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[Rui Zhang, Ying-Chang Liang, and Shuguang Cui]

[A convex optimization perspective]

T

his article provides an overview of the state-of-art results on communication resource allocation over space, time, and frequency for emerging cognitive radio (CR) wireless networks. Focusing on the interference-power/interference-temperature (IT) constraint approach for CRs to protect primary radio transmissions, many new and challenging problems regarding the design of CR systems are formulated, and some of the corresponding solutions are shown to be obtainable by restructuring some classic results known for traditional (non-CR) wireless networks. It is demonstrated that convex optimization plays an essential role in solving these problems, in a both rigorous and efficient way. Promising research directions on interference management for CR and other related multiuser communication systems are discussed. INTRODUCTION In recent years, CR networks, where CRs or the so-called secondary users (SUs) communicate over certain bandwidth originally allocated to a primary network, have drawn great research interests in the academic, industrial, and regulation communities. Accordingly, there is now a rapidly growing awareness that CR technology will play an essential role in enabling dynamic spectrum access for the next generation wireless communications, which could hopefully resolve the spectrum scarcity versus under-utilization dilemma caused by the current static spectrum management polices. Specifically, the users in the primary network, or the so-called primary users (PUs), could be licensed users, who have the absolute right to access their spectrum bands, and yet would be willing to share the spectrum with the unlicensed SUs. Alternatively, both the PUs and SUs could equally coexist in an unlicensed band, where the PUs are regarded as existing active communication links while the SUs are new links to be added. A unique

Digital Object Identifier 10.1109/MSP.2010.936022

? BRAND X PICTURES

IEEE SIGNAL PROCESSING MAGAZINE [102] MAY 2010

1053-5888/10/$26.00?2010IEEE

[8] have provided promising feature of CRs is that they are THERE IS NOW A RAPIDLY GROWING guidelines to approach the able to identify and acquire useAWARENESS THAT CR TECHNOLOGY fundamental limits of such ful environmental information WILL PLAY AN ESSENTIAL ROLE IN networks. However, from a (cognition) across the primary practical viewpoint, the cenand secondary networks, and ENABLING DYNAMIC SPECTRUM tralized design approach for thereby adapt their transmit ACCESS FOR THE NEXT GENERATION PU and SU networks is not strategies to achieve the best WIRELESS COMMUNICATIONS. desirable, since PU and SU sysperformance while maintaining tems usually belong to differa required quality of service for ent operators and thus it is difficult, if not infeasible, for them each coexisting active primary link. Depending on the type of cogto cooperate. Consequently, a decentralized design approach nitive knowledge collected (e.g., on/off statuses of primary links, is more favorable, where the PU network is designed without PU messages, interference power levels at PU receivers, or primathe awareness of the existence of the SU network, while the ry link performance margins) and the primary/secondary network SU network is designed with only partial knowledge (cognimodels of interests (e.g., infrastructure-based versus ad hoc), tion) of the PU network. many new and challenging problems on the design of CR netFollowing this (simplified) decentralized approach, there are works can be formulated, as will be reviewed in this article. furthermore two design paradigms proposed for SS-based CRs. To date, quite a few operation models have been proposed for One is based on the “cognitive relay” concept [9], where the SU CRs; however, there is no consensus yet on the terminology used transmitter allocates only part of its power to deliver the SU for the associated definitions [1]–[3]. Generally speaking, there messages, and uses the remaining power to relay the PU mesare two basic operation models for CRs: opportunistic spectrum sages so as to compensate for the additional SU interference expeaccess (OSA) versus spectrum sharing (SS). In the OSA model, rienced at the PU receiver. However, this technique requires the SUs are allowed to transmit over the band of interest when all noncausal knowledge of the PU messages at the SU transmitter, the PUs are not transmitting at this band. One essential enabling which may be difficult to realize in practice. In contrast, a more technique for OSA-based CRs is spectrum sensing, where the feasible SS design for the SU to protect the PU is to impose a CRs individually or collaboratively detect active PU transmissions constraint on the maximum SU interference power at the PU over the band, and decide to transmit if the sensing results indireceiver, also known as the IT constraint [10], by assuming that cate that all the PU transmitters are inactive at this band with a the SU-to-PU channels are either perfectly known at the SU high probability. Spectrum sensing is now a very active area for transmitters, or can be practically estimated. research; the interested readers may refer to, e.g., [4]–[7] for an In this article, we will focus our study on the IT-based SS overview of the state-of-art results in this area. As a counterpart, model for CRs, namely the IT-SS, as it is a more feasible the SS model allows the SUs to transmit simultaneously with approach compared with other existing ones. In a wireless comPUs at the same band even if they are active, provided that the munication environment, channels are usually subject to spaceSUs know how to control their resultant interference at the PU time-frequency variation due to multipath propagation, receivers such that the performance degradation of each active mobility, and location-dependent shadowing. Thus, dynamic primary link is within a tolerable margin. Thus, OSA and SS can resource allocation (DRA) becomes an essential technique for be regarded as the primary-transmitter-centric and primary-reCRs to optimally deploy their transmit strategies to maximize ceiver-centric dynamic spectrum access techniques, respectively. the secondary network throughput, where the transmit power, Consequently, there will be an inevitable debate on which operabit-rate, bandwidth, and antenna beam should be dynamically tion model, OSA or SS, is better to deploy CRs in practical sysallocated based upon the available channel state information tems; however, a rigorous comparative study for these two (CSI) of the primary and secondary networks. In particular, this models, in terms of spectrum efficiency and implementation article will focus on DRA problem formulations unique to CR complexity tradeoffs, is still open. Generally speaking, SS utilizes systems under the IT-SS model, and the associated solutions the spectrum more efficiently than OSA, since the former supthat are nonobvious in comparison with existing results [11] ports concurrent PU and SU transmissions over the same band known for the traditional (non-CR) wireless networks. More while the latter only allows orthogonal transmissions between importantly, we will emphasize the key role of various convex them. Moreover, the receiver-centric approach for SS is more optimization techniques in solving these problems. effective for CRs to manage the interference to the PU links than the transmitter-centric approach for OSA. NOTATION Hence, the SS model for CRs will be focused in this article. Lowercase and uppercase bold letters denote vectors and It is worth noting that the optimal design approach for matrices, respectively. Rank 1 # 2 , Tr 1 # 2 , | # |, 1 # 2 21, 1 # 2 H, and 1 # 2 1/2 SS-based CR networks should treat all coexisting PU and SU links as a giant interference network and jointly optimize denote the rank of a matrix, trace, determinant, inverse, their transmissions to maximize the SU network throughput Hermitian transpose, and square-root, respectively. I and 0 with a prescribed PU network throughput guarantee. From denote an identity matrix and an all-zero matrix, respectively. this viewpoint, recent advances in network information theory Diag 1 a 2 denotes a diagonal matrix with diagonal elements

