ h B , and α β μi i i=1   M 2 2 2 2 2 2 12 Σ2 = bdiag (σβ σμi + p1 σβ σα + p2 σβ σβ ) I . i=1

When the CSI is imperfect, the total relay transmit power, P , and the interference power to the PU node due to relay transmission, I, are also random variables that depend on the CSI error vectors. Hence, we consider the expected values of these variables in the robust designs, namely, we have HC  + p2 D  HD  +Σ P¯ = Eα,β {P } = zH (p1 C + (p1 σα2 + p2 σβ2 )I)z,

(31)

 1 z2 + p2 D  1 z2 + p1 σ 2 Cz  2 Eα,β,ef {I} = p1 C f  2 + B 3 z2 + σ 2 z2 , +p2 σf2 Dz (32) f

 M 3 = bdiag (σμ2 + p1 σα2 + p2 σ 2 ) 12 f T ⊗ I , where B i β 1

 M M i=1

  = bdiag g  = bdiag h  T ⊗I 1 = iT ⊗I C ,D ,C i i=1 i=1

 M

 M  1 = bdiag  bdiag  , and D . fiT ⊗ gi fiT ⊗ hi i=1 i=1 With the definitions developed above, the problem of robust minimum SMSE relay precoder design can be stated as I¯ =

min

¯1 + ¯2

subject to:

P¯ ≤ PU I¯ ≤ θ.

{Wi },a1 ,a2

(24)

i=1

 M  2 2 2 2 2 2 12 (σα σμi + p2 σα σβ + p1 σα σα ) I . i=1

(25)

min

{Wi },a1 ,a2 ,t1 ,t2

subject to: (26)

The residual self-interference at T2 is given by   √  i + β )s2 − h  Wi h  i s2  + β )Wi (h p 2 a2 (h i i i i M

ζ2 = =

√ p 2 a2

i=1 M  i=1

 i + β Wi β )s2 . (27)  Wi β + β Wi h (h i i i i i

⎧ 2 ⎫ M  ⎬ ⎨   Wi β + β Wi h  i + β Wi β ) = |a2 |2 p2 E  (h i i i i i ⎩  ⎭

min

{Wi },ψ1 ,ψ2 ,τ1 ,τ2

subject to:

(28)

i=1

= |a2 |2 p2

M 



 Wi 2 + σβ2 Wi h  i 2 + σβ2 σβ2 Wi 2 . σβ2 h i

i=1

The sum of first two terms in (26) is given in (29). Then, combining (28) and (29), we obtain the following expression for ¯2 : 2 ¯

=

√  2 2  2 2 2  2 |a2 p1 A 2 z − 1| + |a2 | B2 z + |a2 | p1 σβ Cz  2 + |a2 |2 Σ  2 z2 + |a2 |2 σν2 , +|a2 |2 σβ2 p2 Dz 2

(30)

t1 + t2

(34)

¯1 ≤ t1 ¯2 ≤ t2 P¯ ≤ PU I¯ ≤ θ,

where t1 and t2 are auxiliary optimization variables. After a few algebraic manipulations, we can reformulate the problem in (34) as the following optimization problem:

Based on the above, the last term in (26) can be computed as 2 = E{|ζ2 |2 }

(33)

For facilitating further development, we recast this problem in the epigraph form [25] as follows:

Similarly, for node T2 we have ¯2 = Eα,β,β {2 } =  2 + 2 + 2 .

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where M1 = M2 = M3 = M4 =

√ √ √

T  T √p 1 σ β D  T ΣT , 2p1 σα C T   T2 √p2 σα C T  T ΣT , 2p2 σβ D B T 1 √ T p2 D ΣT (p1 σα2 + p2 σβ2 ) 2 I ,

 T1 B  T1 p2 A  T2 p1 A T p1 C

 √

τ12 + τ22 (35)   [(M1 z − iψ1 )T σν1 ] ≤ ψ1 τ1 ,   [(M2 z − iψ2 )T σν2 ] ≤ ψ2 τ2 ,  M3 z ≤ PU , √ M4 z ≤ θ, 

T 1  T1 √p2 D  T1 √p1 σf C  T √p 2 σ f D T B  T3 σ 2 I , p1 C f

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  1 +

1

M

√   i + β ) − 1|2 ( gi + αi )Wi (h |a1 p2 i

=Eα,β



 2

+ |a1 | Eα,β

i=1

√ =|a1 p2

M  i=1

+ σα2 σβ2 |a1 |2 p2

W2i + |a1 |2

i=1

  2 +

2

=Eα,β

i=1

T ⊗ g i )vec(Wi ) − 1|2 + σβ2 |a1 |2 p2 (h i M 

M  i=1

M  i=1

M

 2

+ |a2 | Eα,β

i=1

√ =|a2 p1

 )vec(Wi ) − 1|2 + σ 2 |a2 |2 p1 ( giT ⊗ h i α

i=1 M 

W2i + |a2 |2

i=1

M 

subject to:

The second sub-problem in the proposed design, which essentially computes the receive scaling factors a1 and a2 that minimize the SMSE for a fixed value of {Wi } can be solved analytically. The optimum values of ai , i = 1, 2 satisfies the following condition: 2 ∂ i=1 ¯i ∂¯ i = = 0, 1 ≤ i ≤ 2. (37) ∂a∗i ∂a∗i

√  p1 (A2 z)∗ a2 = ,  2 z2 + φ2 A

+ |a1 |2 σν21 ,

+ α)

i  H WH Wh h i

i=1

(18)

M 

  σμ2 i (h i

+

 β i )Wi WiH (h i M 

H

+ β)

+ |a2 |2 σν22 ,

iH WH W g gi

i=1

+ σβ2 |a2 |2

M 

σμ2 i Wi 2 + |a2 |2 σν22 .