IEEE SIGNAL PROCESSING MAGAZINE [103] MAY 2010

given in a. E 1 # 2 denotes the statistical expectation. A circularly symmetric complex Gaussian (CSCG) distributed random vector with zero mean and covariance matrix S is denoted by CN 1 0, S 2 . Cm3n denotes the space of m 3 n complex matrices. 7 # 7 denotes the two-norm of a complex vector. Re 1 # 2 and Im 1 # 2 denote the real and imaginary parts of a complex number, respectively. The base of the logarithm function log 1 # 2 is two by default. CR NETWORK MODELS We consider two general types of CR networks, which are of both theoretical and practical interests: One is infrastructurebased, as shown in Figure 1(a), where multiple secondary terminals communicate with a common secondary node referred to as the secondary base station (S-BS); the other is ad hoc, as shown in Figure 1(b), which consists of multiple distributed secondary links. In both types of CR networks, there are coexisting primary terminals operating in the same spectrum band. For the IT-SS model of CRs, the exact operation model of the primary network is not important to our study, provided that all the secondary terminals satisfy the prescribed IT constraints to protect the primary terminals. Without loss of generality, we assume that there are K secondary links and J primary terminals in each type of the CR networks. Consider first the infrastructure-based secondary/CR network with the S-BS coordinating all the CR transmissions, which usually corresponds to one particular cell in a CR cellular network. The uplink transmissions from the SUs to the S-BS are usually modeled by a multiple-access channel (MAC), while the downlink transmissions from the S-BS to different SUs are modeled by a broadcast channel (BC). For the MAC, the equivalent baseband transmission can be represented as y 5 a Hk x k 1 z,

k51 K

z [ CM31 denotes the noise received at S-BS. We assume that xk ’s are independent over k. Similarly, the BC can be represented as yk 5 HH x 1 zk, k 5 1, c, K , k (2)

where yk [ CNk 31 denotes the received signal at the kth SU; for convenience, we have used the Hermitian transposed uplink channel matrix for the corresponding downlink channel matrix, i.e., HH denotes the channel from the S-BS to the k kth SU; x [ CM31 denotes the transmitted signal from S-BS; and zk [ CNk 31 denotes the receiver noise of the kth SU. In the case that x carries information common to all SUs, the associated downlink transmission is usually called multicast, while if x carries independent information for different SUs, it is called unicast. Next, consider the ad hoc secondary/CR network, which is usually modeled as an interference channel (IC). For convenience, we assume that for the k th secondary link, k 5 1, c, K, the transmitter is denoted as SU-TXk and the receiver is denoted as SU-RXk, although in general a secondary terminal can be both a transmitter and a receiver. The baseband transmission of the IC can be represented as | 5H | 1 | | yk kk x k a Hik x i 1 z k, k 5 1, c, K ,

i51, i2k K

(3)

(1)

where y [ CM31 denotes the received signal at the S-BS, with M denoting the number of antennas at S-BS; Hk [ CM3Nk denotes the channel from the kth SU to S-BS, k 5 1, c, K, with Nk denoting the number of antennas at the kth SU; xk [ CNk 31 denotes the transmitted signal of the kth SU; and

S-BS

(a) Secondary Terminal

(b) Primary Terminal

[FIG1] CR networks: (a) infrastructure based and (b) ad hoc.

where |k [ CBk 31 denotes the received signal at SU-RXk, with y Bk denoting the number of receiving antennas; |k [ CAk 31 x denotes the transmitted signal of SU-TXk, with Ak denoting the number of transmitting antennas; Hkk [ CBk 3Ak denotes the direct-link channel from SU-TXk to SU-RXk, while Hik [ CBk 3Ai denotes the cross-link channel from SU-TXi to SU-RXk, i 2 k; and |k [ CBk 31 denotes the noise at SU-RXk. It is assumed that z | ’s are independent over k. xk Furthermore, we assume that the jth PU, j 5 1, c, J, in each type of the CR networks is equipped with Dj antennas, Dj $ 1. We then use Gkj [ CDj 3Nk to denote the channel from the kth SU to the jth PU in the CR MAC, Fj [ CDj 3M to denote the channel from S-BS to the jth PU in the CR BC, and Ekj [ CDj 3Ak to denote the channel from SU-TXk to the jth PU in the CR IC. Moreover, the receiving terminals in the secondary networks may experience interference from active primary transmitters. For simplicity, we assume that such interference is treated as additional noise at the secondary receivers, and the total noise at each secondary receiving terminal is distributed as a CSCG random vector with zero mean and the identity covariance matrix. Note that the (spatial) channels defined in the above CR network models are assumed constant for a fixed transmit dimension such as one time-block in a time-division-multiple-access (TDMA) system or one frequency-bin in an orthogonal-frequency-division-multiplexing (OFDM) system. In a wireless environment, these channels usually change over time and/or frequency dimensions as governed by an underlying joint stochastic process. As such, DRA becomes relevant

IEEE SIGNAL PROCESSING MAGAZINE [104] MAY 2010

first study the case of a single to schedule SUs into different DYNAMIC RESOURCE ALLOCATION transmit dimension 1 L 5 1 2 transmit dimensions based on BECOMES AN ESSENTIAL TECHNIQUE with PTPCs and PIPCs by their CSI. In general, the secFOR CRS TO OPTIMALLY DEPLOY focusing on the spatial-doondary transmitting terminals main transmit optimization for need to satisfy two types of THEIR TRANSMIT STRATEGIES multiantenna CRs in the secpower constraints for DRA: TO MAXIMIZE THE SECONDARY tion “Cognitive Beamforming One is due to their own transNETWORK THROUGHPUT. Optimization,” and then study mit power budgets; the other the general case of L . 1 for is to limit their resulting joint space-time-frequency DRA in CR networks under ATPCs interference level at each PU to be below a prescribed threshand AIPCs in the section “Joint Space-Time-Frequency DRA old. These constraints can be applied over each fixed dimenOptimization.” sion as peak power constraints, or over multiple dimensions as average power constraints. Without loss of generality, we COGNITIVE BEAMFORMING OPTIMIZATION consider DRA for the secondary network over L transmit In this section, we consider the case of L 5 1, where the DRA dimensions with different channel realizations, with L $ 1. for CR networks reduces to the spatial-domain transmit optimiIn total, four different types of power constraints can be zation under PTPCs and PIPCs to maximize the CR network defined for the secondary network. By taking the CR MAC as throughput. We term this practice “cognitive beamforming.” To an example (similarly as for the CR BC/IC), we have investigate the fundamental performance limits of cognitive Peak transmit power constraint (PTPC) ■ beamforming, we study the optimal designs with the availability 1 Sk[l 4 2 # Pk , of perfect knowledge on all the channels in the SU networks, Tr (4) and those from all the secondary transmit terminals to PUs. For 4 denotes the transmit covariance matrix for the convenience, we drop the dimension index l for the rest of this where Sk[l section given L 5 1. l th transmit dimension of the kth SU, l [ 5 1, c, L 6 , First, it is worth noting that the PIPC given in (6) can be k [ 5 1, c, K 6 ; Pk denotes the kth SU’s peak power conunified with the PTPC given in (4) into a form of generalized straint that applies to each of the L transmit dimensions. linear transmit covariance constraint (GLTCC) ■ Average transmit power constraint (ATPC) 1 L a Tr 1 Sk[l 4 2 # Pk , L l51 (5) a Tr 1 Wi Si 2 # w

K

(8)

i51

where Pk denotes the kth SU’s average transmit power constraint over the L transmit dimensions. ■ Peak interference power constraint (PIPC)

H a Tr 1 Gkj 3 l 4 Sk 3 l 4 Gkj 3 l 4 2 # Gj , K

(6)

k51

where Gkj 3 l 4 denotes the realization of channel Gkj for a given l; and Gj denotes the peak interference power constraint for protecting the jth PU, j [ 5 1, c, J 6 , which limits the total interference power caused by all the K SUs across all the receiving antennas of the jth PU, for each of the L transmit dimensions. ■ Average interference power constraint (AIPC) 1 L K H a a Tr 1 Gkj [l 4 Sk [l 4 Gkj [l 4 2 # Gj , L l51 k51 (7)

where Gj denotes the average interference power constraint for the jth PU to limit the total interference power from the K SUs, which is averaged over the L transmit dimensions. Note that DRA for traditional (non-CR) wireless networks under PTPC and/or ATPC has been thoroughly studied in the literature [12], while the study of DRA subject to PIPC and/or AIPC as well as their various combinations with PTPC/ATPC is unique to CR networks and is relatively new. To gain more insights into the optimal DRA designs for CR networks, we will

where Wi ’s and w are constants. For example, with each PIPC given in (6), Wi 5 GH Gij, 4i, and w 5 Gj, while for each PTPC ij given in (4), Wi 5 I if i 5 k and 0 otherwise, with w 5 Pk. Previous studies on transmit optimization for multiantenna or multiple-input multiple-output (MIMO) systems have mostly adopted some special forms of GLTCC such as the user individual power constraints and sum-power constraint. However, it remains unclear whether such existing solutions are applicable to the general form of GLTCC, which is crucial to the problem of CR MIMO transmit optimization with the newly added PIPCs. In the following, we provide an overview of the state-ofart solutions for this problem under different CR network models, while the developed solutions also apply to the case with the general form of GLTCCs as in (8). From a convex optimization perspective, we next divide our discussions into two parts, which deal with the cases of convex and nonconvex problem formulations, respectively. CONVEX PROBLEM FORMULATION First, consider the case where the associated optimization problem in a traditional MIMO system without PIPC is convex. In such cases, since the extra PIPCs are linear over the SU transmit covariance matrices, the resulting transmit covariance optimization problem for CR systems remains convex; and thus, it can be efficiently solved by standard convex optimization techniques.

IEEE SIGNAL PROCESSING MAGAZINE [105] MAY 2010

CR POINT-TO-POINT MIMO CHANNEL We elaborate this case by first considering the CR point-to-point MIMO channel, which can be treated as the special case with only one active SU link in the MAC, BC, or IC-based CR network. Without loss of generality, we will use the notations developed for the CR MAC with K 5 1 in the following discussions. Specifically, the optimal transmit covariance to achieve the CR point-to-point MIMO channel capacity under both the PTPC and PIPCs can be obtained from the following problem [13]: Max. log 0 I 1 HSHH 0 S s. t. Tr 1 S 2 # P Tr 1 Gj SG H 2 # Gj, j 5 1, c, J j S f 0, (P1) where for conciseness we have removed the SU index k in the symbol notations since K 5 1, while S f 0 means that S is a positive semidefinite matrix [14]. We see that (P1) is a convex optimization problem since its objective function is concave over S and its constraints define a convex set over S. Thus, (P1) can be efficiently solved by, e.g., the interior point method [14]. In the special case of CR multiple-input single-output (MISO) channel, i.e., H degrades to a row-vector denoted by h [ C13N, it can be shown by exploiting the KarushKuhn-Tucker (KKT) optimality conditions of (P1) that transmit beamforming is capacity optimal, i.e., Rank 1 S 2 5 1 [13]. Thus, without loss of generality we could write S 5 vvH, where v [ CN31 denotes the precoding vector. Accordingly, (P1) for the special case of MISO CR channel is simplified as (P1-S) [13] Max. 7hv 7

v

the jth PIPC, j 5 1, c, J, while we have the dual function defined as d 1 h 2 ! max log 0 I 1 HSHH 0 2 h0 1 Tr 1 S 2 2 P 2

Sf0

2 a hj 1 Tr 1 GjSGH 2 2 Gj 2 . j

j51

J

(9)

Since (P1) is convex with Slater’s condition satisfied [14], the duality gap between the optimal values of (P1) and (P1-D) is zero, i.e., (P1) can be solved equivalently as (P1-D). Accordingly, an iterative algorithm can be developed to solve (P1-D) by alternating between solving d 1 h 2 for a given h and updating h to minimize d 1 h 2 . At each iteration, h can be updated by a subgradient-based method such as the ellipsoid method [16], according to the subgradients of d 1 h 2 , which can be shown equal to P 2 Tr 1 Sw 2 and Gj 2 Tr 1 Gj Sw GH 2 for h0 j and hj, j 2 0, respectively, where Sw denotes the optimal S to obtain d 1 h 2 for a given h. From (9), it follows that Sw is the solution of the following equivalent problem (by discarding irrelevant constant terms) max log 0 I 1 HSHH 0 2 Tr 1 TS 2 ,

Sf0

(10)

J where T 5 h0I 1 a j51hj 1 G H Gj 2 is a constant matrix for a j given h. To solve (10), we introduce an auxiliary variable: ^ S 5 T1/2ST1/2. Equation (10) is then reexpressed in terms of ^ S as

^ ^ max log 0 I 1 HT21/2 ST 21/2 HH 0 2Tr 1 S 2 .

^ S f0

(11)

s. t. 7v 7 2 # P 7Gjv 7 2 # Gj, j 5 1, c, J,

which is nonconvex due to the nonconcavity of its objective function. However, by observing the fact that if v is the solution of (P1-S), so is ejuv for any arbitrary u, we thus assume without loss of generality that hv is a real number and modify (P1-S) by rewriting its objective function as Re 1 hv 2 and adding an additional linear constraint Im 1 hv 2 5 0. Thereby, (P1-S) can be converted into a second-order cone programming (SOCP) [14] problem, which is convex and thus can be efficiently solved by available convex optimization software [15]. Alternatively, (P1-S) can be shown equivalent to its Lagrange dual problem [13], which is a convex semidefinite programming (SDP) [14] problem and is thus efficiently solvable [15]. For (P1-S) in the case of one single-antenna PU, a closed-form solution for the optimal precoding vector v was derived in [13] via a geometric approach. To reveal the structure of the optimal S for (P1), we consider its Lagrange dual problem defined as (P1-D) Min. d 1 h 2 ,

hf0

where h 5 [h0, h1, c, hJ 4 denotes a vector of dual variables for (P1) with h0 associated with the PTPC, and hj associated with

The above problem can be shown equivalent to the standard point-to-point MIMO channel capacity optimization problem subject to a single sum-power constraint [17], and its solution can be ^ expressed as Sw 5 V SV H, where V is obtained from the singularvalue decomposition (SVD) given as follows: HT 21/2 5 UUVH, with U 5 Diag 1[u 1, c, u T 4 2 and T 5 min 1 M, N 2 , while S 5 Diag 1 [s1, c, sT 4 2 follows the standard water-filling s o l u t i o n [17 ] : si 5 1 1/ln2 2 1/u 2 2 1 , i 5 1, c, T, w i t h i 1 # 2 1 ! max 1 0, # 2 . Thus, the solution of (10) for a given h can be expressed as Sw 5 T 21/2 V SV H T 21/2. Next, we present a heuristic method for solving (P1), which leads to a suboptimal solution in general and could serve as a benchmark to evaluate the effectiveness of the above two approaches based on convex optimization. To gain some intuitions for this method, we first take a look at two special cases of (P1). For the first case, supposing that all the PIPCs are inactive (e.g., by setting Gj 5 `, 4j ) and thus can be removed, (P1) reduces to the standard MIMO channel capacity optimization problem under the PTPC only, for which the optimal solution of S is known to be derivable from the SVD of H [17]. For the second case, assuming that Gj 5 0, 4j, the solution for (P1) is then obtained by the “zeroforcing (ZF)” algorithm [18], which first projects H into the space orthogonal to all Gj’s, and then designs the optimal S based on the SVD of the projected channel. Note that the (nontrivial) ZF-based J solution exists only when N . a j51 Dj. From the above two