(29)

i=1

τ12 + τ22 (36)   [(M1 z − iψ1 )T σν1 ] ≤ ψ1 τ1 ,   [(M2 z − iψ2 )T σν2 ] ≤ ψ2 τ2 ,  M3 z ≤ PU , √ M4 z ≤ θ.

From the optimality conditions given above, we have √  p2 (A1 z)∗ a1 = ,  1 z2 + φ1 A

M 

H

σμ2 i Wi 2 + |a1 |2 σν21 .

 Wi W H h  H + σ 2 |a2 |2 p1 h i i β i

 Wi W H h H σμ2 i h i i i

i=1

+

αi )Wi WiH ( gi

i=1

i=1 M 

ψ1 = |a11 | , ψ2 = |a12 | , and i is the first column of the identity matrix. The MSE constraints in the problem given above are not jointly convex with respect to {{Wi }, {ψi }, {τi }}, and hence this problem is not a convex optimization problem. However, for a fixed value of {ψi }, the objective and the constraints are convex with respect to the remaining optimization variables and vice versa. This observation motivates the approach of solving the non-convex problem in (35) by iteratively solving a pair of sub-problems. The first sub-problem, obtained by keeping the values {ψi }, can be expressed as the following convex optimization program with second-order cone constraints: min

M 

i=1

+ σβ2 σα2 |a2 |2 p1

{Wi },τ1 ,τ2

 σμ2 i ( gi

i Wi WiH g H g + σα2 |a1 |2 p2 i

i Wi WiH g H σμ2 i g + σα2 |a1 |2 i

√   (hi + β i )Wi ( gi + αi ) − 1|2 |a2 p1 M 

M 

(38)

(39)

1 z2 + p1 σ 2 Cz 2 + p2 σ 2 Dz 2 + Σ 1 z2 + where φ1 = B α α 2 z2 +p1 σ 2 Cz 2 +p2 σ 2 Dz 2 +Σ 2 z2 + σν21 and φ2 = B β β

TABLE I: Algorithm for computation of relay precoding matrices {Wi } and receive filters {ai } Select Nmax (maximum number of iterations), Tth (convergence threshold). 0 Initialize x0 = [w10 w20 · · · wM ψ10 ψ20 ]. 1) n = 0 2) while n ≤ Nmax 3) Compute {Win+1 } using {ψin } in (36) 4) Compute {ai }n+1 and hence {ψi }n+1 using {Win+1 } in (38) and (39) n+1 5) xn+1 = [w1n+1 w2n+1 · · · wM ψ1n+1 ψ2n+1 ]  n+1 n  6) if x − x ≤ Tth then 7) break 8) endif 9) n ← n + 1 10) endwhile

σν22 . A solution to the problem in (35) can be obtained by alternately solving the sub-problems described above. The iteration is terminated when the norm of the difference in the results of consecutive iterations are below a threshold or when the maximum number of iterations is reached. The algorithmic form of the proposed iterative scheme is shown in Table I. Although the proposed algorithm converges to a limit as shown below, we note that the convergence to the optimal solution is not guaranteed. Convergence of the iterative algorithm: Let J({Wi }, {ψi }) represent the objective function of the optimization problem in (35). At the (n + 1)th iteration, the value of {{Wi }, {τi }}, denoted by {{Win+1 }, {τin+1 }}, is the solution to sub-problem (36) that minimizes the objective J under the constraints. Hence, we have J({Win+1 }, {ψin }) ≤ J({Win }, {ψin }). Having computed {{Win+1 }, {τin+1 }}, we obtain {ψin+1 } as the solution to the second sub-problem given in (38) and (39). This implies that J({Win+1 }, {ψin+1 }) ≤ J({Win+1 }, {ψin }). From the previous inequalities, we observe that J({Win+1 }, {ψin+1 }) ≤ J({Win }, {ψin }), i.e., the objective function decreases monotonically with the number

UBAIDULLA et al.: ROBUST TWO-WAY COGNITIVE RELAYING: PRECODER DESIGNS UNDER INTERFERENCE CONSTRAINTS AND IMPERFECT CSI

TABLE II: Iterative algorithm for the MSE-balancing relay precoder design Set desired tolerance κ and the interval [bl bu ] that contains the optimum value of the objective. 1) n = 0 2) while bu − bl ≤ κ 3) r ← (bl + bu )/2 4) Solve (42) for τ = r 5) If feasible, bu ← r, else bl ← r. 6) endwhile

of iterations. This observation, coupled with the fact that J({Win }, {ψin }) is lower-bounded, implies that the proposed algorithm converges to a limit as n → ∞. B. Robust MSE-Balancing Precoder Design The problem of robust MSE-balancing cognitive relay precoder design with a constraint on the total relay transmit power can be written as min max(¯ 1 , ¯ 2 ) (40) w,a1 ,a2

subject to:

P¯ ≤ PU , I¯ ≤ θ,

which is equivalent to the following formulation in the epigraph form: min τ (41) w,a1 ,a2 ,τ

subject to:

2 ≤ τ, P¯ ≤ PU , I¯ ≤ θ, ¯1 ≤ τ, ¯

where τ is an auxiliary variable. Based on the developments in the previous subsection, we can reformulate this problem as the following quasi-convex optimization problem: τ (42) min w,ψ1 ,ψ2 ,τ     (M1 z − iψ1 )T σν1  ≤ ψ1 τ, subject to:    (M2 z − iψ2 )T σν1  ≤ ψ2 τ,  M3 z ≤ PU , √ M4 z ≤ θ. This problem can be solved by a bisection search over τ and a feasibility check. Suppose τ ∗ is the optimal solution of the problem in (42). For a fixed value of τ , the problem in (42) is a convex feasibility problem,i.e, to find the set of optimization variables that satisfy all the constraints. If the problem is feasible for a fixed value of τ , then we can see that τ ∗ ≤ τ , otherwise τ ∗ ≥ τ . Based on this observation, we can devise an iterative algorithm to solve the problem in (42). The iterative algorithm given in Table II involves a bisection search in τ , and a solution of a convex feasibility problem. Let [bl bu ] be the interval that contains the the optimal value of the objective. As the objective τ represents MSE which is always non-negative, bl can be initialized as zero, and bu can be initialized as a sufficiently large value. For a given tolerance κ, the size of the interval for search is reduced by half at each step, and so the iteration is guaranteed to converge in log2 ((bu − bl )/κ) steps, where x represents the lowest integer greater than or equal to x. C. Robust Minimum Relay Transmit Power Precoder Design Finally, considering the third design scheme, namely robust minimum relay transmit power precoder design, the corresponding problem can be formulated as

P¯

min

{Wi },a1 ,a2

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(43)

¯1 ≤ η1 , ¯2 ≤ η2 , I¯ ≤ θ.

subject to:

We can reformulate this problem as the following convex optimization problem min t2 (44) {Wi },ψ1 ,ψ2 ,t   [(M1 z − iψ1 )T σν1 ] ≤ ψ1 √η1 , subject to:   [(M2 z − iψ2 )T σν2 ] ≤ ψ2 √η2 , M3 z ≤ t, M4 z ≤ θ, where t is an auxiliary optimization variable. The optimization problem given above consists of second-order cone constraints and it can be solved efficiently [25]. IV. ROBUST C OGNITIVE R ELAY P RECODER D ESIGNS UNDER CSI WITH S PHERICAL U NCERTAINTY R EGION In this section, we consider the relay precoder design when the CSI error is characterized by a spherical uncertainty region around the nominal CSI with a known radius [26]. In such a characterization, the exact distribution of the CSI error is not relevant. We adopt a worst-case design approach to render the performance of the precoder robust in the presence of such CSI errors. The problem of precoder design that minimizes total relay transmit power can be stated as P (45) min {Wi }M i=1 ,a1 ,a2

subject to

1 ≤ η1 , 2 ≤ η2 , I ≤ θ, ∀αi ∈ Rgi , ∀αi ∈ Rαi , ∀βi ∈ Rβi , ∀β i ∈ Rβ , ∀φi ∈ Rφi , i

where R represent the uncertainty regions corresponding to the variable appearing as the subscript. In order to facilitate further development, we rewrite the problem above in its equivalent epigraph form as follows: t (46) min {Wi }M i=1 ,a1 ,a2

subject to

P ≤t 1 ≤ η1 , 2 ≤ η2 , I ≤ θ, ∀αi ∈ Rgi , ∀αi ∈ Rαi , ∀βi ∈ Rβi , ∀β i ∈ Rβ , ∀φi ∈ Rφi , i

where t is an auxiliary variable. The optimization problem given above is a semi-infinite problem since the constraints have to be satisfied for all errors belonging to the uncertainty sets. In general, solution to such semi-infinite optimization problems are mathematically intractable [30]. The presence of multiple uncertainties and the non-linear dependence of the objective and constraint

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functions on the CSI error vectors render the attempt to find an exact solution to the problem in (45) quite difficult. In order to circumvent this difficulty and to obtain a tractable solution, we adopt a cutting-set [31] based approach to solve this problem. The cutting-set method for robust optimization essentially alternates between two steps: the first step consists of an optimization process and the second step consists of the computation of worst-case errors. Assuming that the errors belong to certain known uncertainty region, the optimization step computes the optimal values of the variables for fixed values of the parameters whereas the worst-case analysis computes the worst-case error vectors that maximizes the constraint functions. Employing the cutting-set method to solve the problem at hand, we propose an algorithm which involves alternating sequence of solution of a pair subproblems, i) computation of relay precoders for a fixed set of CSI and ii) computation of the worst-case channels for a fixed set of relay precoders. In the rest of this section, we detail the subproblems and the overall algorithm to solve the robust precoder design. A. Relay Precoder for Fixed CSI Here we consider the first subproblem, i.e., the computation of the relay precoders for a fixed set of CSI. This problem can be formulated as follows: t (47) min