IEEE SIGNAL PROCESSING MAGAZINE [106] MAY 2010

special cases, we observe that as Gj ’s decrease, the optimal S should evolve along with a sequence of subspaces of H with decreasing dimensions as a result of keeping certain orthogonality to Gj’s, which motivates a new design method for cognitive beamforming, named as partial channel projection [13]. Specifically, let Gj 5 Gj/Gj, 4j. Then, define G ! [GT, c, GT 4 T. Denote the SVD 1 J of G as G 5 UGLGVH. Without loss of generality, assume that the G singular values in LG are arranged in a decreasing order. Then, we propose a generalized channel projection operation

1b 1b H' 5 H 1 I 2 VG 2 1 VG 2 2 H 2 , 1 2

K S1, c, SK k51

Max.

a mklog

`I 1 a i51HiSiHH ` i `I 1 a i51 HiSiHH ` i

k21

k

s. t. Tr 1 Sk 2 # Pk, k 5 1, c, K

k51 H a Tr 1 GkjSkGkj 2 # Gj, j 5 1, c , J K

Sk f 0, k 5 1, c, K.

(P2)

(12)

Reordering terms in the objective function of (P2) yields

K21 k51 H a 1 mk 2 mk11 2 log `I 1 a HiSiHi ` i51 k

SU Achievable Rate (b/s/Hz)

where V Gb consists of the first b columns of VG corresponding t o the b largest singular values in LG, 1 # b # J min 1N 2 1, g j51 Dj 2. Note that b could also take a zero value for 1 2 which V G0 ! 0. Now, we are ready to present the transmit covariance matrix for the partial projection method in the form of its eigenvalue decomposition (EVD) as S 5 V'S'VH , where V' is ' obtained from the SVD of the projected channel H', i.e., H' 5 U'L'VH . By substituting this new form of S into (P1), it ' can be shown that the problem reduces to maximizing the sumrate of a set of parallel channels (with channel gains given by L') over their power allocation S' subject to 1 J 1 1 2 linear power constraints, for which the optimal power allocation can be obtained by a generalized “water-filling” algorithm [13]. Note that the partial channel projection works for any values of N and Dj’s. In Figure 2, we plot the achievable rate of a CR MIMO channel under the PTPC and PIPCs with the optimal transmit covariance solution for (P1) via the convex optimization approach, against those with suboptimal covariance solutions via the partial channel projection method with different values of b. The system parameters are given as follows: M 5 N 5 4, J 5 2, D1 5 D2 5 1, and G1 5 G2 5 0.1. The SU achievable rate is plotted versus the SU PTPC, P. It is observed that the optimal covariance solution obtained via the convex optimization approach yields notable rate gains over suboptimal solutions via the heuristic method, for which the optimal value of b (the number of SU-to-PU channel dimensions to be nulled) to maximize the SU achievable rate increases with the SU PTPC. CR MIMO-MAC The solutions proposed for the CR point-to-point MIMO channel shed insight on transmit optimization for the CR MIMO-MAC defined in (1) with K . 1. Assume that in the CR MIMO-MAC, the optimal multiuser detection is deployed at the S-BS to successively decode different SU messages from the received sum-signal. We then consider the problem for jointly optimizing SU transmit covariance matrices to maximize their weighted sum-rate subject to individual PTPCs and joint PIPCs. This problem is referred to as weighted sum-rate maximization (WSRMax). Without loss of generality, we assume that the given user rate weights satisfy that m1 $ m2 $ c$ mK $ 0; thus, the optimal decoding order of users at the S-BS to maximize the weighted sum-rate is in accordance with the reverse user index [19]. Accordingly, the WSRMax for the CR MIMO-MAC can be expressed as

1 mKlog `I 1 a HiSiHH ` . i

i51

K

(13)

From the above new form of the objective function, it can be verified that (P2) is a convex optimization problem over Sk ’s. Thus, similarly as for (P1), (P2) can be solved by an interiorpoint-method-based algorithm or an iterative algorithm via solving the equivalent Lagrange dual problem, for which the details are omitted here for brevity. It is noted that (P2) is for the case with the optimal nonlinear multiuser decoder at the S-BS, while in practice the lowcomplexity linear decoder is usually more preferable. The use of linear instead of nonlinear decoder at the receiver will change the user achievable rates for the CR MIMO-MAC, thus resulting in new problem formulations for transmit optimization. For example, in [20], the authors have considered the CR SIMOMAC (single-antenna for each SU transmitter) with a linear decoder at the receiver, where the power allocation across the SUs is optimized to maximize their signal-to-interference-plusnoise ratios (SINRs) at the receiver subject to both transmit and interference power constraints.

12 10 8 6 4 2 0 0 Optimal Solution Suboptimal Solution (b = 0) Suboptimal Solution (b = 1) Suboptimal Solution (b = 2)

2

4 6 8 10 12 14 16 18 SU Transmit Power Constraint (dB)

20

[FIG2] Comparison of the achievable rates for the CR MIMO channel under the PTPC and PIPCs: The optimal transmit covariance solution for (P1) via the the convex optimization approach versus suboptimal covariance solutions via the partial channel projection method with different values of b is shown.

IEEE SIGNAL PROCESSING MAGAZINE [107] MAY 2010

NONCONVEX PROBLEM FORMULATION Next, we consider the case where the optimization problems in the associated traditional (non-CR) MIMO systems are nonconvex. It thus becomes more challenging whether these nonconvex problems with the addition of convex PIPCs in the corresponding CR MIMO systems can be efficiently solvable. In the following, we present some promising approaches to solve these problems for the CR MIMO-BC and MIMO-IC. CR MIMO-BC First, consider the CR MIMO-BC defined in (2) under both the PTPC at the S-BS and J PIPCs each for one of the J PUs, which can be similarly defined as for the MAC case in (4) and (6), respectively. We focus on the unicast downlink transmission for the CR BC, while for the case of multicast, the interested readers may refer to [21]. For the purpose of exposition, we consider two commonly adopted design criteria for the traditional multiantenna Gaussian BC in the literature: One is for the MIMO-BC deploying the nonlinear “dirty paper coding (DPC)” at the transmitter [22], which maximizes the weighted sum-rate of all the users (i.e., the WSRMax problem); the other is for the MISO-BC (single-antenna for each SU receiver) deploying only linear encoding at the transmitter, which maximizes the minimum SINR among all the users, referred to as “SINR balancing.” Specifically, the WSRMax problem for the CR MIMO-BC can be formulated as

K

problem under a special case of GLTCC: the per-antenna transmit power constraint. However, these existing forms of BC-MAC duality are yet unable to handle the case with arbitrary numbers of GLTCCs, which is the case for (P3) with both the PTPC and PIPCs. In [25], a general method was proposed to solve various MIMO-BC optimization problems under multiple GLTCCs, thus including the CR MIMO-BC WSRMax problem given in (P3). For this method, the first step is to combine all 1 J 1 1 2 power constraints in (P3) into a single GLTCC as shown in the following optimization problem:

K Q1, c, QK

Max.

k51

a mklog

`I 1 HH a a i5kQi bHk ` k `I 1 HH a a i5k11Qi bHk ` k

K

K

s. t. TraA a Qk b # Q

k51

K

Qk f 0, k 5 1, c, K ,

J J

(14)

K Q1, c, QK

Max.

k51 K

a mklog

`I 1 HH a a i5kQi bHk ` k `I 1 HH a a i5k11Qi bHk ` k

K

s. t. a Tr 1 Qk 2 # P TraFj a a Qk b FH b # Gj, j 5 1, c, J j

k51 k51 K

Qk f 0, k 5 1, c, K ,

(P3)

where Qk [ CM3M denotes the covariance matrix for the transmitted signal of S-BS intended for the kth SU, k 5 1, c, K; mk ’s are the given user rate weights; and P denotes the transmit power constraint for the S-BS. Without loss of generality, we assume that m1 $ m2 $ c$ mK $ 0; thus, in (P3) the optimal encoding order of users for DPC to maximize the weighted sum-rate is in accordance with the user index [22]. Note that (P3) is nonconvex with or without the PIPCs due to the fact that the objective function is nonconcave over Qk ’s for K $ 2. As a result, unlike (P1) for the point-to-point CR channel, the standard Lagrange duality method cannot be applied for this problem. For (P3) in the case without the PIPCs, a so-called “BC-MAC duality” relationship was proposed in [23] to transform the nonconvex MIMO-BC problem into an equivalent convex MIMO-MAC problem, which is solvable by efficient convex optimization techniques such as the interior point method. In [24], another form of BC-MAC duality, the socalled “mini-max duality” was explored to solve the MIMO-BC

where A 5 l0I 1 a ljFHFj, and Q 5 l0P 1 a ljGj with j j51 j51 l0, l1, c, lJ being nonnegative constants. For a given set of li ’s, i 5 0, c, J, let the optimal value of the above problem be denoted by F 1 l0, l1, c, lJ 2 . Clearly, F 1 l0, l1, c, lJ 2 is an upper bound on the optimal value of (P3) since any feasible solutions for (P3) must satisfy the constraints of (14) for a given set of li ’s. Interestingly, it can be shown that the optimal value of (P3) is equal to the minimum value of function F 1 l0, l1, c, lJ 2 over all nonnegative li ’s [25]. Therefore, (P3) can be resolved by iteratively solving (14) for a given set of li s and updating li ’s towards their optimal values to minimize function F 1 l0, l1, c, lJ 2 . Specifically, li ’s can be updated via the ellipsoid method according to the subgradients of F 1 l0, l1, c, lJ 2 , which can be shown [25] equal to P 2 a K Tr 1 Qw 2 and Gj 2 Tr1Fj 1 a K Qw 2 FH 2 for l0 and lj k j k51 k51 k 1 j 2 0 2 , respectively, where Qw ’s are the solution of (14) for the k given lk ’s. Furthermore, (14) with a given set of lk ’s can be solved by applying the generalized BC-MAC duality proposed in [25], which extends the existing forms of BC-MAC duality [23], [24] to transform the MIMO-BC problem subject to a single GLTCC as in (14) to an auxiliary (dual) MIMO-MAC problem subject to a corresponding sum-power constraint. Specifically, it is shown in [25] that the MIMO-BC as in (14) and the dual MIMO-MAC, as depicted in Figure 3, have the same achievable rate region. Accordingly, the optimal objective value (weighted sum-rate) of (14) for the primal MIMO-BC can be obtained as that of the following equivalent problem for the dual MIMO-MAC:

K21 S1, c, SK

Max.

k51

H a 1 mk 2 mk11 2 log `A 1 a HiSiHi ` i51 H a HiSiHi ` i51 K

k

1 mKlog `A 1

K

s. t. a Tr 1 Sk 2 # Q

k51

Sk f 0, k 5 1, c, K.

(15)

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Similar to (P2), the above problem is a WSRMax problem for the MIMO-MAC subject to a single sum-power constraint, which is convex and thus can be efficiently solvable by, e.g., the interior point method. After solving (15), the optimal user transmit covariance solutions for the MIMO-MAC, Sw s, can be transk formed to the corresponding ones for the original MIMO-BC, Qw s, via a MAC-BC covariance transformation algorithm given k in [25]. Furthermore, it is worth noting that with K 5 1, the above method can be shown equivalent to that developed for (P1) in the CR point-to-point MIMO channel case based on the Lagrange duality. Consider next the SINR balancing problem for the CR MISOBC, which can be expressed as Max. a ||hHvk||2 k 1 1 a ||hHvi||2 k

i2k 2 a ||vk|| # P K

HH 1 HH 2 . . . HH k (a)

z1 z2

y1

x1

H1 H2 . . .

z y

x

y2 x2

zk

yk xk

Hk (b)

a, v1, c, vK

[FIG3] Generalized MIMO MAC-BC duality. (a) Primal MIMO-BC channel with downlink channels HH and receiver noise vectors k zk , CN 1 0, I 2 , k 5 1, c, K, and a GLTCC: Tr1A a K Qk 2 # Q. k51 (b) Dual MIMO-MAC with uplink channels Hk, k 5 1, c, K and receiver noise vector z , CN 1 0, A 2 , and a sum-power constraint: K a k51 Tr 1 Sk 2 # Q. The MIMO-BC and dual MIMO-MAC have the same achievable rate region [25].

s. t.

$ a, k 5 1, c, K CR MIMO-IC Second, consider the CR MIMO-IC given in (3), subject to both the PTPCs for the K SU-TXs and the PIPCs for the J PUs, which can be similarly defined as for the MAC case in (4) and (6), respectively. From an information-theoretic perspective, the capacity region for the Gaussian IC under PTPCs, which consists of all the simultaneously achievable rates of all the users, still remains unknown in general even for the case of K 5 2 and Ak 5 Bk 5 1, k 5 1, 2 [29]. A pragmatic approach that leads to suboptimal achievable rates in the Gaussian IC is to restrict the system to operate in a decentralized manner, i.e., allowing only single-user encoding and decoding by treating the cochannel interferences from the other users as additional Gaussian noises. For this approach, transmit optimization for the CR MIMO-IC reduces to finding a set of optimal transmit covariance matrices for the K SU links, denoted by Rk [ CAk 3Ak, k 5 1, c, K, to maximize the secondary network throughput under both the PTPCs and PIPCs. More specifically, the WSRMax problem for the CR MIMO-IC can be expressed as

K 21

k51 K

k51

2 a ||Fjvk|| # Gj, j 5 1, c, J ,

(P4)

where a denotes an achievable SINR for all the SUs; vk [ CM31 denotes the precoding vector for the transmitted signal of S-BS intended for the kth SU; and hk represents Hk for the MISO-BC case. Similarly as for (P1-S), by treating hHvk on the left-hand side (LHS) of each SINR constraint in k (P4) as a positive real number [26], it can be shown that (P4) for a given a is equivalent to a SOCP feasibility problem and thus efficiently solvable [15]. For a given a, if the associated SOCP problem is feasible, we know that the optimal solution of (P4) for a, denoted by aw, must satisfy aw $ a; otherwise, aw , a. Based on this fact, aw can be found by a simple bisection search [14]; with aw, the corresponding optimal solution for vk ’s in (P4) can also be obtained. The above technique has also been applied in [27] for (P4) without the PIPCs. The SINR balancing problem for the conventional MISOBC without the PIPCs has also been studied in [28], where an algorithm was proposed using the virtual uplink formulation and a fixed-point iteration. However, this algorithm cannot be extended directly to deal with multiple PIPCs for the case of CR MISO-BC. Similarly as for the previous discussions on the WSRMax problem for the CR MIMO-BC where a generalized MIMO MAC-BC duality holds, a counterpart beamforming duality also holds for the MISO-BC and SIMO-MAC [25]. With this duality result, the SINR balancing problem (P4) for the CR MISO-BC can be converted into an equivalent problem for the dual SIMO-MAC, where the efficient iterative algorithm in [28] can be directly applied. The interested readers may refer to [25] for the details of this method.