M  √    i + β ) − 12 + |a1 |2 σ 2 ( gi + αi )Wi (h 1 =a1 p2 i ν1 i=1

⎧ 2 ⎫ M ⎨  ⎬   i + αi Wi αi ) +|a1 |2 p1 E  ( gi Wi αi + αi Wi g  ⎭ ⎩ i=1

+|a1 |2

i=1

min

subject to:

t2  [(M  1 z − iψ1 )T  [(M  2 z − iψ2 )T

(49)

i=1

⎧ 2 ⎫ M ⎨   ⎬   Wi β + β Wi h  i + β Wi β ) + |a2 |2 p2 E  (h i i i i i  ⎭ ⎩ i=1

M 

+ |a2 |2

i=1

 + β )Wi Σμ WH (h  + β )H . (h i i i i i i

(50)

Note that the third term in the equations above arises due to the inexact self-interference cancellation. The worst-case CSI error corresponding to the MSE constraint at T1 is obtained by solving the following optimization problem: max

{αi ,αi ,β i }M i=1

1

(51)

2 subject to: αi αH i ≤ δαi , 1 ≤ i ≤ M, 2 αH i αi ≤ δαi , 1 ≤ i ≤ M, 2 βH i β i ≤ δβ i , 1 ≤ i ≤ M.

We observe that the solution to this problem is equivalent the solution of the optimization problem in (43) for the special case of zero CSI error variances. Hence, we can reformulate the problem in (47) as the following optimization with secondorder cone constraints: {Wi },ψ1 ,ψ2 ,t

( gi + αi )Wi Σμi WiH ( gi + αi )H ,

and that at T2 as M  √  2  + β )Wi (  2 =  a2 p 1 (h gi + αi ) − 1 + |a2 |2 σν22 i i

{Wi }M i=1 ,a1 ,a2

subject to: P ≤ t 1 ≤ η1 , 2 ≤ η2 , I ≤ θ.

M 

Similarly, the worst-case CSI error corresponding to the MSE constraint at T2 can be computed by solving the following optimization problem: max

{β ,β i ,αi }M i=1

 3 z ≤ t, M  4 z ≤ θ, M  i , 1 ≤ i ≤ 4 are obtained from Mi , 1 ≤ i ≤ 4 by where M setting all CSI error variance to zero. B. Computation of the Worst-case Errors for Given Precoder Having computed the relay precoder and receive filters for a fixed set of CSI, the next sub-problem is to find out the worst-case CSI errors belonging to the uncertainty sets. The worst-case CSI errors are obtained by maximizing the MSE, transmit power and PU interference functions, with respect to the CSI errors. In the rest of this subsection, we describe the computation of these errors. First, we consider the computation of the worst-case channels maximizing the MSE functions. We can express the MSE at T1 under imperfect CSI as

(52)

≤ δβ2 , 1 ≤ i ≤ M, subject to: βi β H i i

2 βH i β i ≤ δβ i , 1 ≤ i ≤ M,

(48)  √ σν1 ] ≤ ψ1 η1 ,  √ σν2 ] ≤ ψ2 η2 ,

2

i

2 αH i αi ≤ δαi , 1 ≤ i ≤ M.

Due to the complexity of the problem, it is difficult to obtain exact solutions to the optimization problems in (51) and (52). However, we can significantly reduce the complexity by considering a first-order approximation of the MSE function in the above problem. Neglecting all the terms containing second and higher order terms of the CSI error in the MSE expression in (49), we arrive at the following approximate expression for the MSE at T1 M  √    i − 12 + |a1 |2 g  i Wi h i Wi Σμi WiH g H g  1 ≈  a1 p 2 i i=1

+

M  i=1

 ih  H WH g H 2 αi |a1 |2 p2 Wi h i i

√ H  i + |a1 |2 Wi Σμ WH g  + |a1 |2 σν21 − a1 p 2 W i h i i i +

M  i=1



√ H  H WH g  2 (|a1 |2 p2 h − a p ) g W β 1 2 i i . i i i i (53)

UBAIDULLA et al.: ROBUST TWO-WAY COGNITIVE RELAYING: PRECODER DESIGNS UNDER INTERFERENCE CONSTRAINTS AND IMPERFECT CSI

A similar approximate expression for 2 can be obtained from (50). Application of Cauchy-Schwarz inequality in the approximate expressions for the MSEs leads to the following worst-case error vectors corresponding to the MSE constraints: αai

√  H (|a2 |2 p1 h  Wi g i − a∗2 p1 ) δαi WiH h i i = , √  H (|a2 |2 p1 h  Wi g i − a∗2 p1 ) WiH h i i

αai = βai =

δαi bH i , bi  i

βai

=

∀i,

i

ci 

i

∀i,

M 

∀i,

M 

i=1

+ p2

σμ2 i Tr(WiH Wi )

i=1

 i + β )H WiH Wi (h  i + β ). (h i i

(58)

i=1

The worst-case channel vectors that maximize this transmit power can be found by solving the following problem: max

αi ,β i

P

(59)

2 subject to: αH i αi ≤ δαi , 1 ≤ i ≤ M,

βH i βi

≤

δβ2i ,

1 ≤ i ≤ M.

In formulating the above, we neglected the last term in (58) as it is not affected by the optimization variables. It is easy to see that the problem above is a separable problem and it can be solved by solving the following set of problems for 1 ≤ i ≤ M: max p1 ( gi + αi )H WiH Wi ( gi + αi ) (60) αi

2 subject to: αH i αi ≤ δαi ,

max βi

 i + β )H WH Wi (h i + β ) p2 (h i i i

(61)

2 subject to: βH i β i ≤ δβ i .