R1, c, RK

Max.

k51

a mklog `I 1 aI 1 a HikRiHik b

i2k K

H

HkkRkHH ` kk

s. t. Tr 1 Rk 2 # Pk, k 5 1, c, K

k51 H a Tr 1 EkjRkEkj 2 # Gj, j 5 1, c, J

Rk f 0, k 5 1, c, K ,

1 P5 2

where mk ’s are the given nonnegative user rate weights. We see that (P5) is nonconvex with or without the PIPCs due to the fact that the objective function is nonconcave over Rk ’s for K . 1. As a result, there are no efficient algorithms yet to obtain the globally optimal solution for this problem. For the same problem setup, there have been recent progresses on characterizing the maximum achievable “degrees of freedom (DoF)” for the user sum-rate (i.e., mk 5 1, 4k ) [30].

IEEE SIGNAL PROCESSING MAGAZINE [109] MAY 2010

vectors (based on the “network Next, we discuss some feasiA PRAGMATIC APPROACH THAT LEADS duality” [39]) or the power ble solutions for (P5). First, it is TO SUBOPTIMAL ACHIEVABLE RATES allocation vectors (by solving worth noting that for (P5) in IN THE GAUSSIAN IC IS TO RESTRICT geometric programming (GP) the case without the PIPCs, a THE SYSTEM TO OPERATE IN A problems [40]), with the others commonly adopted suboptimal DECENTRALIZED MANNER. being fixed. approach is to iteratively optiIt is worth pointing out that mize each user’s transmit covathere are other problem formuriance subject to its individual lations different from (P5) to address the transmit optimization PTPC with the transmit covariances of all the other users fixed. for the CR MIMO-IC. In [41], a new criterion was proposed to This decentralized approach has been first proposed in [31] and design the SU link transmission in a CR MISO-IC via an alter[32] to obtain some local optimal points for (P5) with the PTPCs native decentralized approach, where each SU-TX independentonly, where they differ in that the one in [31] maximizes the ly designs its transmit precoding vector to maximize the ratio user individual rate at each iteration, while the one in [32] maxbetween the received signal power at the desired SU-RX and the imizes the user weighted sum-rate. It is also noted that a paralresulted total interference power at all the PUs, to regulate the lel line of works with similar iterative user optimizations has interference powers at PUs. Moreover, the above discussions are been pursued in the single-antenna but multicarrier-based all based on the assumption that each SU-RX treats the interinterference channels such as the wired discrete-multitone ferences from all the other SU links as additional noises, which (DMT)-based digital subscriber line (DSL) network [33], and the is of practical interest since it simplifies the receiver design for wireless OFDM-based ad hoc network [34]. One important queseach SU link. However, due to independent cross-link channels tion to answer for such iterative algorithms is under what conbetween SU terminals, it may be possible that a SU-RX could ditions the algorithm will guarantee to converge to a local occasionally observe “strong” interference signals from some optimal point. This problem has been addressed in the contexts coexisting SU-TXs and thus be able to decode their messages of both multicarrier- and multiantenna-based interference via multiuser detection techniques and then cancel the associchannels in, e.g., [35] and [36], via game-theoretic approaches. ated interferences. With such “opportunistic” multiuser detecHowever, the above iterative approach cannot be applied tion at each SU-RX, the achievable rate of each SU link becomes directly to solve (P5) with both the PIPCs and PTPCs, since each a function of not only its own transmit covariance, but also PIPC involves all the user transmit covariances and is thus not those of the other SUs as well as their instantaneous transmit separable over the SUs. Thus, a feasible approach for (P5) is to rates. Thus, the corresponding transmit optimization for the decompose each of the J PIPCs into a set of interference-power CR MIMO-IC leads to new and more challenging problem forconstraints over the K SU-TXs, i.e., for the jth PIPC, mulations than (P5); the interested readers may refer to [42] j [ 5 1, c, J 6 , and [43]. Tr 1 EkjRkEH 2 # Gj1k2, k 5 1, c, K , (16) kj JOINT SPACE-TIME-FREQUENCY DRA OPTIMIZATION 1k2 1k2 In the previous section, we have studied DRA for different CR where Gj is a constant, and all Gj ’s, k 5 1, c, K, satisfy 1k2 networks at a single transmit dimension in time/frequency, by a kGj # Gj such that the jth PIPC is guaranteed. Then, the focusing on spatial-domain transmit optimization under the iterative algorithm works here, where each SU link indepenpeak transmit power constraint (PTPC) and peak interference dently optimizes Rk to maximize its achievable rate under its power constraint (PIPC). In this section, we bring the additional PTPC and J interference-power constraints given by (16), with time and/or frequency dimensions into the DRA problem forall other Ri ’s, i 2 k, fixed. It is observed that the resulting probmulations, by applying the average transmit power constraint lem is in the same form of our previously studied (P1) for the (ATPC) and average interference power constraint (AIPC) in CR CR point-point MIMO channel; thus, similar techniques develnetworks. Consider the DRA over L time/frequency dimensions, oped for (P1) can be applied. Note that a suboptimal method for for which all the required channel knowledge is assumed to be this problem in the same spirit of the partial channel projection known. Taking the CR MAC as an example (similar arguments method to reduce the design complexity for each SU transmit can be developed for the CR BC/IC), under both the ATPCs and covariance matrix has also been proposed in [37]. Moreover, it is AIPCs given in (5) and (7), respectively, a generic problem fornoted that Gj1k2 ’s, j 5 1, c, J, k 5 1, c, K, can be searched mulation for DRA optimization can be formulated as over the SUs to further improve their weighted sum-rate. Alternatively, assuming that a centralized optimization is feasible with the global knowledge of all the channels in the SU Max. C 15 Sk[l 462 Sk[l4 f0,4k,l network, as well as those from different SU-TXs to all PUs, s. t. 1 5 2 , 1 7 2 , (P6) another heuristic algorithm for (P5) was proposed in [38]. By rewriting the SU transmit covariance matrices into their equivalent precoding vectors and power allocation vectors, this algorithm iteratively updates the SU transmit precoding where 5 Sk[l 46 denotes the set of Sk[l 4 ’s, k 5 1, c, K, and l 5 1, c, L, while C 1 # 2 is an arbitrary utility function to

IEEE SIGNAL PROCESSING MAGAZINE [110] MAY 2010

measure the CR network performance. We assume that C 1 # 2 is 1 separable over ls, i.e., C 1 5 Sk[l 46 2 5 L a L Ul 1 S1[l 4 , c, SK[l 4 2 l51 with Ul 1 # 2 ’s denoting individual utility functions. Since both the ATPC and AIPC involve L transmit covariance matrices, the Lagrange dual decomposition (see, e.g., a tutorial paper [44]) is a general method to deal with this type of average constraints for optimization over a number of parallel dimensions, which is explained as follows. By introducing a set of dual variables, nk ’s, each for one of the K ATPCs, and dk ’s, each for one of the J AIPCs, the Lagrange dual problem of (P6) can be written as (P6-D)

nf0,df0

Min. d 1 n, d 2

with n 5 [n1, c, nK 4 , d 5 [d1, c, dJ 4 , and the dual function d 1 n,d 2 !

K 1 L max C 15 Sk 3l 462 2 a nk a a Tr 1 Sk 3 l 42 2 Pk b L S [l4 f0,4k,l

k

k51

l51

obtainable, the optimal value of (P6-D) in general only serves as an upper bound on that of (P6). However, in [45] it is pointed out that if a set of so-called “time-sharing” conditions are satisfied by a nonconvex optimization problem, the duality gap for this problem and its dual problem is indeed zero. Furthermore, for the class of DRA problems in the form of (P6), the associated time-sharing conditions are usually satisfied asymptotically as L S ` under some cautious considerations on the continuity of channel distributions [46]. Therefore, the dual decomposition method could still be applied to solve (P6) in the nonconvex case for sufficiently large values of L, provided that the optimal solutions for the subproblems in (18) are obtainable (e.g., a variation of (P3) for the CR MIMO-BC). However, with finite values of L, how to efficiently solve (P6) in the case of nonconcave objective functions is still open. With the above discussions on the general approaches to design joint space-time-frequency DRA for CR networks, we next present some examples of unique interests to CR systems. TDMA/FDMA CONSTRAINED DRA: WHEN IS IT OPTIMAL? Time-/frequency-division multiple-access (TDMA/FDMA), which schedules only one user for transmission at each time/frequency dimension, is usually preferable in practice due to their implementation ease. For the TDMA/FDMA-based CR MAC (similar arguments hold for the CR BC/IC), the optimal DRA over L transmit dimensions to maximize the sum-capacity of the SUs can be formulated as (P6) with properly chosen functions for Ul 1 # 2 ’s, where for any given l, Ul 1 # 2 is expressed as (l is dropped for conciseness) U 1 S1, c SK 2 5 e , log 0 I 1 HkSkHH 0 k 0 Si 5 0, 4i 2 k otherwise. (19)

1 2 a dj a a a Tr 1 Gkj 3l 4 Sk 3l 4 GH 3l 4 2 2 Gj b. kj L

j51 l51 k51

J

L

K

(17)

Since the dual problem (P6-D) is convex regardless of the convexity of the primal problem (P6) [14], (P6-D) can be efficiently solved by the ellipsoid method according to the subgradients of the dual function d 1 n, d 2 , similarly as in our previous discussions, provided that the maximization problem in (17) is solvable for any given set of n and d. It is interesting to observe that this maximization problem can be decomposed into L parallel subproblems each for one of the L dimensions, and all of these subproblems have the same structure and are thus solvable by the same algorithm, a practice known as “dual decomposition.” Without loss of generality, we drop the dimension index l and express each subproblem as max U 1 S1, c, SK 2 2 a Tr 1 Bk 1 nk, d 2 Sk 2 , S f0,4k

k

K

(18)

k51

where Bk 1 nk, d 2 5 nkI 1 a J 1 djGH Gkj 2 is a constant matrix kj j51 for the given nk and d, k 5 1, c, K. We then discuss the next two cases. For the first case, consider that Ul 1 # 2 is a concave function over Sk[l 4 ’s, 4l [e.g., the point-to-point CR channel capacity in (P1), or the weighted sum-rate for the CR MIMO-MAC in (P2)]. Then, (P6) is convex and thus the duality gap between the optimal values of (P6) and (P6-D) is zero, i.e., (P6) and (P6-D) are equivalent problems. Furthermore, each subproblem in (18) is also convex. Thus, the dual decomposition method solves (P6) via its dual problem (P6-D), which is decomposable into L convex subproblems. For the second case, as a counterpart, consider that Ul 1 # 2 is nonconcave over Sk[l 4 ’s [e.g., the weighted sum-rate for the CR MIMOBC/MIMO-IC in (P3)/(P5)]. As a result, (P6) is nonconvex and the duality gap between (P6) and (P6-D) may not be zero. Furthermore, the subproblem (18) is also nonconvex. For this case, even when the optimal solutions of the L subproblems are

Note that U 1 # 2 defined above implies the TDMA/FDMA constraint, i.e., only scheduling one user for transmission at a given dimension with a positive contribution to the sum-capacity. However, it can be shown that U 1 # 2 is nonconcave over Sk ’s in this case and as a result, the corresponding (P6) is nonconvex. Nevertheless, according to our previous discussions, since the time-sharing conditions hold approximately when L S `, the dual decomposition method can be applied to solve (P6) for this case with very large values of L, where the optimal solution of the associated subproblem at each dimension given in (18) can be obtained by finding the SU (selected for transmission) with the largest objective value of the following problem (which is of the same form as (10) and thus solvable in a similar way) max log 0 I 1 HkSkHH 0 2 Tr 1 Bk 1 nk,d 2 Sk 2 . k

Sk f0

(20)

An important question to investigate for TDMA/FDMAbased DRA is how much the performance is degraded as compared with the optimal DRA that allows more than one user to transmit at a given dimension. From an information-theoretic viewpoint, it is thus pertinent to investigate the conditions for

IEEE SIGNAL PROCESSING MAGAZINE [111] MAY 2010

the optimality of TDMA/FDMA, i.e., when they are optimal to achieve the system sum-capacity. For the traditional singleantenna fading MAC under the user ATPCs over time, it has been shown in [47] that TDMA is optimal for achieving the ergodic/long-term sum-capacity. This result has been shown to hold for the fading CR MAC and CR BC under both the ATPCs and AIPCs in [48], where by exploiting the KKT optimality conditions of the associated optimization problems, the optimality conditions for TDMA in other cases of combined peak/average transmit/interference power constraints have been characterized. For the traditional single-antenna IC with interference treated as noise, the optimality of TDMA/ FDMA for the sum capacity has been investigated under the ATPCs in [49] and [50]. It would be interesting to extend these results to the case of CR IC under the additional PIPCs and/or AIPCs. PEAK VERSUS AVERAGE INTERFERENCE POWER CONSTRAINTS: A NEW INTERFERENCE DIVERSITY From a SU’s perspective, it is obvious that the ATPC/AIPC is more flexible than the PTPC/PIPC for DRA under the same power threshold and thus results in a larger SU link capacity. However, from a PU’s perspective, it remains unknown whether the AIPC or PIPC causes more PU link performance degradation. Intuitively speaking, the PIPC should be more favorable than the AIPC since the former limits the interference power at the PU to be below certain threshold at each time/frequency dimension, while the latter results in variations of interference power levels over different dimensions although their average level is kept below the same threshold as that for the PIPC. Somehow surprisingly, in [51] it is shown that for the singleantenna PU fading channel subject to the interference from a SU transmitter, the AIPC is in fact better than its PIPC counterpart under the same average power threshold in terms of minimizing the PU capacity losses, which holds for the cases of both ergodic and outage capacities of the PU channel, with/without power control. To illustrate this result, we consider for simplicity the case without the PU link power control, i.e., the PU transmits with a constant power, Q, over all the fading states. Suppose that the PU link channel power gain is denoted by hp, and that from the SU transmitter to the PU receiver denoted by hsp. Next, consider the following two cases, where the interference power from the SU transmitter at the PU receiver, denoted by Isp 5 hsp ps, with ps denoting the SU transmit power, is fixed over all the fading states in Case I (corresponding to the case of PIPC), and is allowed to be variable in Case II (corresponding to the case of AIPC). For both cases, a constant interference power threshold G is set and is assumed to hold with equality, i.e., for 1I Case I, Isp2 5 G, for all the fading states, while for Case II, 1II2 E 1 Isp 2 5 G. Taking the PU ergodic capacity as an example, which can be expressed as (assuming unit-power receiver Gaussian noise) Cp 5 Ealoga1 1 hpQ 1 1 Isp bb. (21)