We seek a solution to these optimization problems using the optimality conditions. The Lagrangian associated with the problem in (60) is given by H

Li = ( gi + αi )

αH i αi

(57)

( gi + αi )H WiH Wi ( gi + αi ) +

M 

In order to compute the Lagrange multiplier λ, consider the singular value decomposition (SVD) of WH W as WH W = URVH , where U and V are unitary matrices and R is a diagonal matrix containing the singular values, ri , 1 ≤ i ≤ M . Employing this SVD in (64), we can show that

(56)

√ i h  H WH g i + H where bi = |a1 |2 p2 Wi h − a1 p 2 W i h i i H − H i g iH WH h |a1 |2 Wi Σμi WiH g and ci = |a2 |2 p1 Wi g i i √ H . i + |a2 |2 Wi Σμi WiH h a2 p 1 W i g i Next, we consider the worst-case CSI error vectors corresponding to the total relay transmit power. Incorporating the CSI error model, the total relay power transmitted by the relay nodes can be expressed as P =p1

Solving the set of equations above, the worst-case CSI error vectors corresponding to the total relay transmit power are  i if λ > 0 −(WiH Wi + λI)−1 WiH Wi g b (64) αi = i −(WiH Wi )−1 WiH Wi g if λ ≤ 0

1

i

,

(54)

(55)

 i − a∗ √p2 ) H  i Wi h (|a1 |2 p2 g δβi WiH g 1 i ,  i − a∗ √p2 ) H (|a1 |2 p2 g  Wi h WH g δβ cH i

∀i,

WiH Wi ( gi

At the optimal solution, satisfied [25]:

αbi ,

+

αi ) + λ(αH i αi

− δα2 i ).

(62)

the following conditions are ∇Li = 0, λ ≥ 0,

λ(αi H αi − δα2 i ) = 0.

(63)

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=

−2 ζH Rζ i = δα2 i i R(R + λI)

⇒

M  |ζij |2 rj2 − δα2 i = 0, 2 (r + λ) j i=j

(65)

i , and ζij is the jth element of ζ i . The where ζ i = VH g Lagrange multiplier λ can be determined by solving (65). Following the same approach to solve (61), we obtain   i if λ > 0 −(WiH Wi + λ I)−1 WiH Wi h β bi = (66) i if λ ≤ 0, −(WiH Wi )−1 WiH Wi h where the variable λ is obtained by solving

M

 2 2 | rj |ζij i=j (rj +λ )2 − i. ζ i = VH h

 δβ2i = 0 with ζij being the jth element of Finally, we compute the worst-case CSI errors corresponding to the PU interference. Substituting the CSI error model in the expression for the interference to the PU receiver, we have

I =p1

M 

( fi + φi )Wi ( gi + αi )2 +

i=1

+ p2

M 

M 

σμ2 i ( fi + φi )Wi 2

i=1

 i + β )2 . ( fi + φi )Wi (h i

(67)

i=1

In order to simplify the calculation of the worst-case errors, we consider an approximate expression for I involving only the first-order error vectors as given in (68). Using CauchySchwarz inequality, the worst-case error vectors corresponding to the PU interference is obtained as i WiH  fi Wi g fiH δα  , (69) αci = i i WiH   fi Wi g fiH  H  i WH  fi Wi h δβ  i fi β ci = i , (70) H  i WH   fi Wi h i fi  H δφ b (71) φci = i i , i b H i = (p1 Wi g ih  H WH  i g iH WiH  where b fiH + p2 Wi h i i fi + 2 H H σμ Wi Wi fi ).

C. Iterative Procedure for Robust Precoder Design The proposed robust precoder design involves alternating execution of the optimization and worst-case analysis developed in the preceding subsections. Let H be a set of channel vectors. This set is initialized with or estimated   Mnominal  the  M M i channel vectors, viz., { gi }i=1 , h ,  fi . In the i=1

i=1

first step, i.e., the optimization step, Wi , 1 ≤ i ≤ M , a1 , a2 and τ are computed by solving (48), where the constraints should involve all the elements of H. In the second step, the

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2 H  i 2 + σ 2   H WH  i 2 + p2  fiH  fi Wi αi ) + 2p2 (h I =p1  fi Wi g fi Wi h giH WiH  μ fi Wi  + 2p1 ( i i fi fi Wi β i ) i h  H WH  i g H W H  f H + p2 W i h f H + σ 2 Wi W H  f H )). + 2(φ (p1 Wi g i