1 1 Let CpI2 and CpII2 denote the values of Cp in Cases I and II, respectively. The following equalities/inequalities then hold

1 CpI2 5 Ehp aloga1 1

hpQ 11G

bb bb bbb (22)

5 Ehp aloga1 1

hpQ

1II 1 1 E 1 Isp 2 2

# Ehp aEIsp aloga1 1

1 5 CpII2 ,

hpQ

1II 1 1 Isp 2

where (22) is due to the Jensen’s inequality (see, e.g., [17]) and the convexity of the function f 1 x 2 5 log 1 1 1 k/ 1 1 1 x 2 2 where k is any positive constant and x $ 0. Thus, it follows that given the same average power of the interference, G, it is desirable for the PU to have the instantaneous interference power Isp fluctuate over fading states (Case II) rather than stay constant (Case I), to achieve a larger ergodic capacity. In general, the results in [51] reveal a new interference diversity phenomenon for SS-based CR networks, i.e., the randomized interference powers from the secondary network can be more advantageous over deterministic ones across different transmit dimensions over space, time, or frequency for minimizing the resulted primary network capacity losses. Further investigations are required on interference diversity driven DRA for CR or other spectrum sharing systems. BEYOND INTERFERENCE TEMPERATURE: EXPLOITING PRIMARY LINK PERFORMANCE MARGINS So far, we have studied DRA for CR networks based on the IT constraints for protecting the PU transmissions. Given that the IT constraints in general conservatively lead to an upper bound on the PU capacity loss due to the interference from the SUs [13], [52], it would be possible to improve the spectrum sharing capacities for both the SUs and PUs over the IT-based methods if additional cognition on the PU transmissions is available at the CR transmitters. For example, by exploiting CSI of the PU links, the CRs could allocate transmit/interference powers more flexibly over the dimensions where the PU channels exhibit poor conditions, without degrading too much the PU link performances. These PU “null” dimensions could come up in time, frequency, or space. Thus, the IT constraints could be replaced by the more relevant primary link performance margin constraints [52], [53] for the design of DRA in CR networks, to optimally exploit the available primary link performance margins to accommodate the interference from the SUs. Following this new paradigm, many new and challenging DRA problems can be formulated for CR networks. As an example, consider the same setup with a pair of single-antenna PU and SU links over fading channels as in the previous subsection. Instead of applying the conventional AIPC: E 1 hsp ps 2 # G, over the SU power allocation, we may apply the following PU ergodic capacity constraint [52]

IEEE SIGNAL PROCESSING MAGAZINE [112] MAY 2010

Ea1 1

hpQ 1 1 hsp ps

b $ Cp ,

(23)

where Cp is a given threshold for the minimum PU ergodic capacity. Note that the new constraint in (23) is more directly related to the PU transmission than the conventional AIPC. However, it can be verified that the constraint in (23) is nonconvex over ps in general, thus resulting in more challenging SU power allocation problems than that with the convex AIPC. The optimal power allocation rules for the SU link subject to the AIPC versus the newly introduced PU ergodic capacity constraint given by (23) are compared in [52], where it is shown that the new constraint achieves notable rate improvements for both the PU and SU links over the conventional AIPC. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK Dynamic resource allocation (DRA) has become an essential building block in CR networks to exploit various cognitions over both the primary and secondary networks for CR transmit optimization subject to certain required primary protection. In this article, we have presented an extensive list of new, challenging, and unique problems for designing the optimal DRA in CR networks, and demonstrated the key role of various convex optimization techniques in solving the associated design problems. In addition to those open issues as highlighted in our previous discussions, other promising areas of practical and theoretical interests are discussed as follows, which open an avenue for future work. ROBUST COGNITIVE BEAMFORMING In our previous discussions on cognitive beamforming, we have observed that the knowledge of channels from each secondary transmitting terminal to all PUs is essential to the design optimization. However, since the primary and secondary networks usually belong to different operators, it is difficult for the PUs to feed back the required CSI to the CRs. As a result, the SU usually needs to rely on its own observations over the received signals from the primary terminals to extract the required CSI [54]. Nevertheless, the estimated CSI on the SU-to-PU channels may contain errors, which should be taken into account for the design of practical CR systems. This motivates a new and challenging research direction on robust designs for cognitive beamforming to cope with imperfect CSI [55], [56]. More investigations on the robust cognitive beamforming designs for more general CR networks and CSI uncertainty models are appealing. ACTIVE INTERFERENCE-TEMPERATURE CONTROL In this article, we have focused on the design of CR networks subject to the given interference-power constraints for protecting the PUs. We have also discussed some promising rules on how to optimally set the IT constraints in the CR network to achieve the best spectrum sharing throughput. These results lead to a new and universal design paradigm for interference

management in CR or other related multiuser communication systems [57], [58], via appropriately setting the IT levels across the coexisting links. The active IT control approach to interference management for multiuser communication systems is relatively new, and more research endeavors are required along this direction. AUTHORS Rui Zhang (rzhang@i2r.a-star.edu.sg) received the B.Eng. and M.Eng. degrees in electrical and computer engineering from the National University of Singapore (NUS) in 2000 and 2001, respectively, and the Ph.D. degree in electrical engineering from Stanford University, California, in 2007. He is now a senior research fellow with the Institute for Infocomm Research (I2R), Singapore. He is an assistant professor of electrical and computer engineering with NUS. He has authored/coauthored more than 100 refereed international journal and conference papers. His current research interests include cognitive radios, cooperative communications, and multiuser MIMO systems. Ying-Chang Liang (ycliang@i2r.a-star.edu.sg) is a senior scientist at the Institute for Infocomm Research (I2R), Singapore. He also holds adjunct associate professorship positions at Nanyang Technological University (NTU) and NUS. He has been teaching graduate courses at NUS since 2004 and was a visiting scholar with the Department of Electrical Engineering, Stanford University, California, from 2002 to 2003. His research interest includes cognitive radio, dynamic spectrum access, reconfigurable signal processing for broadband communications, space-time wireless communications, wireless networking, information theory, and statistical signal processing. Shuguang Cui (cui@tamu.edu) received the the B.Eng. degree in radio engineering with the highest distinction from Beijing University of Posts and Telecommunications, China, in 1997, the M.Eng degree in electrical engineering from McMaster University, Canada, in 2000, and the Ph.D. degree in electrical engineering from Stanford University in 2005. He is an assistant professor in the Department of Electrical and Computer Engineering at Texas A&M University, College Station. His current research interests include cross-layer optimization for resource-constrained networks and network information theory. He is an associate editor for IEEE Communication Letters and IEEE Transactions on Vehicular Technology. He is an elected member for IEEE Signal Processing Society’s SPCOM Technical Committee. REFERENCES

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