i

i

i

i

i

worst-case error vectors corresponding to various constraints are computed as described in the preceding subsection using (54)-(57), (64), (66), and (69)-(71). In order to describe this process better, consider for instance the errors in gi . The relevant worst-case error vectors are αai , αbi , and αci . Corresponding worst-case channel vectors can be obtained as ia = g i + αai , g ib = g i + αbi , and g ic = g i + αci . Worst-case g channels corresponding to all the links are similarly calculated. The worst-case channels so obtained are then substituted in the constraint functions of (46) to check if they violate the constraints. Those worst-case channels which violate these constraints are appended to H. This implies that, in each iteration, the cardinality of this set may increase or remain the same depending on the number of constraint violations. In each iteration, the precoder design step (i.e., the first step) needs to ensure that the constraints are satisfied for increasing number of channels in H. In order to achieve this, the precoder design problem is solved by incorporating all the design constraints for each channel in this set. In other words, as the size of the set H increases, the number of effective constraints in the precoder design problem increases. The iteration continues till the maximum constraint violation is below a specified threshold. The convergence of the cuttingset method, on which the proposed iterative algorithm is based, is proved in [31]; the iterative procedure converges to the optimal solution if the worst-case analysis admits an exact solution, whereas it converges to a suboptimal solution if the worst-case analysis is only approximate. In the present case, the worst-case analysis makes use of approximations and hence the iteration is not guaranteed to lead to the robust optimal solution. However, the results of numerical simulations demonstrate that, in the presence of CSI errors, the proposed robust design exhibit significant robust performance compared to non-robust designs. We have only considered the robust design that minimizes the total relay transmit power in the context of CSI error characterized by spherical uncertainty region. In this context, it is also possible to apply the technique developed here to address the problems of robust relay precoder designs based on other criteria, like minimum sum MSE and MSE balancing. However, the computational complexity of the latter problems is more than that of the former since the first sub-problem, i.e., the relay precoder design for a fixed CSI, needs to be solved iteratively in the latter case. V. N UMERICAL R ESULTS AND D ISCUSSIONS Hereafter, we illustrate the performance of the proposed cognitive relay precoder designs evaluated through simulations. The fading is modeled as Rayleigh, with the channel vectors, gi , hi and fi , 1 ≤ i ≤ M , comprising i.i.d. samples drawn form complex Gaussian processes with zero mean and unit variance. The noise at each node is assumed to be zero-mean complex Gaussian random variable. In all the

i

μ

i

i

(68)

simulations, we assume σe2 = σα2 = σβ2 = σf2 , σν21 = σν22 and η = η1 = η2 . Since the actual channels are assumed to be of unit variance, it may be noted that the error variance is equivalent to (100σe2 )% of the variance of the actual channel. For a demonstration of the performance of the proposed minimum SMSE cognitive relay precoder design proposed in Sec. III-A, first we consider the effect of CSI error on the achieved MSE at the secondary transceiver nodes. For this purpose, we consider systems with M = 2 and M = 3 relays. In both cases, the relays employ N = 2 antennas. Performance results for the case of perfect CSI, and the results for the performance of the robust and non-robust designs in the case of imperfect CSI are depicted in Fig. 1. The non-robust design is obtained by setting σe = 0 in the robust design. The performance of the non-robust design which assumes the availability of perfect CSI is found to deteriorate as the CSI error variance σe2 increases. However, the proposed robust design is able to achieve lower SMSE compared to the nonrobust design in the presence of CSI error. At higher CSI error variances, the performance gain achieved by the robust design in terms of the average SMSE increases significantly. For instance, the difference in the average SMSE between non-robust and robust design for σe2 = 0.25 is more than 3 times the difference for σe2 = 0.15. It is also found that for less number of relay nodes, the difference between the MSE achieved by robust and non-robust design is more significant. This happens because the system with less available relays requires larger relay transmit power to achieve the same target MSE, which in turn results in larger noise due to the CSI imperfection. Now in terms of convergence, the behavior of the proposed design is shown in Fig. 2. We consider a setup with M = 2 relays equipped with N = 2 or N = 3 antennas. The proposed design with perfect CSI is found to converge in less than 8 iterations. The robust design converges in around 16 iterations in the presence of CSI error with variance σe2 = 0.1. Next, we consider the performance of the MSE-balancing cognitive relay precoder designs presented Sec. III-B. We consider a system with M = 6 relays and interference threshold θ = 0dB. The results are shown in Fig. 3 for the case with perfect CSI, σe2 = 0.1 and σe2 = 0.15. When the CSI is imperfect, the robust design results in lower min-max MSE, min(¯ 1 , ¯2 ), compared to the non-robust design. For a comparison of the performance of the proposed MSE-constrained designs in Sec. III-C and the corresponding non-robust design, we consider the effect of CSI error on the achieved MSE at the secondary transceiver nodes and the interference to the PU receiver. For this purpose, we consider a system with M = 2 relays equipped with N = 2 antennas. The performance results are shown in Fig. 4. It can be observed that the non-robust design that assumes the availability of perfect CSI fails to meet the target MSE at the SU transceiver nodes in the presence of CSI uncertainty.

UBAIDULLA et al.: ROBUST TWO-WAY COGNITIVE RELAYING: PRECODER DESIGNS UNDER INTERFERENCE CONSTRAINTS AND IMPERFECT CSI

1.8

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0.4 Non−robust design, η=0.3 Robust design, η=0.3 Non−robust design, η=0.1 Robust design, η=0.1

Non−robust design, M=N=2

1.6

Robust design, M=N=2 1.4

0.35

Non−robust design, M=3, N=2 Robust design, M=3, N=2

1.2

0.3

Perfect CSI, M=3, N=2 MSE at T1

SMSE

Perfect CSI, M=N=2 1 0.8

0.25

0.6

0.2

0.4 0.15 0.2 0

0

0.05

0.1 0.15 2 CSI error variance σe

0.2

0.1

0.25

Fig. 1. Sum mean-square error versus CSI error variance σe2 for different values of M and N . Interference threshold θ = 0 dB. 0.1

0.005

0.07

Interference power (dB)

SMSE

0.035

0.04

0.045

Non−robust design, θ=−10 dB, η=0.4 Robust design, θ=−10 dB, η=0.4 Non−robust design, θ=−7 dB, η=0.3 Robust design, θ=−7 dB, η=0.3

−4 −5

0.06

0.05

−6 −7 −8

0.04

−9 −10 0

2

4

6

8 10 12 Number of iterations

14

16

18

20

Fig. 2. Convergence behavior of the minimum SMSE relay precoder for different numbers of relays, M , and antennas, N , with the interference threshold θ = −10 dB. 0.05 Non−robust beamformer Robust beamformer Non−robust beamformer Robust beamformer Perfect CSI

0.045

0.04

−11

0

0.02

0.04

0.06

0.08 0.1 0.12 2 CSI error variance σe

0.16

0.18

0.7 δ=0

0.6

0.035

0.5 2 e

σ =0.15 1

δ=0.01

2

σe=0.1

0.025

0.4

1

0.03

0.3 δ=0.03

0.02

0.015

0.14

Fig. 5. Interference power I at the PU receiver versus CSI error variance σe2 . M = N = 2.

Prob(ε > η )

Min−max MSE

0.015 0.02 0.025 0.03 2 CSI error variance σe

−3

0.08

0.03

0.01

Fig. 4. The MSE at the secondary transceiver node T1 versus the CSI error variance σe2 for different values of target MSE η. M = 2, N = 2 and interference threshold θ = 0dB.

Robust design, M=3, N=2 Perfect CSI, M=3, N=2 Robust design, M=3, N=3 Perfect CSI, M=3, N=3

0.09

0

0.2 0

1

2

3

4 5 6 7 Relay transmit power: P (dB)

8

9

10

δ=0.05 0.1

R

Fig. 3. Min-max MSE, min(¯ 1 , ¯2 ), versus total relay transmit power P . M = 6, θ = 0dB.

However, the robust design is able to achieve the target MSE even in the presence of CSI uncertainty. Similar results are observed for both η = 0.3 and η = 0.1. We also consider the performance in terms of the interference power to the PU receiver resulting from the transmission from the SU relay nodes. For this case, we assume the same setup as above. Results for different values of target MSE and target interference thresholds are shown in Fig. 5. The design that assume perfect CSI fails to meet the interference threshold in the presence of CSI errors, whereas our robust design ensures that the interference threshold is not exceeded.

0 0

0.02

0.04

σe

0.06

0.08

0.1

Fig. 6. Outage probability, defined as Prob(1 > η1 ), versus CSI error variance σe for different values of CSI error norm bound δ. Other simulation parameters are: M = N = 3, θ = 1 and target MSE η = 0.05.

Finally, we consider the performance of the robust design Sec. IV that ensures robust performance under imperfect CSI with spherical uncertainty region. We compare the performance of the robust design with that of non-robust design. The performance of the non-robust design is obtained by setting the radius of the uncertainty region or the norm of the CSI error to zero. In all the simulations run for generating these

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50

Total relay transmit power P

45 40 θ = 5 dB, η = 0.07 θ = 1 dB, η = 0.1

35

θ = 5 dB, η = 0.1 30 25 20 15 10 0

0.02

0.04 0.06 CSI error norm δ

0.08

0.1

Fig. 7. Total relay transmit power P versus CSI error norm δ. Number of relays M = 3 and number of antennas per relay node N = 3. 1 0.9 0.8

Prob(I ≤ ω)

0.7 0.6 0.5 0.4

Non−robust δ=0

0.3

Robust δ=0.03

0.2

Robust δ=0.1

Robust δ=0.06

0.1 0 1

1.5

2

2.5

ω

3

3.5

4

4.5

Fig. 8. Cumulative distribution of the PU interference I for different values of CSI error norm bound δ. The upper limit on the PU interference, θ, is set to 3. Other simulation parameters are: M = N = 3, target MSE η = 0.1.

VI. C ONCLUSIONS

0.05 δ=0 δ=0.03 δ=0.06 δ=0.1

0.045

Maximum constraint violation

0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 1

norm less than or equal to δ, and an outage occurs for all error vector with norm exceeding δ. The results are provided for different values of δ. The outage is computed for a system with M = N = 3, PU interference limit θ = 1, and target MSE η1 = η2 = 0.05. The results show that the robust design displays improved outage performance compared to the nonrobust design. The outage probability of the robust design, for a given value of σe , improves with increasing values of δ. For instance, for σe = 0.04, the non-robust design results in an outage with probability of 0.56, whereas the robust design with δ = 0.01 leads to outage probability of 0.25 and that with δ = 0.03 and 0.05 results in a very negligible outage probability. The relay nodes need to expend more transmit power in order to achieve increasing robustness. This trade-off between transmit power and robustness is depicted in Fig. 7. For example, for non-robust design with δ = 0, total transmit power P = 10.6 is expended, whereas for robust design with δ = 0.1 relay transmit power required is P = 16.3. Fig. 8 depicts the effect of the CSI error on the PU interference in terms of cumulative distribution. The robust design is able to keep the PU interference within the limits in the presence of CSI error, whereas the non-robust design results in PU interference beyond the desired limit. For PU interference limit of I = 3, the results show that this limit is violated with a probability of around 0.45 by the non-robust design. The robust design on the other hand keeps the interference below the limit with very high probability. We also study the convergence behavior of the proposed algorithm. The results in Fig. 9 show the variation of the maximum convergence violation with respect to the number of iterations. The maximum constraint violation is found to converge rapidly, in 3 to 4 iterations, for different values of δ.

2

3

4 5 6 Number of iteration

7

8

9

Fig. 9. Convergence behavior of the proposed robust design for different values of CSI error norm bound δ. Other simulation parameters are: M = N = 3, target MSE η = 0.1.

comparisons, we assumed the radius of all the uncertainty regions to be equal and it is represented by δ. In Fig. 6, the performance in terms of outage probability defined as Prob(1 > η1 ) of the SU transceiver is depicted. For this purpose, we consider Gaussian-distributed CSI error with the variance σe2 assumed to be the same for all channels. In the presence of Gaussian-distributed CSI error, the proposed robust design satisfies the constraints for all error vectors with

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UBAIDULLA et al.: ROBUST TWO-WAY COGNITIVE RELAYING: PRECODER DESIGNS UNDER INTERFERENCE CONSTRAINTS AND IMPERFECT CSI

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[29] C. Wang, E. K. S. Au, R. D. Murch, W. H. Mow, R. S. Cheng, and V. Lau, “On the performance of the MIMO zero-forcing receiver in the presence of channel estimation error,” IEEE Trans. Wireless Commun., vol. 6, no. 3, pp. 805–810, Mar. 2007. [30] A. Ben-Tal and A. Nemirovsky, “Selected topics in robust optimization,” Math. Program., vol. 112, pp. 125–158, Feb. 2007. [31] A. Mutapcic and S. Boyd, “Cutting-set methods for robust convex optimization with pessimizing oracles,” Optimization Methods Software, vol. 24, pp. 381–406, June 2009. P. Ubaidulla (S’05, M’12) received the B.Tech. degree in Electronics and Communication Engineering from the National Institute of Technology (NIT), Calicut, in 1997, the M.E. degree in Communication Engineering from NIT, Trichy, in 2001, and the Ph.D. degree in Electrical Communication Engineering from the Indian Institute of Science (IISc), Bangalore, India, in 2011. Prior to joining IISc, he had worked in the industry in the field of radar and sonar signal processing. From 2011 to 2013, he was a Postdoctoral Fellow with the Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) division of King Abdullah University of Science and Technology (KAUST), Saudi Arabia. Currently, he is an Assistant Professor at the International Institute of Information Technology (IIIT), Hyderabad, India. His current research interests are in cognitive radio systems, wireless relay networks, and robust optimization. He is a recipient of the best student paper award at the International Symposium on Wireless Personal Multimedia Communications (WPMC’07), Jaipur, India, 2007. Sonia A¨ıssa (S’93-M’00-SM’03) received her Ph.D. degree in Electrical and Computer Engineering from McGill University, Montreal, QC, Canada, in 1998. Since then, she has been with the Institut National de la Recherche Scientifique-Energy, Materials and Telecommunications Center (INRS-EMT), University of Quebec, Montreal, QC, Canada, where she is a Professor of Telecommunications. From 1996 to 1997, she was a Researcher with the Department of Electronics and Communications of Kyoto University, and with the Wireless Systems Laboratories of NTT, Japan. From 1998 to 2000, she was a Research Associate at INRS-EMT, Montreal. In 2000-2002, while she was an Assistant Professor, she was a Principal Investigator in the major program of personal and mobile communications of the Canadian Institute for Telecommunications Research, leading research in radio resource management for wireless networks. From 2004 to 2007, she was an Adjunct Professor with Concordia University, Montreal. In 2006, she was Visiting Invited Professor with the Graduate School of Informatics, Kyoto University, Japan. Her research interests lie in the area of wireless and mobile communications, and include radio resource management, cross-layer design and optimization, design and analysis of multiple antenna (MIMO) systems, cognitive and cooperative transmission techniques, and performance evaluation, with a focus on Cellular, Ad Hoc, and Cognitive Radio networks. Dr. A¨ıssa is the Founding Chair of the IEEE Women in Engineering Affinity Group in Montreal, 2004-2007; acted or is currently acting as TPC Leading Chair or Cochair of the Wireless Communications Symposium at IEEE ICC in 2006, 2009, 2011 and 2012; PHY/MAC Program Cochair of the 2007 IEEE WCNC; TPC Cochair of the 2013 IEEE VTC-spring; and TPC Symposia Cochair of the 2014 IEEE Globecom. Her main editorial activities include: Editor, IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS, 2004-2012; Technical Editor, IEEE W IRELESS C OMMUNICATIONS M AGA ZINE, 2006-2010; and Associate Editor, Wiley Security and Communication Networks Journal, 2007-2012. She currently serves as Technical Editor for the IEEE C OMMUNICATIONS M AGAZINE. Awards to her credit include the NSERC University Faculty Award in 1999; the Quebec Government FQRNT Strategic Faculty Fellowship in 2001-2006; the INRS-EMT Performance Award multiple times since 2004, for outstanding achievements in research, teaching and service; and the Technical Community Service Award from the FQRNT Centre for Advanced Systems and Technologies in Communications in 2007. She is co-recipient of five IEEE Best Paper Awards and of the 2012 IEICE Best Paper Award; and recipient of NSERC Discovery Accelerator Supplement Award. She is a Distinguished Lecturer of the IEEE Communications Society (ComSoc) and an Elected Member of the ComSoc Board of Governors.